[cig-commits] r13482 - in doc/cigma/manual: . appendix error_analysis examples formats installation integration interpolation introduction license running
luis at geodynamics.org
luis at geodynamics.org
Tue Dec 9 18:11:14 PST 2008
Author: luis
Date: 2008-12-09 18:11:13 -0800 (Tue, 09 Dec 2008)
New Revision: 13482
Added:
doc/cigma/manual/appendix.lyx
doc/cigma/manual/error_analysis.lyx
doc/cigma/manual/examples.lyx
doc/cigma/manual/formats.lyx
doc/cigma/manual/gpl-license.lyx
doc/cigma/manual/installation.lyx
doc/cigma/manual/integration.lyx
doc/cigma/manual/interpolation.lyx
doc/cigma/manual/introduction.lyx
doc/cigma/manual/running.lyx
Removed:
doc/cigma/manual/appendix/appendix.lyx
doc/cigma/manual/error_analysis/error_analysis.lyx
doc/cigma/manual/examples/examples.lyx
doc/cigma/manual/formats/formats.lyx
doc/cigma/manual/installation/installation.lyx
doc/cigma/manual/integration/integration.lyx
doc/cigma/manual/interpolation/interpolation.lyx
doc/cigma/manual/introduction/introduction.lyx
doc/cigma/manual/license/gpl-license.lyx
doc/cigma/manual/running/running.lyx
Modified:
doc/cigma/manual/cigma.lyx
doc/cigma/manual/nocover.lyx
Log:
Preliminary changes
Deleted: doc/cigma/manual/appendix/appendix.lyx
===================================================================
--- doc/cigma/manual/appendix/appendix.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/appendix/appendix.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,2476 +0,0 @@
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-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-quad4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-8
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-12
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Field Variables
-\end_layout
-
-\begin_layout Standard
-A finite element solution basically corresponds to a list of values for
- each degree of freedom in each element.
- Since the shared nodes among adjacent elements have the same value, these
-
-\end_layout
-
-\begin_layout Section
-Integration Rule
-\end_layout
-
-\begin_layout Standard
-As described in Chapter 4, an integration rule is specified by using the
- natural coordinate system of the reference element.
-
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="5">
-<features>
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Weights:
-\begin_inset Formula $(Q,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Points:
-\begin_inset Formula $(Q,n_{dim})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Points:
-\begin_inset Formula $(Q,2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Points:
-\begin_inset Formula $(Q,3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Rule Definition
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{1},\eta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{1},\eta_{1},\zeta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{2},\eta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{2},\eta_{2},\zeta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{Q},\eta_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{Q},\eta_{Q},\zeta_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Shape Function Values
-\end_layout
-
-\begin_layout Standard
-As discussed in Chapter 7, in certain circumstances you may be able to provide
- your custom reference element by simply providing the appropriate
-\begin_inset Formula $n$
-\end_inset
-
- shape function values at the expected integration points.
-\newline
-
-\newline
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="3">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Shape Function Values:
-\begin_inset Formula $(n,Q)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Jacobian Determinant:
-\begin_inset Formula $(Q,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point 1
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{1}),N_{2}(\vec{\xi}_{1}),\ldots,N_{n}(\vec{\xi}_{1})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{1})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point 2
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{2}),N_{2}(\vec{\xi}_{2}),\ldots,N_{n}(\vec{\xi}_{2})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{2})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point
-\begin_inset Formula $Q$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{Q}),N_{2}(\vec{\xi}_{Q}),\ldots,N_{n}(\vec{\xi}_{Q})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{Q})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Integration Points
-\end_layout
-
-\begin_layout Standard
-Given a mesh and an integration rule, we can map the integration points
- on each element into a possibly large set of global points.
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="4">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Points:
-\begin_inset Formula $(n_{pts},n_{dim})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Points:
-\begin_inset Formula $(n_{pts},2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Points:
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Coordinates
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1},z_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2},z_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{pts}}y_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{pts}},y_{n_{pts}},z_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Function Values at Integration Points
-\end_layout
-
-\begin_layout Standard
-The result of an evaluation will depend on the rank of the function being
- evaluated, as indicated in the table below.
- Note that the fastest varying dimension of the array contains the values
- of the function at each point.
-\newline
-
-\newline
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Evaluated Quantity
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Resulting Array Shape
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Scalar
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Tensor
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Tensor
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},6)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Comparison Residuals
-\end_layout
-
-\begin_layout Standard
-Since our norm reduces integrals over cells to a single scalar value, the
- result of a comparison will result in a simple scalar array for each element
- block examined.
- The element blocks used belong to the integration mesh, which may differ
- from the associated mesh belonging to either function.
- Thus, for each element block containing
-\begin_inset Formula $k$
-\end_inset
-
- cells, we have the output array
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(k,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Residual Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{k}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-
-\newline
-These local residual values may be combined in the following form:
-\begin_inset Formula \[
-\varepsilon_{block}=\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\cdots+\varepsilon_{k}^{2}}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The final global residual may be obtained by simply summing over each element
- block in the integration mesh:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\varepsilon_{global}=\sqrt{\sum_{block}\varepsilon_{block}^{2}}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-Numerical Integration
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-Go over numerical integration in 2d and 3d.
-\end_layout
-
-\begin_layout Standard
-
-Required order of numerical integration.
-\end_layout
-
-\begin_layout Standard
-
-Give references to defining your own points here.
- Explain how to use the scripts here.
-\end_layout
-
-\begin_layout Standard
-
-For each element, give the default quadrature rule employed and its order
- of accuracy.
-\end_layout
-
-\begin_layout Standard
-
-Give more examples of quadrature rules.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Mention tabulation of shape function values here.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-References to mention here:
-\end_layout
-
-\begin_layout Standard
-
- FIAT homepage
-\end_layout
-
-\begin_layout Standard
-
- FILIPPA mathematica notebooks
-\end_layout
-
-\begin_layout Standard
-
- Encyclopedia of Cubature Formulas (url?)
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-The core concepts to get across in this chapter are the following:
-\end_layout
-
-\begin_layout Enumerate
-Each element in Cigma is assigned a default quadrature rule
-\end_layout
-
-\begin_layout Enumerate
-Explain briefly how quadrature rules are obtained (product rules)
-\end_layout
-
-\begin_layout Enumerate
-Error bound on integration (dependent on the discretization size h, and
- the order of integration q).
-\end_layout
-
-\begin_layout Enumerate
-How to obtain more quadrature rules (encyclopedia website, FIAT, mathematica
- scripts -- don't forget to mention caveat about using rules that use different
- reference cells -- points transform according to map, while weights are
- multiplied by the appropriate jacobian factor).
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In order to calculate the
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm integral from Chapter 3, we must resort to numerical integration.
- If we integrate over a mesh which does not match the underlying discretization
- of the functions being compared, you may need to increase the accuracy
- of the integration.
- In this chapter we give a brief overview of the quadrature rules available,
- and how you may create your own.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Define integration rule.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In general, we may define an integration rule as a set of points with associated
- weights for which we may approximate an integration procedure by a simple
- weighed sum
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{Q}w_{q}F(\vec{x}_{q})+R\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-It is possible to integrate, but the remainder
-\begin_inset Formula $R$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Gaussian Quadrature
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Hughes, pg 143
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The general Gaussian quadrature rule with
-\begin_inset Formula $Q$
-\end_inset
-
- integration points is given in Hughes, page 143,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}F(\xi)\ d\xi=\sum_{q=1}^{Q}w_{q}F(\xi_{q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-w_{i} & = & \frac{2}{(1-\xi_{i}^{2})P_{Q}'(\xi_{i})^{2}}\\
-R & = & \frac{2^{2Q+1}(Q!)^{4}}{(2Q+1)[(2Q)!]^{3}}F^{(2Q)}(\bar{\xi})\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-where
-\begin_inset Formula $\xi_{i}$
-\end_inset
-
- is the
-\begin_inset Formula $i$
-\end_inset
-
--th zero of the Legendre Polynomial
-\begin_inset Formula $P_{Q}(\xi)$
-\end_inset
-
-, and
-\begin_inset Formula $P_{Q}^{'}$
-\end_inset
-
- denotes the derivative of
-\begin_inset Formula $P_{Q}$
-\end_inset
-
-.
- The Legendre polynomials are defined by
-\begin_inset Formula \[
-P_{Q}(\xi)=\frac{1}{2^{Q}Q!}\frac{d^{Q}}{d\xi^{Q}}(\xi^{2}-1)^{Q}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In multiple dimensions, we simply use a different one-dimensional Gaussian
- rule on each direction.
- For two dimensions, this becomes
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta)\ d\xi d\eta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}w_{q_{1}}w_{q_{2}}F(\xi_{q_{1}},\eta_{q_{2}})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-and similarly for three dimensions
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta,\zeta)\ d\xi d\eta d\zeta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}\sum_{q_{3}=1}^{Q_{3}}w_{q_{1}}w_{q_{2}}w_{q_{3}}F(\xi_{q_{1}},\eta_{q_{2}},\zeta_{q_{3}})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Section
-Quadrilaterals and Hexahedra
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Since these two are very similar, keep them together.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-For the reference quadrilateral and hexahedral elements, we can use the
-
-\end_layout
-
-\begin_layout Standard
-For quadrilaterals, we use the same reference region
-\begin_inset Formula $[-1,1]\times[-1,1]$
-\end_inset
-
- as in Section 4.X.
-\end_layout
-
-\begin_layout Standard
-The reference region for this geometry is
-\begin_inset Formula $[-1,1]\times[-1,1]\times[-1,1]$
-\end_inset
-
-, as in Section 4.X.
-
-\end_layout
-
-\begin_layout Section
-Triangles and Tetrahedra
-\end_layout
-
-\begin_layout Standard
-To easily obtain high order accuracy quadrature rules for triangles, we
- can re-use the same rules for quadrilaterals by collapsing two of the nodes
- of the quadrilateral.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Fig.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Similar to the triangle
-\end_layout
-
-\begin_layout Section
-Other Integration Rules
-\end_layout
-
-\begin_layout Standard
-The integration points given in this section were obtained by using the
- FIAT Python module.
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/appendix.lyx (from rev 13032, doc/cigma/manual/appendix/appendix.lyx)
===================================================================
--- doc/cigma/manual/appendix.lyx (rev 0)
+++ doc/cigma/manual/appendix.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,2476 @@
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+XXX: Content moved to formats.lyx, what else do we need in an appendix?
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Data Formats
+\end_layout
+
+\begin_layout Standard
+Earlier in Chapter 6, we discussed the file formats that Cigma understands.
+ Here, we summarize for your convenience the dimensions of all the major
+ data structures used in Cigma.
+ How you access the data might change depending on the type of input or
+ output file where you store the data values.
+ The actual layout for each file format is discussed in Chapter 6.
+ Currently recognized file extensions include
+\family typewriter
+(*.dat)
+\family default
+ for simple text files,
+\family typewriter
+(*.h5)
+\family default
+ for HDF5 files,
+\family typewriter
+(*.vtk)
+\family default
+ for legacy VTK files,
+\family typewriter
+(*.vtu, *.vts, *.vtr)
+\family default
+ for XML VTK files, and
+\family typewriter
+(*.pvtu, *.pvts, *.pvtr)
+\family default
+ for Parallel XML VTK files.
+\end_layout
+
+\begin_layout Section
+Notation
+\end_layout
+
+\begin_layout Standard
+Throughout this appendix we will refer to
+\end_layout
+
+\begin_layout Section
+Mesh Coordinates
+\end_layout
+
+\begin_layout Standard
+On a mesh with
+\begin_inset Formula $n_{no}$
+\end_inset
+
+ node coordinates, which are specified on a global coordinate system, we
+ have
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Standard
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "50col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{no},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{1},y_{1},z_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{2},y_{2},z_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{n_{no}},y_{n_{no}},z_{n_{no}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "50col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{no},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{1},y_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{2},y_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{n_{no}},y_{n_{no}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Mesh Connectivity
+\end_layout
+
+\begin_layout Standard
+Mesh connectivity is specified at the element-block level.
+ On a given block, we only consider a single element type, which allows
+ us to store the global node ids into a contiguous array of integers.
+ In general, if our block contains
+\begin_inset Formula $k$
+\end_inset
+
+ elements with
+\begin_inset Formula $m$
+\end_inset
+
+-degrees of freedom each, then the connectivity array defining our element
+ block has the form
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(k,m)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Connectivity Data
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $n_{11},n_{12},\ldots,n_{1m}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $n_{21},n_{22},\ldots,n_{2m}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $n_{k1},n_{k2},\ldots,n_{km}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\end_layout
+
+\begin_layout Standard
+For the element types defined in Cigma, the values for
+\begin_inset Formula $m$
+\end_inset
+
+ are as follows.
+ Note that the total number of degrees of freedom depends on the rank of
+ the specific function being represented.
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="4">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Element
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Tensor
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+tet4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+12
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+24
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+tet10
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+10
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+30
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+60
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+hex8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+24
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+48
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+hex20
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+20
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+60
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+120
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+hex27
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+27
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+81
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+162
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\end_layout
+
+\begin_layout Standard
+Similarly for the 2-D elements, we have
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="4">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Element
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Tensor
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+tri3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+6
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+9
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+quad4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+12
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Field Variables
+\end_layout
+
+\begin_layout Standard
+A finite element solution basically corresponds to a list of values for
+ each degree of freedom in each element.
+ Since the shared nodes among adjacent elements have the same value, these
+
+\end_layout
+
+\begin_layout Section
+Integration Rule
+\end_layout
+
+\begin_layout Standard
+As described in Chapter 4, an integration rule is specified by using the
+ natural coordinate system of the reference element.
+
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="5">
+<features>
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Weights:
+\begin_inset Formula $(Q,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Points:
+\begin_inset Formula $(Q,n_{dim})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Points:
+\begin_inset Formula $(Q,2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Points:
+\begin_inset Formula $(Q,3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Rule Definition
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $w_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{\xi}_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{1},\eta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{1},\eta_{1},\zeta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $w_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{\xi}_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{2},\eta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{2},\eta_{2},\zeta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $w_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{\xi}_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{Q},\eta_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\xi_{Q},\eta_{Q},\zeta_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Shape Function Values
+\end_layout
+
+\begin_layout Standard
+As discussed in Chapter 7, in certain circumstances you may be able to provide
+ your custom reference element by simply providing the appropriate
+\begin_inset Formula $n$
+\end_inset
+
+ shape function values at the expected integration points.
+\newline
+
+\newline
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="3">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Shape Function Values:
+\begin_inset Formula $(n,Q)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Jacobian Determinant:
+\begin_inset Formula $(Q,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Integration Point 1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $N_{1}(\vec{\xi}_{1}),N_{2}(\vec{\xi}_{1}),\ldots,N_{n}(\vec{\xi}_{1})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $J(\vec{\xi}_{1})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Integration Point 2
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $N_{1}(\vec{\xi}_{2}),N_{2}(\vec{\xi}_{2}),\ldots,N_{n}(\vec{\xi}_{2})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $J(\vec{\xi}_{2})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Integration Point
+\begin_inset Formula $Q$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $N_{1}(\vec{\xi}_{Q}),N_{2}(\vec{\xi}_{Q}),\ldots,N_{n}(\vec{\xi}_{Q})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $J(\vec{\xi}_{Q})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Integration Points
+\end_layout
+
+\begin_layout Standard
+Given a mesh and an integration rule, we can map the integration points
+ on each element into a possibly large set of global points.
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="4">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Points:
+\begin_inset Formula $(n_{pts},n_{dim})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Points:
+\begin_inset Formula $(n_{pts},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Points:
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{x}_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{1},y_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{1},y_{1},z_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{x}_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{2},y_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{2},y_{2},z_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vec{x}_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{n_{pts}}y_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $x_{n_{pts}},y_{n_{pts}},z_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Function Values at Integration Points
+\end_layout
+
+\begin_layout Standard
+The result of an evaluation will depend on the rank of the function being
+ evaluated, as indicated in the table below.
+ Note that the fastest varying dimension of the array contains the values
+ of the function at each point.
+\newline
+
+\newline
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Evaluated Quantity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Resulting Array Shape
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{pts},1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{pts},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+2-D Tensor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+3-D Tensor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(n_{pts},6)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Comparison Residuals
+\end_layout
+
+\begin_layout Standard
+Since our norm reduces integrals over cells to a single scalar value, the
+ result of a comparison will result in a simple scalar array for each element
+ block examined.
+ The element blocks used belong to the integration mesh, which may differ
+ from the associated mesh belonging to either function.
+ Thus, for each element block containing
+\begin_inset Formula $k$
+\end_inset
+
+ cells, we have the output array
+\newline
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $(k,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Residual Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\varepsilon_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\varepsilon_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Formula $\varepsilon_{k}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\newline
+
+\newline
+These local residual values may be combined in the following form:
+\begin_inset Formula \[
+\varepsilon_{block}=\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\cdots+\varepsilon_{k}^{2}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The final global residual may be obtained by simply summing over each element
+ block in the integration mesh:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\varepsilon_{global}=\sqrt{\sum_{block}\varepsilon_{block}^{2}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Numerical Integration
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+Go over numerical integration in 2d and 3d.
+\end_layout
+
+\begin_layout Standard
+
+Required order of numerical integration.
+\end_layout
+
+\begin_layout Standard
+
+Give references to defining your own points here.
+ Explain how to use the scripts here.
+\end_layout
+
+\begin_layout Standard
+
+For each element, give the default quadrature rule employed and its order
+ of accuracy.
+\end_layout
+
+\begin_layout Standard
+
+Give more examples of quadrature rules.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Mention tabulation of shape function values here.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+References to mention here:
+\end_layout
+
+\begin_layout Standard
+
+ FIAT homepage
+\end_layout
+
+\begin_layout Standard
+
+ FILIPPA mathematica notebooks
+\end_layout
+
+\begin_layout Standard
+
+ Encyclopedia of Cubature Formulas (url?)
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+The core concepts to get across in this chapter are the following:
+\end_layout
+
+\begin_layout Enumerate
+Each element in Cigma is assigned a default quadrature rule
+\end_layout
+
+\begin_layout Enumerate
+Explain briefly how quadrature rules are obtained (product rules)
+\end_layout
+
+\begin_layout Enumerate
+Error bound on integration (dependent on the discretization size h, and
+ the order of integration q).
+\end_layout
+
+\begin_layout Enumerate
+How to obtain more quadrature rules (encyclopedia website, FIAT, mathematica
+ scripts -- don't forget to mention caveat about using rules that use different
+ reference cells -- points transform according to map, while weights are
+ multiplied by the appropriate jacobian factor).
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In order to calculate the
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm integral from Chapter 3, we must resort to numerical integration.
+ If we integrate over a mesh which does not match the underlying discretization
+ of the functions being compared, you may need to increase the accuracy
+ of the integration.
+ In this chapter we give a brief overview of the quadrature rules available,
+ and how you may create your own.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Define integration rule.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In general, we may define an integration rule as a set of points with associated
+ weights for which we may approximate an integration procedure by a simple
+ weighed sum
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{Q}w_{q}F(\vec{x}_{q})+R\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+It is possible to integrate, but the remainder
+\begin_inset Formula $R$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gaussian Quadrature
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Hughes, pg 143
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The general Gaussian quadrature rule with
+\begin_inset Formula $Q$
+\end_inset
+
+ integration points is given in Hughes, page 143,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}F(\xi)\ d\xi=\sum_{q=1}^{Q}w_{q}F(\xi_{q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+w_{i} & = & \frac{2}{(1-\xi_{i}^{2})P_{Q}'(\xi_{i})^{2}}\\
+R & = & \frac{2^{2Q+1}(Q!)^{4}}{(2Q+1)[(2Q)!]^{3}}F^{(2Q)}(\bar{\xi})\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+where
+\begin_inset Formula $\xi_{i}$
+\end_inset
+
+ is the
+\begin_inset Formula $i$
+\end_inset
+
+-th zero of the Legendre Polynomial
+\begin_inset Formula $P_{Q}(\xi)$
+\end_inset
+
+, and
+\begin_inset Formula $P_{Q}^{'}$
+\end_inset
+
+ denotes the derivative of
+\begin_inset Formula $P_{Q}$
+\end_inset
+
+.
+ The Legendre polynomials are defined by
+\begin_inset Formula \[
+P_{Q}(\xi)=\frac{1}{2^{Q}Q!}\frac{d^{Q}}{d\xi^{Q}}(\xi^{2}-1)^{Q}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In multiple dimensions, we simply use a different one-dimensional Gaussian
+ rule on each direction.
+ For two dimensions, this becomes
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta)\ d\xi d\eta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}w_{q_{1}}w_{q_{2}}F(\xi_{q_{1}},\eta_{q_{2}})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+and similarly for three dimensions
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta,\zeta)\ d\xi d\eta d\zeta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}\sum_{q_{3}=1}^{Q_{3}}w_{q_{1}}w_{q_{2}}w_{q_{3}}F(\xi_{q_{1}},\eta_{q_{2}},\zeta_{q_{3}})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Section
+Quadrilaterals and Hexahedra
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Since these two are very similar, keep them together.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For the reference quadrilateral and hexahedral elements, we can use the
+
+\end_layout
+
+\begin_layout Standard
+For quadrilaterals, we use the same reference region
+\begin_inset Formula $[-1,1]\times[-1,1]$
+\end_inset
+
+ as in Section 4.X.
+\end_layout
+
+\begin_layout Standard
+The reference region for this geometry is
+\begin_inset Formula $[-1,1]\times[-1,1]\times[-1,1]$
+\end_inset
+
+, as in Section 4.X.
+
+\end_layout
+
+\begin_layout Section
+Triangles and Tetrahedra
+\end_layout
+
+\begin_layout Standard
+To easily obtain high order accuracy quadrature rules for triangles, we
+ can re-use the same rules for quadrilaterals by collapsing two of the nodes
+ of the quadrilateral.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Fig.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Similar to the triangle
+\end_layout
+
+\begin_layout Section
+Other Integration Rules
+\end_layout
+
+\begin_layout Standard
+The integration points given in this section were obtained by using the
+ FIAT Python module.
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
Modified: doc/cigma/manual/cigma.lyx
===================================================================
--- doc/cigma/manual/cigma.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/cigma.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,5 +1,5 @@
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\begin_layout Author
© California Institute of Technology
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Version 1.0.0
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Global TODO Items
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* Don't forget to change version on cover to version 1.0.0 (done)
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* Change the cover image to use a newer comparison
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* Add description of basic ExodusII format in file_formats.lyx (delay
for version 1.1 -- use Scientific.IO.NetCDF for now)
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* grep for 'Chapter' and 'Section' and make sure that all such numbers
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References:
\end_layout
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\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
* Hughes, Thomas J.R., The Finite Element Method: Linear Static and Dynamic
Finite Element Analysis, Prentice-Hall 1987 (use Dover ref instead)
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
* Zienkiewicz, O.C., Taylor, R.L.
The Finite Element Method: The Basis (Vol.
1)
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
* Akin, J.E., Finite Element Analysis with Error Estimators, Elsevier 2005
(ISBN 0750667222)
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
* Johnson, Claes - Numerical Solutions of Partial Differential Equations
by the Finite Element Method, Cambridge University Press 1987
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
* Carlos Felippa, A Compendium of FEM integration formulas for symbolic
work, Engineering Computation, Volume 21, Number 8, 2004, pages 867-890
\end_layout
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}
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@@ -431,7 +448,8 @@
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label "5"
key "Uesu-etal2005"
Deleted: doc/cigma/manual/error_analysis/error_analysis.lyx
===================================================================
--- doc/cigma/manual/error_analysis/error_analysis.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/error_analysis/error_analysis.lyx 2008-12-10 02:11:13 UTC (rev 13482)
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- and just reference later chapters if we need to.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-In reference to error analysis, we introduce the topics in the following
- order
-\end_layout
-
-\begin_layout Standard
-
- - Quick reference to distance measures, both pointwise and global L2
-\end_layout
-
-\begin_layout Standard
-
- - Define function as used in cigma
-\end_layout
-
-\begin_layout Standard
-
- - Define integration rule in this chapter
-\end_layout
-
-\begin_layout Standard
-
- * in general (1D, 2D, 3D)
-\end_layout
-
-\begin_layout Standard
-
- * as applied to a discretization (sum over cells)
-\end_layout
-
-\begin_layout Standard
-
- * generating own integration points (XXX: moved to integration.lyx) --
- give references
-\end_layout
-
-\begin_layout Standard
-
- (give warning about using a different reference element -- weights,
- not just points, need to be adjusted -- perhaps this warning belongs in
- the next chapter)
-\end_layout
-
-\begin_layout Standard
-
- - Mention the interpolation functions that are available (XXX: moved to
- interpolation.lyx)
-\end_layout
-
-\begin_layout Standard
-
- * global functions (predefined: zero, scalar.one, ...)
-\end_layout
-
-\begin_layout Standard
-
- * FEM on a given discretization (local interpolations on tet4, hex8,
- tri3, quad3, ...
- elements)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
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-
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-\end_layout
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- Change name of chapter to something else (no longer necessary)
-\end_layout
-
-\begin_layout Standard
-
- Give formula in the last section
-\end_layout
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-
- Move discussion of convergence order from examples.lyx and merge it at
- the end of this chapter
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-
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-
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-}
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-
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-name "cha:Error-Analysis"
-
-\end_inset
-
-Error Analysis
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Here, discuss motivation and goals.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-When solving differential equations representing physical systems of interest,
- we are often able to obtain a family of solutions by applying a variety
- of solution techniques.
- Sometimes an analytic method can be found, but most of the time we end
- up resorting to a numerical algorithm, such as the finite element method.
- Assessing the quality of these solutions is an important task, so we would
- like to develop a quantitative measure for indicating just how close our
- solutions approach the exact answer.
-\end_layout
-
-\begin_layout Standard
-The simplest possible quantitative measure of the difference between two
- distinct functions consists of taking the pointwise difference at a common
- set of points.
- While no finite sample of points can perfectly represent a continuum of
- values, valuable information can be inferred from the statistics of the
- resulting set of differences.
- However, the functions we want to compare may not be defined at a common
- set of points.
- Unless we are able to interpolate the two functions on an intermediate
- set of points, this simple pointwise measure becomes inapplicable.
-\end_layout
-
-\begin_layout Standard
-A very useful distance measure we can use is the
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm, defined by the following integral
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{equation}
-\varepsilon=||u-v||_{L_{2}}=\sqrt{\int_{\Omega}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}}\end{equation}
-
-\end_inset
-
-where
-\begin_inset Formula $u$
-\end_inset
-
- and
-\begin_inset Formula $v$
-\end_inset
-
- are the two functions defined on a global coordinate system.
- This gives us a single global estimate
-\begin_inset Formula $\varepsilon$
-\end_inset
-
- representing the distance between the two functions
-\begin_inset Formula $u(\vec{x})$
-\end_inset
-
- and
-\begin_inset Formula $v(\vec{x})$
-\end_inset
-
-.
- Alternatively, you may think of this as the size, or norm, of the
-\series bold
-\emph on
-residual field
-\series default
-\emph default
-
-\begin_inset Formula $\rho(\vec{x})=u(\vec{x})-v(\vec{x})$
-\end_inset
-
-.
- This measure is useful because it is well known that solutions obtained
- with the finite element method converge only in a weak sense, as an average
- over a geometric region.
- In the rest of this chapter we will discuss the details involved in evaluating
- the above integral.
-\end_layout
-
-\begin_layout Standard
-Another quantitative measure, which we won't discuss in this version of
- Cigma, is the energy norm.
- This norm is typically employed in
-\emph on
-a posteriori
-\emph default
- error analysis and is problem dependent.
- In general, the residual field used under this norm is defined in terms
- of the linearized equation from which the solution was obtained.
- Yet another interesting possibility would be to
-\end_layout
-
-\begin_layout Section
-Functions
-\end_layout
-
-\begin_layout Standard
-The functions that we discuss in this manual are assumed to be defined in
- a common region of space denoted by
-\begin_inset Formula $\Omega$
-\end_inset
-
-.
- Elements of
-\begin_inset Formula $\Omega$
-\end_inset
-
- are written as
-\begin_inset Formula $\vec{x}$
-\end_inset
-
-, and the number of components in the corresponding value of
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
-, is said to be the
-\series bold
-\emph on
-rank
-\series default
-\emph default
- of the function
-\begin_inset Formula $f$
-\end_inset
-
-.
- Thus,
-\emph on
-scalar functions
-\emph default
- have a rank of 1,
-\emph on
-vector functions
-\emph default
- would have a rank of 2 or 3, and
-\emph on
-tensor functions
-\emph default
- would have a rank of 6 or 9.
-\end_layout
-
-\begin_layout Section
-Integral Approximations
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Briefer discussion here.
- Move this to Chapter 5.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In order to evaluate the integral for our
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm, we will need to define an integration mesh which partitions the common
- domain
-\begin_inset Formula $\Omega$
-\end_inset
-
- on which our two functions are defined.
- We can then use a numerical approximation, or
-\series bold
-\emph on
-integration rule
-\series default
-\emph default
-, on each of the discrete cell elements
-\begin_inset Formula $\Omega_{i}$
-\end_inset
-
- of our partition.
- Each of these cell integrals of the continuous residual norm
-\begin_inset Formula $||\rho(\vec{x})||^{2}$
-\end_inset
-
- can then be replaced by a weighed sum evaluated at a finite set of points.
- This integral will be valid up to a certain accuracy.
- In Chapter 5, we discuss in more detail how choose the location and values
- for the integration points and weights which give optimal accuracy.
-\end_layout
-
-\begin_layout Standard
-Thus, in general, an integration rule with
-\begin_inset Formula $Q$
-\end_inset
-
- points and weights that allows us to integrate the scalar function
-\begin_inset Formula $F(\vec{x})$
-\end_inset
-
- is given by the equation
-\begin_inset Formula \begin{equation}
-\int_{\Omega_{i}}\ F(\vec{x})\ d\vec{x}\approx\sum_{q=1}^{Q}F(\vec{X_{q}})W_{q}\end{equation}
-
-\end_inset
-
-where the integration points
-\begin_inset Formula $\vec{X}_{q}$
-\end_inset
-
- and integration weights
-\begin_inset Formula $W_{q}$
-\end_inset
-
- depend on the integration region
-\begin_inset Formula $\Omega_{i}$
-\end_inset
-
-.
