[cig-commits] [commit] master: Testing composite image (showing fields for inclusion problem). (5fc0490)
cig_noreply at geodynamics.org
cig_noreply at geodynamics.org
Fri May 9 15:26:01 PDT 2014
Repository : https://github.com/geodynamics/cigma
On branch : master
Link : https://github.com/geodynamics/cigma/compare/65c02138d3ae8b87c088cc14fe4f98e21e3f0805...a26f592c25c89a40622404999ba1effcdf6df9e3
>---------------------------------------------------------------
commit 5fc04902eb4d4ad4ce6d9626e417c022b4407b21
Author: Luis Armendariz <luis>
Date: Sat Mar 28 00:25:49 2009 +0000
Testing composite image (showing fields for inclusion problem).
>---------------------------------------------------------------
5fc04902eb4d4ad4ce6d9626e417c022b4407b21
figures3/bm3/fields_512.png | Bin 0 -> 170423 bytes
main2.lyx | 203 ++++++++++++++++++++++++++++++++++----------
2 files changed, 157 insertions(+), 46 deletions(-)
diff --git a/figures3/bm3/fields_512.png b/figures3/bm3/fields_512.png
new file mode 100644
index 0000000..8db06f8
Binary files /dev/null and b/figures3/bm3/fields_512.png differ
diff --git a/main2.lyx b/main2.lyx
index ff09947..e09586f 100644
--- a/main2.lyx
+++ b/main2.lyx
@@ -1,4 +1,4 @@
-#LyX 1.6.1 created this file. For more info see http://www.lyx.org/
+#LyX 1.6.0 created this file. For more info see http://www.lyx.org/
\lyxformat 345
\begin_document
\begin_header
@@ -1679,8 +1679,8 @@ Since the absolute magnitude of the
Alternatively, we can normalize the global error using the total volume
of the domain
\begin_inset Formula \begin{eqnarray*}
-\varepsilon_{rel} & = & \frac{||u-u_{h}||_{L_{2}}}{V}\\
- & = & \frac{\int_{\Omega}||u(\vec{x})-u_{h}(\vec{x})||^{2}d\vec{x}}{\int_{\Omega}d\vec{x}}\end{eqnarray*}
+\varepsilon_{abs} & = & \frac{||u-u_{h}||_{L_{2}}}{\sqrt{V}}\\
+ & = & \frac{\sqrt{\int_{\Omega}||u(\vec{x})-u_{h}(\vec{x})||^{2}d\vec{x}}}{\sqrt{\int_{\Omega}d\vec{x}}}\end{eqnarray*}
\end_inset
@@ -1690,7 +1690,7 @@ which is the normalization that Cigma uses by default.
\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.
+ error, in absolute units.
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.
@@ -1995,7 +1995,9 @@ For the remainder of this chapter, we will focus on the
\family typewriter
compare
\family default
-command, and the information [TODO this sentence does not end]/
+command, as well as on a few other convenience commands that you can use
+ for querying the fields themselves.
+
\end_layout
\begin_layout Section
@@ -2003,10 +2005,10 @@ Selecting a Dataset
\end_layout
\begin_layout Standard
-All datasets in Cigma are assumed to be simple two dimensional array of
- values.
- As most scientific formats are capable of storing multiple datasets in
- the same file, you will need a consistent way to select a specific array
+In general, you will provide data to Cigma as a simple two dimensional array
+ of values.
+ Now, because most scientific formats are capable of storing multiple datasets
+ in the same file, you will need a consistent way to select a specific array
on a given data file.
Therefore, all option arguments to Cigma that expect a dataset can be specified
in the form
@@ -2044,8 +2046,11 @@ A mesh block is associated with three items of information: (1) geometrical
(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.
+ There are up to three such meshes that might need to be passed as arguments
+ in Cigma, depending on whether the input fields are associated with a particula
+r discretization.
+ These items are determined by the following command line options, arranged
+ in tabular format for easy reference.
\begin_inset Note Note
status open
@@ -2423,12 +2428,13 @@ Option (MC) is required if options (M1) and (M2) are given.
\end_layout
\begin_layout Standard
-It is important to note that Cigma assumes every element in the mesh uses
- the same basis functions.
