[cig-commits] r12676 - in doc/cigma/manual: . error_analysis examples formats integration interpolation running
luis at geodynamics.org
luis at geodynamics.org
Mon Aug 18 14:46:06 PDT 2008
Author: luis
Date: 2008-08-18 14:46:06 -0700 (Mon, 18 Aug 2008)
New Revision: 12676
Modified:
doc/cigma/manual/cigma.lyx
doc/cigma/manual/error_analysis/error_analysis.lyx
doc/cigma/manual/examples/examples.lyx
doc/cigma/manual/formats/formats.lyx
doc/cigma/manual/integration/integration.lyx
doc/cigma/manual/interpolation/interpolation.lyx
doc/cigma/manual/nocover.lyx
doc/cigma/manual/running/running.lyx
doc/cigma/manual/workfile1.lyx
Log:
More changes
Modified: doc/cigma/manual/cigma.lyx
===================================================================
--- doc/cigma/manual/cigma.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/cigma.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -177,7 +177,7 @@
\begin_layout Standard
- * Don't forget to change version on cover to version 1.0.0
+ * Don't forget to change version on cover to version 1.0.0 (done)
\end_layout
\begin_layout Standard
@@ -188,12 +188,13 @@
\begin_layout Standard
* Add description of basic ExodusII format in file_formats.lyx (delay
- this to version 1.1)
+ for version 1.1 -- use Scientific.IO.NetCDF for now)
\end_layout
\begin_layout Standard
- * grep for Chapter and replace all chapter numbers by relative references
+ * grep for 'Chapter' and make sure all chapter numbers are relative
+ references
\end_layout
\begin_layout Standard
@@ -366,7 +367,6 @@
\begin_layout Standard
-
\end_layout
\begin_layout Standard
Modified: doc/cigma/manual/error_analysis/error_analysis.lyx
===================================================================
--- doc/cigma/manual/error_analysis/error_analysis.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/error_analysis/error_analysis.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -636,7 +636,7 @@
\end_layout
\begin_layout Section
-Convergence
+Convergence Rates
\end_layout
\begin_layout Standard
@@ -684,20 +684,28 @@
\end_layout
\begin_layout Standard
-For elliptic problems, we can prove a formula
-\end_layout
-
-\begin_layout Standard
Even if the exact solution
\begin_inset Formula $u(\vec{x})$
\end_inset
- is not known, we may calculate a convergence rate.
- For a given family of solutions
+ is not known, but we have calculated a family of solutions
\begin_inset Formula $u_{h}(\vec{x})$
\end_inset
-, we may
+ on discrete meshes, we may calculate a convergence rate by using the standard
+ error estimate you will find in textbooks:
+\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
\end_layout
\begin_layout Section
@@ -706,7 +714,7 @@
\begin_layout Standard
\begin_inset Note Note
-status collapsed
+status open
\begin_layout Standard
Here we define what we mean by Benchmark, and how we can use.
@@ -722,7 +730,7 @@
\end_layout
\begin_layout Standard
-If
+As seen before, there are
\end_layout
\end_body
Modified: doc/cigma/manual/examples/examples.lyx
===================================================================
--- doc/cigma/manual/examples/examples.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/examples/examples.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -44,7 +44,7 @@
\begin_layout Standard
\begin_inset ERT
-status open
+status collapsed
\begin_layout Standard
@@ -288,7 +288,7 @@
\begin_layout Standard
\begin_inset Note Note
-status open
+status collapsed
\begin_layout Standard
The core concept for this chapter is to show-case specific use cases for
@@ -340,6 +340,19 @@
\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
@@ -364,21 +377,79 @@
\end_layout
\begin_layout Standard
-Figure 8.X
+\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
+\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
@@ -395,5 +466,22 @@
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
Modified: doc/cigma/manual/formats/formats.lyx
===================================================================
--- doc/cigma/manual/formats/formats.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/formats/formats.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -560,13 +560,15 @@
\begin_layout Standard
\begin_inset Note Note
-status collapsed
+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
@@ -591,7 +593,7 @@
\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
@@ -722,24 +724,1642 @@
\end_inset
+\newline
+
+\newline
+The second index varies the fastest, when analyzing the
\end_layout
\begin_layout Subsection
-Mesh
+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 collapsed
+
+\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.
+ 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 Subsection
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 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 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 Subsection
Spherical Harmonics
\end_layout
@@ -789,7 +2409,7 @@
\begin_layout Standard
Now that we have examined what kinds of objects we will be accessing, let's
- discuss the actual layout.
+ discuss the actual layout in the files in which these objects will be stored.
\end_layout
\begin_layout Subsection
@@ -824,8 +2444,11 @@
HDF5 files.
For example, a typical Cigma HDF5 file could have the following structure.
+\end_layout
+
+\begin_layout Standard
\begin_inset Note Note
-status open
+status collapsed
\begin_layout Standard
Mention these assumptions in the corresponding section in running.lyx
@@ -991,9 +2614,12 @@
\begin_layout Standard
The text format is always in table form, with the dimensions of the table
specified in the first line.
- For example, mesh coordinates can be specified in the following format
\end_layout
+\begin_layout Standard
+For example, mesh coordinates can be specified in the following format
+\end_layout
+
\begin_layout LyX-Code
<nno> <nsd>
\end_layout
Modified: doc/cigma/manual/integration/integration.lyx
===================================================================
--- doc/cigma/manual/integration/integration.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/integration/integration.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -163,33 +163,231 @@
\end_inset
norm integral from Chapter 3, we must resort to numerical integration.
- Additionally, we may want to specify a distinct integration mesh which
- does not match the underlying discretization of the functions being compared.