- As we will see in Chapter 7, one possibility is to pre-calculate these
- points and weights explicitly on a specific discretization.
- Thus, as discussed in more detail in Chapters 4 and 5, we will generally
- want to define a reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
- which acts as a template for cells with the same geometric shape.
- We can achieve this by defining a
-\series bold
-\emph on
-reference map
-\series default
-\emph default
-
-\begin_inset Formula $\chi_{i}:\hat{\Omega}\to\Omega_{i}$
-\end_inset
-
- that describes how points in a specific cell
-\begin_inset Formula $\Omega_{i}$
-\end_inset
-
- originate from the reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
- In this sense, the reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
- is said to define a
-\series bold
-\emph on
-local coordinate system
-\series default
-\emph default
- for the
-\begin_inset Formula $i$
-\end_inset
-
--th cell
-\begin_inset Formula $\Omega_{i}.$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-FIGURE refmap.png:
-\backslash
-chi_i:
-\backslash
-hat{
-\backslash
-Omega}
-\backslash
-to
-\backslash
-Omega_i
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Note that a given discretization may contain cells of different shapes,
- which would require us to define an appropriate number of reference cells
- for each type of cell.
- However, for the purposes of this manual we may restrict the discussions
- in this manuals, without any loss of generality, to discretizations consisting
- of a single geometric shape.
- (XXX: reword this last sentence)
-\end_layout
-
-\begin_layout Standard
-The advantage of this approach is readily apparent when one realizes that
- the integration points and weights can be calculated once and for all on
- the reference cell, and then reused for the other cells through the application
- of the corresponding reference map.
- In other words,
-\begin_inset Formula \begin{eqnarray*}
-\vec{X}_{q} & = & \chi_{i}(\vec{\xi}_{q})\\
-W_{q} & = & w_{q}J_{i}(\vec{\xi}_{q})\end{eqnarray*}
-
-\end_inset
-
-where
-\begin_inset Formula $\vec{\xi}_{q}$
-\end_inset
-
- and
-\begin_inset Formula $w_{q}$
-\end_inset
-
- are the optimal integration points and weights for the integration rule
- defined on the reference domain
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
- Here, the factor
-\begin_inset Formula $J_{i}(\vec{\xi})=\det\ \left|\frac{d\chi_{i}}{d\vec{\xi}}\right|$
-\end_inset
-
- is the Jacobian determinant of the transformation
-\begin_inset Formula $\chi_{i}$
-\end_inset
-
-.
- Recall that the index
-\begin_inset Formula $i$
-\end_inset
-
- corresponds to the
-\begin_inset Formula $i$
-\end_inset
-
--th cell
-\begin_inset Formula $\Omega_{i}$
-\end_inset
-
-, and the index
-\begin_inset Formula $q$
-\end_inset
-
- ranges over the usual
-\begin_inset Formula $q=1,\ldots,Q$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Section
-Interpolation Functions
-\end_layout
-
-\begin_layout Standard
-Our two functions
-\begin_inset Formula $u(\vec{x})$
-\end_inset
-
- and
-\begin_inset Formula $v(\vec{x})$
-\end_inset
-
- are given in terms of the point
-\begin_inset Formula $\vec{x}\in\Omega$
-\end_inset
-
-, and are said to be defined on a
-\series bold
-\emph on
-global coordinate system
-\series default
-\emph default
-.
- Based on our discussion in Section 3.1, we see that we only need to know
- the values of
-\begin_inset Formula $u$
-\end_inset
-
- and
-\begin_inset Formula $v$
-\end_inset
-
- at a finite set of global points in order to calculate an approximation
- to the
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm of the residual field
-\begin_inset Formula $\rho=u-v$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Specifically, given a function
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
- we must be able to calculate its values at exactly those integration points.
- If we know a formula or algorithm for
-\begin_inset Formula $f$
-\end_inset
-
-, then our task is easy.
- Alternatively, these values can be given explicitly as a finite list.
- In all other cases, we will need an interpolation scheme that allows us
- to calculate
-\begin_inset Formula $f$
-\end_inset
-
- on any intermediate points.
- This ability is also important because we may want to increase the accuracy
- of our norm by using more integration points, and thus we will need to
- be able to evaluate our function
-\begin_inset Formula $f$
-\end_inset
-
- at the new points.
-\end_layout
-
-\begin_layout Standard
-In general, an interpolation scheme involves a set of known functions
-\begin_inset Formula $\phi_{j}(\vec{x})$
-\end_inset
-
- that we can compute anywhere and a set of parameters
-\begin_inset Formula $c_{j}$
-\end_inset
-
- that define how these known functions are combined.
- Usually, the relation between the parameters and the known functions is
- a linear combination,
-\begin_inset Formula \begin{equation}
-f(\vec{x})=d_{1}\phi_{1}(\vec{x})+d_{2}\phi_{2}(\vec{x})+\cdots+d_{m}\phi_{m}(\vec{x})\end{equation}
-
-\end_inset
-
-where the parameters
-\begin_inset Formula $d_{j}$
-\end_inset
-
-, with
-\begin_inset Formula $j=1,\ldots,m$
-\end_inset
-
-, are also sometimes referred to as the
-\series bold
-\emph on
-degrees of freedom
-\series default
-\emph default
- of the function
-\begin_inset Formula $f$
-\end_inset
-
-.
- As described in Chapter 4, the functions
-\begin_inset Formula $\phi_{j}$
-\end_inset
-
-, also known as
-\series bold
-\emph on
-shape functions
-\series default
-\emph default
-, must satisfy a number of conditions.
- Because the choice of shape functions affects how well the true solution
- is represented, the convergence of the numerical method used to obtain
- the solution, the optimal choice of shape functions is very problem dependent.
-\end_layout
-
-\begin_layout Standard
-If we know the shape functions
-\begin_inset Formula $\phi_{j}$
-\end_inset
-
- directly in terms of the global points
-\begin_inset Formula $\vec{x}\in\Omega$
-\end_inset
-
-, they are said to form a
-\series bold
-\emph on
-global interpolation scheme
-\series default
-\emph default
-, and we may use Eq X directly to find the values of
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
-, and we may refer to the
-\begin_inset Formula $\phi_{j}$
-\end_inset
-
-as
-\emph on
-global shape functions
-\emph default
-.
- Note that since we are working in the global coordinate system, we don't
- need the discretization of the domain
-\begin_inset Formula $\Omega$
-\end_inset
-
-.
- Examples of global shape functions include spherical harmonics, and ...
-\end_layout
-
-\begin_layout Standard
-Perhaps a more typical case is when
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
- is defined piecewise, on each discretization element
-\begin_inset Formula $\Omega_{i}$
-\end_inset
-
-.
- Because these cells partition the original domain by definition, a given
- point in our domain will be found in one and only one cell, which will
- have a local definition.
- That is, for a particular
-\begin_inset Formula $\vec{x}\in\Omega$
-\end_inset
-
- we will be able to find a unique cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-, for some index
-\begin_inset Formula $e$
-\end_inset
-
-.
- We can refer to this as a
-\series bold
-\emph on
-local interpolation scheme
-\series default
-\emph default
-.
-\end_layout
-
-\begin_layout Standard
-See Chapter 5, for more details on the specific interpolation schemes available
- in Cigma.
-\end_layout
-
-\begin_layout Section
-Global Error Measure
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Suppose the domain
-\begin_inset Formula $\Omega$
-\end_inset
-
- into an appropriate set of cells
-\begin_inset Formula $\Omega_{1},\Omega_{2},\ldots,\Omega_{n_{el}}$
-\end_inset
-
-, we can compute the global error
-\begin_inset Formula $\varepsilon^{2}$
-\end_inset
-
- as a sum over localized cell contributions,
-\begin_inset Formula $\varepsilon^{2}=\sum_{e}\varepsilon_{e}^{2}$
-\end_inset
-
-, where each cell residual
-\begin_inset Formula $\varepsilon_{e}^{2}$
-\end_inset
-
- is given by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In general, we won't be able to integrate each cell residual
-\begin_inset Formula $\varepsilon_{e}^{2}$
-\end_inset
-
- exactly since either of the functions
-\begin_inset Formula $u$
-\end_inset
-
- and
-\begin_inset Formula $v$
-\end_inset
-
- may have an incompatible representation relative to the finite element
- space on each domain
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-.
- However, we can still calculate an approximation of each cell residual
- by applying an appropriate quadrature rule with a tolerable truncation
- error
-\begin_inset LatexCommand cite
-key "Encyclopaedia of Cubature Formulas 2005"
-
-\end_inset
-
-.
-
-\end_layout
-
-\begin_layout Standard
-To obtain an approximation to the integral of a function
-\begin_inset Formula $F(\vec{x})$
-\end_inset
-
- over a cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-, we simply apply the quadrature rule with weights
-\begin_inset Formula $w_{e,1},w_{e,2},\ldots,w_{e,n_{Q}}$
-\end_inset
-
- and integration points
-\begin_inset Formula $\vec{x}_{e,1},\vec{x}_{e,2},\ldots,\vec{x}_{e,n_{Q}}$
-\end_inset
-
- appropriate for the physical element
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Applying this quadrature rule directly over the entire physical domain
-\begin_inset Formula $\Omega=\cup\Omega_{e}$
-\end_inset
-
- gives us
-\begin_inset Formula \[
-\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{e=1}^{n_{el}}\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-For efficiency reasons, it is undesirable in finite element applications
- to perform calculations in a global coordinate system.
- To avoid duplication of work, shape function evaluations may be performed
- once on a reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
- and then transformed back into the corresponding physical cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- as needed.
-\end_layout
-
-\begin_layout Standard
-To compute integrals of
-\begin_inset Formula $F$
-\end_inset
-
- in a reference coordinate system, we need to apply a change of variables:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}\]
-
-\end_inset
-
-where
-\color none
-the additional factor
-\begin_inset Formula $J_{e}(\vec{\xi})=\det\left[\frac{d\vec{x}_{e}}{d\vec{\xi}}\right]$
-\end_inset
-
- is the Jacobian determinant of the reference map
-\begin_inset Formula $\vec{x}_{e}(\vec{\xi}):\hat{\Omega}\to\Omega_{e}$
-\end_inset
-
-.
- This map describes how the physical points
-\begin_inset Formula $\vec{x}\in\Omega_{e}$
-\end_inset
-
- are transformed from the reference points
-\begin_inset Formula $\vec{\xi}\in\hat{\Omega}$
-\end_inset
-
-.
- Put another way, the inverse reference map
-\color inherit
-
-\begin_inset Formula $\vec{\xi}=\vec{x}_{e}^{-1}(\vec{x})$
-\end_inset
-
- tells us how the physical domain
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- maps into the reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-At this point, we can assume without loss of generality that every physical
- cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- can be derived from a single reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-, so that our quadrature rule becomes simply a set of weights
-\begin_inset Formula $w_{1},\ldots,w_{n_{Q}}$
-\end_inset
-
- and points
-\begin_inset Formula $\vec{\xi}_{1},\ldots,\vec{\xi}_{n_{Q}}$
-\end_inset
-
- over
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
- After changing variables, we end up with the final form of the quadrature
- rule that we can use to integrate the global residual field:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}=\sum_{q=1}^{n_{Q}}w_{q}F(\vec{x}_{e}(\vec{\xi}_{q}))J_{e}(\vec{\xi}_{q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-If we let
-\begin_inset Formula $F(\vec{x})=||u(\vec{x})-v(\vec{x})||^{2}$
-\end_inset
-
- in the above expression, we can find the cell residual contribution over
-
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- from
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}\ ||u(\vec{x})-v(\vec{x})||^{2}\ d\vec{x}\\
- & = & \int_{\hat{\Omega}}\ ||u(\vec{x}_{e}(\vec{\xi}))-v(\vec{x}_{e}(\vec{\xi}))||^{2}J_{e}(\vec{\xi})\ d\vec{\xi}\\
- & = & \sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The global error
-\begin_inset Formula $\varepsilon=\sqrt{\sum_{e}\varepsilon_{e}^{2}}$
-\end_inset
-
- is then approximated by the expression
-\end_layout
-
-\begin_layout Standard
-\begin_inset Box Boxed
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "100text%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\newline
-
-\end_layout
-
-\begin_layout Standard
-We can make a number of observations based on our final expression for the
- global error.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In this section we show the formula used by Cigma to compute the
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm of the residual field
-\begin_inset Formula $\rho$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-Suppose the domain
-\begin_inset Formula $\Omega$
-\end_inset
-
- into an appropriate set of cells
-\begin_inset Formula $\Omega_{1},\Omega_{2},\ldots,\Omega_{n_{el}}$
-\end_inset
-
-, we can compute the global error
-\begin_inset Formula $\varepsilon^{2}$
-\end_inset
-
- as a sum over localized cell contributions,
-\begin_inset Formula $\varepsilon^{2}=\sum_{e}\varepsilon_{e}^{2}$
-\end_inset
-
-, where each cell residual
-\begin_inset Formula $\varepsilon_{e}^{2}$
-\end_inset
-
- is given by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In general, we won't be able to integrate each cell residual
-\begin_inset Formula $\varepsilon_{e}^{2}$
-\end_inset
-
- exactly since either of the functions
-\begin_inset Formula $u$
-\end_inset
-
- and
-\begin_inset Formula $v$
-\end_inset
-
- may have an incompatible representation relative to the finite element
- space on each domain
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-.
- However, we can still calculate an approximation of each cell residual
- by applying an appropriate quadrature rule with a tolerable truncation
- error
-\begin_inset LatexCommand cite
-key "Encyclopaedia of Cubature Formulas 2005"
-
-\end_inset
-
-.
-
-\end_layout
-
-\begin_layout Standard
-To obtain an approximation to the integral of a function
-\begin_inset Formula $F(\vec{x})$
-\end_inset
-
- over a cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-, we simply apply the quadrature rule with weights
-\begin_inset Formula $w_{e,1},w_{e,2},\ldots,w_{e,n_{Q}}$
-\end_inset
-
- and integration points
-\begin_inset Formula $\vec{x}_{e,1},\vec{x}_{e,2},\ldots,\vec{x}_{e,n_{Q}}$
-\end_inset
-
- appropriate for the physical element
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Applying this quadrature rule directly over the entire physical domain
-\begin_inset Formula $\Omega=\cup\Omega_{e}$
-\end_inset
-
- gives us
-\begin_inset Formula \[
-\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{e=1}^{n_{el}}\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-For efficiency reasons, it is undesirable in finite element applications
- to perform calculations in a global coordinate system.
- To avoid duplication of work, shape function evaluations may be performed
- once on a reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
- and then transformed back into the corresponding physical cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- as needed.
-\end_layout
-
-\begin_layout Standard
-To compute integrals of
-\begin_inset Formula $F$
-\end_inset
-
- in a reference coordinate system, we need to apply a change of variables:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}\]
-
-\end_inset
-
-where
-\color none
-the additional factor
-\begin_inset Formula $J_{e}(\vec{\xi})=\det\left[\frac{d\vec{x}_{e}}{d\vec{\xi}}\right]$
-\end_inset
-
- is the Jacobian determinant of the reference map
-\begin_inset Formula $\vec{x}_{e}(\vec{\xi}):\hat{\Omega}\to\Omega_{e}$
-\end_inset
-
-.
- This map describes how the physical points
-\begin_inset Formula $\vec{x}\in\Omega_{e}$
-\end_inset
-
- are transformed from the reference points
-\begin_inset Formula $\vec{\xi}\in\hat{\Omega}$
-\end_inset
-
-.
- Put another way, the inverse reference map
-\color inherit
-
-\begin_inset Formula $\vec{\xi}=\vec{x}_{e}^{-1}(\vec{x})$
-\end_inset
-
- tells us how the physical domain
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- maps into the reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Standard
-At this point, we can assume without loss of generality that every physical
- cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- can be derived from a single reference cell
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-, so that our quadrature rule becomes simply a set of weights
-\begin_inset Formula $w_{1},\ldots,w_{n_{Q}}$
-\end_inset
-
- and points
-\begin_inset Formula $\vec{\xi}_{1},\ldots,\vec{\xi}_{n_{Q}}$
-\end_inset
-
- over
-\begin_inset Formula $\hat{\Omega}$
-\end_inset
-
-.
- After changing variables, we end up with the final form of the quadrature
- rule that we can use to integrate the global residual field:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}=\sum_{q=1}^{n_{Q}}w_{q}F(\vec{x}_{e}(\vec{\xi}_{q}))J_{e}(\vec{\xi}_{q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-If we let
-\begin_inset Formula $F(\vec{x})=||u(\vec{x})-v(\vec{x})||^{2}$
-\end_inset
-
- in the above expression, we can find the cell residual contribution over
-
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
- from
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}\ ||u(\vec{x})-v(\vec{x})||^{2}\ d\vec{x}\\
- & = & \int_{\hat{\Omega}}\ ||u(\vec{x}_{e}(\vec{\xi}))-v(\vec{x}_{e}(\vec{\xi}))||^{2}J_{e}(\vec{\xi})\ d\vec{\xi}\\
- & = & \sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The global error
-\begin_inset Formula $\varepsilon=\sqrt{\sum_{e}\varepsilon_{e}^{2}}$
-\end_inset
-
- is then approximated by the expression
-\end_layout
-
-\begin_layout Standard
-\begin_inset Box Boxed
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "100text%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\newline
-We use this final form to calculate the global and localized errors on arbitrary
- discretizations.
-\end_layout
-
-\begin_layout Section
-Comparing Residuals
-\end_layout
-
-\begin_layout Standard
-For comparing errors of different solutions, you can normalize the global
- error by the norm of the exact solution.
- For a family of solutions
-\begin_inset Formula $u_{h}(\vec{x})$
-\end_inset
-
-, parametrized by the the maximum element size
-\begin_inset Formula $h$
-\end_inset
-
- of the underlying discretization, and an exact solution
-\begin_inset Formula $u(\vec{x})$
-\end_inset
-
-, this normalized error is given by
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{rel} & = & \frac{||u-u_{h}||_{L_{2}}}{||u||_{L_{2}}}\\
- & = & \frac{\int_{\Omega}||u(\vec{x})-u_{h}(\vec{x})||^{2}d\vec{x}}{\int_{\Omega}||u(\vec{x})||^{2}d\vec{x}}\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-This normalized error can be interpreted as the average error in the physical
- quantity being evaluated, so that a value of 0.01 corresponds to a 1% averaged
- error.
-\end_layout
-
-\begin_layout Standard
-Even if the exact solution is not currently known, this normalized error
- may be used to test the accuracy between two or more numerical solutions,
- defined on successively refined meshes.
-\end_layout
-
-\begin_layout Section
-Convergence Rates
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-TODO: Move discussion of convergence from old running.lyx into this section.
-\end_layout
-
-\begin_layout Standard
-Standard error estimate (Hughes, pg 190; Johnson, pg 97)
-\end_layout
-
-\begin_layout Standard
-Give formula for
-\begin_inset Formula $\alpha$
-\end_inset
-
- for when we know the exact solution (only need two levels), and we are
- testing our numerical code.
-\end_layout
-
-\begin_layout Standard
-Give formula for
-\begin_inset Formula $\alpha$
-\end_inset
-
- for when we have three levels (same formula), and we are testing our code
- for convergence.
-\end_layout
-
-\begin_layout Standard
-Talk about least squares fit for when we have more than three levels.
- (mention scipy script, and reproduce it here, since it's short)
-\end_layout
-
-\begin_layout Standard
-SciPy script for optimizing least squares fit into the formula y = a*x^b
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Even if the exact solution
-\begin_inset Formula $u(\vec{x})$
-\end_inset
-
- is not known, but we have calculated a family of solutions
-\begin_inset Formula $u_{h}(\vec{x})$
-\end_inset
-
- on discrete meshes, we may calculate a convergence rate by using the standard
- error estimate
-\begin_inset Formula $||u-u_{h}||_{p}\leq Ch^{\alpha},$
-\end_inset
-
-where
-\begin_inset Formula $C$
-\end_inset
-
- is independent of both
-\begin_inset Formula $h$
-\end_inset
-
- and
-\begin_inset Formula $u$
-\end_inset
-
-.
- This error bound holds problems
-\end_layout
-
-\begin_layout Section
-Uses in Benchmarking
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Here we define what we mean by Benchmark, and how we can use.
-\end_layout
-
-\begin_layout Standard
-Give URLs for benchmark repositories on CIG svn tree.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-As we saw in Section 3.5, it is easier to calculate the convergence rate
- on a sequence of meshes if we have access to the exact solution.
-
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Since one of our goals is to provide an easy means of calculating these
- comparisons,
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/error_analysis.lyx (from rev 13032, doc/cigma/manual/error_analysis/error_analysis.lyx)
===================================================================
--- doc/cigma/manual/error_analysis.lyx (rev 0)
+++ doc/cigma/manual/error_analysis.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,1484 @@
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\use_default_options false
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Plain Layout
+
+Purpose of this chapter is to provide a high-level description of cigma,
+ and just reference later chapters if we need to.
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+In reference to error analysis, we introduce the topics in the following
+ order
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Quick reference to distance measures, both pointwise and global L2
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Define function as used in cigma
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Define integration rule in this chapter
+\end_layout
+
+\begin_layout Plain Layout
+
+ * in general (1D, 2D, 3D)
+\end_layout
+
+\begin_layout Plain Layout
+
+ * as applied to a discretization (sum over cells)
+\end_layout
+
+\begin_layout Plain Layout
+
+ * generating own integration points (XXX: moved to integration.lyx) --
+ give references
+\end_layout
+
+\begin_layout Plain Layout
+
+ (give warning about using a different reference element -- weights,
+ not just points, need to be adjusted -- perhaps this warning belongs in
+ the next chapter)
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Mention the interpolation functions that are available (XXX: moved to
+ interpolation.lyx)
+\end_layout
+
+\begin_layout Plain Layout
+
+ * global functions (predefined: zero, scalar.one, ...)
+\end_layout
+
+\begin_layout Plain Layout
+
+ * FEM on a given discretization (local interpolations on tet4, hex8,
+ tri3, quad3, ...
+ elements)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Other tasks:
+\end_layout
+
+\begin_layout Plain Layout
+
+ Change name of chapter to something else (no longer necessary)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Give formula in the last section
+\end_layout
+
+\begin_layout Plain Layout
+
+ Move discussion of convergence order from examples.lyx and merge it at
+ the end of this chapter
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Error-Analysis"
+
+\end_inset
+
+Error Analysis
+\end_layout
+
+\begin_layout Standard
+When solving differential equations representing physical systems of interest,
+ we are often able to obtain a family of solutions by applying a variety
+ of solution techniques.
+ Sometimes an analytic method can be found, but most of the time we end
+ up resorting to a numerical algorithm, such as the finite element method.
+ Assessing the quality of these solutions is an important task, so we would
+ like to develop a quantitative measure for indicating just how close our
+ solutions approach the exact answer.
+\end_layout
+
+\begin_layout Standard
+The simplest possible quantitative measure of the difference between two
+ distinct functions consists of taking the pointwise difference at a common
+ set of points.
+ While no finite sample of points can perfectly represent a continuum of
+ values, valuable information can be inferred from the statistics of the
+ resulting set of differences.
+ However, the functions we want to compare may not be defined at a common
+ set of points.
+ Unless we are able to interpolate the two functions on an intermediate
+ set of points, this simple pointwise measure becomes inapplicable.
+\end_layout
+
+\begin_layout Standard
+A very useful distance measure we can use is the
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm, defined by the following integral
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{equation}
+\varepsilon=||u-v||_{L_{2}}=\sqrt{\int_{\Omega}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}}\end{equation}
+
+\end_inset
+
+where
+\begin_inset Formula $u$
+\end_inset
+
+ and
+\begin_inset Formula $v$
+\end_inset
+
+ are the two functions defined on a global coordinate system.
+ This gives us a single global estimate
+\begin_inset Formula $\varepsilon$
+\end_inset
+
+ representing the distance between the two functions
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ and
+\begin_inset Formula $v(\vec{x})$
+\end_inset
+
+.
+ Alternatively, you may think of this as the size, or norm, of the
+\series bold
+\emph on
+residual field
+\series default
+\emph default
+
+\begin_inset Formula $\rho(\vec{x})=u(\vec{x})-v(\vec{x})$
+\end_inset
+
+.
+ This measure is useful because it is well known that solutions obtained
+ with the finite element method converge only in a weak sense, as an average
+ over a geometric region.
+ In the rest of this chapter we will discuss the details involved in evaluating
+ the above integral.
+\end_layout
+
+\begin_layout Standard
+Another quantitative measure, which we won't discuss in this version of
+ Cigma, is the energy norm.
+ This norm is typically employed in
+\emph on
+a posteriori
+\emph default
+ error analysis and is problem dependent.
+ In general, the residual field used under this norm is defined in terms
+ of the linearized equation from which the solution was obtained.
+ Yet another interesting possibility would be to
+\end_layout
+
+\begin_layout Section
+Functions
+\end_layout
+
+\begin_layout Standard
+The functions that we discuss in this manual are assumed to be defined in
+ a common region of space denoted by
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Elements of
+\begin_inset Formula $\Omega$
+\end_inset
+
+ are written as
+\begin_inset Formula $\vec{x}$
+\end_inset
+
+, and the number of components in the corresponding value of
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+, is said to be the
+\series bold
+\emph on
+rank
+\series default
+\emph default
+ of the function
+\begin_inset Formula $f$
+\end_inset
+
+.
+ Thus,
+\emph on
+scalar functions
+\emph default
+ have a rank of 1,
+\emph on
+vector functions
+\emph default
+ would have a rank of 2 or 3, and
+\emph on
+tensor functions
+\emph default
+ would have a rank of 6 or 9.
+\end_layout
+
+\begin_layout Section
+Integral Approximations
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Briefer discussion here.
+ Move this to Chapter 5.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In order to evaluate the integral for our
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm, we will need to define an integration mesh which partitions the common
+ domain
+\begin_inset Formula $\Omega$
+\end_inset
+
+ on which our two functions are defined.
+ We can then use a numerical approximation, or
+\series bold
+\emph on
+integration rule
+\series default
+\emph default
+, on each of the discrete cell elements
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+ of our partition.
+ Each of these cell integrals of the continuous residual norm
+\begin_inset Formula $||\rho(\vec{x})||^{2}$
+\end_inset
+
+ can then be replaced by a weighed sum evaluated at a finite set of points.
+ This integral will be valid up to a certain accuracy.
+ In Chapter 5, we discuss in more detail how choose the location and values
+ for the integration points and weights which give optimal accuracy.
+\end_layout
+
+\begin_layout Standard
+Thus, in general, an integration rule with
+\begin_inset Formula $Q$
+\end_inset
+
+ points and weights that allows us to integrate the scalar function
+\begin_inset Formula $F(\vec{x})$
+\end_inset
+
+ is given by the equation
+\begin_inset Formula \begin{equation}
+\int_{\Omega_{i}}\ F(\vec{x})\ d\vec{x}\approx\sum_{q=1}^{Q}F(\vec{X_{q}})W_{q}\end{equation}
+
+\end_inset
+
+where the integration points
+\begin_inset Formula $\vec{X}_{q}$
+\end_inset
+
+ and integration weights
+\begin_inset Formula $W_{q}$
+\end_inset
+
+ depend on the integration region
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+.
+ As we will see in Chapter 7, one possibility is to pre-calculate these
+ points and weights explicitly on a specific discretization.
+ Thus, as discussed in more detail in Chapters 4 and 5, we will generally
+ want to define a reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ which acts as a template for cells with the same geometric shape.
+ We can achieve this by defining a
+\series bold
+\emph on
+reference map
+\series default
+\emph default
+
+\begin_inset Formula $\chi_{i}:\hat{\Omega}\to\Omega_{i}$
+\end_inset
+
+ that describes how points in a specific cell
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+ originate from the reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ In this sense, the reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ is said to define a
+\series bold
+\emph on
+local coordinate system
+\series default
+\emph default
+ for the
+\begin_inset Formula $i$
+\end_inset
+
+-th cell
+\begin_inset Formula $\Omega_{i}.$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+FIGURE refmap.png:
+\backslash
+chi_i:
+\backslash
+hat{
+\backslash
+Omega}
+\backslash
+to
+\backslash
+Omega_i
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Note that a given discretization may contain cells of different shapes,
+ which would require us to define an appropriate number of reference cells
+ for each type of cell.
+ However, for the purposes of this manual we may restrict the discussions
+ in this manuals, without any loss of generality, to discretizations consisting
+ of a single geometric shape.
+ (XXX: reword this last sentence)
+\end_layout
+
+\begin_layout Standard
+The advantage of this approach is readily apparent when one realizes that
+ the integration points and weights can be calculated once and for all on
+ the reference cell, and then reused for the other cells through the application
+ of the corresponding reference map.
+ In other words,
+\begin_inset Formula \begin{eqnarray*}
+\vec{X}_{q} & = & \chi_{i}(\vec{\xi}_{q})\\
+W_{q} & = & w_{q}J_{i}(\vec{\xi}_{q})\end{eqnarray*}
+
+\end_inset
+
+where
+\begin_inset Formula $\vec{\xi}_{q}$
+\end_inset
+
+ and
+\begin_inset Formula $w_{q}$
+\end_inset
+
+ are the optimal integration points and weights for the integration rule
+ defined on the reference domain
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ Here, the factor
+\begin_inset Formula $J_{i}(\vec{\xi})=\det\ \left|\frac{d\chi_{i}}{d\vec{\xi}}\right|$
+\end_inset
+
+ is the Jacobian determinant of the transformation
+\begin_inset Formula $\chi_{i}$
+\end_inset
+
+.
