[cig-commits] r7144 - cs/cigma/trunk/doc/manual
sue at geodynamics.org
sue at geodynamics.org
Mon Jun 11 18:22:09 PDT 2007
Author: sue
Date: 2007-06-11 18:22:08 -0700 (Mon, 11 Jun 2007)
New Revision: 7144
Modified:
cs/cigma/trunk/doc/manual/cigma.lyx
Log:
changes as discussed with luis; prob with image that prevents pdf creation (still working on it)
Modified: cs/cigma/trunk/doc/manual/cigma.lyx
===================================================================
--- cs/cigma/trunk/doc/manual/cigma.lyx 2007-06-12 01:01:49 UTC (rev 7143)
+++ cs/cigma/trunk/doc/manual/cigma.lyx 2007-06-12 01:22:08 UTC (rev 7144)
@@ -161,9 +161,9 @@
\begin_layout Standard
Cigma can do analysis in two ways.
It can take a random sampling of points inside a domain of interest and
- analyze the point use differences between physical fields.
- A second way it can operate is to perform an integration over a discretized
- version of the domain.
+ analyze the pointwise differences between physical fields.
+ A second way it can operate is to perform an integration of the errors
+ over a discretized version of the domain.
\end_layout
\begin_layout Standard
@@ -292,11 +292,153 @@
With ...
\end_layout
+\begin_layout Section
+Shape Functions
+\end_layout
+
+\begin_layout Subparagraph*
+Hex 8
+\end_layout
+
\begin_layout Standard
-(TODO: move some of the images/material from appendix into this section)
+\begin_inset Formula $\begin{array}{c}
+a:\, HN_{0}\left(\vec{\xi}\right)=\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\\
+b:\, HN_{1}\left(\vec{\xi}\right)=\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\\
+c:\, HN_{2}\left(\vec{\xi}\right)=\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\\
+d:\, HN_{3}\left(\vec{\xi}\right)=\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\\
+e:\, HN_{4}\left(\vec{\xi}\right)=\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\\
+f:\, HN_{5}\left(\vec{\xi}\right)=\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\\
+g:\, HN_{6}\left(\vec{\xi}\right)=\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\\
+h:\, HN_{7}\left(\vec{\xi}\right)=\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\end{array}$
+\end_inset
+
+
\end_layout
+\begin_layout Subparagraph*
+Tet 4
+\end_layout
+
\begin_layout Standard
+\begin_inset Formula $\begin{array}{c}
+a:\, TN_{0}\left(\vec{\xi}\right)=\frac{1}{2}\left(-1-\xi-\eta-\zeta\right)\\
+b:\, TN_{1}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\xi\right)\\
+c:\, TN_{2}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\eta\right)\\
+d:\, TN_{3}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\zeta\right)\end{array}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Quadrature Rules
+\end_layout
+
+\begin_layout Standard
+(TODO: Define quadrature rules)
+\end_layout
+
+\begin_layout Standard
+\noindent
+\begin_inset Graphics
+ filename figures/quadrature-rule-diagram.jpg
+ lyxscale 25
+ scale 25
+ rotateOrigin leftBottom
+
+\end_inset
+
+
+\begin_inset Formula \[
+\varphi_{K}^{e}\left(\vec{x}\left(\vec{\xi}\right)\right)\]
+
+\end_inset
+
+set of points
+\begin_inset Formula $\vec{q}_{i}$
+\end_inset
+
+ \InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+\InsetSpace ~
+
+\begin_inset Formula $Q=\left\{ 1,\,\ldots\,,\left|Q\right|\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+set of weights
+\begin_inset Formula $w_{1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\noindent
+quadrature rule of degree
+\begin_inset Formula $\left|Q\right|$
+\end_inset
+
+ \InsetSpace ~
+(
+\begin_inset Formula $\left|Q\right|^{d}$
+\end_inset
+
+ points and weights)
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+F\left(\xi\right),\rightarrow\int_{K_{ref}}\,\, F\left(\vec{\xi}\right)d\vec{\xi}=\begin{array}{c}
+\intop\intop\intop\\
+K_{ref}\end{array}\, F\left(\xi_{1},\xi_{2}\,,\xi_{3}\right)d\xi,\, d\xi_{2},\, d\xi_{3}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+=\sum_{i\epsilon Q}\, F\left(\vec{q}_{i}\right)\cdot w_{i}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\int_{K_{e}}f\left(\vec{x}\right)d\vec{x}=\int_{K_{ref}}f\left(\vec{\xi}\right)\cdot\left|\mathrm{J^{-1}}\left(\vec{\xi}\right)\right|d\vec{\xi}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+=\sum_{i\epsilon Q}\left(f\left(\vec{q}_{i}\right)\left|\mathrm{J^{-1}\left(\vec{q}_{i}\right)}\right|\right)\cdot w_{i}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
(TODO: discussion of error aliasing that arises from undersampling the integrand
)
\end_layout
@@ -310,13 +452,24 @@
\end_layout
\begin_layout Standard
-Cigma was developed with the following use cases in mind
+Cigma was developed with the following use cases in mind:
\end_layout
\begin_layout Itemize
-...
