[cig-commits] [commit] master: Fixed various sections, including images (should use relative paths to the image, instead of absolute) (70e9de0)
cig_noreply at geodynamics.org
cig_noreply at geodynamics.org
Fri May 9 15:25:54 PDT 2014
Repository : https://github.com/geodynamics/cigma
On branch : master
Link : https://github.com/geodynamics/cigma/compare/65c02138d3ae8b87c088cc14fe4f98e21e3f0805...a26f592c25c89a40622404999ba1effcdf6df9e3
>---------------------------------------------------------------
commit 70e9de03777cf927995447aeeb28255cde99f776
Author: Luis Armendariz <luis>
Date: Tue Mar 17 20:33:23 2009 +0000
Fixed various sections, including images (should use relative paths to the image,
instead of absolute)
>---------------------------------------------------------------
70e9de03777cf927995447aeeb28255cde99f776
main2.lyx | 561 +++++++++++++++++++++++++++++++++++++++++++++++++++++++-------
1 file changed, 499 insertions(+), 62 deletions(-)
diff --git a/main2.lyx b/main2.lyx
index 38bbf01..0dc3e23 100644
--- a/main2.lyx
+++ b/main2.lyx
@@ -1,4 +1,4 @@
-#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.1 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
@@ -43,18 +43,6 @@
\begin_body
\begin_layout Chapter
-\begin_inset ERT
-status collapsed
-
-\begin_layout Plain Layout
-
-
-\backslash
-raggedbottom
-\end_layout
-
-\end_inset
-
Introduction
\end_layout
@@ -661,9 +649,9 @@ $ make install
\end_layout
\begin_layout Standard
-CppUnit is available for download from
+CppUnit is available for download from its SourceForge project site
\begin_inset Flex URL
-status collapsed
+status open
\begin_layout Plain Layout
@@ -1294,7 +1282,8 @@ If we know the shape functions
global interpolation scheme
\series default
\emph default
-, and we may use Eq 3 directly to find the values of
+, and we may use Eq.
+ 3.3 directly to find the values of
\begin_inset Formula $f(\vec{x})$
\end_inset
@@ -1302,7 +1291,7 @@ global interpolation scheme
\begin_inset Formula $\phi_{j}$
\end_inset
-as
+ as
\emph on
global shape functions
\emph default
@@ -1605,7 +1594,7 @@ If we let
\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*}
+ & = & \sum_{q=1}^{\mathrm{n_{Q}}}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}w_{q}J_{e}(\vec{\xi}_{q})\end{eqnarray*}
\end_inset
@@ -1621,7 +1610,7 @@ The global error
\end_layout
\begin_layout Standard
-\begin_inset Box Boxed
+\begin_inset Box Frameless
position "t"
hor_pos "c"
has_inner_box 1
@@ -1635,7 +1624,7 @@ 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*}
+\varepsilon & = & \sqrt{\sum_{e=1}^{\mathrm{n_{el}}}\sum_{q=1}^{\mathrm{n_{Q}}}||u(\vec{x}_{e}(\vec{\xi}_{q}))-v(\vec{x}_{e}(\vec{\xi}_{q}))||^{2}w_{q}J_{e}(\vec{\xi}_{q})}\end{eqnarray*}
\end_inset
@@ -1770,7 +1759,7 @@ Once we have calculated a family of solutions
\end_layout
\begin_layout Standard
-For a single refinement level, we have two discretizations
+For a single refinement level we have two discretizations
\begin_inset Formula $\Omega_{1}$
\end_inset
@@ -1787,7 +1776,7 @@ For a single refinement level, we have two discretizations
\end_inset
.
- The convergence rate can the be estimated from the two approximate bounds
+ The convergence rate can then be estimated from the two approximate bounds
\begin_inset Formula \begin{eqnarray*}
\varepsilon_{1} & \sim & Ch_{1}^{\alpha}\\
@@ -1795,17 +1784,11 @@ For a single refinement level, we have two discretizations
\end_inset
-which can be combined into
-\begin_inset Formula \[
-\left(\frac{\varepsilon_{2}}{\varepsilon_{1}}\right)\sim\left(\frac{h_{2}}{h_{1}}\right)^{\alpha}\]
-
-\end_inset
-
-or, solving for
+by taking their ratio and solving for
\begin_inset Formula $\alpha$
\end_inset
-, we arrive at the simple equation
+, giving us the equation
\begin_inset Formula \[
\alpha\sim\frac{\log(\varepsilon_{2}/\varepsilon_{1})}{\log(h_{2}/h_{1})}\]
@@ -2058,12 +2041,26 @@ Mesh Options
\begin_layout Standard
A mesh block is associated with three items of information: (1) geometrical
information for a number of nodes defined on a global coordinate system,
- and (2) topological information describing how those nodes are connected
- to each other to form elements, and (3) an element type associated with
- the cell.
+ (2) topological information describing how those nodes are connected to
+ each other to form elements, and (3) an element type associated with the
+ cell.