+ Recall that the index
+\begin_inset Formula $i$
+\end_inset
+
+ corresponds to the
+\begin_inset Formula $i$
+\end_inset
+
+-th cell
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+, and the index
+\begin_inset Formula $q$
+\end_inset
+
+ ranges over the usual
+\begin_inset Formula $q=1,\ldots,Q$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Interpolation Functions
+\end_layout
+
+\begin_layout Standard
+Our two functions
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ and
+\begin_inset Formula $v(\vec{x})$
+\end_inset
+
+ are given in terms of the point
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+, and are said to be defined on a
+\series bold
+\emph on
+global coordinate system
+\series default
+\emph default
+.
+ Based on our discussion in Section 3.1, we see that we only need to know
+ the values of
+\begin_inset Formula $u$
+\end_inset
+
+ and
+\begin_inset Formula $v$
+\end_inset
+
+ at a finite set of global points in order to calculate an approximation
+ to the
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm of the residual field
+\begin_inset Formula $\rho=u-v$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Specifically, given a function
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ we must be able to calculate its values at exactly those integration points.
+ If we know a formula or algorithm for
+\begin_inset Formula $f$
+\end_inset
+
+, then our task is easy.
+ Alternatively, these values can be given explicitly as a finite list.
+ In all other cases, we will need an interpolation scheme that allows us
+ to calculate
+\begin_inset Formula $f$
+\end_inset
+
+ on any intermediate points.
+ This ability is also important because we may want to increase the accuracy
+ of our norm by using more integration points, and thus we will need to
+ be able to evaluate our function
+\begin_inset Formula $f$
+\end_inset
+
+ at the new points.
+\end_layout
+
+\begin_layout Standard
+In general, an interpolation scheme involves a set of known functions
+\begin_inset Formula $\phi_{j}(\vec{x})$
+\end_inset
+
+ that we can compute anywhere and a set of parameters
+\begin_inset Formula $c_{j}$
+\end_inset
+
+ that define how these known functions are combined.
+ Usually, the relation between the parameters and the known functions is
+ a linear combination,
+\begin_inset Formula \begin{equation}
+f(\vec{x})=d_{1}\phi_{1}(\vec{x})+d_{2}\phi_{2}(\vec{x})+\cdots+d_{m}\phi_{m}(\vec{x})\end{equation}
+
+\end_inset
+
+where the parameters
+\begin_inset Formula $d_{j}$
+\end_inset
+
+, with
+\begin_inset Formula $j=1,\ldots,m$
+\end_inset
+
+, are also sometimes referred to as the
+\series bold
+\emph on
+degrees of freedom
+\series default
+\emph default
+ of the function
+\begin_inset Formula $f$
+\end_inset
+
+.
+ As described in Chapter 4, the functions
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+, also known as
+\series bold
+\emph on
+shape functions
+\series default
+\emph default
+, must satisfy a number of conditions.
+ Because the choice of shape functions affects how well the true solution
+ is represented, the convergence of the numerical method used to obtain
+ the solution, the optimal choice of shape functions is very problem dependent.
+\end_layout
+
+\begin_layout Standard
+If we know the shape functions
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+ directly in terms of the global points
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+, they are said to form a
+\series bold
+\emph on
+global interpolation scheme
+\series default
+\emph default
+, and we may use Eq X directly to find the values of
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+, and we may refer to the
+\begin_inset Formula $\phi_{j}$
+\end_inset
+
+as
+\emph on
+global shape functions
+\emph default
+.
+ Note that since we are working in the global coordinate system, we don't
+ need the discretization of the domain
+\begin_inset Formula $\Omega$
+\end_inset
+
+.
+ Examples of global shape functions include spherical harmonics, and ...
+\end_layout
+
+\begin_layout Standard
+Perhaps a more typical case is when
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ is defined piecewise, on each discretization element
+\begin_inset Formula $\Omega_{i}$
+\end_inset
+
+.
+ Because these cells partition the original domain by definition, a given
+ point in our domain will be found in one and only one cell, which will
+ have a local definition.
+ That is, for a particular
+\begin_inset Formula $\vec{x}\in\Omega$
+\end_inset
+
+ we will be able to find a unique cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, for some index
+\begin_inset Formula $e$
+\end_inset
+
+.
+ We can refer to this as a
+\series bold
+\emph on
+local interpolation scheme
+\series default
+\emph default
+.
+\end_layout
+
+\begin_layout Standard
+See Chapter 5, for more details on the specific interpolation schemes available
+ in Cigma.
+\end_layout
+
+\begin_layout Section
+Global Error Measure
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Suppose the domain
+\begin_inset Formula $\Omega$
+\end_inset
+
+ into an appropriate set of cells
+\begin_inset Formula $\Omega_{1},\Omega_{2},\ldots,\Omega_{n_{el}}$
+\end_inset
+
+, we can compute the global error
+\begin_inset Formula $\varepsilon^{2}$
+\end_inset
+
+ as a sum over localized cell contributions,
+\begin_inset Formula $\varepsilon^{2}=\sum_{e}\varepsilon_{e}^{2}$
+\end_inset
+
+, where each cell residual
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ is given by
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+In general, we won't be able to integrate each cell residual
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ exactly since either of the functions
+\begin_inset Formula $u$
+\end_inset
+
+ and
+\begin_inset Formula $v$
+\end_inset
+
+ may have an incompatible representation relative to the finite element
+ space on each domain
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+.
+ However, we can still calculate an approximation of each cell residual
+ by applying an appropriate quadrature rule with a tolerable truncation
+ error
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Encyclopaedia of Cubature Formulas 2005"
+
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Plain Layout
+To obtain an approximation to the integral of a function
+\begin_inset Formula $F(\vec{x})$
+\end_inset
+
+ over a cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, we simply apply the quadrature rule with weights
+\begin_inset Formula $w_{e,1},w_{e,2},\ldots,w_{e,n_{Q}}$
+\end_inset
+
+ and integration points
+\begin_inset Formula $\vec{x}_{e,1},\vec{x}_{e,2},\ldots,\vec{x}_{e,n_{Q}}$
+\end_inset
+
+ appropriate for the physical element
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+Applying this quadrature rule directly over the entire physical domain
+\begin_inset Formula $\Omega=\cup\Omega_{e}$
+\end_inset
+
+ gives us
+\begin_inset Formula \[
+\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{e=1}^{n_{el}}\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+For efficiency reasons, it is undesirable in finite element applications
+ to perform calculations in a global coordinate system.
+ To avoid duplication of work, shape function evaluations may be performed
+ once on a reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ and then transformed back into the corresponding physical cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ as needed.
+\end_layout
+
+\begin_layout Plain Layout
+To compute integrals of
+\begin_inset Formula $F$
+\end_inset
+
+ in a reference coordinate system, we need to apply a change of variables:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}\]
+
+\end_inset
+
+where
+\color none
+the additional factor
+\begin_inset Formula $J_{e}(\vec{\xi})=\det\left[\frac{d\vec{x}_{e}}{d\vec{\xi}}\right]$
+\end_inset
+
+ is the Jacobian determinant of the reference map
+\begin_inset Formula $\vec{x}_{e}(\vec{\xi}):\hat{\Omega}\to\Omega_{e}$
+\end_inset
+
+.
+ This map describes how the physical points
+\begin_inset Formula $\vec{x}\in\Omega_{e}$
+\end_inset
+
+ are transformed from the reference points
+\begin_inset Formula $\vec{\xi}\in\hat{\Omega}$
+\end_inset
+
+.
+ Put another way, the inverse reference map
+\color inherit
+
+\begin_inset Formula $\vec{\xi}=\vec{x}_{e}^{-1}(\vec{x})$
+\end_inset
+
+ tells us how the physical domain
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ maps into the reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Plain Layout
+At this point, we can assume without loss of generality that every physical
+ cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ can be derived from a single reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+, so that our quadrature rule becomes simply a set of weights
+\begin_inset Formula $w_{1},\ldots,w_{n_{Q}}$
+\end_inset
+
+ and points
+\begin_inset Formula $\vec{\xi}_{1},\ldots,\vec{\xi}_{n_{Q}}$
+\end_inset
+
+ over
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ After changing variables, we end up with the final form of the quadrature
+ rule that we can use to integrate the global residual field:
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula \[
+\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}=\sum_{q=1}^{n_{Q}}w_{q}F(\vec{x}_{e}(\vec{\xi}_{q}))J_{e}(\vec{\xi}_{q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+If we let
+\begin_inset Formula $F(\vec{x})=||u(\vec{x})-v(\vec{x})||^{2}$
+\end_inset
+
+ in the above expression, we can find the cell residual contribution over
+
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ from
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}\ ||u(\vec{x})-v(\vec{x})||^{2}\ d\vec{x}\\
+ & = & \int_{\hat{\Omega}}\ ||u(\vec{x}_{e}(\vec{\xi}))-v(\vec{x}_{e}(\vec{\xi}))||^{2}J_{e}(\vec{\xi})\ d\vec{\xi}\\
+ & = & \sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+The global error
+\begin_inset Formula $\varepsilon=\sqrt{\sum_{e}\varepsilon_{e}^{2}}$
+\end_inset
+
+ is then approximated by the expression
+\end_layout
+
+\begin_layout Plain Layout
+\begin_inset Box Boxed
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100text%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Plain Layout
+We can make a number of observations based on our final expression for the
+ global error.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In this section we show the formula used by Cigma to compute the
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm of the residual field
+\begin_inset Formula $\rho$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Suppose the domain
+\begin_inset Formula $\Omega$
+\end_inset
+
+ into an appropriate set of cells
+\begin_inset Formula $\Omega_{1},\Omega_{2},\ldots,\Omega_{n_{el}}$
+\end_inset
+
+, we can compute the global error
+\begin_inset Formula $\varepsilon^{2}$
+\end_inset
+
+ as a sum over localized cell contributions,
+\begin_inset Formula $\varepsilon^{2}=\sum_{e}\varepsilon_{e}^{2}$
+\end_inset
+
+, where each cell residual
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ is given by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}||u(\vec{x})-v(\vec{x})||^{2}d\vec{x}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In general, we won't be able to integrate each cell residual
+\begin_inset Formula $\varepsilon_{e}^{2}$
+\end_inset
+
+ exactly since either of the functions
+\begin_inset Formula $u$
+\end_inset
+
+ and
+\begin_inset Formula $v$
+\end_inset
+
+ may have an incompatible representation relative to the finite element
+ space on each domain
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+.
+ However, we can still calculate an approximation of each cell residual
+ by applying an appropriate quadrature rule with a tolerable truncation
+ error
+\begin_inset CommandInset citation
+LatexCommand cite
+key "Encyclopaedia of Cubature Formulas 2005"
+
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+To obtain an approximation to the integral of a function
+\begin_inset Formula $F(\vec{x})$
+\end_inset
+
+ over a cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, we simply apply the quadrature rule with weights
+\begin_inset Formula $w_{e,1},w_{e,2},\ldots,w_{e,n_{Q}}$
+\end_inset
+
+ and integration points
+\begin_inset Formula $\vec{x}_{e,1},\vec{x}_{e,2},\ldots,\vec{x}_{e,n_{Q}}$
+\end_inset
+
+ appropriate for the physical element
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Applying this quadrature rule directly over the entire physical domain
+\begin_inset Formula $\Omega=\cup\Omega_{e}$
+\end_inset
+
+ gives us
+\begin_inset Formula \[
+\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{e=1}^{n_{el}}\sum_{q=1}^{n_{Q}}w_{e,q}F(\vec{x}_{e,q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For efficiency reasons, it is undesirable in finite element applications
+ to perform calculations in a global coordinate system.
+ To avoid duplication of work, shape function evaluations may be performed
+ once on a reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+ and then transformed back into the corresponding physical cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ as needed.
+\end_layout
+
+\begin_layout Standard
+To compute integrals of
+\begin_inset Formula $F$
+\end_inset
+
+ in a reference coordinate system, we need to apply a change of variables:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega_{e}}\ F(\vec{x})\ d\vec{x}=\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}\]
+
+\end_inset
+
+where
+\color none
+the additional factor
+\begin_inset Formula $J_{e}(\vec{\xi})=\det\left[\frac{d\vec{x}_{e}}{d\vec{\xi}}\right]$
+\end_inset
+
+ is the Jacobian determinant of the reference map
+\begin_inset Formula $\vec{x}_{e}(\vec{\xi}):\hat{\Omega}\to\Omega_{e}$
+\end_inset
+
+.
+ This map describes how the physical points
+\begin_inset Formula $\vec{x}\in\Omega_{e}$
+\end_inset
+
+ are transformed from the reference points
+\begin_inset Formula $\vec{\xi}\in\hat{\Omega}$
+\end_inset
+
+.
+ Put another way, the inverse reference map
+\color inherit
+
+\begin_inset Formula $\vec{\xi}=\vec{x}_{e}^{-1}(\vec{x})$
+\end_inset
+
+ tells us how the physical domain
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ maps into the reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+At this point, we can assume without loss of generality that every physical
+ cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ can be derived from a single reference cell
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+, so that our quadrature rule becomes simply a set of weights
+\begin_inset Formula $w_{1},\ldots,w_{n_{Q}}$
+\end_inset
+
+ and points
+\begin_inset Formula $\vec{\xi}_{1},\ldots,\vec{\xi}_{n_{Q}}$
+\end_inset
+
+ over
+\begin_inset Formula $\hat{\Omega}$
+\end_inset
+
+.
+ After changing variables, we end up with the final form of the quadrature
+ rule that we can use to integrate the global residual field:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\hat{\Omega}}\ F(\vec{x}_{e}(\vec{\xi}))J_{e}(\vec{\xi})\ d\vec{\xi}=\sum_{q=1}^{n_{Q}}w_{q}F(\vec{x}_{e}(\vec{\xi}_{q}))J_{e}(\vec{\xi}_{q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+If we let
+\begin_inset Formula $F(\vec{x})=||u(\vec{x})-v(\vec{x})||^{2}$
+\end_inset
+
+ in the above expression, we can find the cell residual contribution over
+
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+ from
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{e}^{2} & = & \int_{\Omega_{e}}\ ||u(\vec{x})-v(\vec{x})||^{2}\ d\vec{x}\\
+ & = & \int_{\hat{\Omega}}\ ||u(\vec{x}_{e}(\vec{\xi}))-v(\vec{x}_{e}(\vec{\xi}))||^{2}J_{e}(\vec{\xi})\ d\vec{\xi}\\
+ & = & \sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The global error
+\begin_inset Formula $\varepsilon=\sqrt{\sum_{e}\varepsilon_{e}^{2}}$
+\end_inset
+
+ is then approximated by the expression
+\end_layout
+
+\begin_layout Standard
+\begin_inset Box Boxed
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100text%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}w_{q}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+We use this final form to calculate the global and localized errors on arbitrary
+ discretizations.
+\end_layout
+
+\begin_layout Section
+Comparing Residuals
+\end_layout
+
+\begin_layout Standard
+For comparing errors of different solutions, you can normalize the global
+ error by the norm of the exact solution.
+ For a family of solutions
+\begin_inset Formula $u_{h}(\vec{x})$
+\end_inset
+
+, parametrized by the the maximum element size
+\begin_inset Formula $h$
+\end_inset
+
+ of the underlying discretization, and an exact solution
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+, this normalized error is given by
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+\varepsilon_{rel} & = & \frac{||u-u_{h}||_{L_{2}}}{||u||_{L_{2}}}\\
+ & = & \frac{\int_{\Omega}||u(\vec{x})-u_{h}(\vec{x})||^{2}d\vec{x}}{\int_{\Omega}||u(\vec{x})||^{2}d\vec{x}}\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This normalized error can be interpreted as the average error in the physical
+ quantity being evaluated, so that a value of 0.01 corresponds to a 1% averaged
+ error.
+\end_layout
+
+\begin_layout Standard
+Even if the exact solution is not currently known, this normalized error
+ may be used to test the accuracy between two or more numerical solutions,
+ defined on successively refined meshes.
+\end_layout
+
+\begin_layout Section
+Convergence Rates
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+TODO: Move discussion of convergence from old running.lyx into this section.
+\end_layout
+
+\begin_layout Plain Layout
+Standard error estimate (Hughes, pg 190; Johnson, pg 97)
+\end_layout
+
+\begin_layout Plain Layout
+Give formula for
+\begin_inset Formula $\alpha$
+\end_inset
+
+ for when we know the exact solution (only need two levels), and we are
+ testing our numerical code.
+\end_layout
+
+\begin_layout Plain Layout
+Give formula for
+\begin_inset Formula $\alpha$
+\end_inset
+
+ for when we have three levels (same formula), and we are testing our code
+ for convergence.
+\end_layout
+
+\begin_layout Plain Layout
+Talk about least squares fit for when we have more than three levels.
+ (mention scipy script, and reproduce it here, since it's short)
+\end_layout
+
+\begin_layout Plain Layout
+SciPy script for optimizing least squares fit into the formula y = a*x^b
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Even if the exact solution
+\begin_inset Formula $u(\vec{x})$
+\end_inset
+
+ is not known, but we have calculated a family of solutions
+\begin_inset Formula $u_{h}(\vec{x})$
+\end_inset
+
+ on discrete meshes, we may calculate a convergence rate by using the standard
+ error estimate
+\begin_inset Formula $||u-u_{h}||_{p}\leq Ch^{\alpha},$
+\end_inset
+
+where
+\begin_inset Formula $C$
+\end_inset
+
+ is independent of both
+\begin_inset Formula $h$
+\end_inset
+
+ and
+\begin_inset Formula $u$
+\end_inset
+
+.
+ This error bound holds problems
+\end_layout
+
+\begin_layout Section
+Uses in Benchmarking
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Here we define what we mean by Benchmark, and how we can use.
+\end_layout
+
+\begin_layout Plain Layout
+Give URLs for benchmark repositories on CIG svn tree.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+As we saw in Section 3.5, it is easier to calculate the convergence rate
+ on a sequence of meshes if we have access to the exact solution.
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Since one of our goals is to provide an easy means of calculating these
+ comparisons,
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/examples/examples.lyx
===================================================================
--- doc/cigma/manual/examples/examples.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/examples/examples.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,487 +0,0 @@
-#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-\newcommand{\mynote}[1]{}
-\end_preamble
-\language english
-\inputencoding auto
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\paperfontsize default
-\spacing single
-\papersize default
-\use_geometry false
-\use_amsmath 1
-\use_esint 1
-\cite_engine basic
-\use_bibtopic false
-\paperorientation portrait
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\defskip medskip
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\author ""
-\author ""
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-Here are the examples discussed in this section and the intended motivation
- for introducing them
-\end_layout
-
-\begin_layout Standard
-
- * Strikeslip Benchmark (without gravity)
-\end_layout
-
-\begin_layout Standard
-
- - PyLith vs PyLith (hex8, tet4 elts...higher order available?)
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate convergence order using sequence of meshes -- generated
- using cubit
-\end_layout
-
-\begin_layout Standard
-
- - Analytic (okada) vs Pylith -- initial timestep
-\end_layout
-
-\begin_layout Standard
-
- - Variables compared: displacement, velocity
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate vtk visualization techniques - cell-to-point filter, then
- isosurface
-\end_layout
-
-\begin_layout Standard
-
- - Same case but with gravity?
-\end_layout
-
-\begin_layout Standard
-
- - http://www.geodynamics.org/cig/workinggroups/short/workarea/benchmarks/
-\end_layout
-
-\begin_layout Standard
-
- * Poisson solution on unit cube
-\end_layout
-
-\begin_layout Standard
-
- - Analytic vs deal.II vs libMesh
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of quadratic elements vs.
- linear
-\end_layout
-
-\begin_layout Standard
-
- - tet{4,10} convergence, hex{8,20,27} convergence -- where possible
-\end_layout
-
-\begin_layout Standard
-
- - take a planar slice and plot differences there
-\end_layout
-
-\begin_layout Standard
-
- - inter-element solution comparisons (tet4 vs hex8, tet10 vs.
- hex8)
-\end_layout
-
-\begin_layout Standard
-
- * Cylinder viscoelastic extension
-\end_layout
-
-\begin_layout Standard
-
- - Analytic (cylinder_extension.*.*) vs.
- GeoFEST(tet4) vs.
- PyLith (tet4,hex8)
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of predefined function
-\end_layout
-
-\begin_layout Standard
-
- - Maxwell material (cylinder_extension.maxwell.{displacement,velocity,stress})
-\end_layout
-
-\begin_layout Standard
-
- - Non-newtonian material (cylinder_extension.non_newtonian.{displacement,veloc
-ity,stress})
-\end_layout
-
-\begin_layout Standard
-
- * Cylinder viscoelastic relaxation
-\end_layout
-
-\begin_layout Standard
-
- - Analytic (cylinder_relaxation.*.*) vs.
- GeoFEST (tet4) vs.
- PyLith (tet4,hex8)
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of extract/eval -- on integration mesh
-\end_layout
-
-\begin_layout Standard
-
- - Effect of varying the integration order
-\end_layout
-
-\begin_layout Standard
-
- - Maxwell material (cylinder_relaxation.maxwell.{displacement,velocity,stress}
-)
-\end_layout
-
-\begin_layout Standard
-
- - Non-newtonian material (cylinder_relaxation.non_newtonian.{displacement,velo
-city,stress})
-\end_layout
-
-\begin_layout Standard
-
- * Gale Circular Inclusion benchmark
-\end_layout
-
-\begin_layout Standard
-
- - Analytic (circular_inclusion.pressure) vs.
- Gale
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of the Parallel VTK Format (PVTS structured grid)
-\end_layout
-
-\begin_layout Standard
-
- - limitations of FEM description (discontinuity near boundary of circle)
-\end_layout
-
-\begin_layout Standard
-
- - need discontinuous elements, perhaps?
-\end_layout
-
-\begin_layout Standard
-
- * Mantle Convection -- Cartesian
-\end_layout
-
-\begin_layout Standard
-
- - CitcomCU convergence order
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of Parallel VTK Format (PVTR files)
-\end_layout
-
-\begin_layout Standard
-
- - http://www.geodynamics.org/cig/workinggroups/mc/workarea/benchmark/
-\end_layout
-
-\begin_layout Standard
-
- * Mantle Convection -- Spherical
-\end_layout
-
-\begin_layout Standard
-
- - CitcomS vs HC
-\end_layout
-
-\begin_layout Standard
-
- - Demonstrate use of global basis functions (spherical harmonics)
-\end_layout
-
-\begin_layout Standard
-
- * MADDS
-\end_layout
-
-\begin_layout Standard
-
- - Run SpecRidge example
-\end_layout
-
-\begin_layout Standard
-
- - Compare spectral solution against higher order elements
-\end_layout
-
-\begin_layout Standard
-
- - http://www.geodynamics.org/cig/workinggroups/magma/workarea/benchmark/
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-The core concept for this chapter is to show-case specific use cases for
- cigma.
-\end_layout
-
-\begin_layout Standard
-All the necessary data should be available in the
-\family typewriter
-examples/
-\family default
- directory.
-\newline
-
-\end_layout
-
-\begin_layout Standard
-The following points should be emphasized for each example:
-\end_layout
-
-\begin_layout Enumerate
-Problem definition and method used to obtain the solutions.
-\end_layout
-
-\begin_layout Enumerate
-Pre-processing step for preparing the data
-\end_layout
-
-\begin_layout Enumerate
-Cigma command line usage
-\end_layout
-
-\begin_layout Enumerate
-Post-processing step for visualizing the resulting residuals, or for calculating
- the required convergence rate.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-Examples
-\end_layout
-
-\begin_layout Section
-Strikeslip Benchmark
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Reference: Taken from pylith.pdf page 123.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-This benchmark problem computes the viscoelastic (Maxwell) relaxation of
- stresses from a single, finite, strike-slip earthquake in 3D without gravity.
- Dirichlet boundary conditions equal to the analytical elastic solution
- are imposed on the sides of a cube with sides of length 24 km.
- Anti-plane strain boundary conditions are imposed at
-\begin_inset Formula $y=0$
-\end_inset
-
-, so the solution is equivalent to that for a domain with a 48 km length
- in the
-\begin_inset Formula $y$
-\end_inset
-
- direction.
- In the initial time-step, we can use the analytical solution both to apply
- the boundary conditions, and to compare against the numerically-computed
- elastic solution.
-\end_layout
-
-\begin_layout Subsection
-Problem Description
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Do we need a subsection? Try this, but use a single section for the rest
- of examples.
- Settle into whichever style you like better.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Figure 8.X refers to the
-\end_layout
-
-\begin_layout Section
-Cylinder Extension
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Purpose: Compares solutions from physical problem obtained using two different
- codes, and checked against actual analytical solution.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-This benchmark problem computes the viscoelastic displacements and velocities
- resulting from extending a unit cylinder.
-
-\end_layout
-
-\begin_layout Section
-Cylinder Relaxation
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Purpose: Compares solutions from physical problem obtained using two different
- codes, and checked against known analytical solution
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-This problem is similar to the one in the previous section, where we have
- taken.
-\end_layout
-
-\begin_layout Section
-Circular Inclusion
-\end_layout
-
-\begin_layout Standard
-Description from gale
-\end_layout
-
-\begin_layout Section
-Mantle Convection on Rectangular Box
-\end_layout
-
-\begin_layout Standard
-Given the benchmark files in /examples/mc1/
-\end_layout
-
-\begin_layout Section
-Mantle Convection on a Sphere
-\end_layout
-
-\begin_layout Standard
-Here we use the output from CitcomS and the spherical.
-\end_layout
-
-\begin_layout Section
-Waves
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Example of a 1-D comparison.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/examples.lyx (from rev 13032, doc/cigma/manual/examples/examples.lyx)
===================================================================
--- doc/cigma/manual/examples.lyx (rev 0)
+++ doc/cigma/manual/examples.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,487 @@
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+Here are the examples discussed in this section and the intended motivation
+ for introducing them
+\end_layout
+
+\begin_layout Standard
+
+ * Strikeslip Benchmark (without gravity)
+\end_layout
+
+\begin_layout Standard
+
+ - PyLith vs PyLith (hex8, tet4 elts...higher order available?)
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate convergence order using sequence of meshes -- generated
+ using cubit
+\end_layout
+
+\begin_layout Standard
+
+ - Analytic (okada) vs Pylith -- initial timestep
+\end_layout
+
+\begin_layout Standard
+
+ - Variables compared: displacement, velocity
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate vtk visualization techniques - cell-to-point filter, then
+ isosurface
+\end_layout
+
+\begin_layout Standard
+
+ - Same case but with gravity?
+\end_layout
+
+\begin_layout Standard
+
+ - http://www.geodynamics.org/cig/workinggroups/short/workarea/benchmarks/
+\end_layout
+
+\begin_layout Standard
+
+ * Poisson solution on unit cube
+\end_layout
+
+\begin_layout Standard
+
+ - Analytic vs deal.II vs libMesh
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of quadratic elements vs.
+ linear
+\end_layout
+
+\begin_layout Standard
+
+ - tet{4,10} convergence, hex{8,20,27} convergence -- where possible
+\end_layout
+
+\begin_layout Standard
+
+ - take a planar slice and plot differences there
+\end_layout
+
+\begin_layout Standard
+
+ - inter-element solution comparisons (tet4 vs hex8, tet10 vs.
+ hex8)
+\end_layout
+
+\begin_layout Standard
+
+ * Cylinder viscoelastic extension
+\end_layout
+
+\begin_layout Standard
+
+ - Analytic (cylinder_extension.*.*) vs.
+ GeoFEST(tet4) vs.
+ PyLith (tet4,hex8)
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of predefined function
+\end_layout
+
+\begin_layout Standard
+
+ - Maxwell material (cylinder_extension.maxwell.{displacement,velocity,stress})
+\end_layout
+
+\begin_layout Standard
+
+ - Non-newtonian material (cylinder_extension.non_newtonian.{displacement,veloc
+ity,stress})
+\end_layout
+
+\begin_layout Standard
+
+ * Cylinder viscoelastic relaxation
+\end_layout
+
+\begin_layout Standard
+
+ - Analytic (cylinder_relaxation.*.*) vs.
+ GeoFEST (tet4) vs.
+ PyLith (tet4,hex8)
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of extract/eval -- on integration mesh
+\end_layout
+
+\begin_layout Standard
+
+ - Effect of varying the integration order
+\end_layout
+
+\begin_layout Standard
+
+ - Maxwell material (cylinder_relaxation.maxwell.{displacement,velocity,stress}
+)
+\end_layout
+
+\begin_layout Standard
+
+ - Non-newtonian material (cylinder_relaxation.non_newtonian.{displacement,velo
+city,stress})
+\end_layout
+
+\begin_layout Standard
+
+ * Gale Circular Inclusion benchmark
+\end_layout
+
+\begin_layout Standard
+
+ - Analytic (circular_inclusion.pressure) vs.
+ Gale
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of the Parallel VTK Format (PVTS structured grid)
+\end_layout
+
+\begin_layout Standard
+
+ - limitations of FEM description (discontinuity near boundary of circle)
+\end_layout
+
+\begin_layout Standard
+
+ - need discontinuous elements, perhaps?
+\end_layout
+
+\begin_layout Standard
+
+ * Mantle Convection -- Cartesian
+\end_layout
+
+\begin_layout Standard
+
+ - CitcomCU convergence order
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of Parallel VTK Format (PVTR files)
+\end_layout
+
+\begin_layout Standard
+
+ - http://www.geodynamics.org/cig/workinggroups/mc/workarea/benchmark/
+\end_layout
+
+\begin_layout Standard
+
+ * Mantle Convection -- Spherical
+\end_layout
+
+\begin_layout Standard
+
+ - CitcomS vs HC
+\end_layout
+
+\begin_layout Standard
+
+ - Demonstrate use of global basis functions (spherical harmonics)
+\end_layout
+
+\begin_layout Standard
+
+ * MADDS
+\end_layout
+
+\begin_layout Standard
+
+ - Run SpecRidge example
+\end_layout
+
+\begin_layout Standard
+
+ - Compare spectral solution against higher order elements
+\end_layout
+
+\begin_layout Standard
+
+ - http://www.geodynamics.org/cig/workinggroups/magma/workarea/benchmark/
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+The core concept for this chapter is to show-case specific use cases for
+ cigma.