+Error Analysis: To assist scientists in comparing two different codes, possibly
+ on the same mesh
\end_layout
+\begin_layout Itemize
+Benchmarking: To help the geodynamics community agree on a standard solution
+ to a specific problem
+\end_layout
+
+\begin_layout Itemize
+Verification: For developers to test software changes so as to ensure that
+ those changes don't affect the consistency of the results
+\end_layout
+
\begin_layout Section
Components
\end_layout
@@ -1006,9 +1159,38 @@
\end_layout
\begin_layout Standard
-(TODO: Insert cigma-meshA-diagram.jpg here)
+TODO: figure out why this graphic will not appear (get
+\begin_inset Quotes sld
+\end_inset
+
+error loading file into memory
+\begin_inset Quotes srd
+\end_inset
+
+)
\end_layout
+\begin_layout Standard
+\begin_inset Float figure
+wide false
+sideways false
+status open
+
+\begin_layout Caption
+\begin_inset Graphics
+ filename figures/cigma-meshA-diagram.jpg
+ scale 75
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Chapter
Numerical Results
\end_layout
@@ -1297,6 +1479,10 @@
Shape Functions
\end_layout
+\begin_layout Section
+Linear Hexahedral Element
+\end_layout
+
\begin_layout Standard
\begin_inset Float figure
placement H
@@ -1311,6 +1497,7 @@
\begin_layout Standard
\begin_inset Graphics
filename figures/cigma-a2h-cube.jpg
+ scale 50
\end_inset
@@ -1482,7 +1669,7 @@
\begin_layout Standard
\begin_inset Formula \[
\begin{array}{cc}
-x\left(\xi\eta\zeta\right) & =\alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta+\alpha_{4}\xi\eta+\alpha_{5}\xi\zeta+\alpha_{6}\eta\zeta+\alpha_{7}\xi\eta\zeta\\
+x\left(\xi,\eta,\zeta\right) & =\alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta+\alpha_{4}\xi\eta+\alpha_{5}\xi\zeta+\alpha_{6}\eta\zeta+\alpha_{7}\xi\eta\zeta\\
& =\frac{1}{8}\left[x_{0}\cdot\left(1-\xi-\eta-\zeta+\xi\eta+\xi\zeta+\eta\zeta-\xi\eta\zeta\right)\right.\\
& +x_{1}\cdot\left(1+\xi-\eta-\zeta-\xi\eta-\xi\zeta+\eta\zeta+\xi\eta\zeta\right)\\
& +x_{2}\cdot\left(1-\xi+\eta-\zeta-\xi\eta+\xi\zeta-\eta\zeta+\xi\eta\zeta\right)\\
@@ -1681,6 +1868,10 @@
\end_layout
+\begin_layout Section
+Linear Tetrahedral Element
+\end_layout
+
\begin_layout LyX-Code
\begin_inset Tabular
<lyxtabular version="3" rows="1" columns="2">
@@ -1932,7 +2123,7 @@
\end_layout
-\begin_layout Paragraph*
+\begin_layout Section
Jacobian Matrix
\end_layout
@@ -2351,7 +2542,7 @@
\begin_layout Standard
\begin_inset Formula \[
-\mathrm{J}=\left|\frac{2\vec{x}}{2\vec{\xi}}\right|=\left|\begin{array}{c}
+\mathrm{J}=\left|\frac{\partial\vec{x}}{\partial\vec{\xi}}\right|=\left|\begin{array}{c}
\frac{1}{2}\left(x_{1}-x_{0}\right)\,\,\frac{1}{2}\left(x_{2}-x_{0}\right)\,\,\frac{1}{2}\left(x_{3}-x_{0}\right)\\
\frac{1}{2}\left(y_{1}-y_{0}\right)\,\,\frac{1}{2}\left(y_{2}-y_{0}\right)\,\,\frac{1}{2}\left(y_{3}-y_{0}\right)\\
