[cig-commits] r6992 - cs/cigma/trunk/doc/manual

[sue at geodynamics.org]

Author: sue
Date: 2007-05-29 17:03:15 -0700 (Tue, 29 May 2007)
New Revision: 6992

Modified:
   cs/cigma/trunk/doc/manual/cigma.lyx
Log:
first batch of equations

Modified: cs/cigma/trunk/doc/manual/cigma.lyx
===================================================================
--- cs/cigma/trunk/doc/manual/cigma.lyx	2007-05-29 16:25:26 UTC (rev 6991)
+++ cs/cigma/trunk/doc/manual/cigma.lyx	2007-05-30 00:03:15 UTC (rev 6992)
@@ -307,15 +307,370 @@
 Code
 \end_layout
 
+\begin_layout Standard
+\begin_inset Formula \[
+\begin{array}{ccccc}
+(-1, & -1, & -1) & \mapsto & a\\
+(+1, & -1, & -1) & \mapsto & b\\
+(-1, & +1, & -1) & \mapsto & c\\
+(+1, & +1, & -1) & \mapsto & d\end{array}\,\,\,\begin{array}{ccccc}
+(-1, & -1, & +1) & \mapsto & e\\
+(+1, & -1, & +1) & \mapsto & f\\
+(-1, & +1, & +1) & \mapsto & g\\
+(+1, & +1, & +1) & \mapsto & h\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
 \begin_layout LyX-Code
+\begin_inset Formula \[
+\vec{x}=\phi(\vec{\xi)}\Longleftrightarrow x_{i}=\phi_{i}\left(\overrightarrow{\xi}\right)\]
 
+\end_inset
+
+
 \end_layout
 
 \begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{ccc}
+\phi_{x} & = & \alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta+\alpha_{4}\xi\eta+\alpha_{5}\xi\zeta+\alpha_{6}\eta\zeta+\alpha_{7}\xi\eta\zeta\\
+\phi_{y} & = & \beta_{0}+\beta_{1}\xi+\beta_{2}\eta+\beta_{3}\zeta+\beta_{4}\xi\eta+\beta_{5}\xi\zeta+\beta_{6}\eta\zeta+\beta_{7}\xi\eta\zeta\\
+\phi_{z} & = & \gamma_{0}+\gamma_{1}\xi+\gamma_{2}\eta+\gamma_{3}\zeta+\gamma_{4}\xi\eta+\gamma_{5}\xi\zeta+\gamma_{6}\eta\zeta+\gamma_{7}\xi\eta\zeta\end{array}\]
 