+\end_layout
+
+\begin_layout Standard
+All the necessary data should be available in the
+\family typewriter
+examples/
+\family default
+ directory.
+\newline
+
+\end_layout
+
+\begin_layout Standard
+The following points should be emphasized for each example:
+\end_layout
+
+\begin_layout Enumerate
+Problem definition and method used to obtain the solutions.
+\end_layout
+
+\begin_layout Enumerate
+Pre-processing step for preparing the data
+\end_layout
+
+\begin_layout Enumerate
+Cigma command line usage
+\end_layout
+
+\begin_layout Enumerate
+Post-processing step for visualizing the resulting residuals, or for calculating
+ the required convergence rate.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Examples
+\end_layout
+
+\begin_layout Section
+Strikeslip Benchmark
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Reference: Taken from pylith.pdf page 123.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This benchmark problem computes the viscoelastic (Maxwell) relaxation of
+ stresses from a single, finite, strike-slip earthquake in 3D without gravity.
+ Dirichlet boundary conditions equal to the analytical elastic solution
+ are imposed on the sides of a cube with sides of length 24 km.
+ Anti-plane strain boundary conditions are imposed at
+\begin_inset Formula $y=0$
+\end_inset
+
+, so the solution is equivalent to that for a domain with a 48 km length
+ in the
+\begin_inset Formula $y$
+\end_inset
+
+ direction.
+ In the initial time-step, we can use the analytical solution both to apply
+ the boundary conditions, and to compare against the numerically-computed
+ elastic solution.
+\end_layout
+
+\begin_layout Subsection
+Problem Description
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Do we need a subsection? Try this, but use a single section for the rest
+ of examples.
+ Settle into whichever style you like better.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Figure 8.X refers to the
+\end_layout
+
+\begin_layout Section
+Cylinder Extension
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Purpose: Compares solutions from physical problem obtained using two different
+ codes, and checked against actual analytical solution.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This benchmark problem computes the viscoelastic displacements and velocities
+ resulting from extending a unit cylinder.
+
+\end_layout
+
+\begin_layout Section
+Cylinder Relaxation
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Purpose: Compares solutions from physical problem obtained using two different
+ codes, and checked against known analytical solution
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+This problem is similar to the one in the previous section, where we have
+ taken.
+\end_layout
+
+\begin_layout Section
+Circular Inclusion
+\end_layout
+
+\begin_layout Standard
+Description from gale
+\end_layout
+
+\begin_layout Section
+Mantle Convection on Rectangular Box
+\end_layout
+
+\begin_layout Standard
+Given the benchmark files in /examples/mc1/
+\end_layout
+
+\begin_layout Section
+Mantle Convection on a Sphere
+\end_layout
+
+\begin_layout Standard
+Here we use the output from CitcomS and the spherical.
+\end_layout
+
+\begin_layout Section
+Waves
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Example of a 1-D comparison.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/formats/formats.lyx
===================================================================
--- doc/cigma/manual/formats/formats.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/formats/formats.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,3349 +0,0 @@
-#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-\newcommand{\mynote}[1]{}
-\end_preamble
-\language english
-\inputencoding auto
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\paperfontsize default
-\spacing single
-\papersize default
-\use_geometry false
-\use_amsmath 1
-\use_esint 1
-\cite_engine basic
-\use_bibtopic false
-\paperorientation portrait
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\defskip medskip
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\author ""
-\author ""
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-TODO: Rename file_formats.lyx to formats.lyx (done)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-TODO: What about NodeNumberMap?
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Define the dimensions for every object in the file
-\end_layout
-
-\begin_layout Standard
-
- Mesh Coordinates
-\end_layout
-
-\begin_layout Standard
-
- Mesh Connectivity (list of Element Blocks)
-\end_layout
-
-\begin_layout Standard
-
- Field Variables (Global, Nodal, Element)
-\end_layout
-
-\begin_layout Standard
-
- Element Tabulation (Shape function, Shape Function Derivatives)
-\end_layout
-
-\begin_layout Standard
-
- Integration Rule (Weights and Local Points)
-\end_layout
-
-\begin_layout Standard
-
- Global Integration Points
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Explain structure of each file format:
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- Text Format
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- All arrays specified explicitly in several text files containing a list
- of floating point numbers.
-\end_layout
-
-\begin_layout Standard
-
- The first two numbers indicate the dimensions of the array, followed
- by a list of floats
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --integration-mesh-coordinates=imesh_coords.dat
-\end_layout
-
-\begin_layout Standard
-
- --integration-mesh-connectivity=imesh_connect.dat
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --integration-rule-points=tet4_order10_points.dat
-\end_layout
-
-\begin_layout Standard
-
- --integration-rule-weights=tet4_order10_weights.dat
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --first-mesh-coordinates=first_coords.dat
-\end_layout
-
-\begin_layout Standard
-
- --first-mesh-connectivity=first_connect.dat
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --second-mesh-coordinates=first_coords.dat
-\end_layout
-
-\begin_layout Standard
-
- --second-mesh-connectivity=first_connect.dat
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --first-dofs=first_var_stepN.dat
-\end_layout
-
-\begin_layout Standard
-
- --second-dofs=second_var_stepN.dat
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- --integration-rule=tet4_order10.dat <-- (weights in first column)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- Simple Arrays (HDF5)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- Structure of Model Definition
-\end_layout
-
-\begin_layout Standard
-
- root group at /model
-\end_layout
-
-\begin_layout Standard
-
- mesh data at /model/mesh
-\end_layout
-
-\begin_layout Standard
-
- node coordinates as specified on global coordinate system at /model/mesh
-/coordinates
-\end_layout
-
-\begin_layout Standard
-
- list of element blocks at /model/mesh/connectivity
-\end_layout
-
-\begin_layout Standard
-
- note that each element block is tagged by element type in the FE
- attribute
-\end_layout
-
-\begin_layout Standard
-
- degrees-of-freedom data
-\end_layout
-
-\begin_layout Standard
-
- organized by variable at /model/vars/var, or even just /model/var
-\end_layout
-
-\begin_layout Standard
-
- list of steps: /model/var/step{0,1,...,N}
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- Structure of file defining an Integration Rule
-\end_layout
-
-\begin_layout Standard
-
- root group is / by default, but may be specified directly (eg, as
- /rule)
-\end_layout
-
-\begin_layout Standard
-
- root group is tagged with FE string indicating the source element
- (tet4, ...)
-\end_layout
-
-\begin_layout Standard
-
- integration points contained at /rule/points
-\end_layout
-
-\begin_layout Standard
-
- integration points contained at /rule/weights
-\end_layout
-
-\begin_layout Standard
-
- Structure of file with Residuals
-\end_layout
-
-\begin_layout Standard
-
- root group is / by default, but it must exist if specified
-\end_layout
-
-\begin_layout Standard
-
-
-\end_layout
-
-\begin_layout Standard
-
- VTK
-\end_layout
-
-\begin_layout Standard
-
- Basic structure:
-\end_layout
-
-\begin_layout Standard
-
- VTK Grid
-\end_layout
-
-\begin_layout Standard
-
- VTK Dataset consists of the mesh information (node coordinates
- and associated connectivity relations)
-\end_layout
-
-\begin_layout Standard
-
- VTK Data Arrays come in two types (Point Data and Cell Data)
-\end_layout
-
-\begin_layout Standard
-
- VTK Point Data - scalars, vectors, tensors as defined on the
- nodes
-\end_layout
-
-\begin_layout Standard
-
- VTK Cell Data - scalars, vectors, tensors as defined on each
- element
-\end_layout
-
-\begin_layout Standard
-
- Structure of different types
-\end_layout
-
-\begin_layout Standard
-
- Unstructured Grid
-\end_layout
-
-\begin_layout Standard
-
- mesh points: given explicitly in a simple array
-\end_layout
-
-\begin_layout Standard
-
- mesh connectivity: given explicitly in a list of simple arrays
-\end_layout
-
-\begin_layout Standard
-
- element types: Line, Quad, Tetra, Hexahedron, Quadratic Triangle,
- Quadratic Quad, Quadratic Tetra, Quadratic Hexahedron
-\end_layout
-
-\begin_layout Standard
-
- Structured Grid
-\end_layout
-
-\begin_layout Standard
-
- mesh points: given explicitly in a simple array
-\end_layout
-
-\begin_layout Standard
-
- mesh connectivity: derived from dimensions (nx, ny, nz)
-\end_layout
-
-\begin_layout Standard
-
- element types: Quad, Pixel, Hex, Voxel
-\end_layout
-
-\begin_layout Standard
-
- Rectilinear Grid
-\end_layout
-
-\begin_layout Standard
-
- mesh dimensions: (nx, ny, nz)
-\end_layout
-
-\begin_layout Standard
-
- mesh points: derived from 'double xcoords[nx], ycoords[ny], zcoords[n
-z]' arrays
-\end_layout
-
-\begin_layout Standard
-
- element types: Pixel, Voxel
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- UCD
-\end_layout
-
-\begin_layout Standard
-
- Format described in (LINK-TO-DOCUMENT).
-\end_layout
-
-\begin_layout Standard
-
- Using VTK API (vtkAVSucdReader) to read (*.inp) files
-\end_layout
-
-\begin_layout Standard
-
- Just extracts the data into an Unstructured Grid
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- ExodusII
-\end_layout
-
-\begin_layout Standard
-
- This type of mesh can be generated by other programs, such as CUBIT
-\end_layout
-
-\begin_layout Standard
-
- Currently only used to specify an integration mesh, but it is capable
- of storing degree-of-freedom data as well.
-\end_layout
-
-\begin_layout Standard
-
- Utility script for converting *.exo into the appropriate *.h5 file (need
- PyTables, Numpy, Scientific.IO.NetCDF)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
---------------
-\end_layout
-
-\begin_layout Standard
-
-From running.lyx:
-\end_layout
-
-\begin_layout Standard
-
-The underlying data storage format for Cigma is the HDF5 format, due to
- its flexibility for storing and organizing large amounts of data.
- The Hierarchical Data Format (HDF) is designed for storing, retrieving,
- analyzing, visualizing, and converting scientific data.
- It uses a hierarchical structure that provides users a host of options
- for organizing how their data is stored in HDF5 files.
- Using HDF5 datasets in Cigma allows us to avoid having to convert between
- too many distinct formats.
- Moreover, due to the amount of disk I/O, large finite element meshes can
- be handled more efficiently in binary format.
-
-\end_layout
-
-\begin_layout Standard
-
-Another popular format for providing mesh and field inputs.You can easily
- examine the structure of an input file by using the cigma list command,
- which will simply reveal the names and dimensions of all datasets inside
- the specified file.
-\end_layout
-
-\begin_layout Standard
-
-Inputs can be either HDF5 or VTK.
- Specifying the complete path to a dataset consists of the special form
- filepath:dataset, a colon-delimited pair of file path and dataset path.
-\end_layout
-
-\begin_layout Standard
-
-Mesh inputs are currently only unstructured grids.
-\end_layout
-
-\begin_layout Standard
-
-Only exception are the residuals, which are written in legacy VTK format
- as scalar cell data over an unstructured grid.
-\end_layout
-
-\begin_layout Standard
-
-Because Cigma relies on the ability of the user to specify dataset paths,
- we have provided a command called list for viewing the structure of an
- input file.
- Its usage is very simple.
-
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-This chapter may be summarized by simply stating that most input data can
- be specified as a two-dimensional array.
- The meaning of this array, of course, is dependent on context.
- That is the main concept that we have to get accross in this chapter.
- If the content is similar enough, perhaps the appendix could just be incorporat
-ed into this section.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-Data and File Formats
-\end_layout
-
-\begin_layout Standard
-As we've seen in the previous chapters, there are a number of data objects
- involved in making any particular comparison.
- Here, we discuss the variety of formats available, as well as the layout
- within the corresponding data file.
-\end_layout
-
-\begin_layout Section
-Data Formats
-\end_layout
-
-\begin_layout Standard
-The basic data structure is a two-dimensional array of values, stored in
- a contiguous format as shown below
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(m,n)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Data
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $a_{11},a_{12},\ldots,a_{1n}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $a_{21},a_{22},\ldots,a_{2n}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\cdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $a_{m1},a_{m2},\ldots,a_{mn}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-The second index in this array varies the fastest, so that data often referenced
- together remains together in memory as well.
-\end_layout
-
-\begin_layout Subsection
-Node Coordinates
-\end_layout
-
-\begin_layout Standard
-One of the more efficient
-\end_layout
-
-\begin_layout Standard
-On a mesh with
-\begin_inset Formula $n_{no}$
-\end_inset
-
- node coordinates, which are specified on a global coordinate system, we
- have.
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Box Frameless
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "100col%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Box Frameless
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "50col%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{no},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Coordinates
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1},z_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2},z_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{no}},y_{n_{no}},z_{n_{no}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\begin_inset Box Frameless
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "50col%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{no},2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Coordinates
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{no}},y_{n_{no}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Element Connectivity
-\end_layout
-
-\begin_layout Standard
-Mesh connectivity is specified at the element-block level.
- On a given block, we only consider a single element type, which allows
- us to store the global node ids into a contiguous array of integers.
- Multiple blocks are specified separately, and they all reference the same
- node ids into the node coordinates table.
-\end_layout
-
-\begin_layout Standard
-In general, if our block contains
-\begin_inset Formula $k$
-\end_inset
-
- elements with
-\begin_inset Formula $m$
-\end_inset
-
--degrees of freedom each, then the connectivity array defining our element
- block has the form
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(k,m)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Connectivity Data
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $n_{11},n_{12},\ldots,n_{1m}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $n_{21},n_{22},\ldots,n_{2m}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $n_{k1},n_{k2},\ldots,n_{km}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\end_layout
-
-\begin_layout Standard
-For the element types defined in Cigma, the values for
-\begin_inset Formula $m$
-\end_inset
-
- are as follows.
- Note that the total number of degrees of freedom depends on the rank of
- the specific function being represented.
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="4">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Element
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Scalar
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Tensor
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-tet4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-12
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-24
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-tet10
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-10
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-30
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-60
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-hex8
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-8
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-24
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-48
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-hex20
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-20
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-60
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-120
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-hex27
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-27
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-81
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-162
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\end_layout
-
-\begin_layout Standard
-Similarly for the 2-D elements, we have
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="4">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Element
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Scalar
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Tensor
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-tri3
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-6
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-9
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-quad4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-4
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-8
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-12
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Field Variables
-\end_layout
-
-\begin_layout Standard
-A field variable represents one of the functions that we can evaluate, and
- is typically stored over a series of timesteps.
- The data for a field variable consists of snapshots through time, which
- are stored as separate arrays.
- A value for each degree of freedom is provided on the global list of nodes
- in the mesh.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-arrays are subject to interpretation based on their corresponding tag:
-\end_layout
-
-\begin_layout Standard
-by default, interpreted as shape function coefficients (from text and vtk)
-\end_layout
-
-\begin_layout Standard
-can also specify spherical harmonics coefficients.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Shape Function Values
-\end_layout
-
-\begin_layout Standard
-In certain circumstances you may be able to provide your custom reference
- element by simply providing the appropriate
-\begin_inset Formula $n$
-\end_inset
-
- shape function values to be used at the expected integration points.
- This is most useful when you are using the same mesh for both the integration
- mesh and the array.
-\newline
-
-\newline
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="3">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Shape Function Values:
-\begin_inset Formula $(n,Q)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Jacobian Determinant:
-\begin_inset Formula $(Q,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point 1
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{1}),N_{2}(\vec{\xi}_{1}),\ldots,N_{n}(\vec{\xi}_{1})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{1})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point 2
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{2}),N_{2}(\vec{\xi}_{2}),\ldots,N_{n}(\vec{\xi}_{2})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{2})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Integration Point
-\begin_inset Formula $Q$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $N_{1}(\vec{\xi}_{Q}),N_{2}(\vec{\xi}_{Q}),\ldots,N_{n}(\vec{\xi}_{Q})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $J(\vec{\xi}_{Q})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Integration Rule
-\end_layout
-
-\begin_layout Standard
-As described in Chapter 4, an integration rule is specified by a list of
- points and associated weights.
- The points are given on natural coordinate system of the reference element.
-
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="5">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Weights:
-\begin_inset Formula $(Q,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Points:
-\begin_inset Formula $(Q,n_{dim})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Points:
-\begin_inset Formula $(Q,2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Points:
-\begin_inset Formula $(Q,3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Rule Definition
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{1},\eta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{1},\eta_{1},\zeta_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{2},\eta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{2},\eta_{2},\zeta_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $w_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{\xi}_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{Q},\eta_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\xi_{Q},\eta_{Q},\zeta_{Q}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-A typical point
-\begin_inset Formula $\vec{\xi}_{i}$
-\end_inset
-
- in 2-D is typically written as
-\begin_inset Formula $(\xi_{i},\eta_{i})$
-\end_inset
-
-, and in 3-D we write
-\begin_inset Formula $(\xi_{i},\eta_{i},\zeta_{i})$
-\end_inset
-
-.
- In special cases, integration points may be specified in
-\end_layout
-
-\end_inset
-
-
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Note: It is important to realize that the integration weights need to be
- adjusted by the appropriate factor when trying to compare rules defined
- on different reference regions.
- It is not sufficient to simply transform the points, as the weights need
- to be adjusted by the jacobian factor as well (see the change of variables
- formula for integrals).
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Integration Points and Values
-\end_layout
-
-\begin_layout Standard
-Given a mesh and an integration rule, we can map the integration points
- on each element into a possibly large set of global points.
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="4">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Points:
-\begin_inset Formula $(n_{pts},n_{dim})$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Points:
-\begin_inset Formula $(n_{pts},2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Points:
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Coordinates
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{1},y_{1},z_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{2},y_{2},z_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vec{x}_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{pts}}y_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $x_{n_{pts}},y_{n_{pts}},z_{n_{pts}}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-The result of an evaluation will depend on the rank of the function being
- evaluated, as indicated in the table below.
- Note that the fastest varying dimension of the array contains the components
- of the function value at each point.
-\newline
-
-\newline
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Evaluated Quantity
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Resulting Array Shape
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Scalar
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},2)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Vector
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-2-D Tensor
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},3)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-3-D Tensor
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(n_{pts},6)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Comparison Residuals
-\end_layout
-
-\begin_layout Standard
-Since our norm reduces integrals over cells to a single scalar value, the
- result of a comparison will result in a simple scalar array for each element
- block examined.
- Note that element blocks used in the integration process belong to the
- integration mesh, which may differ from the associated mesh to either function.
-\end_layout
-
-\begin_layout Standard
-Thus, for each element block containing
-\begin_inset Formula $k$
-\end_inset
-
- cells, we have the output array
-\newline
-
-\newline
-
-\begin_inset Tabular
-<lyxtabular version="3" rows="5" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Array Shape
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $(k,1)$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Residual Value
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{1}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{2}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\vdots$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-\begin_inset Formula $\varepsilon_{k}$
-\end_inset
-
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-These residual values may be combined in the following form:
-\begin_inset Formula \[
-\varepsilon_{block}=\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\cdots+\varepsilon_{k}^{2}}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The final global residual norm maybe obtained by simply summing over all
- element blocks that partition the domain of interest:
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\varepsilon_{global}=\sqrt{\sum_{block}\varepsilon_{block}^{2}}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-File Formats
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-XXX: The input/output file dichotomy does not seem as necessary now that
- we've added the Data Format section.
- Get rid of those subsection headings.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-A number of file formats are supported for storing and retrieving the above
- data structures to and from disk.
- The following file extensions are used for these purposes:
-\newline
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-File Format
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Recognized Extensions
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-HDF5
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-*.h5
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-VTK
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-*.vtk
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-XML VTK
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-*.vtu, *.vts, *.vtr
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Parallel XML VTK
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-*.pvtu, *.pvts, *.pvtr
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Text
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-*.txt, *.dat
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Input Files
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-There are five different objects you will want to use as input to Cigma,
- all of which can be represented as two-dimensional datasets.
- These are the
-\end_layout
-
-\begin_layout Itemize
-mesh coordinates
-\end_layout
-
-\begin_layout Itemize
-mesh connectivity
-\end_layout
-
-\begin_layout Itemize
-field coefficients (degrees of freedom)
-\end_layout
-
-\begin_layout Itemize
-quadrature rule points
-\end_layout
-
-\begin_layout Itemize
-quadrature rule weights
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In Section 1.1, we examined what kinds of objects we will be accessing, so
- now let's discuss the actual layout in the files in which these objects
- will be stored.
-\end_layout
-
-\begin_layout Subsection
-HDF5
-\end_layout
-
-\begin_layout Standard
-HDF5 files are binary files encoded in a data format designed to store large
- amounts of scientific data in a portable and self-describing way.
-\end_layout
-
-\begin_layout Standard
-The internal organization of a typical HDF5 file consists of a hierarchical
- structure similar to a UNIX file system.
- Two types of primary objects,
-\emph on
-groups
-\emph default
- and
-\emph on
-datasets
-\emph default
-, are stored in this structure.
- A group contains instances of zero or more groups or datasets, while a
- dataset stores a multi-dimensional array of data elements.
- In a sense, datasets are analogous to files in a traditional filesystem,
- and groups are analagous to folders.
- One important difference, however, is that you can attach supporting metadata
- to both kinds of objects.
- The metadata objects are called
-\emph on
-attribute
-\emph default
-s.
-\end_layout
-
-\begin_layout Standard
-A dataset is physically stored in two parts: a header and a data array.
- The header contains miscellaneous metadata describing the dataset as well
- as information that is needed to interpret the array portion of the dataset.
- Essentially, it includes the name, datatype, dataspace, and storage layout
- of the dataset.
- The name is a text string identifying the dataset.
- The datatype describes the type of the data array elements.
- The dataspace defines the dimensionality of the dataset, i.e., the size and
- shape of the multi-dimensional array.
-\end_layout
-
-\begin_layout Standard
-In Cigma you may always provide an explicit path to every dataset, so you
- have a fair amount of flexibility for how you organize your datasets inside
- HDF5 files.
- For example, a typical Cigma HDF5 file could have the following structure.
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Mention these assumptions in the corresponding section in running.lyx
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout LyX-Code
-model.h5
-\end_layout
-
-\begin_layout LyX-Code
-
-\backslash
-__ model
-\end_layout
-
-\begin_layout LyX-Code
- |__ mesh
-\end_layout
-
-\begin_layout LyX-Code
- | |__ coordinates [nno x nsd]
-\end_layout
-
-\begin_layout LyX-Code
- |
-\backslash
-__ connectivity [nel x ndof]
-\end_layout
-
-\begin_layout LyX-Code
-
-\backslash
-__ variables
-\end_layout
-
-\begin_layout LyX-Code
- |__ temperature
-\end_layout
-
-\begin_layout LyX-Code
- | |__ step00000 [nno x 1]
-\end_layout
-
-\begin_layout LyX-Code
- | |__ step00010 [nno x 1]
-\end_layout
-
-\begin_layout LyX-Code
- |
-\backslash
-...
-\end_layout
-
-\begin_layout LyX-Code
- |__ displacement
-\end_layout
-
-\begin_layout LyX-Code
- | |__ step00000 [nno x 3]
-\end_layout
-
-\begin_layout LyX-Code
- | |__ step00010 [nno x 3]
-\end_layout
-
-\begin_layout LyX-Code
- |
-\backslash
-...
-\end_layout
-
-\begin_layout LyX-Code
-
-\backslash
-__ velocity
-\end_layout
-
-\begin_layout LyX-Code
- |__ step00000 [nno x 3]
-\end_layout
-
-\begin_layout LyX-Code
- |__ step00010 [nno x 3]
-\end_layout
-
-\begin_layout LyX-Code
-
-\backslash
-...
-\end_layout
-
-\begin_layout Standard
-Generally, Cigma will only require you to specify the path to a specific
- field dataset.
- If a small amount of metadata is present in your field dataset, the rest
- of the required information, such as which mesh and finite elements to
- use, will be deduced from that metadata.
-\end_layout
-
-\begin_layout Description
-MeshID an identifier assigned for use in linking child datasets to their
- parent mesh.
-\end_layout
-
-\begin_layout Description
-MeshLocation points to the HDF5 group which contains the appropriate coordinates
- and connectivity datasets.
-\end_layout
-
-\begin_layout Description
-FunctionSpace string identifier to determine which shape functions to use
- for interpolating values inside the element (e.g., tet4, hex8, quad4, tri3,
- ...).
-\end_layout
-
-\begin_layout Description
-DatasetType string identifier for classifying the type of data contained
- in the dataset (e.g., points, connectivity, degrees of freedom, quadrature
- rules, global quadrature points, global field values).
-\end_layout
-
-\begin_layout Subsection
-VTK
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-VTK files are typically designed for visualization purposes,
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Note that while you typically provide a path (or name) for every dataset,
- this is not necessary when specifying a VTK mesh, since this data is taken
- from the special Points and Cells arrays, which you cannot rename.
- However, you will still need to provide a name when referring to the field
- coefficients, which are assumed to be stored as Point Data in the input
- VTK file.
-\end_layout
-
-\begin_layout Standard
-For more information, you may want to refer to Visualization ToolKit's
-\begin_inset LatexCommand htmlurl
-name "File Formats"
-target "www.vtk.org/pdf/file-formats.pdf"
-
-\end_inset
-
- document.
-
-\end_layout
-
-\begin_layout Subsection
-Text
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Don't forget about the comment character '#' (which skips everything until
- the next newline)
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The text files we use in Cigma consist of a simple list of numbers in ASCII
- format whose layout corresponds exactly to the simple array layout discussed
- earlier in Section 1.1.
- The first line of the file contains the dimensions of the array, just two
- integers separated by whitespace.
- The rest of the file, describing the array data, must contain as many numbers
- as the product of the two dimensions given in the first line.
- These numbers are formatted as integer or floating point, depending on
- context, and are all separated by whitespace.
-\end_layout
-
-\begin_layout Standard
-Some restrictions apply when providing data in text format, mostly because
- you can only specify a single element block in a text file containing connectiv
-ity information.
- Operations involving a larger number of blocks must use data in the format
- described in Section 1.3.1.
-\end_layout
-
-\begin_layout Standard
-Another point worthy of mention is that arrays must be two-dimensional,
- so two integers are always expected in the first line.
- When specifying a one-dimensional array, such as a list of weights correspondin
-g to an integration rule, one of the array dimensions must be 1.
-\end_layout
-
-\begin_layout Section
-Output Files
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Depending on the particular operation you perform, there are three different
- kinds of output files.
- The output format for residuals consists simply of a list of scalars for
- each element in the discretization.
- If you specify a
-\family typewriter
-.vtk
-\family default
- extension for your output file, this will result in an scalar dataset named
- epsilon stored using the legacy VTK file format in the Cell Data section
- of the output file.
- Note that this output consists of the squared values of the local residuals,
- so further post-processing will be necessary.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Depending on the particular operation you perform, there are three different
- kinds of output files.
- The output format for residuals consists simply of a list of scalars for
- each element in the discretization, stored using the legacy VTK file format
- in the Cell Data section of the output file.
- Note that this output consists of the squared values of the local residuals,
- so further post-processing will be necessary.
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/formats.lyx (from rev 13032, doc/cigma/manual/formats/formats.lyx)
===================================================================
--- doc/cigma/manual/formats.lyx (rev 0)
+++ doc/cigma/manual/formats.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,3405 @@
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\use_default_options false
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Plain Layout
+
+TODO: Rename file_formats.lyx to formats.lyx (done)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+TODO: What about NodeNumberMap?
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Define the dimensions for every object in the file
+\end_layout
+
+\begin_layout Plain Layout
+
+ Mesh Coordinates
+\end_layout
+
+\begin_layout Plain Layout
+
+ Mesh Connectivity (list of Element Blocks)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Field Variables (Global, Nodal, Element)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Element Tabulation (Shape function, Shape Function Derivatives)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Integration Rule (Weights and Local Points)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Global Integration Points
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Explain structure of each file format:
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ Text Format
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ All arrays specified explicitly in several text files containing a list
+ of floating point numbers.
+\end_layout
+
+\begin_layout Plain Layout
+
+ The first two numbers indicate the dimensions of the array, followed
+ by a list of floats
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --integration-mesh-coordinates=imesh_coords.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+ --integration-mesh-connectivity=imesh_connect.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --integration-rule-points=tet4_order10_points.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+ --integration-rule-weights=tet4_order10_weights.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --first-mesh-coordinates=first_coords.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+ --first-mesh-connectivity=first_connect.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --second-mesh-coordinates=first_coords.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+ --second-mesh-connectivity=first_connect.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --first-dofs=first_var_stepN.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+ --second-dofs=second_var_stepN.dat
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ --integration-rule=tet4_order10.dat <-- (weights in first column)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ Simple Arrays (HDF5)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ Structure of Model Definition
+\end_layout
+
+\begin_layout Plain Layout
+
+ root group at /model
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh data at /model/mesh
+\end_layout
+
+\begin_layout Plain Layout
+
+ node coordinates as specified on global coordinate system at /model/mesh
+/coordinates
+\end_layout
+
+\begin_layout Plain Layout
+
+ list of element blocks at /model/mesh/connectivity
+\end_layout
+
+\begin_layout Plain Layout
+
+ note that each element block is tagged by element type in the FE
+ attribute
+\end_layout
+
+\begin_layout Plain Layout
+
+ degrees-of-freedom data
+\end_layout
+
+\begin_layout Plain Layout
+
+ organized by variable at /model/vars/var, or even just /model/var
+\end_layout
+
+\begin_layout Plain Layout
+
+ list of steps: /model/var/step{0,1,...,N}
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ Structure of file defining an Integration Rule
+\end_layout
+
+\begin_layout Plain Layout
+
+ root group is / by default, but may be specified directly (eg, as
+ /rule)
+\end_layout
+
+\begin_layout Plain Layout
+
+ root group is tagged with FE string indicating the source element
+ (tet4, ...)