\frac{1}{2}\left(z_{1}-z_{0}\right)\,\,\frac{1}{2}\left(z_{2}-z_{0}\right)\,\,\frac{1}{2}\left(z_{3}-z_{0}\right)\end{array}\right|\]
@@ -2945,114 +3136,10 @@
\end_layout
\begin_layout Chapter
-Quadrature Rules
+Mapping From Reference/Mapping To Reference
\end_layout
\begin_layout Standard
-\noindent
-\begin_inset Graphics
- filename quadrature-rule-diagram.jpg
- lyxscale 25
- scale 25
- rotateOrigin leftBottom
-
-\end_inset
-
-
-\begin_inset Formula \[
-\varphi_{K}^{e}\left(\vec{x}\left(\vec{\xi}\right)\right)\]
-
-\end_inset
-
-set of points
-\begin_inset Formula $\vec{q}_{i}$
-\end_inset
-
- \InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-\InsetSpace ~
-
-\begin_inset Formula $Q=\left\{ 1,\,\ldots\,,\left|Q\right|\right\} $
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-set of weights
-\begin_inset Formula $w_{1}$
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\noindent
-quadrature rule of degree
-\begin_inset Formula $\left|Q\right|$
-\end_inset
-
- \InsetSpace ~
-(
-\begin_inset Formula $\left|Q\right|^{d}$
-\end_inset
-
- points and weights)
-\end_layout
-
-\begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-F\left(\xi\right),\rightarrow\int_{K_{ref}}\,\, F\left(\vec{\xi}\right)d\vec{\xi}=\begin{array}{c}
-\intop\intop\intop\\
-K_{ref}\end{array}\, F\left(\xi_{1},\xi_{2}\,,\xi_{3}\right)d\xi,\, d\xi_{2},\, d\xi_{3}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-=\sum_{i\epsilon Q}\, F\left(\vec{q}_{i}\right)\cdot w_{i}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-\int_{K_{e}}f\left(\vec{x}\right)d\vec{x}=\int_{K_{ref}}f\left(\vec{\xi}\right)\cdot\left|\mathrm{J^{-1}}\left(\vec{\xi}\right)\right|d\vec{\xi}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-=\sum_{i\epsilon Q}\left(f\left(\vec{q}_{i}\right)\left|\mathrm{J^{-1}\left(\vec{q}_{i}\right)}\right|\right)\cdot w_{i}\]
-
-\end_inset
-
-
-\end_layout
-
-\begin_layout Section
-Mapping from reference/Mapping to reference
-\end_layout
-
-\begin_layout Standard
Reference elements are on the figures on the left.
\end_layout
@@ -3063,11 +3150,12 @@
placement H
wide false
sideways false
-status open
+status collapsed
\begin_layout Standard
\begin_inset Graphics
filename figures/mapping1.gif
+ scale 60
\end_inset
@@ -3090,11 +3178,12 @@
placement H
wide false
sideways false
-status open
+status collapsed
\begin_layout Standard
\begin_inset Graphics
filename figures/mapping2.gif
+ scale 60
\end_inset
@@ -3114,6 +3203,7 @@
\noindent
\align center
\begin_inset Float figure
+placement H
wide false
sideways false
status open
@@ -3121,6 +3211,7 @@
\begin_layout Standard
\begin_inset Graphics
filename figures/mapping3.gif
+ scale 60
\end_inset
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