+It is important to note that currently Cigma assumes that every element
+ in the mesh is defined over the same basis functions.
+
\end_layout
\begin_layout Subsection
-Mesh Files
+Data Layout in a Mesh File
\end_layout
\begin_layout Standard
@@ -2474,6 +2480,7 @@ CellType
\series default
\emph default
with the appropriate value.
+ A typical hierarchy that specifies a mesh would look like this:
\end_layout
\begin_layout LyX-Code
@@ -2489,7 +2496,7 @@ filename
\emph on
\emph default
- Group
+`Group
\emph on
\bar under
@@ -2830,11 +2837,11 @@ Function Options
Cigma will accept two kinds of function arguments: (1) an analytic function
chosen from a pre-defined list, or (2) an array of coefficients describing
a finite element field.
- Both of these [TODO this sentence does not end]
+ Of these two, only the second is associated with a specific discretization.
\end_layout
\begin_layout Subsection
-By an Analytic Function
+Analytic Functions
\end_layout
\begin_layout Standard
@@ -2855,11 +2862,11 @@ FunctionB
Two such examples can be found in the Cigma source code.
A simple analytic benchmark is available in the
\family typewriter
-fn_gale2.h
+fn_inclusion.h
\family default
and
\family typewriter
-fn_gale2.cpp
+fn_inclusion.cpp
\family default
source files, while a more complex benchmark which calls external procedures
is available in
@@ -2871,12 +2878,12 @@ fn_disloc3d.h
fn_disloc3d.cpp
\family default
.
- For detailed help on the exact steps involved in registering your own functions
-, you may refer to Appendix B.
+ If you wish to define your own analytic functions, it might be instructive
+ to use either of these two examples as a template.
\end_layout
\begin_layout Subsection
-By a Finite Element Field
+Finite Element Fields
\end_layout
\begin_layout Standard
@@ -3045,6 +3052,7 @@ ng a local error integral by a finite weighed sum over a set of quadrature
\end_inset
per quadrature point evaluation.
+
\end_layout
\begin_layout Standard
@@ -3308,15 +3316,17 @@ Verifying the Results
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 is using the wrong
- node numbering for a particular Cigma element, in which case you might
+ node numbering for a particular Cigma element, in which case you will probably
encounter cells with negative or zero volumes, and incorrect results for
the inverse reference map
-\begin_inset Formula $\chi_{e}^{-1}$
+\begin_inset Formula $\vec{x}_{e}^{-1}(\vec{x})$
\end_inset
.
Also, if the degrees of freedom are specified using a different node ordering
than the mesh, the interpolation will yield different results than expected.
+ Querying your model at a number of pre-selected points will probably curtail
+ most of these mistakes when preparing your input files.
\end_layout
\begin_layout Subsection
@@ -3712,7 +3722,10 @@ cigma function-info
reference-cells.h5
\family default
.
- For example, querying
+\end_layout
+
+\begin_layout Standard
+For example, querying
\begin_inset Formula $N_{0}$
\end_inset
@@ -4019,12 +4032,16 @@ err_6_5
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
sideways false
-status collapsed
+status open
\begin_layout Plain Layout
\begin_inset Graphics
@@ -4051,7 +4068,7 @@ Caption for 5.4 here
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
@@ -4083,7 +4100,7 @@ Caption for 5.4 here
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
@@ -4115,7 +4132,7 @@ Caption for 5.4 here
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
@@ -4147,7 +4164,7 @@ Caption for 5.4 here
\end_layout
-\begin_layout Standard
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
@@ -4179,6 +4196,11 @@ Caption for 5.4 here
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
\begin_inset space ~
\end_inset
@@ -4201,7 +4223,11 @@ status open
\begin_inset Caption
\begin_layout Plain Layout
-Caption for 5.4 here
+Convergence of
+\begin_inset Formula $L_{2}$
+\end_inset
+
+ Global Error for Laplace Problem.