+\end_layout
+
+\begin_layout Plain Layout
+
+ integration points contained at /rule/points
+\end_layout
+
+\begin_layout Plain Layout
+
+ integration points contained at /rule/weights
+\end_layout
+
+\begin_layout Plain Layout
+
+ Structure of file with Residuals
+\end_layout
+
+\begin_layout Plain Layout
+
+ root group is / by default, but it must exist if specified
+\end_layout
+
+\begin_layout Plain Layout
+
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK
+\end_layout
+
+\begin_layout Plain Layout
+
+ Basic structure:
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK Grid
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK Dataset consists of the mesh information (node coordinates
+ and associated connectivity relations)
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK Data Arrays come in two types (Point Data and Cell Data)
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK Point Data - scalars, vectors, tensors as defined on the
+ nodes
+\end_layout
+
+\begin_layout Plain Layout
+
+ VTK Cell Data - scalars, vectors, tensors as defined on each
+ element
+\end_layout
+
+\begin_layout Plain Layout
+
+ Structure of different types
+\end_layout
+
+\begin_layout Plain Layout
+
+ Unstructured Grid
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh points: given explicitly in a simple array
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh connectivity: given explicitly in a list of simple arrays
+\end_layout
+
+\begin_layout Plain Layout
+
+ element types: Line, Quad, Tetra, Hexahedron, Quadratic Triangle,
+ Quadratic Quad, Quadratic Tetra, Quadratic Hexahedron
+\end_layout
+
+\begin_layout Plain Layout
+
+ Structured Grid
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh points: given explicitly in a simple array
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh connectivity: derived from dimensions (nx, ny, nz)
+\end_layout
+
+\begin_layout Plain Layout
+
+ element types: Quad, Pixel, Hex, Voxel
+\end_layout
+
+\begin_layout Plain Layout
+
+ Rectilinear Grid
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh dimensions: (nx, ny, nz)
+\end_layout
+
+\begin_layout Plain Layout
+
+ mesh points: derived from 'double xcoords[nx], ycoords[ny], zcoords[n
+z]' arrays
+\end_layout
+
+\begin_layout Plain Layout
+
+ element types: Pixel, Voxel
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ UCD
+\end_layout
+
+\begin_layout Plain Layout
+
+ Format described in (LINK-TO-DOCUMENT).
+\end_layout
+
+\begin_layout Plain Layout
+
+ Using VTK API (vtkAVSucdReader) to read (*.inp) files
+\end_layout
+
+\begin_layout Plain Layout
+
+ Just extracts the data into an Unstructured Grid
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ ExodusII
+\end_layout
+
+\begin_layout Plain Layout
+
+ This type of mesh can be generated by other programs, such as CUBIT
+\end_layout
+
+\begin_layout Plain Layout
+
+ Currently only used to specify an integration mesh, but it is capable
+ of storing degree-of-freedom data as well.
+\end_layout
+
+\begin_layout Plain Layout
+
+ Utility script for converting *.exo into the appropriate *.h5 file (need
+ PyTables, Numpy, Scientific.IO.NetCDF)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+--------------
+\end_layout
+
+\begin_layout Plain Layout
+
+From running.lyx:
+\end_layout
+
+\begin_layout Plain Layout
+
+The underlying data storage format for Cigma is the HDF5 format, due to
+ its flexibility for storing and organizing large amounts of data.
+ The Hierarchical Data Format (HDF) is designed for storing, retrieving,
+ analyzing, visualizing, and converting scientific data.
+ It uses a hierarchical structure that provides users a host of options
+ for organizing how their data is stored in HDF5 files.
+ Using HDF5 datasets in Cigma allows us to avoid having to convert between
+ too many distinct formats.
+ Moreover, due to the amount of disk I/O, large finite element meshes can
+ be handled more efficiently in binary format.
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Another popular format for providing mesh and field inputs.You can easily
+ examine the structure of an input file by using the cigma list command,
+ which will simply reveal the names and dimensions of all datasets inside
+ the specified file.
+\end_layout
+
+\begin_layout Plain Layout
+
+Inputs can be either HDF5 or VTK.
+ Specifying the complete path to a dataset consists of the special form
+ filepath:dataset, a colon-delimited pair of file path and dataset path.
+\end_layout
+
+\begin_layout Plain Layout
+
+Mesh inputs are currently only unstructured grids.
+\end_layout
+
+\begin_layout Plain Layout
+
+Only exception are the residuals, which are written in legacy VTK format
+ as scalar cell data over an unstructured grid.
+\end_layout
+
+\begin_layout Plain Layout
+
+Because Cigma relies on the ability of the user to specify dataset paths,
+ we have provided a command called list for viewing the structure of an
+ input file.
+ Its usage is very simple.
+
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+This chapter may be summarized by simply stating that most input data can
+ be specified as a two-dimensional array.
+ The meaning of this array, of course, is dependent on context.
+ That is the main concept that we have to get accross in this chapter.
+ If the content is similar enough, perhaps the appendix could just be incorporat
+ed into this section.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Data and File Formats
+\end_layout
+
+\begin_layout Standard
+As we've seen in the previous chapters, there are a number of data objects
+ involved in making any particular comparison.
+ Here, we discuss the variety of formats available, as well as the layout
+ within the corresponding data file.
+\end_layout
+
+\begin_layout Section
+Data Formats
+\end_layout
+
+\begin_layout Standard
+The basic data structure is a two-dimensional array of values, stored in
+ a contiguous format as shown below
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(m,n)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Data
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $a_{11},a_{12},\ldots,a_{1n}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $a_{21},a_{22},\ldots,a_{2n}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\cdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $a_{m1},a_{m2},\ldots,a_{mn}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+The second index in this array varies the fastest, so that data often referenced
+ together remains together in memory as well.
+\end_layout
+
+\begin_layout Subsection
+Node Coordinates
+\end_layout
+
+\begin_layout Standard
+One of the more efficient
+\end_layout
+
+\begin_layout Standard
+On a mesh with
+\begin_inset Formula $n_{no}$
+\end_inset
+
+ node coordinates, which are specified on a global coordinate system, we
+ have.
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "50col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{no},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{1},y_{1},z_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{2},y_{2},z_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{n_{no}},y_{n_{no}},z_{n_{no}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "50col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Plain Layout
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{no},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{1},y_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{2},y_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{n_{no}},y_{n_{no}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Element Connectivity
+\end_layout
+
+\begin_layout Standard
+Mesh connectivity is specified at the element-block level.
+ On a given block, we only consider a single element type, which allows
+ us to store the global node ids into a contiguous array of integers.
+ Multiple blocks are specified separately, and they all reference the same
+ node ids into the node coordinates table.
+\end_layout
+
+\begin_layout Standard
+In general, if our block contains
+\begin_inset Formula $k$
+\end_inset
+
+ elements with
+\begin_inset Formula $m$
+\end_inset
+
+-degrees of freedom each, then the connectivity array defining our element
+ block has the form
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(k,m)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Connectivity Data
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $n_{11},n_{12},\ldots,n_{1m}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $n_{21},n_{22},\ldots,n_{2m}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $n_{k1},n_{k2},\ldots,n_{km}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For the element types defined in Cigma, the values for
+\begin_inset Formula $m$
+\end_inset
+
+ are as follows.
+ Note that the total number of degrees of freedom depends on the rank of
+ the specific function being represented.
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="4">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Element
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Tensor
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+tet4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+12
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+24
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+tet10
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+10
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+30
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+60
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+hex8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+24
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+48
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+hex20
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+20
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+60
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+120
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+hex27
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+27
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+81
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+162
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Similarly for the 2-D elements, we have
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="4">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Element
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Tensor
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+tri3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+6
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+9
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+quad4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+8
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+12
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Field Variables
+\end_layout
+
+\begin_layout Standard
+A field variable represents one of the functions that we can evaluate, and
+ is typically stored over a series of timesteps.
+ The data for a field variable consists of snapshots through time, which
+ are stored as separate arrays.
+ A value for each degree of freedom is provided on the global list of nodes
+ in the mesh.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+arrays are subject to interpretation based on their corresponding tag:
+\end_layout
+
+\begin_layout Plain Layout
+by default, interpreted as shape function coefficients (from text and vtk)
+\end_layout
+
+\begin_layout Plain Layout
+can also specify spherical harmonics coefficients.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Shape Function Values
+\end_layout
+
+\begin_layout Standard
+In certain circumstances you may be able to provide your custom reference
+ element by simply providing the appropriate
+\begin_inset Formula $n$
+\end_inset
+
+ shape function values to be used at the expected integration points.
+ This is most useful when you are using the same mesh for both the integration
+ mesh and the array.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="3">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Shape Function Values:
+\begin_inset Formula $(n,Q)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Jacobian Determinant:
+\begin_inset Formula $(Q,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Integration Point 1
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{1}(\vec{\xi}_{1}),N_{2}(\vec{\xi}_{1}),\ldots,N_{n}(\vec{\xi}_{1})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $J(\vec{\xi}_{1})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Integration Point 2
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{1}(\vec{\xi}_{2}),N_{2}(\vec{\xi}_{2}),\ldots,N_{n}(\vec{\xi}_{2})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $J(\vec{\xi}_{2})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Integration Point
+\begin_inset Formula $Q$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $N_{1}(\vec{\xi}_{Q}),N_{2}(\vec{\xi}_{Q}),\ldots,N_{n}(\vec{\xi}_{Q})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $J(\vec{\xi}_{Q})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Integration Rule
+\end_layout
+
+\begin_layout Standard
+As described in Chapter 4, an integration rule is specified by a list of
+ points and associated weights.
+ The points are given on natural coordinate system of the reference element.
+
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="5">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Weights:
+\begin_inset Formula $(Q,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Points:
+\begin_inset Formula $(Q,n_{dim})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Points:
+\begin_inset Formula $(Q,2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Points:
+\begin_inset Formula $(Q,3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Rule Definition
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $w_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{\xi}_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{1},\eta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{1},\eta_{1},\zeta_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $w_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{\xi}_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{2},\eta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{2},\eta_{2},\zeta_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $w_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{\xi}_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{Q},\eta_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\xi_{Q},\eta_{Q},\zeta_{Q}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+A typical point
+\begin_inset Formula $\vec{\xi}_{i}$
+\end_inset
+
+ in 2-D is typically written as
+\begin_inset Formula $(\xi_{i},\eta_{i})$
+\end_inset
+
+, and in 3-D we write
+\begin_inset Formula $(\xi_{i},\eta_{i},\zeta_{i})$
+\end_inset
+
+.
+ In special cases, integration points may be specified in
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Note: It is important to realize that the integration weights need to be
+ adjusted by the appropriate factor when trying to compare rules defined
+ on different reference regions.
+ It is not sufficient to simply transform the points, as the weights need
+ to be adjusted by the jacobian factor as well (see the change of variables
+ formula for integrals).
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Integration Points and Values
+\end_layout
+
+\begin_layout Standard
+Given a mesh and an integration rule, we can map the integration points
+ on each element into a possibly large set of global points.
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="4">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Points:
+\begin_inset Formula $(n_{pts},n_{dim})$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Points:
+\begin_inset Formula $(n_{pts},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Points:
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Coordinates
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{x}_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{1},y_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{1},y_{1},z_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{x}_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{2},y_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{2},y_{2},z_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vec{x}_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{n_{pts}}y_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $x_{n_{pts}},y_{n_{pts}},z_{n_{pts}}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+The result of an evaluation will depend on the rank of the function being
+ evaluated, as indicated in the table below.
+ Note that the fastest varying dimension of the array contains the components
+ of the function value at each point.
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Evaluated Quantity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Resulting Array Shape
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Scalar
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{pts},1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{pts},2)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Vector
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+2-D Tensor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{pts},3)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+3-D Tensor
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(n_{pts},6)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Comparison Residuals
+\end_layout
+
+\begin_layout Standard
+Since our norm reduces integrals over cells to a single scalar value, the
+ result of a comparison will result in a simple scalar array for each element
+ block examined.
+ Note that element blocks used in the integration process belong to the
+ integration mesh, which may differ from the associated mesh to either function.
+\end_layout
+
+\begin_layout Standard
+Thus, for each element block containing
+\begin_inset Formula $k$
+\end_inset
+
+ cells, we have the output array
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Tabular
+<lyxtabular version="3" rows="5" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Array Shape
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $(k,1)$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Residual Value
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\varepsilon_{1}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\varepsilon_{2}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\vdots$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+\begin_inset Formula $\varepsilon_{k}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+These residual values may be combined in the following form:
+\begin_inset Formula \[
+\varepsilon_{block}=\sqrt{\varepsilon_{1}^{2}+\varepsilon_{2}^{2}+\cdots+\varepsilon_{k}^{2}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The final global residual norm maybe obtained by simply summing over all
+ element blocks that partition the domain of interest:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\varepsilon_{global}=\sqrt{\sum_{block}\varepsilon_{block}^{2}}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+File Formats
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+XXX: The input/output file dichotomy does not seem as necessary now that
+ we've added the Data Format section.
+ Get rid of those subsection headings.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+A number of file formats are supported for storing and retrieving the above
+ data structures to and from disk.
+ The following file extensions are used for these purposes:
+\begin_inset Newline newline
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+File Format
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Recognized Extensions
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+HDF5
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+*.h5
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+VTK
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+*.vtk
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+XML VTK
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+*.vtu, *.vts, *.vtr
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Parallel XML VTK
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+*.pvtu, *.pvts, *.pvtr
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+Text
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+
+\family typewriter
+*.txt, *.dat
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Input Files
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+There are five different objects you will want to use as input to Cigma,
+ all of which can be represented as two-dimensional datasets.
+ These are the
+\end_layout
+
+\begin_layout Itemize
+mesh coordinates
+\end_layout
+
+\begin_layout Itemize
+mesh connectivity
+\end_layout
+
+\begin_layout Itemize
+field coefficients (degrees of freedom)
+\end_layout
+
+\begin_layout Itemize
+quadrature rule points
+\end_layout
+
+\begin_layout Itemize
+quadrature rule weights
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In Section 1.1, we examined what kinds of objects we will be accessing, so
+ now let's discuss the actual layout in the files in which these objects
+ will be stored.
+\end_layout
+
+\begin_layout Subsection
+HDF5
+\end_layout
+
+\begin_layout Standard
+HDF5 files are binary files encoded in a data format designed to store large
+ amounts of scientific data in a portable and self-describing way.
+\end_layout
+
+\begin_layout Standard
+The internal organization of a typical HDF5 file consists of a hierarchical
+ structure similar to a UNIX file system.
+ Two types of primary objects,
+\emph on
+groups
+\emph default
+ and
+\emph on
+datasets
+\emph default
+, are stored in this structure.
+ A group contains instances of zero or more groups or datasets, while a
+ dataset stores a multi-dimensional array of data elements.
+ In a sense, datasets are analogous to files in a traditional filesystem,
+ and groups are analagous to folders.
+ One important difference, however, is that you can attach supporting metadata
+ to both kinds of objects.
+ The metadata objects are called
+\emph on
+attribute
+\emph default
+s.
+\end_layout
+
+\begin_layout Standard
+A dataset is physically stored in two parts: a header and a data array.
+ The header contains miscellaneous metadata describing the dataset as well
+ as information that is needed to interpret the array portion of the dataset.
+ Essentially, it includes the name, datatype, dataspace, and storage layout
+ of the dataset.
+ The name is a text string identifying the dataset.
+ The datatype describes the type of the data array elements.
+ The dataspace defines the dimensionality of the dataset, i.e., the size and
+ shape of the multi-dimensional array.
+\end_layout
+
+\begin_layout Standard
+In Cigma you may always provide an explicit path to every dataset, so you
+ have a fair amount of flexibility for how you organize your datasets inside
+ HDF5 files.
+ For example, a typical Cigma HDF5 file could have the following structure.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Mention these assumptions in the corresponding section in running.lyx
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+model.h5
+\end_layout
+
+\begin_layout LyX-Code
+
+\backslash
+__ model
+\end_layout
+
+\begin_layout LyX-Code
+ |__ mesh
+\end_layout
+
+\begin_layout LyX-Code
+ | |__ coordinates [nno x nsd]
+\end_layout
+
+\begin_layout LyX-Code
+ |
+\backslash
+__ connectivity [nel x ndof]
+\end_layout
+
+\begin_layout LyX-Code
+
+\backslash
+__ variables
+\end_layout
+
+\begin_layout LyX-Code
+ |__ temperature
+\end_layout
+
+\begin_layout LyX-Code
+ | |__ step00000 [nno x 1]
+\end_layout
+
+\begin_layout LyX-Code
+ | |__ step00010 [nno x 1]
+\end_layout
+
+\begin_layout LyX-Code
+ |
+\backslash
+...
+\end_layout
+
+\begin_layout LyX-Code
+ |__ displacement
+\end_layout
+
+\begin_layout LyX-Code
+ | |__ step00000 [nno x 3]
+\end_layout
+
+\begin_layout LyX-Code
+ | |__ step00010 [nno x 3]
+\end_layout
+
+\begin_layout LyX-Code
+ |
+\backslash
+...
+\end_layout
+
+\begin_layout LyX-Code
+
+\backslash
+__ velocity
+\end_layout
+
+\begin_layout LyX-Code
+ |__ step00000 [nno x 3]
+\end_layout
+
+\begin_layout LyX-Code
+ |__ step00010 [nno x 3]
+\end_layout
+
+\begin_layout LyX-Code
+
+\backslash
+...
+\end_layout
+
+\begin_layout Standard
+Generally, Cigma will only require you to specify the path to a specific
+ field dataset.
+ If a small amount of metadata is present in your field dataset, the rest
+ of the required information, such as which mesh and finite elements to
+ use, will be deduced from that metadata.
+\end_layout
+
+\begin_layout Description
+MeshID an identifier assigned for use in linking child datasets to their
+ parent mesh.
+\end_layout
+
+\begin_layout Description
+MeshLocation points to the HDF5 group which contains the appropriate coordinates
+ and connectivity datasets.
+\end_layout
+
+\begin_layout Description
+FunctionSpace string identifier to determine which shape functions to use
+ for interpolating values inside the element (e.g., tet4, hex8, quad4, tri3,
+ ...).
+\end_layout
+
+\begin_layout Description
+DatasetType string identifier for classifying the type of data contained
+ in the dataset (e.g., points, connectivity, degrees of freedom, quadrature
+ rules, global quadrature points, global field values).
+\end_layout
+
+\begin_layout Subsection
+VTK
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+VTK files are typically designed for visualization purposes,
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Note that while you typically provide a path (or name) for every dataset,
+ this is not necessary when specifying a VTK mesh, since this data is taken
+ from the special Points and Cells arrays, which you cannot rename.
+ However, you will still need to provide a name when referring to the field
+ coefficients, which are assumed to be stored as Point Data in the input
+ VTK file.
+\end_layout
+
+\begin_layout Standard
+For more information, you may want to refer to Visualization ToolKit's File
+ Formats
+\begin_inset Flex URL
+status collapsed
+
+\begin_layout Plain Layout
+
+www.vtk.org/pdf/file-formats.pdf
+\end_layout
+
+\end_inset
+
+ document.
+
+\end_layout
+
+\begin_layout Subsection
+Text
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+Don't forget about the comment character '#' (which skips everything until
+ the next newline)
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The text files we use in Cigma consist of a simple list of numbers in ASCII
+ format whose layout corresponds exactly to the simple array layout discussed
+ earlier in Section 1.1.
+ The first line of the file contains the dimensions of the array, just two
+ integers separated by whitespace.
+ The rest of the file, describing the array data, must contain as many numbers
+ as the product of the two dimensions given in the first line.
+ These numbers are formatted as integer or floating point, depending on
+ context, and are all separated by whitespace.
+\end_layout
+
+\begin_layout Standard
+Some restrictions apply when providing data in text format, mostly because
+ you can only specify a single element block in a text file containing connectiv
+ity information.
+ Operations involving a larger number of blocks must use data in the format
+ described in Section 1.3.1.
+\end_layout
+
+\begin_layout Standard
+Another point worthy of mention is that arrays must be two-dimensional,
+ so two integers are always expected in the first line.
+ When specifying a one-dimensional array, such as a list of weights correspondin
+g to an integration rule, one of the array dimensions must be 1.
+\end_layout
+
+\begin_layout Section
+Output Files
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
+Depending on the particular operation you perform, there are three different
+ kinds of output files.
+ The output format for residuals consists simply of a list of scalars for
+ each element in the discretization.
+ If you specify a
+\family typewriter
+.vtk
+\family default
+ extension for your output file, this will result in an scalar dataset named
+ epsilon stored using the legacy VTK file format in the Cell Data section
+ of the output file.
+ Note that this output consists of the squared values of the local residuals,
+ so further post-processing will be necessary.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Depending on the particular operation you perform, there are three different
+ kinds of output files.
+ The output format for residuals consists simply of a list of scalars for
+ each element in the discretization, stored using the legacy VTK file format
+ in the Cell Data section of the output file.
+ Note that this output consists of the squared values of the local residuals,
+ so further post-processing will be necessary.
+\end_layout
+
+\end_body
+\end_document
Copied: doc/cigma/manual/gpl-license.lyx (from rev 13032, doc/cigma/manual/license/gpl-license.lyx)
===================================================================
--- doc/cigma/manual/gpl-license.lyx (rev 0)
+++ doc/cigma/manual/gpl-license.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,660 @@
+#LyX 1.5.4 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Chapter
+License
+\end_layout
+
+\begin_layout Standard
+
+\series bold
+GNU GENERAL PUBLIC LICENSE Version 2, June 1991.
+ Copyright (C) 1989, 1991 Free Software Foundation, Inc.
+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+\series default
+
+\newline
+
+\series medium
+Everyone is permitted to copy and distribute verbatim copies of this license
+ document, but changing it is not allowed.
+\end_layout
+
+\begin_layout Section*
+Preamble
+\end_layout
+
+\begin_layout Standard
+The licenses for most software are designed to take away your freedom to
+ share and change it.
+ By contrast, the GNU General Public License is intended to guarantee your
+ freedom to share and change free software -- to make sure the software
+ is free for all its users.
+ This General Public License applies to most of the Free Software Foundation's
+ software and to any other program whose authors commit to using it.
+ (Some other Free Software Foundation software is covered by the GNU Library
+ General Public License instead.) You can apply it to your programs, too.
+\end_layout
+
+\begin_layout Standard
+When we speak of free software, we are referring to freedom, not price.
+ Our General Public Licenses are designed to make sure that you have the
+ freedom to distribute copies of free software (and charge for this service
+ if you wish), that you receive source code or can get it if you want it,
+ that you can change the software or use pieces of it in new free programs;
+ and that you know you can do these things.
+\end_layout
+
+\begin_layout Standard
+To protect your rights, we need to make restrictions that forbid anyone
+ to deny you these rights or to ask you to surrender the rights.
+ These restrictions translate to certain responsibilities for you if you
+ distribute copies of the software, or if you modify it.
+\end_layout
+
+\begin_layout Standard
+For example, if you distribute copies of such a program, whether gratis
+ or for a fee, you must give the recipients all the rights that you have.
+ You must make sure that they, too, receive or can get the source code.
+ And you must show them these terms so they know their rights.
+\end_layout
+
+\begin_layout Standard
+We protect your rights with two steps:
+\end_layout
+
+\begin_layout Enumerate
+Copyright the software, and
+\end_layout
+
+\begin_layout Enumerate
+Offer you this license which gives you legal permission to copy, distribute
+ and/or modify the software.
+\end_layout
+
+\begin_layout Standard
+Also, for each author's protection and ours, we want to make certain that
+ everyone understands that there is no warranty for this free software.
+ If the software is modified by someone else and passed on, we want its
+ recipients to know that what they have is not the original, so that any
+ problems introduced by others will not reflect on the original authors'
+ reputations.
+\end_layout
+
+\begin_layout Standard
+Finally, any free program is threatened constantly by software patents.
+ We wish to avoid the danger that redistributors of a free program will
+ individually obtain patent licenses, in effect making the program proprietary.
+ To prevent this, we have made it clear that any patent must be licensed
+ for everyone's free use or not licensed at all.
+
+\end_layout
+
+\begin_layout Standard
+The precise terms and conditions for copying, distribution and modification
+ follow.
+\end_layout
+
+\begin_layout Section*
+GNU GENERAL PUBLIC LICENSE TERMS AND CONDITIONS FOR COPYING, DISTRIBUTION
+ AND MODIFICATION
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+begin{itemize}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+item[0.]
+\end_layout
+
+\end_inset
+
+This License applies to any program or other work which contains a notice
+ placed by the copyright holder saying it may be distributed under the terms
+ of this General Public License.
+ The "Program" below refers to any such program or work, and a "work based
+ on the Program" means either the Program or any derivative work under copyright
+ law: that is to say, a work containing the Program or a portion of it,
+ either verbatim or with modifications and/or translated into another language.
+ (Hereinafter, translation is included without limitation in the term "modificat
+ion.") Each licensee is addressed as "you."
+\newline
+
+\newline
+Activities other than copying,
+ distribution and modification are not covered by this License; they are
+ outside its scope.
+ The act of running the Program is not restricted, and the output from the
+ Program is covered only if its contents constitute a work based on the
+ Program (independent of having been made by running the Program).
+ Whether that is true depends on what the Program does.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+end{itemize}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+You may copy and distribute verbatim copies of the Program's source code
+ as you receive it, in any medium, provided that you conspicuously and appropria
+tely publish on each copy an appropriate copyright notice and disclaimer
+ of warranty; keep intact all the notices that refer to this License and
+ to the absence of any warranty; and give any other recipients of the Program
+ a copy of this License along with the Program.
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+You may charge a fee for the physical act of transferring a copy, and you
+ may at your option offer warranty protection in exchange for a fee.
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+You may modify your copy or copies of the Program or any portion of it,
+ thus forming a work based on the Program, and copy and distribute such
+ modifications or work under the terms of Section 1 above, provided that
+ you also meet all of these conditions:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+You must cause the modified files to carry prominent notices stating that
+ you changed the files and the date of any change.
+
+\end_layout
+
+\begin_layout Enumerate
+You must cause any work that you distribute or publish, that in whole or
+ in part contains or is derived from the Program or any part thereof, to
+ be licensed as a whole at no charge to all third parties under the terms
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+
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+
+\begin_layout Enumerate
+If the modified program normally reads commands interactively when run,
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+ most ordinary way, to print or display an announcement including an appropriate
+ copyright notice and a notice that there is no warranty (or else, saying
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+ (Exception: if the Program itself is interactive but does not normally
+ print such an announcement, your work based on the Program is not required
+ to print an announcement.)
+\end_layout
+
+\begin_layout Standard
+These requirements apply to the modified work as a whole.
+ If identifiable sections of that work are not derived from the Program,
+ and can be reasonably considered independent and separate works in themselves,
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+
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+Thus, it is not the intent of this section to claim rights or contest your
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+
+\end_layout
+
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+In addition, mere aggregation of another work not based on the Program with
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+\begin_layout Enumerate
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+\begin_layout Enumerate
+Accompany it with the complete corresponding machine-readable source code,
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+
+\begin_layout Enumerate
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+\begin_layout Enumerate
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+ (This alternative is allowed only for noncommercial distribution and only
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+ an offer, in accord with Subsection b above.)
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+
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+The source code for a work means the preferred form of the work for making
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+ For an executable work, complete source code means all the source code
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+If distribution of executable or object code is made by offering access
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+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+You may not copy, modify, sublicense, or distribute the Program except as
+ expressly provided under this License.
+ Any attempt otherwise to copy, modify, sublicense or distribute the Program
+ is void, and will automatically terminate your rights under this License.
+ However, parties who have received copies, or rights, from you under this
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+ remain in full compliance.
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+
+\begin_layout Enumerate
+You are not required to accept this License, since you have not signed it.
+ However, nothing else grants you permission to modify or distribute the
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+ Therefore, by modifying or distributing the Program (or any work based
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+
+\begin_layout Enumerate
+If, as a consequence of a court judgment or allegation of patent infringement
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+ the conditions of this License, they do not excuse you from the conditions
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+ If you cannot distribute so as to satisfy simultaneously your obligations
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+ For example, if a patent license would not permit royalty-free redistribution
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+
+\begin_deeper
+\begin_layout Standard
+If any portion of this section is held invalid or unenforceable under any
+ particular circumstance, the balance of the section is intended to apply
+ and the section as a whole is intended to apply in other circumstances.
+\end_layout
+
+\begin_layout Standard
+It is not the purpose of this section to induce you to infringe any patents
+ or other property right claims or to contest validity of any such claims;
+ this section has the sole purpose of protecting the integrity of the free
+ software distribution system, which is implemented by public license practices.
+ Many people have made generous contributions to the wide range of software
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+ that system; it is up to the author/donor to decide if he or she is willing
+ to distribute software through any other system and a licensee cannot impose
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+
+\end_layout
+
+\begin_layout Standard
+This section is intended to make thoroughly clear what is believed to be
+ a consequence of the rest of this License.
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+If the distribution and/or use of the Program is restricted in certain countries
+ either by patents or by copyrighted interfaces, the original copyright
+ holder who places the Program under this License may add an explicit geographic
+al distribution limitation excluding those countries, so that distribution
+ is permitted only in or among countries not thus excluded.
+ In such case, this License incorporates the limitation as if written in
+ the body of this License.
+
+\end_layout
+
+\begin_layout Enumerate
+The Free Software Foundation may publish revised and/or new versions of
+ the General Public License from time to time.
+ Such new versions will be similar in spirit to the present version, but
+ may differ in detail to address new problems or concerns.
+
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Each version is given a distinguishing version number.
+ If the Program specifies a version number of this License which applies
+ to it and "any later version," you have the option of following the terms
+ and conditions either of that version or of any later version published
+ by the Free Software Foundation.
+ If the Program does not specify a version number of this License, you may
+ choose any version ever published by the Free Software Foundation.