In Cigma, these items are determined by the following command line options,
arranged in tabular format for easy reference.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO: How in the world do we insert newlines here.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Newline newline
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -2163,6 +2160,7 @@ M
\begin_layout Plain Layout
\family typewriter
+\size small
--first-mesh
\end_layout
@@ -2174,6 +2172,7 @@ M
\begin_layout Plain Layout
\family typewriter
+\size small
--second-mesh
\end_layout
@@ -2185,6 +2184,7 @@ M
\begin_layout Plain Layout
\family typewriter
+\size small
--mesh
\end_layout
@@ -2216,6 +2216,7 @@ M1
\begin_layout Plain Layout
\family typewriter
+\size small
--first-mesh-coords
\end_layout
@@ -2227,6 +2228,7 @@ M1
\begin_layout Plain Layout
\family typewriter
+\size small
--second-mesh-coords
\end_layout
@@ -2238,6 +2240,7 @@ M1
\begin_layout Plain Layout
\family typewriter
+\size small
--mesh-coords
\end_layout
@@ -2269,6 +2272,7 @@ M2
\begin_layout Plain Layout
\family typewriter
+\size small
--first-mesh-connect
\end_layout
@@ -2280,6 +2284,7 @@ M2
\begin_layout Plain Layout
\family typewriter
+\size small
--second-mesh-connect
\end_layout
@@ -2291,6 +2296,7 @@ M2
\begin_layout Plain Layout
\family typewriter
+\size small
--mesh-connect
\end_layout
@@ -2322,6 +2328,7 @@ MC
\begin_layout Plain Layout
\family typewriter
+\size small
--first-cell
\end_layout
@@ -2333,6 +2340,7 @@ MC
\begin_layout Plain Layout
\family typewriter
+\size small
--second-cell
\end_layout
@@ -2344,6 +2352,7 @@ MC
\begin_layout Plain Layout
\family typewriter
+\size small
--mesh-cell
\end_layout
@@ -2373,6 +2382,10 @@ MC
\end_inset
+\begin_inset Newline newline
+\end_inset
+
+
\end_layout
\begin_layout Standard
@@ -2515,10 +2528,8 @@ connectivity
\end_layout
\begin_layout Standard
-[TODO i know you mean the data above to lead in your sentence but how can
- we refer to it, to make this a real sentence beginning? ] may specify options
- (M1) and (M2) separately by pointing directly to the arrays corresponding
- to the coordinates and connectivity information.
+One may specify options (M1) and (M2) separately by pointing directly to
+ the arrays corresponding to the coordinates and connectivity information.
The node ordering on the connectivity dataset is described in Appendix
A.
\end_layout
@@ -2538,7 +2549,7 @@ POINTS
\family typewriter
CELLS
\family default
- , which are required by the VTK file format.
+, which are required by the VTK file format.
The value of the option (MC) is determined from the first entry in the
\family typewriter
@@ -2828,8 +2839,8 @@ By an Analytic Function
\begin_layout Standard
Sometimes you will be able to express a function in terms of a general formula
- or algorithm, in which case would like to be able to refer to such a function
- by using a simple name when specifying either of the
+ or algorithm, in which case you would like to be able to refer to such
+ a function by using a simple name when specifying either of the
\bar under
FunctionA
\bar default
@@ -3184,8 +3195,8 @@ B
\end_layout
\begin_layout Standard
-In this case, the exit code will indicate failure (return a non-zero value),
- whenever the specified threshold condition is not met.
+In this case, the exit code will indicate failure (i.e., return a non-zero
+ value) whenever the specified threshold condition is not met.
This allows you to set up automated regression scripts that can constantly
compare output from your numerical codes against a series of known benchmark
solutions.
@@ -3220,7 +3231,7 @@ raw
, so that errors accumulated over smaller cells have more visual influence
than errors of the same magnitude accumulated over larger cells.
- It may be also be useful visually, to output the logarithm of the residual
+ It may also be useful visually, to output the logarithm of the residual
values.
Using a logarithmic scale will accentuate the contrast betwen the orders
of magnitude in the local residuals.
@@ -3296,10 +3307,10 @@ Verifying the Results
\begin_layout Standard
The rest of the Cigma commands are there to help you query your model, allowing
you to determine whether the input files are being interpreted properly.
- A common problem in specifying a finite element mesh would be using the
- wrong node numbering for a particular Cigma element, in which case you
- might encounter cells with negative or zero volumes, incorrect results
- for the inverse reference map
+ A common problem in specifying a finite element mesh is using the wrong
+ node numbering for a particular Cigma element, in which case you might
+ encounter cells with negative or zero volumes, and incorrect results for
+ the inverse reference map
\begin_inset Formula $\chi_{e}^{-1}$
\end_inset
@@ -3414,7 +3425,7 @@ You can also query any function accepted by the
\family typewriter
cigma compare
\family default
- command, not just builtin ones, at any arbitrary point in its function
+ command, not just built-in ones, at any arbitrary point in its function
domain,
\end_layout
@@ -4019,7 +4030,7 @@ status collapsed
\begin_layout Plain Layout
\begin_inset Graphics
- filename /Users/sue/cigma/figures2/log_res_citcomcu_velocity_32_08.png
+ filename figures2/log_res_citcomcu_velocity_32_08.png
lyxscale 40
scale 40
@@ -4052,7 +4063,7 @@ status collapsed
\begin_layout Plain Layout
\begin_inset Graphics
- filename /Users/sue/cigma/figures2/log_res_citcomcu_velocity_32_16.png
+ filename figures2/log_res_citcomcu_velocity_32_16.png
lyxscale 40
scale 40
@@ -4090,7 +4101,6 @@ Circular Inclusion Benchmark
\begin_layout Standard
We begin by analyzing a two-dimensional example benchmark problem for which
we know an exact analytical solution.
