[cig-commits] r7006 - cs/cigma/trunk/doc/manual

[sue at geodynamics.org]

Author: sue
Date: 2007-05-30 16:44:08 -0700 (Wed, 30 May 2007)
New Revision: 7006

Modified:
   cs/cigma/trunk/doc/manual/cigma.lyx
Log:
rework equation format and add more

Modified: cs/cigma/trunk/doc/manual/cigma.lyx
===================================================================
--- cs/cigma/trunk/doc/manual/cigma.lyx	2007-05-30 21:04:00 UTC (rev 7005)
+++ cs/cigma/trunk/doc/manual/cigma.lyx	2007-05-30 23:44:08 UTC (rev 7006)
@@ -523,8 +523,6 @@
 \end_layout
 
 \begin_layout Standard
-\noindent
-\align center
 where 
 \begin_inset Formula $\xi_{a}=\pm1,\,\eta_{b}=\pm1,\,\zeta_{c}\pm1$
 \end_inset
@@ -545,66 +543,50 @@
 \begin_inset Text
 
 \begin_layout Standard
-\begin_inset Formula \[
-x\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)x_{i}\]
-
+\begin_inset Formula $\begin{array}{c}
+x\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)x_{i}\\
+y\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)y_{i}\\
+z\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)z_{i}\end{array}$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Standard
-\begin_inset Formula \[
-y\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)y_{i}\]
-
 \end_inset
-
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula \[
-z\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)z_{i}\]
-
-\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 $\vec{x}=x\hat{x}+y\hat{y}+z\hat{z}$
+\begin_inset Formula $\begin{array}{c}
+{\textstyle \vec{x}=x\hat{x}+y\hat{y}+z\hat{z}}\\
+=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)\left(x_{i}\hat{x}+y_{i}\hat{y}+z_{i}\hat{z}\right)\\
+=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)\vec{x}_{i}\end{array}$
 \end_inset
 
 
 \end_layout
 
-\begin_layout Standard
-\begin_inset Formula $=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)\left(x_{i}\hat{x}+y_{i}\hat{y}+z_{i}\hat{z}\right)$
 \end_inset
+</cell>
+</row>
+</lyxtabular>
 
-
-\end_layout
-
-\begin_layout Standard
-\begin_inset Formula $=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)\vec{x}_{i}\,\,\,\,\,\mathrm{tabulate\, the}\, N_{i}\left(\vec{\xi}\right)\mathrm{'s\, for\, each\, required}\,\vec{\xi}$
 \end_inset
 
 
 \end_layout
 
+\begin_layout Standard
+Above right, last equation, tabulate the 
+\begin_inset Formula $N_{i}\left(\vec{\xi}\right)$
 \end_inset
-</cell>
-</row>
-</lyxtabular>
 
+'s for each required 
+\begin_inset Formula $\vec{\xi}$
 \end_inset
 
-
+.
 \end_layout
 
 \begin_layout LyX-Code
@@ -612,10 +594,6 @@
 \end_layout
 
 \begin_layout LyX-Code
-
-\end_layout
-
-\begin_layout LyX-Code
 \begin_inset Formula \[
 x\left(\vec{\xi}_{R}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)x_{i}=\left[N_{0}\left(\vec{\xi}_{R}\right)N_{1}\left(\vec{\xi}_{R}\right)N_{2}\left(\vec{\xi}_{R}\right)\ldots\ldots N_{7}\left(\vec{\xi}_{R}\right)\right]\left[\begin{array}{c}
 x_{0}\\
@@ -683,6 +661,261 @@
 