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+If you wish to incorporate parts of the Program into other free programs
+ whose distribution conditions are different, write to the author to ask
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+ For software which is copyrighted by the Free Software Foundation, write
+ to the Free Software Foundation; we sometimes make exceptions for this.
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+
+\end_layout
+
+\begin_layout Section*
+NO WARRANTY
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+begin{itemize}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+item[11.]
+\end_layout
+
+\end_inset
+
+BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR
+ THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW.
+ EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER
+ PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER
+ EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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+ THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH
+ YOU.
+ SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY
+ SERVICING, REPAIR OR CORRECTION.
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+item[12.]
+\end_layout
+
+\end_inset
+
+IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL
+ ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE
+ THE PROGRAM AS PERMITTED ABOVE, BE LIABLE TO YOU FOR DAMAGES, INCLUDING
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+ OF DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR
+ THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+ EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY
+ OF SUCH DAMAGES.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+end{itemize}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+END OF TERMS AND CONDITIONS
+\end_layout
+
+\begin_layout Standard
+How to Apply These Terms to Your New Programs
+\end_layout
+
+\begin_layout Standard
+If you develop a new program, and you want it to be of the greatest possible
+ use to the public, the best way to achieve this is to make it free software
+ which everyone can redistribute and change under these terms.
+
+\end_layout
+
+\begin_layout Standard
+To do so, attach the following notices to the program.
+ It is safest to attach them to the start of each source file to most effectivel
+y convey the exclusion of warranty; and each file should have at least the
+ "copyright" line and a pointer to where the full notice is found.
+ For example:
+\end_layout
+
+\begin_layout Standard
+One line to give the program's name and a brief idea of what it does.
+ Copyright
+\size footnotesize
+© (
+\size default
+year) (name of author)
+\end_layout
+
+\begin_layout Standard
+This program is free software; you can redistribute it and/or modify it
+ under the terms of the GNU General Public License as published by the Free
+ Software Foundation; either version 2 of the License, or (at your option)
+ any later version.
+
+\end_layout
+
+\begin_layout Standard
+This program is distributed in the hope that it will be useful, but WITHOUT
+ ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
+ FOR A PARTICULAR PURPOSE.
+ See the GNU General Public License for more details.
+
+\end_layout
+
+\begin_layout Standard
+You should have received a copy of the GNU General Public License along
+ with this program; if not, write to the Free Software Foundation, Inc.,
+ 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
+\end_layout
+
+\begin_layout Standard
+Also add information on how to contact you by electronic and paper mail.
+
+\end_layout
+
+\begin_layout Standard
+If the program is interactive, make it output a short notice like this when
+ it starts in an interactive mode:
+\end_layout
+
+\begin_layout Standard
+Gnomovision version 69, Copyright © year name of author Gnomovision comes
+ with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it under certain
+ conditions; type `show c' for details.
+
+\end_layout
+
+\begin_layout Standard
+The hypothetical commands `show w' and `show c' should show the appropriate
+ parts of the General Public License.
+ Of course, the commands you use may be called something other than `show
+ w' and `show c'; they could even be mouse-clicks or menu items -- whatever
+ suits your program.
+
+\end_layout
+
+\begin_layout Standard
+You should also get your employer (if you work as a programmer) or your
+ school, if any, to sign a "copyright disclaimer" for the program, if necessary.
+ Here is a sample; alter the names:
+\end_layout
+
+\begin_layout Standard
+Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovisi
+on' (which makes passes at compilers) written by James Hacker.
+
+\end_layout
+
+\begin_layout Standard
+(signature of Ty Coon)
+\newline
+1 April 1989
+\newline
+Ty Coon, President of Vice
+\end_layout
+
+\begin_layout Standard
+This General Public License does not permit incorporating your program into
+ proprietary programs.
+ If your program is a subroutine library, you may consider it more useful
+ to permit linking proprietary applications with the library.
+ If this is what you want to do, use the GNU Library General Public License
+ instead of this License.
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/installation/installation.lyx
===================================================================
--- doc/cigma/manual/installation/installation.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/installation/installation.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,605 +0,0 @@
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-
-Things to mention:
-\end_layout
-
-\begin_layout Standard
-
- Mention installation from binary (test the binaries from buildbot and
- make sure they install properly)
-\end_layout
-
-\begin_layout Standard
-
- Mention how to obtain a copy of the development tree
-\end_layout
-
-\begin_layout Standard
-
- Add Scientific.IO.NetCDF to list of optional tools
-\end_layout
-
-\begin_layout Standard
-
- Add SciPy to the list of optional tools
-\end_layout
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-
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-
-Portions of Boost that we use
-\end_layout
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- boost::filesystem
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- boost::program_options
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- boost::regex (parsing parameters from source strings)
-\end_layout
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- boost::python (for wrapping the core library objects)
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-}
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-\begin_layout Chapter
-Installation and Getting Help
-\end_layout
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-\begin_layout Section
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-\end_layout
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-\begin_inset LatexCommand url
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-
-\end_inset
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-.
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- If you encounter any bugs or have problems installing Cigma, please submit
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-
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- our utility scripts and how they depend on each other, so that you can
- decide which of these to install.
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-The ScientificPython module is a Python library for common tasks in scientific
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-\end_inset
-
-.
- To install this extension from source, download the latest version and
- proceed as follows:
-\end_layout
-
-\begin_layout LyX-Code
-$ tar xvfz ScientificPython-2.x.y.tar.gz
-\end_layout
-
-\begin_layout LyX-Code
-$ cd ScientificPython-2.x.y
-\end_layout
-
-\begin_layout LyX-Code
-$ python setup.py install
-\end_layout
-
-\begin_layout Standard
-To test whether the Scientific.IO.NetCDF module was installed correctly, you
- can run `
-\family typewriter
-import Scientific.IO.NetCDF
-\family default
-' from within your standard Python interpreter shell.
-\end_layout
-
-\begin_layout Subsection
-SciPy
-\end_layout
-
-\begin_layout Standard
-SciPy is a Python module for use in mathematics, science, and engineering.
- It contains algorithms for optimization, linear algebra, and other common
- scientific tasks.
- SciPy is available for download from
-\begin_inset LatexCommand url
-target "scipy.org"
-
-\end_inset
-
-.
-\end_layout
-
-\begin_layout Subsection
-FIAT
-\end_layout
-
-\begin_layout Standard
-The Finite Element Automatic Tabulator (FIAT) Python module supports generation
- of arbitrary order instances of the Lagrange elements, as well as arbitrary
- order instances of the Jacobi-type quadrature rules on the same element
- shapes.
- We can also use it to evaluate the shape function values at the required
- integration points.
-\end_layout
-
-\begin_layout Standard
-FIAT is available from
-\begin_inset LatexCommand htmlurl
-name "The FEniCS Project"
-target "www.fenics.org/wiki/FIAT"
-
-\end_inset
-
-.
- To install this module, download it and issue the following command inside
- the FIAT source directory:
-\end_layout
-
-\begin_layout LyX-Code
-$ python setup.py install
-\end_layout
-
-\begin_layout Standard
-You should now be able to run `
-\family typewriter
-import FIAT
-\family default
-' from within your standard Python interpreter shell.
-\end_layout
-
-\begin_layout Subsection
-HDFView
-\end_layout
-
-\begin_layout Standard
-NCSA HDFView is a graphical user interface tool for accessing data in your
- HDF5 files.
- You can use it for viewing the internal group hierarchy, adding new datasets,
- and modifying or deleting existing datasets, or altering metadata on groups
- and datasets.
- You can download it from the
-\begin_inset LatexCommand htmlurl
-name "HDFView home page"
-target "hdf.ncsa.uiuc.edu/hdf-java-html/hdfview"
-
-\end_inset
-
-.
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/installation.lyx (from rev 13032, doc/cigma/manual/installation/installation.lyx)
===================================================================
--- doc/cigma/manual/installation.lyx (rev 0)
+++ doc/cigma/manual/installation.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,605 @@
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+Things to mention:
+\end_layout
+
+\begin_layout Standard
+
+ Mention installation from binary (test the binaries from buildbot and
+ make sure they install properly)
+\end_layout
+
+\begin_layout Standard
+
+ Mention how to obtain a copy of the development tree
+\end_layout
+
+\begin_layout Standard
+
+ Add Scientific.IO.NetCDF to list of optional tools
+\end_layout
+
+\begin_layout Standard
+
+ Add SciPy to the list of optional tools
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Portions of Boost that we use
+\end_layout
+
+\begin_layout Standard
+
+ boost::filesystem
+\end_layout
+
+\begin_layout Standard
+
+ boost::program_options
+\end_layout
+
+\begin_layout Standard
+
+ boost::regex (parsing parameters from source strings)
+\end_layout
+
+\begin_layout Standard
+
+ boost::python (for wrapping the core library objects)
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Installation and Getting Help
+\end_layout
+
+\begin_layout Section
+Getting Help
+\end_layout
+
+\begin_layout Standard
+For help, send an e-mail to the
+\begin_inset LatexCommand url
+name "CIG Computational Science Mailing List"
+target "cig-cs at geodynamics.org"
+
+\end_inset
+
+.
+ You can subscribe to the
+\family typewriter
+cig-cs
+\family default
+ mailing list and view archived discussions at the
+\begin_inset LatexCommand htmlurl
+name "CIG Mail Lists web page"
+target "geodynamics.org/cig/lists"
+
+\end_inset
+
+.
+ If you encounter any bugs or have problems installing Cigma, please submit
+ a report to the
+\begin_inset LatexCommand htmlurl
+name "CIG Bug Tracker"
+target "geodynamics.org/bugs"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Installing from Source
+\end_layout
+
+\begin_layout Standard
+To install Cigma, download the source package from the
+\begin_inset LatexCommand htmlurl
+name "CIG Cigma web page"
+target "geodynamics.org/cig/software/packages/cs/cigma"
+
+\end_inset
+
+.
+ You will need the GNU C++ compiler for this step.
+ The Boost, HDF5, and VTK libraries, described later in this section, are
+ also required components.
+\end_layout
+
+\begin_layout Standard
+After installing the necessary dependencies, simply unpack the source tar
+ file and issue the following commands
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz cigma-1.0.0.tar.gz
+\end_layout
+
+\begin_layout LyX-Code
+$ cd cigma-1.0.0
+\end_layout
+
+\begin_layout LyX-Code
+$ ./configure --with-boost=$BOOST_PREFIX
+\end_layout
+
+\begin_layout LyX-Code
+ --with-hdf5=$HDF5_PREFIX
+\end_layout
+
+\begin_layout LyX-Code
+ --with-vtk=$VTK_PREFIX
+\end_layout
+
+\begin_layout LyX-Code
+$ make
+\end_layout
+
+\begin_layout LyX-Code
+$ make install
+\end_layout
+
+\begin_layout Standard
+Both HDF5 and VTK prefixes are required if you have installed these libraries
+ in a custom location.
+ After these steps are complete, you should be able to run the
+\family typewriter
+cigma
+\family default
+ binary.
+\end_layout
+
+\begin_layout Subsection
+Boost
+\end_layout
+
+\begin_layout Standard
+The Boost C++ libraries are used in Cigma for parsing command-line options,
+ ensuring cross-platform access to the filesystem, and providing Python
+ bindings to certain Cigma C++ functions and classes.
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz boost_1_35_0.tar.gz
+\end_layout
+
+\begin_layout LyX-Code
+$ cd boost_1_35_0
+\end_layout
+
+\begin_layout LyX-Code
+$ ./configure
+\end_layout
+
+\begin_layout LyX-Code
+$ make
+\end_layout
+
+\begin_layout LyX-Code
+$ make install
+\end_layout
+
+\begin_layout Standard
+Without a custom installation prefix, Boost will target your
+\family typewriter
+/usr/local
+\family default
+ directory.
+\end_layout
+
+\begin_layout Subsection
+HDF5
+\end_layout
+
+\begin_layout Standard
+The Hierarchical Data Format (HDF5) library is available for download from
+
+\begin_inset LatexCommand htmlurl
+name "The HDF Group"
+target "hdfgroup.org/HDF5"
+
+\end_inset
+
+.
+ Binaries can be obtained at
+\begin_inset LatexCommand htmlurl
+target "hdfgroup.org/HDF5/release/obtain5.html"
+
+\end_inset
+
+.
+ To install from source, download the latest stable 1.6 version of this library
+ (1.6.7, for example) and issue the following commands
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz hdf5-1.6.7.tar.gz
+\end_layout
+
+\begin_layout LyX-Code
+$ cd hdf5-1.6.7
+\end_layout
+
+\begin_layout LyX-Code
+$ ./configure --prefix=$HOME/opt/hdf5/1.6.7
+\end_layout
+
+\begin_layout LyX-Code
+$ make
+\end_layout
+
+\begin_layout LyX-Code
+$ make install
+\end_layout
+
+\begin_layout Standard
+Without the custom prefix, this installation procedure will target your
+
+\family typewriter
+/usr
+\family default
+ directory.
+\end_layout
+
+\begin_layout Subsection
+VTK
+\end_layout
+
+\begin_layout Standard
+The Visualization Toolkit (VTK) library is a popular open source graphics
+ library for scientific visualizations.
+ We recommend that you obtain the binaries from a package manager, making
+ sure to install the associated development headers along with the library.
+ The source for the VTK library is available from
+\begin_inset LatexCommand htmlurl
+name "Kitware, Inc."
+target "www.vtk.org/get-software.php"
+
+\end_inset
+
+.
+ If you decide to compile VTK from source, you will also need the CMake
+ build environment, available from
+\begin_inset LatexCommand htmlurl
+name "CMake"
+target "cmake.org"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+After downloading the source package, you can issue the following commands
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz vtk-5.0.4.tar.gz
+\end_layout
+
+\begin_layout LyX-Code
+$ cd VTK
+\end_layout
+
+\begin_layout LyX-Code
+$ mkdir build
+\end_layout
+
+\begin_layout LyX-Code
+$ cd build
+\end_layout
+
+\begin_layout LyX-Code
+$ ccmake ..
+\end_layout
+
+\begin_layout LyX-Code
+$ make
+\end_layout
+
+\begin_layout LyX-Code
+$ make install
+\end_layout
+
+\begin_layout Standard
+You can set the installation prefix during the
+\family typewriter
+ccmake
+\family default
+ step above, which defaults to your
+\family typewriter
+/usr/local
+\family default
+ directory.
+\end_layout
+
+\begin_layout Section
+Additional Software (optional)
+\end_layout
+
+\begin_layout Standard
+The following software packages are not strictly required, but are listed
+ here for completeness since they are used by various utility scripts included
+ in the Cigma source tree.
+ These utilities are meant for creating and manipulating Cigma input/output
+ files.
+ We include a short description for how we use these software packages in
+ our utility scripts and how they depend on each other, so that you can
+ decide which of these to install.
+ You should be able to find binary packages for each of these Python modules
+ in the respective project home pages.
+ Note that you will need a C and Fortran compiler to install these from
+ source, as well as the appropriate libraries and development headers.
+\end_layout
+
+\begin_layout Subsection
+NumPy
+\end_layout
+
+\begin_layout Standard
+NumPy is a Python module which adds support for large multi-dimensional
+ arrays and matrices, and is available for download from the
+\begin_inset LatexCommand htmlurl
+name "NumPy Home Page"
+target "numpy.scipy.org "
+
+\end_inset
+
+.
+
+\end_layout
+
+\begin_layout Standard
+To install this extension module from source, download it and issue the
+ following command in the NumPy source directory
+\end_layout
+
+\begin_layout LyX-Code
+$ python setup.py install
+\end_layout
+
+\begin_layout Standard
+To verify the installation, the command `
+\family typewriter
+import numpy
+\family default
+' should run successfully from within your standard Python interpreter shell.
+\end_layout
+
+\begin_layout Subsection
+PyTables
+\end_layout
+
+\begin_layout Standard
+PyTables is a Python extension module that builds on top of the HDF5 library.
+ It provides a convenient scripting interface to manipulate HDF5 files,
+ which can be used to manipulate the input files needed by Cigma.
+ PyTables is available for download from
+\begin_inset LatexCommand htmlurl
+name "PyTables"
+target "www.pytables.org"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+To install this extension from source, download the latest stable version
+ (currently 2.0.2) and issue the following commands
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz pytables-2.0.2
+\end_layout
+
+\begin_layout LyX-Code
+$ cd pytables-2.0.2
+\end_layout
+
+\begin_layout LyX-Code
+$ python setup.py install
+\end_layout
+
+\begin_layout Standard
+To verify the installation, the command `
+\family typewriter
+import tables
+\family default
+' should run successfully from within your standard Python interpreter shell.
+\end_layout
+
+\begin_layout Subsection
+ScientificPython
+\end_layout
+
+\begin_layout Standard
+The ScientificPython module is a Python library for common tasks in scientific
+ computing.
+ Primarily, we use it to convert the ExodusII mesh files generated by the
+ Sandia National Laboratories CUBIT package.
+ Prior to installing this Python module, you must also install the NetCDF
+ development libraries and headers in order for the Scientific.IO.NetCDF module
+ to work properly.
+\end_layout
+
+\begin_layout Standard
+ScientificPython is available for download from
+\begin_inset LatexCommand htmlurl
+target "dirac.cnrs-orleans.fr/plone/software/scientificpython"
+
+\end_inset
+
+.
+ To install this extension from source, download the latest version and
+ proceed as follows:
+\end_layout
+
+\begin_layout LyX-Code
+$ tar xvfz ScientificPython-2.x.y.tar.gz
+\end_layout
+
+\begin_layout LyX-Code
+$ cd ScientificPython-2.x.y
+\end_layout
+
+\begin_layout LyX-Code
+$ python setup.py install
+\end_layout
+
+\begin_layout Standard
+To test whether the Scientific.IO.NetCDF module was installed correctly, you
+ can run `
+\family typewriter
+import Scientific.IO.NetCDF
+\family default
+' from within your standard Python interpreter shell.
+\end_layout
+
+\begin_layout Subsection
+SciPy
+\end_layout
+
+\begin_layout Standard
+SciPy is a Python module for use in mathematics, science, and engineering.
+ It contains algorithms for optimization, linear algebra, and other common
+ scientific tasks.
+ SciPy is available for download from
+\begin_inset LatexCommand url
+target "scipy.org"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+FIAT
+\end_layout
+
+\begin_layout Standard
+The Finite Element Automatic Tabulator (FIAT) Python module supports generation
+ of arbitrary order instances of the Lagrange elements, as well as arbitrary
+ order instances of the Jacobi-type quadrature rules on the same element
+ shapes.
+ We can also use it to evaluate the shape function values at the required
+ integration points.
+\end_layout
+
+\begin_layout Standard
+FIAT is available from
+\begin_inset LatexCommand htmlurl
+name "The FEniCS Project"
+target "www.fenics.org/wiki/FIAT"
+
+\end_inset
+
+.
+ To install this module, download it and issue the following command inside
+ the FIAT source directory:
+\end_layout
+
+\begin_layout LyX-Code
+$ python setup.py install
+\end_layout
+
+\begin_layout Standard
+You should now be able to run `
+\family typewriter
+import FIAT
+\family default
+' from within your standard Python interpreter shell.
+\end_layout
+
+\begin_layout Subsection
+HDFView
+\end_layout
+
+\begin_layout Standard
+NCSA HDFView is a graphical user interface tool for accessing data in your
+ HDF5 files.
+ You can use it for viewing the internal group hierarchy, adding new datasets,
+ and modifying or deleting existing datasets, or altering metadata on groups
+ and datasets.
+ You can download it from the
+\begin_inset LatexCommand htmlurl
+name "HDFView home page"
+target "hdf.ncsa.uiuc.edu/hdf-java-html/hdfview"
+
+\end_inset
+
+.
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/integration/integration.lyx
===================================================================
--- doc/cigma/manual/integration/integration.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/integration/integration.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,393 +0,0 @@
-#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-\newcommand{\mynote}[1]{}
-\end_preamble
-\language english
-\inputencoding auto
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\paperfontsize default
-\spacing single
-\papersize default
-\use_geometry false
-\use_amsmath 1
-\use_esint 1
-\cite_engine basic
-\use_bibtopic false
-\paperorientation portrait
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\defskip medskip
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\author ""
-\author ""
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-Go over numerical integration in 2d and 3d.
-\end_layout
-
-\begin_layout Standard
-
-Required order of numerical integration.
-\end_layout
-
-\begin_layout Standard
-
-Give references to defining your own points here.
- Explain how to use the scripts here.
-\end_layout
-
-\begin_layout Standard
-
-For each element, give the default quadrature rule employed and its order
- of accuracy.
-\end_layout
-
-\begin_layout Standard
-
-Give more examples of quadrature rules.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Mention tabulation of shape function values here.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-References to mention here:
-\end_layout
-
-\begin_layout Standard
-
- FIAT homepage
-\end_layout
-
-\begin_layout Standard
-
- FILIPPA mathematica notebooks
-\end_layout
-
-\begin_layout Standard
-
- Encyclopedia of Cubature Formulas (url?)
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-The core concepts to get across in this chapter are the following:
-\end_layout
-
-\begin_layout Enumerate
-Each element in Cigma is assigned a default quadrature rule
-\end_layout
-
-\begin_layout Enumerate
-Explain briefly how quadrature rules are obtained (product rules)
-\end_layout
-
-\begin_layout Enumerate
-Error bound on integration (dependent on the discretization size h, and
- the order of integration q).
-\end_layout
-
-\begin_layout Enumerate
-How to obtain more quadrature rules (encyclopedia website, FIAT, mathematica
- scripts -- don't forget to mention caveat about using rules that use different
- reference cells -- points transform according to map, while weights are
- multiplied by the appropriate jacobian factor).
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-Numerical Integration
-\end_layout
-
-\begin_layout Standard
-In order to calculate the
-\begin_inset Formula $L_{2}$
-\end_inset
-
- norm integral from Chapter 3, we must resort to numerical integration.
- If we integrate over a mesh which does not match the underlying discretization
- of the functions being compared, you may need to increase the accuracy
- of the integration.
- In this chapter we give a brief overview of the quadrature rules available,
- and how you may create your own.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Define integration rule.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In general, we may define an integration rule as a set of points with associated
- weights for which we may approximate an integration procedure by a simple
- weighed sum
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{Q}w_{q}F(\vec{x}_{q})+R\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-It is possible to integrate, but the remainder
-\begin_inset Formula $R$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Gaussian Quadrature
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Hughes, pg 143
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-The general Gaussian quadrature rule with
-\begin_inset Formula $Q$
-\end_inset
-
- integration points is given in Hughes, page 143,
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}F(\xi)\ d\xi=\sum_{q=1}^{Q}w_{q}F(\xi_{q})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-w_{i} & = & \frac{2}{(1-\xi_{i}^{2})P_{Q}'(\xi_{i})^{2}}\\
-R & = & \frac{2^{2Q+1}(Q!)^{4}}{(2Q+1)[(2Q)!]^{3}}F^{(2Q)}(\bar{\xi})\end{eqnarray*}
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-where
-\begin_inset Formula $\xi_{i}$
-\end_inset
-
- is the
-\begin_inset Formula $i$
-\end_inset
-
--th zero of the Legendre Polynomial
-\begin_inset Formula $P_{Q}(\xi)$
-\end_inset
-
-, and
-\begin_inset Formula $P_{Q}^{'}$
-\end_inset
-
- denotes the derivative of
-\begin_inset Formula $P_{Q}$
-\end_inset
-
-.
- The Legendre polynomials are defined by
-\begin_inset Formula \[
-P_{Q}(\xi)=\frac{1}{2^{Q}Q!}\frac{d^{Q}}{d\xi^{Q}}(\xi^{2}-1)^{Q}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In multiple dimensions, we simply use a different one-dimensional Gaussian
- rule on each direction.
- For two dimensions, this becomes
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta)\ d\xi d\eta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}w_{q_{1}}w_{q_{2}}F(\xi_{q_{1}},\eta_{q_{2}})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-and similarly for three dimensions
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta,\zeta)\ d\xi d\eta d\zeta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}\sum_{q_{3}=1}^{Q_{3}}w_{q_{1}}w_{q_{2}}w_{q_{3}}F(\xi_{q_{1}},\eta_{q_{2}},\zeta_{q_{3}})\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Quadrilaterals and Hexahedra
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Since these two are very similar, keep them together.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-For the reference quadrilateral and hexahedral elements, we can use the
-
-\end_layout
-
-\begin_layout Standard
-For quadrilaterals, we use the same reference region
-\begin_inset Formula $[-1,1]\times[-1,1]$
-\end_inset
-
- as in Section 4.X.
-\end_layout
-
-\begin_layout Standard
-The reference region for this geometry is
-\begin_inset Formula $[-1,1]\times[-1,1]\times[-1,1]$
-\end_inset
-
-, as in Section 4.X.
-
-\end_layout
-
-\begin_layout Section
-Triangles and Tetrahedra
-\end_layout
-
-\begin_layout Standard
-To easily obtain high order accuracy quadrature rules for triangles, we
- can re-use the same rules for quadrilaterals by collapsing two of the nodes
- of the quadrilateral.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Fig.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Other Integration Rules
-\end_layout
-
-\begin_layout Standard
-The integration points given in this section were obtained by using the
- FIAT Python module.
-
-\end_layout
-
-\begin_layout Standard
-(XXX: here, insert and explain the example script
-\family typewriter
-src/fiat_rules.py
-\family default
-)
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/integration.lyx (from rev 13032, doc/cigma/manual/integration/integration.lyx)
===================================================================
--- doc/cigma/manual/integration.lyx (rev 0)
+++ doc/cigma/manual/integration.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,393 @@
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+Go over numerical integration in 2d and 3d.
+\end_layout
+
+\begin_layout Standard
+
+Required order of numerical integration.
+\end_layout
+
+\begin_layout Standard
+
+Give references to defining your own points here.
+ Explain how to use the scripts here.
+\end_layout
+
+\begin_layout Standard
+
+For each element, give the default quadrature rule employed and its order
+ of accuracy.
+\end_layout
+
+\begin_layout Standard
+
+Give more examples of quadrature rules.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Mention tabulation of shape function values here.
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+References to mention here:
+\end_layout
+
+\begin_layout Standard
+
+ FIAT homepage
+\end_layout
+
+\begin_layout Standard
+
+ FILIPPA mathematica notebooks
+\end_layout
+
+\begin_layout Standard
+
+ Encyclopedia of Cubature Formulas (url?)
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+The core concepts to get across in this chapter are the following:
+\end_layout
+
+\begin_layout Enumerate
+Each element in Cigma is assigned a default quadrature rule
+\end_layout
+
+\begin_layout Enumerate
+Explain briefly how quadrature rules are obtained (product rules)
+\end_layout
+
+\begin_layout Enumerate
+Error bound on integration (dependent on the discretization size h, and
+ the order of integration q).
+\end_layout
+
+\begin_layout Enumerate
+How to obtain more quadrature rules (encyclopedia website, FIAT, mathematica
+ scripts -- don't forget to mention caveat about using rules that use different
+ reference cells -- points transform according to map, while weights are
+ multiplied by the appropriate jacobian factor).
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Numerical Integration
+\end_layout
+
+\begin_layout Standard
+In order to calculate the
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ norm integral from Chapter 3, we must resort to numerical integration.
+ If we integrate over a mesh which does not match the underlying discretization
+ of the functions being compared, you may need to increase the accuracy
+ of the integration.
+ In this chapter we give a brief overview of the quadrature rules available,
+ and how you may create your own.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Define integration rule.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In general, we may define an integration rule as a set of points with associated
+ weights for which we may approximate an integration procedure by a simple
+ weighed sum
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{\Omega}\ F(\vec{x})\ d\vec{x}=\sum_{q=1}^{Q}w_{q}F(\vec{x}_{q})+R\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+It is possible to integrate, but the remainder
+\begin_inset Formula $R$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Gaussian Quadrature
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Hughes, pg 143
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+The general Gaussian quadrature rule with
+\begin_inset Formula $Q$
+\end_inset
+
+ integration points is given in Hughes, page 143,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}F(\xi)\ d\xi=\sum_{q=1}^{Q}w_{q}F(\xi_{q})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \begin{eqnarray*}
+w_{i} & = & \frac{2}{(1-\xi_{i}^{2})P_{Q}'(\xi_{i})^{2}}\\
+R & = & \frac{2^{2Q+1}(Q!)^{4}}{(2Q+1)[(2Q)!]^{3}}F^{(2Q)}(\bar{\xi})\end{eqnarray*}
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+where
+\begin_inset Formula $\xi_{i}$
+\end_inset
+
+ is the
+\begin_inset Formula $i$
+\end_inset
+
+-th zero of the Legendre Polynomial
+\begin_inset Formula $P_{Q}(\xi)$
+\end_inset
+
+, and
+\begin_inset Formula $P_{Q}^{'}$
+\end_inset
+
+ denotes the derivative of
+\begin_inset Formula $P_{Q}$
+\end_inset
+
+.
+ The Legendre polynomials are defined by
+\begin_inset Formula \[
+P_{Q}(\xi)=\frac{1}{2^{Q}Q!}\frac{d^{Q}}{d\xi^{Q}}(\xi^{2}-1)^{Q}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+In multiple dimensions, we simply use a different one-dimensional Gaussian
+ rule on each direction.
+ For two dimensions, this becomes
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta)\ d\xi d\eta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}w_{q_{1}}w_{q_{2}}F(\xi_{q_{1}},\eta_{q_{2}})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+and similarly for three dimensions
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{-1}^{1}\int_{-1}^{1}\int_{-1}^{1}\ F(\xi,\eta,\zeta)\ d\xi d\eta d\zeta\cong\sum_{q_{1}=1}^{Q_{1}}\sum_{q_{2}=1}^{Q_{2}}\sum_{q_{3}=1}^{Q_{3}}w_{q_{1}}w_{q_{2}}w_{q_{3}}F(\xi_{q_{1}},\eta_{q_{2}},\zeta_{q_{3}})\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Quadrilaterals and Hexahedra
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Since these two are very similar, keep them together.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For the reference quadrilateral and hexahedral elements, we can use the
+
+\end_layout
+
+\begin_layout Standard
+For quadrilaterals, we use the same reference region
+\begin_inset Formula $[-1,1]\times[-1,1]$
+\end_inset
+
+ as in Section 4.X.