+ 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 open
+
+\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
-Reference Regions
+Gaussian Quadrature
\end_layout
\begin_layout Standard
+\begin_inset Note Note
+status open
+\begin_layout Standard
+Hughes, pg 143
\end_layout
-\begin_layout Subsection
-Line Segment Rules
+\end_inset
+
+
\end_layout
-\begin_layout Subsection
+\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
\end_layout
-\begin_layout Subsection
-Quadrilaterals
+\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 Subsection
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Standard
+Fig.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
Tetrahedra
\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_body
\end_document
Modified: doc/cigma/manual/interpolation/interpolation.lyx
===================================================================
--- doc/cigma/manual/interpolation/interpolation.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/interpolation/interpolation.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -44,7 +44,7 @@
\begin_layout Standard
\begin_inset ERT
-status collapsed
+status open
\begin_layout Standard
@@ -244,8 +244,7 @@
\end_layout
\begin_layout Standard
-Because Cigma will perform comparisons on a common domain, we will typically
- have defined.
+Because we wish to integrate
\end_layout
\begin_layout Section
@@ -265,14 +264,118 @@
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 function can be.
+\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
+\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
+
+
+\end_layout
+
+\begin_layout Subsection
+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
+Hexahedron
+\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
@@ -747,7 +850,11 @@
Finally, the third class of functions consists of example benchmark cases
that you may use to define your own.
\newline
-Basic functions
+
+\newline
+There are two basic functions that are
+ available, listed in the table below.
+
\end_layout
\begin_layout Standard
@@ -808,7 +915,11 @@
\newline
-Test functions
+
+\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
@@ -911,6 +1022,8 @@
\newline
+
+\newline
Cylinder extension case (maxwell and non-newtonian materials)
\end_layout
@@ -1003,6 +1116,8 @@
\newline
+
+\newline
Cylinder relaxation case (maxwell and non-newtonian materials)
\end_layout
@@ -1095,7 +1210,9 @@
\newline
-Circular inclusion
+
+\newline
+Using Gale, we may compute various
\end_layout
\begin_layout Standard
@@ -1122,7 +1239,11 @@
\newline
-Strike-slip benchmark (no gravity)
+
+\newline
+We can also calculate the initial step of the strike-slip benchmark (no
+ gravity) discussed in Chapter 8.
+ This is based on the Okada (1992) dislocation code.
\end_layout
\begin_layout Standard
@@ -1174,6 +1295,10 @@
\end_inset
+\newline
+
+\newline
+
\end_layout
\begin_layout Subsection
@@ -1182,10 +1307,10 @@
\begin_layout Standard
\begin_inset Note Note
-status collapsed
+status open
\begin_layout Standard
-Show a skeleton definition for a subclass of cigma::Function.
+Show example subclass of cigma::Function.
\end_layout
\end_inset
Modified: doc/cigma/manual/nocover.lyx
===================================================================
--- doc/cigma/manual/nocover.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/nocover.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -48,6 +48,8 @@
\begin_layout Standard
This document exists for preview purposes.
+\newline
+
\end_layout
\begin_layout Standard
Modified: doc/cigma/manual/running/running.lyx
===================================================================
--- doc/cigma/manual/running/running.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/running/running.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -345,6 +345,15 @@
\end_layout
\begin_layout Section
+Data Paths
+\end_layout
+
+\begin_layout Standard
+In the previous chapter we studied the layout.
+ Now throughout this chapter we will reference..
+\end_layout
+
+\begin_layout Section
Command Line Interface
\end_layout
Modified: doc/cigma/manual/workfile1.lyx
===================================================================
--- doc/cigma/manual/workfile1.lyx 2008-08-18 18:53:45 UTC (rev 12675)
+++ doc/cigma/manual/workfile1.lyx 2008-08-18 21:46:06 UTC (rev 12676)
@@ -1,4 +1,4 @@
-#LyX 1.5.1 created this file. For more info see http://www.lyx.org/
+#LyX 1.5.5 created this file. For more info see http://www.lyx.org/
\lyxformat 276
\begin_document
\begin_header
@@ -63,7 +63,7 @@
The Linear Tetrahedron
\end_layout
-\begin_layout Section
+\begin_layout Section*
15.2 The Linear Tetrahedron
\end_layout
@@ -71,7 +71,7 @@
The linear tetrahedron, shown in figure 15.1(a), is not used often for stress
analysis because of its poor performance.
\begin_inset Foot
-status open
+status collapsed
\begin_layout Standard
Derivative of shape functions are constant over the element volume.
@@ -97,7 +97,7 @@
Figure 15.1a and 15.1b
\end_layout
-\begin_layout Subsection
+\begin_layout Section*
15.2.1 Tetrahedron Geometry
\end_layout
@@ -123,7 +123,7 @@
\begin_layout Standard
The volume measure of the tetrahedron is denoted
\begin_inset Foot
-status open
+status collapsed
\begin_layout Standard
This symbol (Upsilon) is used to avoid confusion with
@@ -192,7 +192,7 @@
.
\begin_inset Foot
-status open
+status collapsed
\begin_layout Standard
The tetrahedron volume can be zero only if the four corners are coplanar.
@@ -209,7 +209,7 @@
Figure 15.2a and 15.2b
\end_layout
-\begin_layout Subsection
+\begin_layout Section*
15.2.2 Tetrahedral Coordinates
\end_layout
@@ -312,7 +312,7 @@
\end_layout
-\begin_layout Subsection
+\begin_layout Subsection*
15.2.3 Coordinate Transformations
\end_layout
@@ -572,7 +572,7 @@
coordinates,'' will not be used here.
\end_layout
-\begin_layout Subsection
+\begin_layout Subsection*
15.2.5 Partial Derivatives
\end_layout
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