 \end_layout
 
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Tabular
+<lyxtabular version="3" rows="1" columns="2">
+<features>
+<column alignment="center" valignment="top" width="0">
+<column alignment="center" valignment="top" width="0">
+<row>
+<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
+\begin_inset Text
+
+\begin_layout Standard
+\begin_inset Graphics
+	filename cigma-abcd-triangle.jpg
+	lyxscale 25
+	scale 25
+
+\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 $\begin{array}{ccccc}
+ &  &  & \phi\\
+(-1, & -1, & -1) & \mapsto & a\\
+(+1, & -1, & -1) & \mapsto & b\\
+(-1, & +1, & -1) & \mapsto & c\\
+(+1, & +1, & -1) & \mapsto & d\end{array}$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+</cell>
+</row>
+</lyxtabular>
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{c}
+x\left(\vec{\xi}\right)=\alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta\\
+y\left(\vec{\xi}\right)=\beta_{0}+\beta_{1}\xi+\beta_{2}\eta+\beta_{3}\zeta\\
+z\left(\vec{\xi}\right)=\gamma_{0}+\gamma_{1}\xi+\gamma_{2}\eta+\gamma_{3}\zeta\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{c}
+a\\
+b\\
+c\\
+d\end{array}\left[\begin{array}{cccc}
+1 & (-1) & (-1) & (-1)\\
+1 & (+1) & (-1) & (-1)\\
+1 & (-1) & (+1) & (-1)\\
+1 & (-1) & (-1) & (-1)\end{array}\right]\left[\begin{array}{c}
+\alpha_{0}\\
+\alpha_{1}\\
+\alpha_{2}\\
+\alpha_{3}\end{array}\right]\left[\begin{array}{c}
+x_{0}\\
+x_{1}\\
+x_{2}\\
+x_{3}\end{array}\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\left[\begin{array}{cccc}
+1 & -1 & -1 & -1\\
+1 & +1 & -1 & -1\\
+1 & -1 & +1 & -1\\
+1 & -1 & -1 & +1\end{array}\right]\left[\begin{array}{ccc}
+\alpha_{0} & \beta_{0} & \gamma_{0}\\
+\alpha_{1} & \beta_{1} & \gamma_{1}\\
+\alpha_{2} & \beta_{2} & \gamma_{2}\\
+\alpha_{3} & \beta_{3} & \gamma_{3}\end{array}\right]=\left[\begin{array}{ccc}
+x_{0} & y_{0} & z_{0}\\
+x_{1} & y_{1} & z_{1}\\
+x_{2} & y_{2} & z_{2}\\
+x_{3} & y_{3} & z_{3}\end{array}\right]\]
+
+\end_inset
+
+
+\begin_inset Formula \[
+\left[\begin{array}{ccc}
+\alpha_{0} & \beta_{0} & \gamma_{0}\\
+\alpha_{1} & \beta_{1} & \gamma_{1}\\
+\alpha_{2} & \beta_{2} & \gamma_{2}\\
+\alpha_{3} & \beta_{3} & \gamma_{3}\end{array}\right]=\frac{1}{2}\left[\begin{array}{cccc}
+-1 & 1 & 1 & 1\\
+-1 & 1 & 0 & 0\\
+-1 & 0 & 1 & 0\\
+-1 & 0 & 0 & 1\end{array}\right]\left[\begin{array}{ccc}
+x_{0} & y_{0} & z_{0}\\
+x_{1} & y_{1} & z_{1}\\
+x_{2} & y_{2} & z_{2}\\
+x_{3} & y_{3} & z_{3}\end{array}\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{cc}
+x\left(\vec{\xi}\right) & =\alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta\\
+ & =\left[\frac{1}{2}\left(-x_{0}+x_{1}+x_{2}+x_{3}\right)\right]\\
+ & +\left[\frac{1}{2}\left(-x_{0}+x_{1}\right)\right]\xi\\
+ & +\left[\frac{1}{2}\left(-x_{0}+x_{2}\right)\right]\eta\\
+ & +\left[\frac{1}{2}\left(-x_{0}+x_{3}\right)\right]\zeta\\
+\\ & =\left[\frac{1}{2}\left(-1-\xi-\eta-\zeta\right)\right]\times0\\
+ & +\left[\frac{1}{2}\left(1+\xi\right)\right]\times1\\
+ & +\left[\frac{1}{2}\left(1+\eta\right)\right]\times2\\
+ & +\left[\frac{1}{2}\left(1+\zeta\right)\right]\times3\\
+\\\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\begin{array}{c}
+x\left(\vec{\xi}\right)=N_{0}\left(\vec{\xi}\right)\cdot x_{0}+N_{1}\left(\vec{\xi}\right)\cdot x_{1}+N_{2}\left(\vec{\xi}\right)\cdot x_{2}+N_{3}\left(\vec{\xi}\right)\cdot x_{3}\\
+y\left(\vec{\xi}\right)=N_{0}\left(\vec{\xi}\right)\cdot y_{0}+N_{1}\left(\vec{\xi}\right)\cdot y_{1}+N_{2}\left(\vec{\xi}\right)\cdot y_{2}+N_{3}\left(\vec{\xi}\right)\cdot y_{3}\\
+z\left(\vec{\xi}\right)=N_{0}\left(\vec{\xi}\right)\cdot z_{0}+N_{1}\left(\vec{\xi}\right)\cdot z_{1}+N_{2}\left(\vec{\xi}\right)\cdot z_{2}+N_{3}\left(\vec{\xi}\right)\cdot z_{3}\\
+\\\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula \[
+\begin{array}{cc}
+where & N_{0}\left(\vec{\xi}\right)=\frac{1}{2}\left(-1-\xi-\eta-\zeta\right)\\
+ & N_{1}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\xi\right)\\
+ & N_{2}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\eta\right)\\
+ & N_{3}\left(\vec{\xi}\right)=\frac{1}{2}\left(1+\zeta\right)\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\noindent
+\align center
+
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\left[\begin{array}{ccc}
+x\left(\vec{\xi}_{0}\right) & y\left(\vec{\xi}_{0}\right) & z\left(\vec{\xi}_{0}\right)\\
+x\left(\vec{\xi}_{1}\right) & y\left(\vec{\xi}_{1}\right) & z\left(\vec{\xi}_{1}\right)\\
+\vdots & \vdots & \vdots\\
+x\left(\vec{\xi}_{q-1}\right) & y\left(\vec{\xi}_{q-1}\right) & z\left(\vec{\xi}_{q-1}\right)\end{array}\right]=\left[\begin{array}{ccccc}
+N_{0}\left(\vec{\xi}_{0}\right) & N_{1}\left(\vec{\xi}_{0}\right) & N_{2}\left(\vec{\xi}_{0}\right) & N_{3}\left(\vec{\xi}_{0}\right)\\
+N_{0}\left(\vec{\xi}_{1}\right) & N_{1}\left(\vec{\xi}_{1}\right) & N_{2}\left(\vec{\xi}_{1}\right) & N_{3}\left(\vec{\xi}_{1}\right)\\
+\vdots & \vdots & \vdots & \vdots\\
+N_{0}\left(\vec{\xi}_{q-1}\right) & N_{1}\left(\vec{\xi}_{q-1}\right) & N_{2}\left(\vec{\xi}_{q-1}\right) & N_{3}\left(\vec{\xi}_{q-1}\right)\end{array}\right]\left[\begin{array}{ccc}
+x{}_{0} & y_{0} & z_{0}\\
+x_{1} & y_{1} & z_{1}\\
+x_{2} & y_{2} & z_{2}\\
+x_{3} & y_{3} & z_{3}\end{array}\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\left[q\times3\right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[q\times8\right]\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[4\times3\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph*
+Summary:
+\end_layout
+
+\begin_layout Subparagraph*
+Hex 8
+\end_layout
+
+\begin_layout Standard
+\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 Chapter
 \begin_inset Formula $ $
 \end_inset