+\end_layout
+
+\begin_layout Standard
+The reference region for this geometry is
+\begin_inset Formula $[-1,1]\times[-1,1]\times[-1,1]$
+\end_inset
+
+, as in Section 4.X.
+
+\end_layout
+
+\begin_layout Section
+Triangles and Tetrahedra
+\end_layout
+
+\begin_layout Standard
+To easily obtain high order accuracy quadrature rules for triangles, we
+ can re-use the same rules for quadrilaterals by collapsing two of the nodes
+ of the quadrilateral.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Fig.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Other Integration Rules
+\end_layout
+
+\begin_layout Standard
+The integration points given in this section were obtained by using the
+ FIAT Python module.
+
+\end_layout
+
+\begin_layout Standard
+(XXX: here, insert and explain the example script
+\family typewriter
+src/fiat_rules.py
+\family default
+)
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/interpolation/interpolation.lyx
===================================================================
--- doc/cigma/manual/interpolation/interpolation.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/interpolation/interpolation.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,1421 +0,0 @@
-#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
-\begin_document
-\begin_header
-\textclass book
-\begin_preamble
-\newcommand{\mynote}[1]{}
-\end_preamble
-\language english
-\inputencoding auto
-\font_roman default
-\font_sans default
-\font_typewriter default
-\font_default_family default
-\font_sc false
-\font_osf false
-\font_sf_scale 100
-\font_tt_scale 100
-\graphics default
-\paperfontsize default
-\spacing single
-\papersize default
-\use_geometry false
-\use_amsmath 1
-\use_esint 1
-\cite_engine basic
-\use_bibtopic false
-\paperorientation portrait
-\secnumdepth 3
-\tocdepth 3
-\paragraph_separation indent
-\defskip medskip
-\quotes_language english
-\papercolumns 1
-\papersides 1
-\paperpagestyle default
-\tracking_changes false
-\output_changes false
-\author ""
-\author ""
-\end_header
-
-\begin_body
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-TODO:
-\end_layout
-
-\begin_layout Standard
-
- * Move discussion from error_analysis.lyx and running.lyx here (done)
-\end_layout
-
-\begin_layout Standard
-
- * Rename document to interpolation.lyx (done)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Note that we place the interpolation chapter before the integration chapter
- because we wish to define the isoparametric elements in here (interpolation.lyx)
-, and simply refer back to them in integration.lyx (for listing the integration
- rules)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Random Notes:
-\end_layout
-
-\begin_layout Standard
-
-For parametrizing the predefined functions we would need to use boost::regex
- when initializing the function from a user specified source string
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Give list of predefined functions here
-\end_layout
-
-\begin_layout Standard
-
- - basic:
-\end_layout
-
-\begin_layout Standard
-
- * zero
-\end_layout
-
-\begin_layout Standard
-
- * scalar.one
-\end_layout
-
-\begin_layout Standard
-
- - benchmarks:
-\end_layout
-
-\begin_layout Standard
-
- * strikeslip.okada.disloc3d
-\end_layout
-
-\begin_layout Standard
-
- * cylinder_extension.{maxwell,non_newtonian}.{displacement,velocity,stress}
-\end_layout
-
-\begin_layout Standard
-
- * cylinder_relaxation.{maxwell,non_newtonian}.{displacement,velocity,stress}
-\end_layout
-
-\begin_layout Standard
-
- * circular_inclusion.pressure
-\end_layout
-
-\begin_layout Standard
-
- * test.{cube,sphere,square,circle}
-\end_layout
-
-\begin_layout Standard
-
- - Explain what is needed to define own function
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Important things to discuss:
-\end_layout
-
-\begin_layout Standard
-
- - Natural coordinates
-\end_layout
-
-\begin_layout Standard
-
- - Restricting discussion to isoparametric elements
-\end_layout
-
-\begin_layout Standard
-
- - Available elements: tet4, tet10, hex8, hex20, hex27, tri3, quad4
-\end_layout
-
-\begin_layout Standard
-
- - Node orderings for each element type (use figures to illustrate)
-\end_layout
-
-\begin_layout Standard
-
- - Shape functions
-\end_layout
-
-\begin_layout Standard
-
- - Shape function derivatives
-\end_layout
-
-\begin_layout Standard
-
- - Calculation of Jacobian
-\end_layout
-
-\begin_layout Standard
-
- - Calculation of Volumes
-\end_layout
-
-\begin_layout Standard
-
- - Deciding if a given point is contained in element
-\end_layout
-
-\begin_layout Standard
-
- - Default integration rule for each element type
-\end_layout
-
-\begin_layout Standard
-
- - Mesh Locator for finding element given an arbitrary point -- returns
- list of likely elements
-\end_layout
-
-\begin_layout Standard
-
- - Deciding if a given global point is contained in element (need inverse
- reference map)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-Interpolation Functions
-\end_layout
-
-\begin_layout Standard
-As noted in Chapter 3, we will often need to evaluate a function
-\begin_inset Formula $f(\vec{x})$
-\end_inset
-
- at arbitrary points on a global coordinate system.
-\end_layout
-
-\begin_layout Section
-Local Functions
-\end_layout
-
-\begin_layout Standard
-Typically, as a result of numerical simulations, your function will be defined
- locally on a discretization of the domain of interest.
- Each discretization element defines a local coordinate system.
- Since the integration points are defined on a global coordinate system,
- we might have to invert the appropriate reference map after running a search
- through each discretization cell, followed by a newton-iteration to convert
- into the local coordinates, followed by an interior test.
- (XXX: Insert workfile1.lyx in this section)
-\end_layout
-
-\begin_layout Subsection
-Domain Discretization
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Definition of mesh here
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-A typical discretization will partition the domain of interest into a large
- number of cells.
-
-\end_layout
-
-\begin_layout Standard
-A set of cells which partition the domain.
- These cells are defined geometrically by a number of node points, which
- together with a set of connectivity rules.
- This set of node points will have a number of degrees of freedom associated
- with them.
- An interpolation
-\end_layout
-
-\begin_layout Subsection
-Interpolation Step
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Shape functions.
- Interpolant function.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-Once we arrive at a solution on a cell
-\begin_inset Formula $\Omega_{e}$
-\end_inset
-
-, we can define an interpolant function that will give us the value of our
- function on any point in the interior of the cell.
-\end_layout
-
-\begin_layout Subsection
-Available Elements
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Reference domain.
-
-\end_layout
-
-\end_inset
-
-Fig.
- elts.jpg
-\end_layout
-
-\begin_layout Subsection
-Triangles and Tetrahedra
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Reference domain.
- Shape functions.
- Jacobian determinant and volume.
- Interior test.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Quadrilateral and Hexahedral Elements
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-Reference cell.
- Shape functions.
- Jacobian determinant & volume.
- Interior test.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Global Functions
-\end_layout
-
-\begin_layout Standard
-These functions are given explicitly on a domain.
- They may be defined explicitly, in the source code, or may also be defined
- by a global set of basis functions (such as spherical harmonics).
- Either way, they will be associated with a callback function that Cigma
- will evaluate whenever necessary.
-\end_layout
-
-\begin_layout Subsection
-Available Global Functions
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-mynote{
-\end_layout
-
-\begin_layout Standard
-
-TODO: do we really need to compile every function?
-\end_layout
-
-\begin_layout Standard
-
-If the number of functions becomes unwieldy, maybe this should become a
- configure-time option.
-\end_layout
-
-\begin_layout Standard
-
-For example: we could configure cigma with the following flags:
-\end_layout
-
-\begin_layout Standard
-
- --with-all-functions=yes, --with-gale-functions=no, --with-test-functions,
- --with-foo-functions, etc..
-\end_layout
-
-\begin_layout Standard
-
-TODO: add more to the description of each function
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-TODO: Is there a way to get rid of the internal horizontal rules in the
- second column of a table?
-\end_layout
-
-\begin_layout Standard
-\begin_inset Box Frameless
-position "t"
-hor_pos "c"
-has_inner_box 1
-inner_pos "t"
-use_parbox 0
-width "100col%"
-special "none"
-height "1in"
-height_special "totalheight"
-status open
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="18" columns="2">
-<features>
-<column alignment="center" valignment="top" leftline="true" width="0">
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Name
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Description
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-zero
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-scalar.one
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-strikeslip.disloc3d.displacement
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Strike-slip benchmark calculation with Okada disloc3d routine
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-strikeslip.disloc3d.velocity
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-strikeslip.disloc3d.stress
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-cylinder_ext.maxwell_mat.displacement
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Extension of Cylinder
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-cylinder_ext.maxwell_mat.velocity
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-cylinder_ext.maxwell_mat.stress
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-
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-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
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-\begin_inset Text
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-\begin_layout Standard
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-\end_layout
-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
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-
-\end_inset
-</cell>
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-
-\end_inset
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-
-\end_inset
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-
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-
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-
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-
-\end_inset
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-
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-\begin_inset Text
-
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-cylinder_rel.non_newt_mat.velocity
-\end_layout
-
-\end_inset
-</cell>
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-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
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-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-cylinder_rel.non_newt_mat.stress
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-By default, we compile a number of functions into Cigma.
- We subdivide these into three categories.
- The first class contains only two functions, zero and one, with special
- initialization rules.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-By default, we compile a number of functions into Cigma.
- We subdivide these into three categories.
- The first class contains only two functions, zero and one.
- The second class of functions consists of test functions intended to verify
- the operation of Cigma.
- Finally, the third class of functions consists of example benchmark cases
- that you may use as a base for defining your own functions.
-\newline
-
-\newline
-There are two
- basic functions that are available, listed in the table below.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="2" columns="2">
-<features>
-<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
-<column alignment="left" valignment="top" rightline="true" width="0">
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-\begin_inset Text
-
-\begin_layout Standard
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-\family typewriter
-zero
-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Always evaluates to 0.
- Compatible with scalar, vector, and tensor functions.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
-Always evaluates to 1.
- Only compatible with scalar functions.
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-The following functions have short definitions and are available for internal
- testing purposes.
- Mainly, these are used to make sure Cigma returns consistent results
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="4" columns="2">
-<features>
-<column alignment="left" valignment="top" leftline="true" width="0">
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-\begin_layout Standard
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-
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-\begin_inset Text
-
-\begin_layout Standard
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-\end_layout
-
-\end_inset
-</cell>
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-<row topline="true">
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
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-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Vector field in 3-dimensions
-\end_layout
-
-\end_inset
-</cell>
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-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
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-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Scalar field in 2-dimensions
-\end_layout
-
-\end_inset
-</cell>
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-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
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-\end_layout
-
-\end_inset
-</cell>
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-Vector field in 2-dimensions
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-We can also calculate the initial step of the strike-slip benchmark (no
- gravity) discussed in Chapter X.
- This is based on the Okada (1992) dislocation code[?].
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="3" columns="1">
-<features>
-<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
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-\begin_layout Standard
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
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-
-\end_inset
-</cell>
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
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-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-The full list of all pre-defined functions which are available by name are
- given in the appendix.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Cylinder extension case (maxwell and non-newtonian materials)
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="1">
-<features>
-<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
-
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-
-\end_inset
-</cell>
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-\begin_layout Standard
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-
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-\begin_inset Text
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-\begin_layout Standard
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_ext.non_newt_mat.velocity
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-
-\end_inset
-</cell>
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_ext.non_newt_mat.stress
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-Cylinder relaxation case (maxwell and non-newtonian materials)
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="6" columns="1">
-<features>
-<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true">
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-\begin_inset Text
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-\begin_layout Standard
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-\family typewriter
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-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_rel.maxwell_mat.velocity
-\end_layout
-
-\end_inset
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-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
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-
-\end_inset
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-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_rel.non_newt_mat.displacement
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_rel.non_newt_mat.velocity
-\end_layout
-
-\end_inset
-</cell>
-</row>
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-cylinder_rel.non_newt_mat.stress
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\newline
-
-\newline
-Using Gale, we may compute various
-\end_layout
-
-\begin_layout Standard
-\begin_inset Tabular
-<lyxtabular version="3" rows="1" columns="1">
-<features>
-<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
-<row topline="true" bottomline="true">
-<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
-\begin_inset Text
-
-\begin_layout Standard
-
-\family typewriter
-circular_inclusion.pressure
-\end_layout
-
-\end_inset
-</cell>
-</row>
-</lyxtabular>
-
-\end_inset
-
-
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Defining Your Own Global Function
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status collapsed
-
-\begin_layout Standard
-Show example subclass of cigma::Function.
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-There are three things you need to do in order to register a new function.
- These are the steps you need to follow in order to let Cigma know about
- your function.
-\end_layout
-
-\begin_layout Enumerate
-Drop the definition of your global function in a subdirectory of the
-\family typewriter
-src/core/functions/
-\family default
- directory, and define an appropriate subclass of
-\family typewriter
-cigma::Function
-\family default
- which interfaces to your function.
-\end_layout
-
-\begin_layout Enumerate
-Add the new source and header files into the
-\family typewriter
-libcigma_a_SOURCES
-\family default
- variable in
-\family typewriter
-src/Makefile.am
-\family default
-, following the examples given in the section that refers to the
-\family typewriter
-ZeroFunction.h
-\family default
- and
-\family typewriter
-ZeroFunction.cpp
-\family default
- files.
-\end_layout
-
-\begin_layout Enumerate
-Instantiate your function and insert it into the appropriate map object
- in
-\family typewriter
-src/core/functions/FunctionRegistry.cpp
-\family default
-, again following the examples already in the file.
- It is in this file where you will give a unique name to your function,
- for use in the command line.
-\end_layout
-
-\begin_layout Standard
-Other examples to examine are located at
-\family typewriter
-TestFunctions.cpp
-\family default
- and in
-\family typewriter
-StrikeslipFunctions.cpp
-\family default
- in the
-\family typewriter
-src/core/functions/
-\family default
- directory.
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/interpolation.lyx (from rev 13032, doc/cigma/manual/interpolation/interpolation.lyx)
===================================================================
--- doc/cigma/manual/interpolation.lyx (rev 0)
+++ doc/cigma/manual/interpolation.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,1421 @@
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
+\lyxformat 276
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\paperfontsize default
+\spacing single
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+TODO:
+\end_layout
+
+\begin_layout Standard
+
+ * Move discussion from error_analysis.lyx and running.lyx here (done)
+\end_layout
+
+\begin_layout Standard
+
+ * Rename document to interpolation.lyx (done)
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Note that we place the interpolation chapter before the integration chapter
+ because we wish to define the isoparametric elements in here (interpolation.lyx)
+, and simply refer back to them in integration.lyx (for listing the integration
+ rules)
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Random Notes:
+\end_layout
+
+\begin_layout Standard
+
+For parametrizing the predefined functions we would need to use boost::regex
+ when initializing the function from a user specified source string
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Give list of predefined functions here
+\end_layout
+
+\begin_layout Standard
+
+ - basic:
+\end_layout
+
+\begin_layout Standard
+
+ * zero
+\end_layout
+
+\begin_layout Standard
+
+ * scalar.one
+\end_layout
+
+\begin_layout Standard
+
+ - benchmarks:
+\end_layout
+
+\begin_layout Standard
+
+ * strikeslip.okada.disloc3d
+\end_layout
+
+\begin_layout Standard
+
+ * cylinder_extension.{maxwell,non_newtonian}.{displacement,velocity,stress}
+\end_layout
+
+\begin_layout Standard
+
+ * cylinder_relaxation.{maxwell,non_newtonian}.{displacement,velocity,stress}
+\end_layout
+
+\begin_layout Standard
+
+ * circular_inclusion.pressure
+\end_layout
+
+\begin_layout Standard
+
+ * test.{cube,sphere,square,circle}
+\end_layout
+
+\begin_layout Standard
+
+ - Explain what is needed to define own function
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+Important things to discuss:
+\end_layout
+
+\begin_layout Standard
+
+ - Natural coordinates
+\end_layout
+
+\begin_layout Standard
+
+ - Restricting discussion to isoparametric elements
+\end_layout
+
+\begin_layout Standard
+
+ - Available elements: tet4, tet10, hex8, hex20, hex27, tri3, quad4
+\end_layout
+
+\begin_layout Standard
+
+ - Node orderings for each element type (use figures to illustrate)
+\end_layout
+
+\begin_layout Standard
+
+ - Shape functions
+\end_layout
+
+\begin_layout Standard
+
+ - Shape function derivatives
+\end_layout
+
+\begin_layout Standard
+
+ - Calculation of Jacobian
+\end_layout
+
+\begin_layout Standard
+
+ - Calculation of Volumes
+\end_layout
+
+\begin_layout Standard
+
+ - Deciding if a given point is contained in element
+\end_layout
+
+\begin_layout Standard
+
+ - Default integration rule for each element type
+\end_layout
+
+\begin_layout Standard
+
+ - Mesh Locator for finding element given an arbitrary point -- returns
+ list of likely elements
+\end_layout
+
+\begin_layout Standard
+
+ - Deciding if a given global point is contained in element (need inverse
+ reference map)
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+Interpolation Functions
+\end_layout
+
+\begin_layout Standard
+As noted in Chapter 3, we will often need to evaluate a function
+\begin_inset Formula $f(\vec{x})$
+\end_inset
+
+ at arbitrary points on a global coordinate system.
+\end_layout
+
+\begin_layout Section
+Local Functions
+\end_layout
+
+\begin_layout Standard
+Typically, as a result of numerical simulations, your function will be defined
+ locally on a discretization of the domain of interest.
+ Each discretization element defines a local coordinate system.
+ Since the integration points are defined on a global coordinate system,
+ we might have to invert the appropriate reference map after running a search
+ through each discretization cell, followed by a newton-iteration to convert
+ into the local coordinates, followed by an interior test.
+ (XXX: Insert workfile1.lyx in this section)
+\end_layout
+
+\begin_layout Subsection
+Domain Discretization
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Definition of mesh here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+A typical discretization will partition the domain of interest into a large
+ number of cells.
+
+\end_layout
+
+\begin_layout Standard
+A set of cells which partition the domain.
+ These cells are defined geometrically by a number of node points, which
+ together with a set of connectivity rules.
+ This set of node points will have a number of degrees of freedom associated
+ with them.
+ An interpolation
+\end_layout
+
+\begin_layout Subsection
+Interpolation Step
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Shape functions.
+ Interpolant function.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Once we arrive at a solution on a cell
+\begin_inset Formula $\Omega_{e}$
+\end_inset
+
+, we can define an interpolant function that will give us the value of our
+ function on any point in the interior of the cell.
+\end_layout
+
+\begin_layout Subsection
+Available Elements
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Reference domain.
+
+\end_layout
+
+\end_inset
+
+Fig.
+ elts.jpg
+\end_layout
+
+\begin_layout Subsection
+Triangles and Tetrahedra
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Reference domain.
+ Shape functions.
+ Jacobian determinant and volume.
+ Interior test.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Quadrilateral and Hexahedral Elements
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Reference cell.
+ Shape functions.
+ Jacobian determinant & volume.
+ Interior test.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Global Functions
+\end_layout
+
+\begin_layout Standard
+These functions are given explicitly on a domain.
+ They may be defined explicitly, in the source code, or may also be defined
+ by a global set of basis functions (such as spherical harmonics).
+ Either way, they will be associated with a callback function that Cigma
+ will evaluate whenever necessary.
+\end_layout
+
+\begin_layout Subsection
+Available Global Functions
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Standard
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Standard
+
+TODO: do we really need to compile every function?
+\end_layout
+
+\begin_layout Standard
+
+If the number of functions becomes unwieldy, maybe this should become a
+ configure-time option.
+\end_layout
+
+\begin_layout Standard
+
+For example: we could configure cigma with the following flags:
+\end_layout
+
+\begin_layout Standard
+
+ --with-all-functions=yes, --with-gale-functions=no, --with-test-functions,
+ --with-foo-functions, etc..
+\end_layout
+
+\begin_layout Standard
+
+TODO: add more to the description of each function
+\end_layout
+
+\begin_layout Standard
+
+}
+\end_layout
+
+\end_inset
+
+
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+TODO: Is there a way to get rid of the internal horizontal rules in the
+ second column of a table?
+\end_layout
+
+\begin_layout Standard
+\begin_inset Box Frameless
+position "t"
+hor_pos "c"
+has_inner_box 1
+inner_pos "t"
+use_parbox 0
+width "100col%"
+special "none"
+height "1in"
+height_special "totalheight"
+status open
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="18" columns="2">
+<features>
+<column alignment="center" valignment="top" leftline="true" width="0">
+<column alignment="center" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Name
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Description
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+zero
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+scalar.one
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+strikeslip.disloc3d.displacement
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Strike-slip benchmark calculation with Okada disloc3d routine
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+strikeslip.disloc3d.velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+strikeslip.disloc3d.stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.maxwell_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Extension of Cylinder
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.maxwell_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.maxwell_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.non_newt_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Extending a cylinder with a non-newtonian material
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.non_newt_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_ext.non_newt_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.maxwell_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Gravity relaxation of
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.maxwell_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.maxwell_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.non_newt_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.non_newt_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+cylinder_rel.non_newt_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+By default, we compile a number of functions into Cigma.
+ We subdivide these into three categories.
+ The first class contains only two functions, zero and one, with special
+ initialization rules.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+By default, we compile a number of functions into Cigma.
+ We subdivide these into three categories.
+ The first class contains only two functions, zero and one.
+ The second class of functions consists of test functions intended to verify
+ the operation of Cigma.
+ Finally, the third class of functions consists of example benchmark cases
+ that you may use as a base for defining your own functions.
+\newline
+
+\newline
+There are two
+ basic functions that are available, listed in the table below.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="2" columns="2">
+<features>
+<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
+<column alignment="left" valignment="top" rightline="true" width="0">
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+zero
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Always evaluates to 0.
+ Compatible with scalar, vector, and tensor functions.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+one
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Always evaluates to 1.
+ Only compatible with scalar functions.
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\newline
+The following functions have short definitions and are available for internal
+ testing purposes.
+ Mainly, these are used to make sure Cigma returns consistent results
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="4" columns="2">
+<features>
+<column alignment="left" valignment="top" leftline="true" width="0">
+<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+test.cube
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Scalar field in 3-dimensions
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+test.sphere
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Vector field in 3-dimensions
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+test.square
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Scalar field in 2-dimensions
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+test.circle
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+Vector field in 2-dimensions
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\newline
+We can also calculate the initial step of the strike-slip benchmark (no
+ gravity) discussed in Chapter X.
+ This is based on the Okada (1992) dislocation code[?].
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="3" columns="1">
+<features>
+<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+strikeslip.disloc3d.displacement
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+strikeslip.disloc3d.velocity
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+strikeslip.disloc3d.stress
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\newline
+The full list of all pre-defined functions which are available by name are
+ given in the appendix.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Standard
+Cylinder extension case (maxwell and non-newtonian materials)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="1">
+<features>
+<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.maxwell_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.maxwell_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.maxwell_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.non_newt_mat.displacement
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.non_newt_mat.velocity
+\end_layout
+
+\end_inset
+</cell>
+</row>
+<row topline="true" bottomline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+
+\family typewriter
+cylinder_ext.non_newt_mat.stress
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\newline
+
+\newline
+Cylinder relaxation case (maxwell and non-newtonian materials)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Tabular
+<lyxtabular version="3" rows="6" columns="1">
+<features>
+<column alignment="left" valignment="top" leftline="true" rightline="true" width="0">
+<row topline="true">
+<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
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+
+\newline
+Using Gale, we may compute various
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+
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+
+\begin_layout Subsection
+Defining Your Own Global Function
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
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+
+\begin_layout Standard
+Show example subclass of cigma::Function.
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+
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+
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+There are three things you need to do in order to register a new function.
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+\end_layout
+
+\begin_layout Enumerate
+Drop the definition of your global function in a subdirectory of the
+\family typewriter
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+\family default
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+\family typewriter
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+\family default
+ which interfaces to your function.
+\end_layout
+
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+Add the new source and header files into the
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+\family default
+, following the examples given in the section that refers to the
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+ and
+\family typewriter
+ZeroFunction.cpp
+\family default
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+\end_layout
+
+\begin_layout Enumerate
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+ in
+\family typewriter
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+\family default
+, again following the examples already in the file.
+ It is in this file where you will give a unique name to your function,
+ for use in the command line.
+\end_layout
+
+\begin_layout Standard
+Other examples to examine are located at
+\family typewriter
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+\family default
+ and in
+\family typewriter
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+\family default
+ in the
+\family typewriter
+src/core/functions/
+\family default
+ directory.
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/introduction/introduction.lyx
===================================================================
--- doc/cigma/manual/introduction/introduction.lyx 2008-12-09 23:41:45 UTC (rev 13481)
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- This comparison can take place even when the underlying discretizations
- do not overlap.
- In benchmarking, Cigma can help the geodynamics community agree on a standard
- solution to specific problems by facilitating the process of comparing
- different numerical codes against each other.
- Lastly, as an automated tool, Cigma can help application developers create
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-\begin_layout Standard
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- EAR-0406751.
- Any opinions, findings, and conclusions or recommendations expressed in
- this material are those of the authors and do not necessarily reflect the
- views of the National Science Foundation.
- The code is being released under the GNU General Public License.
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Copied: doc/cigma/manual/introduction.lyx (from rev 13032, doc/cigma/manual/introduction/introduction.lyx)
===================================================================
--- doc/cigma/manual/introduction.lyx (rev 0)
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+About Cigma
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+\begin_layout Standard
+The CIG Model Analyzer (Cigma) consists of a suite of tools for comparing
+ numerical models.
+\end_layout
+
+\begin_layout Standard
+The current version of Cigma is intended for the calculation of
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ residuals in finite element models, but can be extended to compare arbitrary
+ functions.
+ This software was designed for performing the following three main tasks:
+ (1) error analysis, (2) benchmarking, and (3) code verification.
+ In error analysis, Cigma can calculate a global measure of the differences
+ between two solutions by performing an integration over a discretized version
+ of the common domain.
+ This comparison can take place even when the underlying discretizations
+ do not overlap.
+ In benchmarking, Cigma can help the geodynamics community agree on a standard
+ solution to specific problems by facilitating the process of comparing
+ different numerical codes against each other.
+ Lastly, as an automated tool, Cigma can help application developers create
+ regression tests to ensure that software changes do not affect the consistency
+ of the results.
+\end_layout
+
+\begin_layout Standard
+CIG has developed Cigma in response to demand from the short-term tectonics
+ community for a simple tool that can perform rigorous error analysis on
+ their finite element codes.
+ The long-term goal for Cigma, however, is for it to be used for nearly
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+ The FInite element Automatic Tabulator (FIAT)
+\begin_inset Flex URL
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+\begin_layout Plain Layout
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+www.cs.umd.edu/~mount/ANN/
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+\end_inset
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+\end_layout
+
+\begin_layout Standard
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+\begin_inset Flex URL
+status collapsed
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+\begin_layout Plain Layout
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+www.cs.umd.edu/~mount/ANN/
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+\end_inset
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+\end_layout
+
+\begin_layout Plain Layout
+Both of these libraries extend and generalize Cigma's functionality so it
+ can handle other types of elements, and provide the ability to compare
+ vector fields.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Citation
+\end_layout
+
+\begin_layout Standard
+Computational Infrastructure for Geodynamics (CIG) is making this source
+ code available to you in the hope that the software will enhance your research
+ in geophysics.
+ This is a brand-new code and at present no papers are published or press.
+ Please cite this manual as follows:
+\end_layout
+
+\begin_layout Itemize
+Armendariz, L., and S.
+ Kientz.
+
+\emph on
+Cigma User Manual.
+
+\emph default
+ Pasadena, CA: Computational Infrastructure of Geodynamics, 2008.
+ URL: geodynamics.org/cig/software/cs/cigma/cigma.pdf
+\end_layout
+
+\begin_layout Standard
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+geodynamics.org
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+
+\end_inset
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+.
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+\end_layout
+
+\begin_layout Standard
+Cigma development is based upon work supported by the National Science Foundatio
+n under Grant No.
+ EAR-0406751.
+ Any opinions, findings, and conclusions or recommendations expressed in
+ this material are those of the authors and do not necessarily reflect the
+ views of the National Science Foundation.
+ The code is being released under the GNU General Public License.
+\end_layout
+
+\end_body
+\end_document
Deleted: doc/cigma/manual/license/gpl-license.lyx
===================================================================
--- doc/cigma/manual/license/gpl-license.lyx 2008-12-09 23:41:45 UTC (rev 13481)
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-
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- in part contains or is derived from the Program or any part thereof, to
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-\begin_deeper
-\begin_layout Standard
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-
-\begin_layout Standard
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-
-\end_layout
-
-\begin_layout Standard
-This section is intended to make thoroughly clear what is believed to be
- a consequence of the rest of this License.
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-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-If the distribution and/or use of the Program is restricted in certain countries
- either by patents or by copyrighted interfaces, the original copyright
- holder who places the Program under this License may add an explicit geographic
-al distribution limitation excluding those countries, so that distribution
- is permitted only in or among countries not thus excluded.
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-
-\end_layout
-
-\begin_layout Enumerate
-The Free Software Foundation may publish revised and/or new versions of
- the General Public License from time to time.
- Such new versions will be similar in spirit to the present version, but
- may differ in detail to address new problems or concerns.
-
-\end_layout
-
-\begin_deeper
-\begin_layout Standard
-Each version is given a distinguishing version number.
- If the Program specifies a version number of this License which applies
- to it and "any later version," you have the option of following the terms
- and conditions either of that version or of any later version published
- by the Free Software Foundation.
- If the Program does not specify a version number of this License, you may
- choose any version ever published by the Free Software Foundation.