+\end_inset
+
+
 \end_layout
 
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{c}
+a\\
+b\\
+c\\
+d\\
+e\\
+f\\
+g\\
+h\end{array}\left[\begin{array}{cccccccc}
+1 & (-1) & (-1) & (-1) & (-1)(-1) & (-1)(-1) & (-1)(-1) & (-1)(-1)(-1)\\
+1 & (+1) & (-1) & (-1) & (+1)(-1) & (+1)(-1) & (-1)(-1) & (+1)(-1)(-1)\\
+1 & (-1) & (+1) & (-1) & (-1)(+1) & (-1)(-1) & (+1)(-1) & (-1)(+1)(-1)\\
+1 & (+1) & (+1) & (-1) & (+1)(+1) & (+1)(-1) & (+1)(-1) & (+1)(+1)(-1)\\
+1 & (-1) & (-1) & (+1) & (-1)(-1) & (-1)(+1) & (-1)(+1) & (-1)(-1)(+1)\\
+1 & (+1) & (-1) & (+1) & (+1)(-1) & (+1)(+1) & (-1)(+1) & (+1)(-1)(+1)\\
+1 & (-1) & (+1) & (+1) & (-1)(+1) & (-1)(+1) & (+1)(+1) & (-1)(+1)(+1)\\
+1 & (+1) & (+1) & (+1) & (+1)(+1) & (+1)(+1) & (+1)(+1) & (+1)(+1)(+1)\end{array}\right]\left[\begin{array}{c}
+\alpha_{0}\\
+\alpha_{1}\\
+\alpha_{2}\\
+\alpha_{3}\\
+\alpha_{4}\\
+\alpha_{5}\\
+\alpha_{6}\\
+\alpha_{7}\end{array}\right]=\left[\begin{array}{c}
+x_{0}\\
+x_{1}\\
+x_{2}\\
+x_{3}\\
+x_{4}\\
+x_{5}\\
+x_{6}\\
+x_{7}\end{array}\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\left[\begin{array}{c}
+\alpha_{0}\\
+\alpha_{1}\\
+\alpha_{2}\\
+\alpha_{3}\\
+\alpha_{4}\\
+\alpha_{5}\\
+\alpha_{6}\\
+\alpha_{7}\end{array}\begin{array}{c}
+\beta_{0}\\
+\beta_{1}\\
+\beta_{2}\\
+\beta_{3}\\
+\beta_{4}\\
+\beta_{5}\\
+\beta_{6}\\
+\beta_{7}\end{array}\begin{array}{c}
+\gamma_{0}\\
+\gamma_{1}\\
+\gamma_{2}\\
+\gamma_{3}\\
+\gamma_{4}\\
+\gamma_{5}\\
+\gamma_{6}\\
+\gamma_{7}\end{array}\right]=\frac{1}{8}\,\,\left[\begin{array}{cccccccc}
++1 & +1 & +1 & +1 & +1 & +1 & +1 & +1\\
+-1 & +1 & -1 & +1 & -1 & +1 & -1 & +1\\
+-1 & -1 & +1 & +1 & -1 & -1 & +1 & +1\\
+-1 & -1 & -1 & -1 & +1 & +1 & +1 & +1\\
++1 & -1 & -1 & +1 & +1 & -1 & -1 & +1\\
++1 & -1 & +1 & -1 & -1 & +1 & -1 & +1\\
++1 & +1 & -1 & -1 & -1 & -1 & +1 & +1\\
+-1 & +1 & +1 & -1 & +1 & -1 & -1 & +1\end{array}\right]\left[\begin{array}{c}
+x_{0}\\
+x_{1}\\
+x_{2}\\
+x_{3}\\
+x_{4}\\
+x_{5}\\
+x_{6}\\
+x_{7}\end{array}\begin{array}{c}
+y_{0}\\
+y_{1}\\
+y_{2}\\
+y_{3}\\
+y_{4}\\
+y_{5}\\
+y_{6}\\
+y_{7}\end{array}\begin{array}{c}
+z_{0}\\
+z_{1}\\
+z_{2}\\
+z_{3}\\
+z_{4}\\
+z_{5}\\
+z_{6}\\
+z_{7}\end{array}\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+x\left(\xi\eta\zeta\right)=\alpha_{0}+\alpha_{1}\xi+\alpha_{2}\eta+\alpha_{3}\zeta+\alpha_{4}\xi\eta+\alpha_{5}\xi\zeta+\alpha_{6}\eta\zeta+\alpha_{7}\xi\eta\zeta\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+=\frac{1}{8}\left[x_{0}\cdot\left(1-\xi-\eta-\zeta+\xi\eta+\xi\zeta+\eta\zeta-\xi\eta\zeta\right)\right.\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{1}\cdot\left(1+\xi-\eta-\zeta-\xi\eta-\xi\zeta+\eta\zeta+\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{2}\cdot\left(1-\xi+\eta-\zeta-\xi\eta+\xi\zeta-\eta\zeta+\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{3}\cdot\left(1+\xi+\eta-\zeta+\xi\eta-\xi\zeta-\eta\zeta-\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{4}\cdot\left(1-\xi-\eta+\zeta+\xi\eta-\xi\zeta-\eta\zeta+\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{5}\cdot\left(1+\xi-\eta+\zeta-\xi\eta+\xi\zeta-\eta\zeta-\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++x_{6}\cdot\left(1-\xi+\eta+\zeta-\xi\eta-\xi\zeta+\eta\zeta-\xi\eta\zeta\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\left.+x_{7}\cdot\left(1+\xi+\eta+\zeta+\xi\eta+\xi\zeta+\eta\zeta+\xi\eta\zeta\right)\right]\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+=\left[\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\right]\cdot x_{0}\,\,\,\,\,\,\,\,\,\,\left(-1,-1,-1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1-\zeta\right)\right]\cdot x_{1}\,\,\,\,\,\,\,\,\,\,\left(+1,-1,-1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\right]\cdot x_{2}\,\,\,\,\,\,\,\,\,\,\left(-1,+1,-1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1-\zeta\right)\right]\cdot x_{3}\,\,\,\,\,\,\,\,\,\,\left(+1,+1,-1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1-\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\right]\cdot x_{4}\,\,\,\,\,\,\,\,\,\,\left(-1,-1,+1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1+\xi\right)\left(1-\eta\right)\left(1+\zeta\right)\right]\cdot x_{5}\,\,\,\,\,\,\,\,\,\,\left(+1,-1,+1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1-\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\right]\cdot x_{6}\,\,\,\,\,\,\,\,\,\,\left(-1,+1,+1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
++\left[\frac{1}{8}\left(1+\xi\right)\left(1+\eta\right)\left(1+\zeta\right)\right]\cdot x_{7}\,\,\,\,\,\,\,\,\,\,\left(+1,+1,+1\right)\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+\begin{array}{cc}
+= & N_{0}\left(\vec{\xi}\right)\cdot x_{0}+N_{1}\left(\vec{\xi}\right)\cdot x_{1}+N_{2}\left(\vec{\xi}\right)\cdot x_{2}+N_{3}\left(\vec{\xi}\right)\cdot x_{3}+N_{4}\left(\vec{\xi}\right)\cdot x_{4}+...\\
+=\end{array}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+N_{i}=\frac{1}{8}\left(1+\xi\xi_{a}\right)\left(1+\eta\eta_{b}\right)\left(1+\zeta\zeta_{c}\right)\,\,\,\,\,,\,\, i=0,..,7\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+where 
+\begin_inset Formula $\xi_{a}=\pm1,\,\eta_{b}=\pm1,\,\zeta_{c}\pm1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout LyX-Code
+\begin_inset Formula \[
+x\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)x_{i}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout LyX-Code
+\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 LyX-Code
+\begin_inset Formula \[
+z\left(\vec{\xi}\right)=\sum_{i=0}^{7}N_{i}\left(\vec{\xi}\right)z_{i}\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\end_layout
+
 \begin_layout Chapter
+\begin_inset Formula $ $
+\end_inset
+
 Benchmarks
 \end_layout