-
\end_layout
\begin_layout LyX-Code
@@ -4150,7 +4160,7 @@ status collapsed
\begin_layout Plain Layout
\begin_inset Graphics
- filename /Users/sue/cigma/figures2/log_res_circular_inclusion_256_064.png
+ filename figures2/log_res_circular_inclusion_256_064.png
lyxscale 40
scale 18
@@ -4183,7 +4193,7 @@ status collapsed
\begin_layout Plain Layout
\begin_inset Graphics
- filename /Users/sue/cigma/figures2/log_res_circular_inclusion_256_128.png
+ filename figures2/log_res_circular_inclusion_256_128.png
lyxscale 40
scale 18
@@ -4348,17 +4358,443 @@ Laplace Problem
\end_layout
\begin_layout Standard
-Here we obtain a sequence of solutions by solving a Laplace problem inside
- a cubic domain.
+Here we obtain a sequence of solutions over five refinement levels by solving
+ the Laplace problem
+\begin_inset Formula $\nabla^{2}\phi(x,y)=4x^{4}+4y^{4}$
+\end_inset
+
+ inside
+\begin_inset Formula $\Omega=[-1,1]^{2}$
+\end_inset
+
+, subject to
+\begin_inset Formula $\phi(x_{0},y_{0})=x_{0}^{2}+y_{0}^{2}$
+\end_inset
+
+ for points
+\begin_inset Formula $(x_{0},y_{0})$
+\end_inset
+
+ in the boundary
+\begin_inset Formula $\partial\Omega$
+\end_inset
+
+.
+ This problem can easily be solved using the Deal.II library, and is available
+ under Step 4 in the documentation.
+\end_layout
+
+\begin_layout LyX-Code
+$ levels=
+\begin_inset Quotes erd
+\end_inset
+
+2 3 4 5
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+$ for i in ${levels}; do
+\end_layout
+
+\begin_layout LyX-Code
+ cigma compare
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ square6.vtk square${i}.vtk
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -o square.h5:/error_6_${i}
+\end_layout
+
+\begin_layout LyX-Code
+ vtk-residuals --output-log-values
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -m square6.vtk
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -i square.h5:/error_6_${i}
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -o log_error_square_6_${i}.vtk:log_error
+\end_layout
+
+\begin_layout LyX-Code
+ done
+\end_layout
+
+\begin_layout Standard
+From output of the above commands we can collect the following table
\end_layout
\begin_layout Standard
-\begin_inset Note Note
+\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
+Case
+\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
+\begin_inset Formula $h$
+\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
+
+\begin_inset Formula $L_{2}$
+\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
+err_6_2
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0.70717
+\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
+0.08280
+\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
+err_6_3
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0.35355
+\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
+0.02462
+\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
+err_6_4
+\end_layout
+
+\end_inset
+</cell>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Plain Layout
+0.17677
+\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
+0.00631
+\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
+err_6_5
+\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
+0.08838
+\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
+0.00137
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures2/laplace_square6.png
+ lyxscale 80
+ scale 25
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures2/log_res_square_6_2.png
+ lyxscale 40
+ scale 18
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures2/log_res_square_6_3.png
+ lyxscale 40
+ scale 18
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures2/log_res_square_6_4.png
+ lyxscale 40
+ scale 18
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status collapsed
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures2/log_res_square_6_5.png
+ lyxscale 40
+ scale 18
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset space ~
+\end_inset
+
+
+\begin_inset Float figure
+placement H
+wide false
+sideways false
status open
\begin_layout Plain Layout
-Discuss higher order elements here.
- For now, just show how power-plot works
+\begin_inset Graphics
+ filename figures2/alpha_square.png
+ scale 30
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Caption for 5.4 here
\end_layout
\end_inset
@@ -4366,7 +4802,8 @@ Discuss higher order elements here.
\end_layout
-\begin_layout LyX-Code
+\end_inset
+
\end_layout
@@ -4661,7 +5098,7 @@ Data Formats
\begin_layout Standard
The basic data structure is a two-dimensional array of values, stored in
- a contiguous format as shown below
+ a contiguous format as shown below:
\begin_inset Newline newline
\end_inset
@@ -5926,8 +6363,8 @@ Integration Rule
\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 [TODO - the?] natural coordinate system of the
- reference element.
+ The points should be specified on the natural coordinate system used by
+ the corresponding reference element.
\begin_inset Newline newline
\end_inset
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