\end_layout
\end_inset
@@ -4398,13 +4424,15 @@ From these two results, we can estimate how fast we are converging to a
\end_layout
\begin_layout Standard
-\begin_inset Formula \begin{eqnarray*}
-\alpha & \sim & \frac{\log(\varepsilon_{b}/\varepsilon_{a})}{\log(h_{b}/h_{a})}\\
- & \sim & \frac{\log(0.01/0.05)}{\log(0.108/0.217)}\\
- & \sim & 2.3\end{eqnarray*}
+\begin_inset Formula \[
+\alpha\sim\frac{\log(\varepsilon_{b}/\varepsilon_{a})}{\log(h_{b}/h_{a})}\sim\frac{\log(0.01/0.05)}{\log(0.108/0.217)}\sim2.3\]
\end_inset
+
+\end_layout
+
+\begin_layout Standard
where we have assumed that the highest resolution field is equivalent to
the exact solution in our approximation for
\begin_inset Formula $\alpha$
@@ -4475,6 +4503,10 @@ $ vtk-residuals
\end_layout
\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
@@ -4539,6 +4571,11 @@ Caption for 5.2 here
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
In the figures above, we show three cross sections of the error in the velocity
field.
@@ -4552,34 +4589,99 @@ Circular Inclusion Benchmark
\end_layout
\begin_layout Standard
-We begin by analyzing a two-dimensional example benchmark problem for which
- we know an exact analytical solution.
+Here, we analyze a two-dimensional problem for which we know an exact analytical
+ solution.
+ Using the Gale code, we obtain an approximate solution to a circular inclusion
+ under simple shear.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+ filename figures3/bm3/fields_512.png
+ lyxscale 40
+ scale 18
+
+\end_inset
+
+
+\begin_inset Caption
+
+\begin_layout Plain Layout
+Pressure and Velocity fields for Circular Inclusion Problem.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+To compare against the analytical solution, we can run the command,
\end_layout
\begin_layout LyX-Code
+$ cigma compare
+\backslash
\end_layout
\begin_layout LyX-Code
+ -a 512_8/fields.00000.pvts:PressureField
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -b
+\bar under
+bm.circular_inclusion.pressure
+\bar default
+
+\backslash
+
+\end_layout
+
+\begin_layout LyX-Code
+ -o inclusion.h5:/errror_pressure_512
+\end_layout
+
+\begin_layout Standard
+You can also compare how well the Gale solutions approach each other by
+ comparing the fields against the highest resolution field.
+\end_layout
+
+\begin_layout LyX-Code
$ cigma compare
\backslash
\end_layout
\begin_layout LyX-Code
- -a p256.vts:PressureField
+ -a 512_8/fields.00000.vts:PressureField
\backslash
\end_layout
\begin_layout LyX-Code
- -b p128.vts:PressureField
+ -b 128_8/fields.00000.vts:PressureField
\backslash
\end_layout
\begin_layout LyX-Code
- -o circ_inc.h5:/pressure_256_128
+ -o inclusion.h5:/error_pressure_512_128
\end_layout
\begin_layout LyX-Code
@@ -4589,27 +4691,31 @@ $ cigma compare
\end_layout
\begin_layout LyX-Code
- -a p256.vts:PressureField
+ -a 512_8/fields.00000.vts:PressureField
\backslash
\end_layout
\begin_layout LyX-Code
- -b p64.vts:PressureField
+ -b 256_8/fields.00000.vts:PressureField
\backslash
\end_layout
\begin_layout LyX-Code
- -o circ_inc.h5:/pressure_256_064
+ -o circ_inc.h5:/error_pressure_512_256
\end_layout
\begin_layout Standard
+\begin_inset Note Note
+status collapsed
+
+\begin_layout Plain Layout
\begin_inset Float figure
placement H
wide false
sideways false
-status collapsed
+status open
\begin_layout Plain Layout
\begin_inset Graphics
@@ -4669,6 +4775,11 @@ Caption for 5.4 here
\end_layout
+\end_inset
+
+
+\end_layout
+
\begin_layout Standard
The analytic solution is registered under the somewhat verbose name
\end_layout
@@ -4703,7 +4814,7 @@ $ cigma compare ${bm}.pressure p128.vts:${p} -o circ_inc.h5:/pressure_128
$ cigma compare ${bm}.pressure p64.vts:${p} -o circ_inc.h5:/pressure_064
\end_layout
-\begin_layout Plain Layout
+\begin_layout Standard
\begin_inset Note Note
status open
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