-\end_layout
-
-\end_deeper
-\begin_layout Enumerate
-If you wish to incorporate parts of the Program into other free programs
- whose distribution conditions are different, write to the author to ask
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- For software which is copyrighted by the Free Software Foundation, write
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-
-\end_layout
-
-\begin_layout Section*
-NO WARRANTY
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-begin{itemize}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-item[11.]
-\end_layout
-
-\end_inset
-
-BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE, THERE IS NO WARRANTY FOR
- THE PROGRAM, TO THE EXTENT PERMITTED BY APPLICABLE LAW.
- EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND/OR OTHER
- PARTIES PROVIDE THE PROGRAM "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER
- EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
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- THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH
- YOU.
- SHOULD THE PROGRAM PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY
- SERVICING, REPAIR OR CORRECTION.
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
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-
-\backslash
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-
-\end_inset
-
-IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRITING WILL
- ANY COPYRIGHT HOLDER, OR ANY OTHER PARTY WHO MAY MODIFY AND/OR REDISTRIBUTE
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- EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY
- OF SUCH DAMAGES.
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status collapsed
-
-\begin_layout Standard
-
-
-\backslash
-end{itemize}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-END OF TERMS AND CONDITIONS
-\end_layout
-
-\begin_layout Standard
-How to Apply These Terms to Your New Programs
-\end_layout
-
-\begin_layout Standard
-If you develop a new program, and you want it to be of the greatest possible
- use to the public, the best way to achieve this is to make it free software
- which everyone can redistribute and change under these terms.
-
-\end_layout
-
-\begin_layout Standard
-To do so, attach the following notices to the program.
- It is safest to attach them to the start of each source file to most effectivel
-y convey the exclusion of warranty; and each file should have at least the
- "copyright" line and a pointer to where the full notice is found.
- For example:
-\end_layout
-
-\begin_layout Standard
-One line to give the program's name and a brief idea of what it does.
- Copyright
-\size footnotesize
-© (
-\size default
-year) (name of author)
-\end_layout
-
-\begin_layout Standard
-This program is free software; you can redistribute it and/or modify it
- under the terms of the GNU General Public License as published by the Free
- Software Foundation; either version 2 of the License, or (at your option)
- any later version.
-
-\end_layout
-
-\begin_layout Standard
-This program is distributed in the hope that it will be useful, but WITHOUT
- ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
- FOR A PARTICULAR PURPOSE.
- See the GNU General Public License for more details.
-
-\end_layout
-
-\begin_layout Standard
-You should have received a copy of the GNU General Public License along
- with this program; if not, write to the Free Software Foundation, Inc.,
- 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
-\end_layout
-
-\begin_layout Standard
-Also add information on how to contact you by electronic and paper mail.
-
-\end_layout
-
-\begin_layout Standard
-If the program is interactive, make it output a short notice like this when
- it starts in an interactive mode:
-\end_layout
-
-\begin_layout Standard
-Gnomovision version 69, Copyright © year name of author Gnomovision comes
- with ABSOLUTELY NO WARRANTY; for details type `show w'.
- This is free software, and you are welcome to redistribute it under certain
- conditions; type `show c' for details.
-
-\end_layout
-
-\begin_layout Standard
-The hypothetical commands `show w' and `show c' should show the appropriate
- parts of the General Public License.
- Of course, the commands you use may be called something other than `show
- w' and `show c'; they could even be mouse-clicks or menu items -- whatever
- suits your program.
-
-\end_layout
-
-\begin_layout Standard
-You should also get your employer (if you work as a programmer) or your
- school, if any, to sign a "copyright disclaimer" for the program, if necessary.
- Here is a sample; alter the names:
-\end_layout
-
-\begin_layout Standard
-Yoyodyne, Inc., hereby disclaims all copyright interest in the program `Gnomovisi
-on' (which makes passes at compilers) written by James Hacker.
-
-\end_layout
-
-\begin_layout Standard
-(signature of Ty Coon)
-\newline
-1 April 1989
-\newline
-Ty Coon, President of Vice
-\end_layout
-
-\begin_layout Standard
-This General Public License does not permit incorporating your program into
- proprietary programs.
- If your program is a subroutine library, you may consider it more useful
- to permit linking proprietary applications with the library.
- If this is what you want to do, use the GNU Library General Public License
- instead of this License.
-\end_layout
-
-\end_body
-\end_document
Modified: doc/cigma/manual/nocover.lyx
===================================================================
--- doc/cigma/manual/nocover.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/nocover.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,11 +1,12 @@
-#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
-\lyxformat 276
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
\begin_document
\begin_header
\textclass book
\begin_preamble
\newcommand{\mynote}[1]{}
\end_preamble
+\use_default_options false
\language english
\inputencoding auto
\font_roman default
@@ -16,9 +17,11 @@
\font_osf false
\font_sf_scale 100
\font_tt_scale 100
+
\graphics default
\paperfontsize default
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+\use_hyperref false
\papersize default
\use_geometry false
\use_amsmath 1
@@ -44,15 +47,17 @@
\begin_layout Standard
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+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
This document exists for preview purposes.
-\newline
+\begin_inset Newline newline
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+
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
When rebuilding the original cigma.lyx, don't forget to double-check the
following for errors:
\end_layout
@@ -79,8 +84,9 @@
\end_layout
\begin_layout Standard
-\begin_inset Include \include{introduction/introduction.lyx}
-preview false
+\begin_inset CommandInset include
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@@ -88,8 +94,9 @@
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\begin_layout Standard
-\begin_inset Include \include{installation/installation.lyx}
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@@ -97,8 +104,9 @@
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\begin_layout Standard
-\begin_inset Include \include{error_analysis/error_analysis.lyx}
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@@ -106,8 +114,9 @@
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@@ -124,8 +134,9 @@
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\begin_layout Standard
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@@ -133,8 +144,9 @@
\end_layout
\begin_layout Standard
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+\begin_inset CommandInset include
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@@ -142,8 +154,9 @@
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@@ -155,7 +168,7 @@
\begin_inset ERT
status collapsed
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%dummy comment inserted to ensure this paragraph is not empty
\end_layout
@@ -166,8 +179,9 @@
\end_layout
\begin_layout Standard
-\begin_inset Include \include{appendix/appendix.lyx}
-preview false
+\begin_inset CommandInset include
+LatexCommand include
+filename "appendix.lyx"
\end_inset
@@ -176,9 +190,9 @@
\begin_layout Standard
\begin_inset Note Note
-status open
+status collapsed
-\begin_layout Standard
+\begin_layout Plain Layout
Include appendix.lyx, and gpl-license.lyx
\end_layout
Deleted: doc/cigma/manual/running/running.lyx
===================================================================
--- doc/cigma/manual/running/running.lyx 2008-12-09 23:41:45 UTC (rev 13481)
+++ doc/cigma/manual/running/running.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -1,697 +0,0 @@
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-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Do not forget to mention the following:
-\end_layout
-
-\begin_layout Standard
-
- - Default behaviour if a flag or option is not specified (eg., what if
- we omit --output?)
-\end_layout
-
-\begin_layout Standard
-
- - Common flags:
-\end_layout
-
-\begin_layout Standard
-
- Misc options: --quiet and --verbose (misc option)
-\end_layout
-
-\begin_layout Standard
-
- Hidden options: --debug flag to enable logging
-\end_layout
-
-\begin_layout Standard
-
- - Turn off logging if --disable-logging is given to ./configure script
- (logging on by default)
-\end_layout
-
-\begin_layout Standard
-
- - The --output option needs an override flag so we can force the rewriting
- of specific datasets
-\end_layout
-
-\begin_layout Standard
-
- * Default behavior? Exit if output dataset already exists
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-
-\begin_layout Standard
-
- * Possible names if default behavior is to exit: --force-writes, --force-ou
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-
-\begin_layout Standard
-
- * Possible names if default is to allow overwrites: --dont-overwrite
-\end_layout
-
-\begin_layout Standard
-
- - Addition modifiers to compare command?
-\end_layout
-
-\begin_layout Standard
-
- * Try --weighed-residuals={0,1} flag in order to divide the cell residuals
- by the volume
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Given the context of the previous chapter (on file formats), define the
-\end_layout
-
-\begin_layout Standard
-
-'Dataset Path' when discussing the location of datasets within a file
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-XXX: Split the discussion of datasets and I/O into their corresponding chapters
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
- Ensure that most instances of the world 'Field' are replaced by 'Function'
-\end_layout
-
-\begin_layout Standard
-
- Ensure that 'Quadrature Rule' is changed into 'Integration Rule'
-\end_layout
-
-\begin_layout Standard
-
- Reference datasets in HDF5 files as 'Simple Arrays'
-\end_layout
-
-\begin_layout Standard
-
- Discuss ExodusII format
-\end_layout
-
-\begin_layout Standard
-
- Discuss VTK formats: Legacy VTK (.vtk), Unstructured Grid (.vtu), Structured
- Grid (.vts), Rectilinear Grid (.vtr)
-\end_layout
-
-\begin_layout Standard
-
- Discuss Parallel VTK formats: *.pvtu, *.pvts, *.pvtr
-\end_layout
-
-\begin_layout Standard
-
- Discuss UCD format: *.inp, *.ucd -- uses class vtkAVSucdReader from the
- VTK API
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Mention man pages as well -- which also provide an EXAMPLES section
-\end_layout
-
-\begin_layout Standard
-
- Usage string from 'cigma help' displays short form of command -- using
- the positional arguments
-\end_layout
-
-\begin_layout Standard
-
- Commonly used long options are also provided
-\end_layout
-
-\begin_layout Standard
-
- For more advanced usage, 'man cigma-compare' would bring up more examples.
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Don't forget to mention that every positional argument can also be specified
- through the use of a long option
-\end_layout
-
-\begin_layout Standard
-
-Mention common flags, such as --quiet and --verbose for tuning the amount
- of output, and --debug for turning on logging (off by default)
-\end_layout
-
-\begin_layout Standard
-
-In general, --verbose should turn on the progress meter for long-running
- operations, while --quiet will emit the minimal amount of output (none
- for most commands, but the compare command should output just the (L2 norm,
- Linf norm, time in seconds))
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-Command for tracking memory usage of a running process?
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-----------------------------
-\end_layout
-
-\begin_layout Standard
-
-The default data storage format for Cigma is the Hierarchical Data Format
- (HDF5), a portable file format developed at the $X .
- The HDF5 is designed for storing multi-dimensional arrays together with
- meta-data in a portable self-describing format.
- X = [National Center for Supercomputing Applications (NCSA)](hdf.ncsa.uiuc.edu/HDF
-5)
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
------------------------------------------------------
-\end_layout
-
-\begin_layout Standard
-
-\end_layout
-
-\begin_layout Standard
-
-}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Chapter
-\begin_inset LatexCommand label
-name "cha:Running-Cigma"
-
-\end_inset
-
-Running Cigma
-\end_layout
-
-\begin_layout Standard
-As mentioned in the first chapter, Cigma is primarily designed for calculating
- error estimates between arbitrary functions.
- Therefore its primary usage revolves around the following operation
-\end_layout
-
-\begin_layout LyX-Code
-$
-\series bold
-cigma compare
-\emph on
-\bar under
-FunctionA
-\emph default
-\bar default
-
-\emph on
-\bar under
-FunctionB
-\emph default
-\bar default
- -o residuals.vtk
-\end_layout
-
-\begin_layout Standard
-In general, you will provide two paths pointing to the location of the data
- describing the two functions you wish to compare, although in special cases
- you will be able to refer to specific functions by name instead.
-\end_layout
-
-\begin_layout Standard
-You may also need to provide further information by using additional command
- line options.
- For example, you may want to use a different domain discretization for
- the numerical integration step.
- or you may need to increase the order of the quadrature rule in order to
- improve accuracy of the integration.
-\end_layout
-
-\begin_layout Section
-Data Paths
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-XXX
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-In Chapter 6, we described the layout used in the various file formats.
- To refer to a particular array on the command line you can use the string
-
-\series bold
-A:B
-\series default
-, to refer to stored in a file
-\series bold
-A
-\series default
-, you can refer to a particular named array
-\series bold
-B
-\series default
- by using the string
-\series bold
-A:B
-\series default
-, with the colon character `:' serving as a separator between the two paths.
-\end_layout
-
-\begin_layout Standard
-\begin_inset Note Note
-status open
-
-\begin_layout Standard
-For example, a VTK file named
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Command Line Interface
-\end_layout
-
-\begin_layout Standard
-\begin_inset ERT
-status open
-
-\begin_layout Standard
-
-
-\backslash
-mynote{Summarize commands.
- Explain 'cigma help foo' and 'man cigma-foo'.
- Include flow diagram showing the relationship between the various commands.}
-\end_layout
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Subsection
-Comparing Two Finite Element Fields
-\end_layout
-
-\begin_layout Standard
-Comparing two arbitrary finite element fields can be accomplished with the
-
-\family typewriter
-cigma compare
-\family default
- command-line utility.
- By default, the comparison will involve a numerical integration over each
- of the elements in the mesh associated with the first field.
- If this discretization is inadequate, you may also specify an alternative
- discretization over which to perform the integration.
- The comparison operation will output the local residual values are into
- the specified output file.
-\end_layout
-
-\begin_layout Standard
-A basic comparison can be as simple as specifying the following arguments:
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare --first=field1.h5:/field1/stepN
-\end_layout
-
-\begin_layout LyX-Code
-
-\series bold
-
-\series default
- --second=field2.h5:/field2/stepN
-\end_layout
-
-\begin_layout LyX-Code
- --output=residuals.vtk
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout Subsection
-Comparing against Known Values
-\end_layout
-
-\begin_layout Standard
-A finite element description might not always be available for one of the
- fields.
- However, you can break the comparison into several steps if you have a
- means to compute that field on any of the required points.
-\end_layout
-
-\begin_layout Standard
-First, extract the global coordinates of the integration points.
- This will result in an explicit list of points over which to evaluate your
- field.
-\end_layout
-
-\begin_layout LyX-Code
-cigma extract field1.h5:/model/mesh/ -o points.h5:/projected_points
-\end_layout
-
-\begin_layout Standard
-Now you can evaluate your function at the designated points.
- You can provide the path to the explicit set of values with
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare --first=field1.h5:/field1/stepN
-\end_layout
-
-\begin_layout LyX-Code
- --second-points=points.h5:/projected_points
-\end_layout
-
-\begin_layout LyX-Code
- --second-values=values.h5:/projected_values
-\end_layout
-
-\begin_layout LyX-Code
- --output=residuals.vtk
-\end_layout
-
-\begin_layout Subsection
-\begin_inset LatexCommand label
-name "sub:Comparing-against-a"
-
-\end_inset
-
-Comparing against a Known Function
-\end_layout
-
-\begin_layout Standard
-If one of your fields is easily described by an analytic expression, then
- you also have the option to compile your analytic function as a builtin
- Cigma function.
- This will enable you to reference your function by name when using the
-
-\family typewriter
-compare
-\family default
- command.
- For example,
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare --first=field1.h5:/vars/displacement/step0
-\end_layout
-
-\begin_layout LyX-Code
-
-\series bold
-
-\series default
- --second=
-\bar under
-disloc3d
-\end_layout
-
-\begin_layout LyX-Code
- --output=residuals.vtk
-\end_layout
-
-\begin_layout Standard
-You may also interact with your analytic function by using the
-\family typewriter
-cigma
-\family default
-
-\family typewriter
-eval
-\family default
- command, and obtain a set of values which may then be passed back to the
-
-\family typewriter
-compare
-\family default
- command.
-
-\end_layout
-
-\begin_layout LyX-Code
-cigma eval --function=
-\bar under
-disloc3d
-\end_layout
-
-\begin_layout LyX-Code
- --points=points.h5:/projected_points
-\end_layout
-
-\begin_layout LyX-Code
- --values=values.h5:/disloc3d_values
-\end_layout
-
-\begin_layout Standard
-Once these values have been obtained, we can use it to resume the calculation
- of
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare --output=residuals.vtk
-\end_layout
-
-\begin_layout LyX-Code
- --first=field1.h5:/vars/displacement/step0
-\end_layout
-
-\begin_layout LyX-Code
- --second-points=points.h5:/projected_points
-\end_layout
-
-\begin_layout LyX-Code
- --second-values=values.h5:/disloc3d_values
-\end_layout
-
-\begin_layout Standard
-Another built-in function you might find useful is the
-\family typewriter
-\bar under
-zero
-\family default
-\bar default
- function, which is defined to return 0 everywhere.
- You can use this function for estimating the norm of your field.
-\end_layout
-
-\begin_layout Subsection
-Specifying a Integration Mesh
-\end_layout
-
-\begin_layout Standard
-To override the mesh used in the integration, you can specify an extra argument
- providing the location of the mesh,
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare [...]
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-mesh=mesh.h5:/model/mesh/
-\end_layout
-
-\begin_layout Standard
-Alternatively, you can also specify the coordinates and connectivity arrays
- separately, in case they reside in separate files.
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare [...]
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-mesh-coordinates=file1.h5:/model/mesh/coordinates
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-mesh-connectivity=file2.h5:/model/mesh/connectivity
-\end_layout
-
-\begin_layout Subsection
-Specifying a Quadrature Rule
-\end_layout
-
-\begin_layout Standard
-To specify a quadrature rule, you will have to provide the quadrature weights
- and points in the appropriate reference element.
- This can be done with the following additional argument:
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare [...]
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-rule=quadrature-rules.h5:/path/to/rule
-\end_layout
-
-\begin_layout Standard
-You may also specify the location of the points and weights separately:
-\end_layout
-
-\begin_layout LyX-Code
-cigma compare [...]
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-rule-points=file.h5:/path/to/rule/points
-\end_layout
-
-\begin_layout LyX-Code
- --quadrature-rule-weights=file.h5:/path/to/rule/weights
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout Standard
-In the
-\family typewriter
-src/
-\family default
- directory of the Cigma distribution, you can find a Python script called
-
-\family typewriter
-rules.py
-\family default
- that uses FIAT for calculating an appropriate set of quadrature rules for
- use in Cigma.
-\end_layout
-
-\end_body
-\end_document
Copied: doc/cigma/manual/running.lyx (from rev 13032, doc/cigma/manual/running/running.lyx)
===================================================================
--- doc/cigma/manual/running.lyx (rev 0)
+++ doc/cigma/manual/running.lyx 2008-12-10 02:11:13 UTC (rev 13482)
@@ -0,0 +1,689 @@
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
+\lyxformat 345
+\begin_document
+\begin_header
+\textclass book
+\begin_preamble
+\newcommand{\mynote}[1]{}
+\end_preamble
+\use_default_options false
+\language english
+\inputencoding auto
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_default_family default
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+
+\graphics default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_amsmath 1
+\use_esint 1
+\cite_engine basic
+\use_bibtopic false
+\paperorientation portrait
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\defskip medskip
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\author ""
+\author ""
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status collapsed
+
+\begin_layout Plain Layout
+
+
+\backslash
+mynote{
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+In general, verify that the output of the commands, whenever given, matches
+ the actual output when running a similar command
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Do not forget to mention the following:
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Default behaviour if a flag or option is not specified (eg., what if
+ we omit --output?)
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Common flags:
+\end_layout
+
+\begin_layout Plain Layout
+
+ Misc options: --quiet and --verbose (misc option)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Hidden options: --debug flag to enable logging
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Turn off logging if --disable-logging is given to ./configure script
+ (logging on by default)
+\end_layout
+
+\begin_layout Plain Layout
+
+ - The --output option needs an override flag so we can force the rewriting
+ of specific datasets
+\end_layout
+
+\begin_layout Plain Layout
+
+ * Default behavior? Exit if output dataset already exists
+\end_layout
+
+\begin_layout Plain Layout
+
+ * Possible names if default behavior is to exit: --force-writes, --force-ou
+tput, --overwrite-output
+\end_layout
+
+\begin_layout Plain Layout
+
+ * Possible names if default is to allow overwrites: --dont-overwrite
+\end_layout
+
+\begin_layout Plain Layout
+
+ - Addition modifiers to compare command?
+\end_layout
+
+\begin_layout Plain Layout
+
+ * Try --weighed-residuals={0,1} flag in order to divide the cell residuals
+ by the volume
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Given the context of the previous chapter (on file formats), define the
+\end_layout
+
+\begin_layout Plain Layout
+
+'Dataset Path' when discussing the location of datasets within a file
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+XXX: Split the discussion of datasets and I/O into their corresponding chapters
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+ Ensure that most instances of the world 'Field' are replaced by 'Function'
+\end_layout
+
+\begin_layout Plain Layout
+
+ Ensure that 'Quadrature Rule' is changed into 'Integration Rule'
+\end_layout
+
+\begin_layout Plain Layout
+
+ Reference datasets in HDF5 files as 'Simple Arrays'
+\end_layout
+
+\begin_layout Plain Layout
+
+ Discuss ExodusII format
+\end_layout
+
+\begin_layout Plain Layout
+
+ Discuss VTK formats: Legacy VTK (.vtk), Unstructured Grid (.vtu), Structured
+ Grid (.vts), Rectilinear Grid (.vtr)
+\end_layout
+
+\begin_layout Plain Layout
+
+ Discuss Parallel VTK formats: *.pvtu, *.pvts, *.pvtr
+\end_layout
+
+\begin_layout Plain Layout
+
+ Discuss UCD format: *.inp, *.ucd -- uses class vtkAVSucdReader from the
+ VTK API
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Mention man pages as well -- which also provide an EXAMPLES section
+\end_layout
+
+\begin_layout Plain Layout
+
+ Usage string from 'cigma help' displays short form of command -- using
+ the positional arguments
+\end_layout
+
+\begin_layout Plain Layout
+
+ Commonly used long options are also provided
+\end_layout
+
+\begin_layout Plain Layout
+
+ For more advanced usage, 'man cigma-compare' would bring up more examples.
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Don't forget to mention that every positional argument can also be specified
+ through the use of a long option
+\end_layout
+
+\begin_layout Plain Layout
+
+Mention common flags, such as --quiet and --verbose for tuning the amount
+ of output, and --debug for turning on logging (off by default)
+\end_layout
+
+\begin_layout Plain Layout
+
+In general, --verbose should turn on the progress meter for long-running
+ operations, while --quiet will emit the minimal amount of output (none
+ for most commands, but the compare command should output just the (L2 norm,
+ Linf norm, time in seconds))
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+Command for tracking memory usage of a running process?
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+----------------------------
+\end_layout
+
+\begin_layout Plain Layout
+
+The default data storage format for Cigma is the Hierarchical Data Format
+ (HDF5), a portable file format developed at the $X .
+ The HDF5 is designed for storing multi-dimensional arrays together with
+ meta-data in a portable self-describing format.
+ X = [National Center for Supercomputing Applications (NCSA)](hdf.ncsa.uiuc.edu/HDF
+5)
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+-----------------------------------------------------
+\end_layout
+
+\begin_layout Plain Layout
+
+\end_layout
+
+\begin_layout Plain Layout
+
+}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Chapter
+\begin_inset CommandInset label
+LatexCommand label
+name "cha:Running-Cigma"
+
+\end_inset
+
+Running Cigma
+\end_layout
+
+\begin_layout Standard
+As mentioned in the first chapter, Cigma is primarily designed for calculating
+ error estimates between arbitrary functions.
+ Therefore its primary usage revolves around the following operation
+\end_layout
+
+\begin_layout LyX-Code
+$
+\series bold
+cigma compare
+\emph on
+\bar under
+FunctionA
+\emph default
+\bar default
+
+\emph on
+\bar under
+FunctionB
+\emph default
+\bar default
+ -o residuals.vtk
+\end_layout
+
+\begin_layout Standard
+In general, you will provide two paths pointing to the location of the data
+ describing the two functions you wish to compare, although in special cases
+ you will be able to refer to specific functions by name instead.
+\end_layout
+
+\begin_layout Standard
+You may also need to provide further information by using additional command
+ line options.
+ For example, you may want to use a different domain discretization for
+ the numerical integration step.
+ or you may need to increase the order of the quadrature rule in order to
+ improve accuracy of the integration.
+\end_layout
+
+\begin_layout Section
+Data Paths
+\end_layout
+
+\begin_layout Standard
+In Chapter 6, we described the layout used in the various file formats.
+ To refer to a particular array on the command line you can use the string
+
+\series bold
+A:B
+\series default
+, to refer to stored in a file
+\series bold
+A
+\series default
+, you can refer to a particular named array
+\series bold
+B
+\series default
+ by using the string
+\series bold
+A:B
+\series default
+, with the colon character `:' serving as a separator between the two paths.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+For example, a VTK file named
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Command Line Interface
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+mynote{Summarize commands.
+ Explain 'cigma help foo' and 'man cigma-foo'.
+ Include flow diagram showing the relationship between the various commands.}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Comparing Two Finite Element Fields
+\end_layout
+
+\begin_layout Standard
+Comparing two arbitrary finite element fields can be accomplished with the
+
+\family typewriter
+cigma compare
+\family default
+ command-line utility.
+ By default, the comparison will involve a numerical integration over each
+ of the elements in the mesh associated with the first field.
+ If this discretization is inadequate, you may also specify an alternative
+ discretization over which to perform the integration.
+ The comparison operation will output the local residual values are into
+ the specified output file.
+\end_layout
+
+\begin_layout Standard
+A basic comparison can be as simple as specifying the following arguments:
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare --first=field1.h5:/field1/stepN
+\end_layout
+
+\begin_layout LyX-Code
+
+\series bold
+
+\series default
+ --second=field2.h5:/field2/stepN
+\end_layout
+
+\begin_layout LyX-Code
+ --output=residuals.vtk
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout Subsection
+Comparing against Known Values
+\end_layout
+
+\begin_layout Standard
+A finite element description might not always be available for one of the
+ fields.
+ However, you can break the comparison into several steps if you have a
+ means to compute that field on any of the required points.
+\end_layout
+
+\begin_layout Standard
+First, extract the global coordinates of the integration points.
+ This will result in an explicit list of points over which to evaluate your
+ field.
+\end_layout
+
+\begin_layout LyX-Code
+cigma extract field1.h5:/model/mesh/ -o points.h5:/projected_points
+\end_layout
+
+\begin_layout Standard
+Now you can evaluate your function at the designated points.
+ You can provide the path to the explicit set of values with
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare --first=field1.h5:/field1/stepN
+\end_layout
+
+\begin_layout LyX-Code
+ --second-points=points.h5:/projected_points
+\end_layout
+
+\begin_layout LyX-Code
+ --second-values=values.h5:/projected_values
+\end_layout
+
+\begin_layout LyX-Code
+ --output=residuals.vtk
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "sub:Comparing-against-a"
+
+\end_inset
+
+Comparing against a Known Function
+\end_layout
+
+\begin_layout Standard
+If one of your fields is easily described by an analytic expression, then
+ you also have the option to compile your analytic function as a builtin
+ Cigma function.
+ This will enable you to reference your function by name when using the
+
+\family typewriter
+compare
+\family default
+ command.
+ For example,
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare --first=field1.h5:/vars/displacement/step0
+\end_layout
+
+\begin_layout LyX-Code
+
+\series bold
+
+\series default
+ --second=
+\bar under
+disloc3d
+\end_layout
+
+\begin_layout LyX-Code
+ --output=residuals.vtk
+\end_layout
+
+\begin_layout Standard
+You may also interact with your analytic function by using the
+\family typewriter
+cigma
+\family default
+
+\family typewriter
+eval
+\family default
+ command, and obtain a set of values which may then be passed back to the
+
+\family typewriter
+compare
+\family default
+ command.
+
+\end_layout
+
+\begin_layout LyX-Code
+cigma eval --function=
+\bar under
+disloc3d
+\end_layout
+
+\begin_layout LyX-Code
+ --points=points.h5:/projected_points
+\end_layout
+
+\begin_layout LyX-Code
+ --values=values.h5:/disloc3d_values
+\end_layout
+
+\begin_layout Standard
+Once these values have been obtained, we can use it to resume the calculation
+ of
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare --output=residuals.vtk
+\end_layout
+
+\begin_layout LyX-Code
+ --first=field1.h5:/vars/displacement/step0
+\end_layout
+
+\begin_layout LyX-Code
+ --second-points=points.h5:/projected_points
+\end_layout
+
+\begin_layout LyX-Code
+ --second-values=values.h5:/disloc3d_values
+\end_layout
+
+\begin_layout Standard
+Another built-in function you might find useful is the
+\family typewriter
+\bar under
+zero
+\family default
+\bar default
+ function, which is defined to return 0 everywhere.
+ You can use this function for estimating the norm of your field.
+\end_layout
+
+\begin_layout Subsection
+Specifying a Integration Mesh
+\end_layout
+
+\begin_layout Standard
+To override the mesh used in the integration, you can specify an extra argument
+ providing the location of the mesh,
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare [...]
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-mesh=mesh.h5:/model/mesh/
+\end_layout
+
+\begin_layout Standard
+Alternatively, you can also specify the coordinates and connectivity arrays
+ separately, in case they reside in separate files.
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare [...]
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-mesh-coordinates=file1.h5:/model/mesh/coordinates
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-mesh-connectivity=file2.h5:/model/mesh/connectivity
+\end_layout
+
+\begin_layout Subsection
+Specifying a Quadrature Rule
+\end_layout
+
+\begin_layout Standard
+To specify a quadrature rule, you will have to provide the quadrature weights
+ and points in the appropriate reference element.
+ This can be done with the following additional argument:
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare [...]
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-rule=quadrature-rules.h5:/path/to/rule
+\end_layout
+
+\begin_layout Standard
+You may also specify the location of the points and weights separately:
+\end_layout
+
+\begin_layout LyX-Code
+cigma compare [...]
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-rule-points=file.h5:/path/to/rule/points
+\end_layout
+
+\begin_layout LyX-Code
+ --quadrature-rule-weights=file.h5:/path/to/rule/weights
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout Standard
+In the
+\family typewriter
+src/
+\family default
+ directory of the Cigma distribution, you can find a Python script called
+
+\family typewriter
+rules.py
+\family default
+ that uses FIAT for calculating an appropriate set of quadrature rules for
+ use in Cigma.
+\end_layout
+
+\end_body
+\end_document
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