Commit f82c37a4 authored by Matthieu Constant's avatar Matthieu Constant
Browse files

members of matrices

parent 7574a8d3
\documentclass[12pt]{paper}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{bm}
\usepackage[a1paper,landscape,total={32in,20in}]{geometry}
\usepackage[dvipsnames]{xcolor}
\makeatletter
\newcount\bracketnum
\newcommand\makecolorlist[1]{%
\bracketnum0\relax
\makecolorlist@#1,.%
\bracketnum0\relax
}
\def\makecolorlist@#1,{%
\advance\bracketnum1\relax
\expandafter\def\csname bracketcolor\the\bracketnum\endcsname{\color{#1}}%
\@ifnextchar.{\@gobble}{\makecolorlist@}%
}
\let\oldleft\left
\let\oldright\right
\def\left#1{%
\global\advance\bracketnum1\relax
\colorlet{temp}{.}%
\csname bracketcolor\the\bracketnum\endcsname
\oldleft#1%
\color{temp}%
}
\def\right#1{%
\colorlet{temp}{.}%
\csname bracketcolor\the\bracketnum\endcsname
\oldright#1%
\global\advance\bracketnum-1\relax
\color{temp}%
}
\makeatother
\makecolorlist{black,red,blue,ForestGreen,orange,Magenta}
\usepackage{graphics}
\newcommand{\langlC}{\thicklines\begin{picture}(7,7)
\put(1.1,2.5){\rotatebox{60}{\line(1,0){15}}}
\put(1.1,2.5){\rotatebox{300}{\line(1,0){15}}}
\end{picture}}
\newcommand{\ranglC}{\thicklines\begin{picture}(7,7)
\put(-2,2.5){\rotatebox{120}{\line(1,0){15}}}
\put(-2,2.5){\rotatebox{240}{\line(1,0){15}}}
\end{picture}}
\newcommand{\langlCp}{\thicklines\begin{picture}(7,7)
\put(1.1,2.5){\rotatebox{60}{\line(1,0){7}}}
\put(1.1,2.5){\rotatebox{300}{\line(1,0){7}}}
\end{picture}}
\newcommand{\ranglCp}{\thicklines\begin{picture}(7,7)
\put(.2,2.5){\rotatebox{120}{\line(1,0){7}}}
\put(.2,2.5){\rotatebox{240}{\line(1,0){7}}}
\end{picture}}
\newcommand{\langlen}{\Huge\langle}
\newcommand {\ub} {\mathbf u}
\newcommand {\Fb} {\mathbf F}
\newcommand {\gb} {\tilde{\mathbf {g}}}
\newcommand {\nablab} {\bm{\nabla}}
\newcommand {\dphi}[2]{\dfrac{\partial\phi_{#1}}{\partial #2}}
\newcommand{\dcdt}{ \langlC\dfrac{C_k^{old}-C_k}{\Delta t}\phi_k\ranglC}
\newcommand{\dudt}{ \langlC\dfrac{U_k^{old}-U_k}{\Delta t}\phi_k\ranglC}
\newcommand{\dvdt}{ \langlC\dfrac{V_k^{old}-V_k}{\Delta t}\phi_k\ranglC}
\newcommand{\cn}{\langlCp C_k\phi_k\ranglCp}
\newcommand{\dcdx}{\langlC\dphi{k}{x}C_k\ranglC}
\newcommand{\dcdy}{\langlC\dphi{k}{y}C_k\ranglC}
\newcommand{\qn}{\langlCp Q_k\phi_k\ranglCp}
\newcommand{\pn}{\langlCp P_k\phi_k\ranglCp}
\newcommand{\dpn}[1]{\langlC\dphi{k}{#1}P_k\ranglC}
\newcommand{\un}{\langlCp U_k\phi_k\ranglCp}
\newcommand{\vn}{\langlCp V_k\phi_k\ranglCp}
\newcommand{\dudx}{\langlC \dphi{k}{x}U_k\ranglC}
\newcommand{\dudy}{\langlC \dphi{k}{y}U_k\ranglC}
\newcommand{\dvdx}{\langlC \dphi{k}{x}V_k\ranglC}
\newcommand{\dvdy}{\langlC\dphi{k}{y}V_k\ranglC}
\newcommand{\divu}{\langlC \dphi{k}{x}U_k+\dphi{k}{y}V_k\ranglC}
\newcommand{\utaux}{\un \left(\dudx+ \dcdx \dfrac{\un}{\cn}\right)+\vn \left(\dudy+ \dcdy \dfrac{\un}{\cn}\right)}
\newcommand{\utauy}{\un \left(\dvdx+ \dcdx \dfrac{\vn}{\cn}\right)+\vn \left(\dvdy+ \dcdy \dfrac{\vn}{\cn}\right)}
\begin{document}
\renewcommand*{\arraystretch}{4}
\[\text{LOCAL\_VECTOR}=\left(\begin{aligned}
&jw\left\lbrace\mu\left[\dphi{0}{x}\left(2\left(\dudx+\dcdx\dfrac{\un}{\cn}\right)\right)+\dphi{0}{y}\left(\dudy+\dcdy\dfrac{\un}{\cn}+\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right]\right.& +\left.\rho\phi_0\left(\dudt+\dfrac{\un}{\cn}\divu+\dfrac{1}{\cn}\left(\utaux\right)\right)\right.& -\left.\dphi{0}{x}\pn\right\rbrace\\
&jw\left\lbrace\mu\left[\dphi{1}{x}\left(2\left(\dudx+\dcdx\dfrac{\un}{\cn}\right)\right)+\dphi{1}{y}\left(\dudy+\dcdy\dfrac{\un}{\cn}+\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right]\right.& +\left.\rho\phi_1\left(\dudt+\dfrac{\un}{\cn}\divu+\dfrac{1}{\cn}\left(\utaux\right)\right)\right.& -\left.\dphi{1}{x}\pn\right\rbrace&\qquad\bm{U}\\
&jw\left\lbrace\mu\left[\dphi{2}{x}\left(2\left(\dudx+\dcdx\dfrac{\un}{\cn}\right)\right)+\dphi{2}{y}\left(\dudy+\dcdy\dfrac{\un}{\cn}+\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right]\right.& +\left.\rho\phi_2\left(\dudt+\dfrac{\un}{\cn}\divu+\dfrac{1}{\cn}\left(\utaux\right)\right)\right.& -\left.\dphi{2}{x}\pn\right\rbrace\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
&jw\left\lbrace\mu\left[\dphi{0}{x}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}+\dudy+\dcdy\dfrac{\un}{\cn}\right)+\dphi{0}{y}\left(2\left(\dvdy+\dcdy\dfrac{\vn}{\cn}\right)\right)\right]\right.& +\left.\rho\phi_0\left(\dvdt+\dfrac{\vn}{\cn}\divu+\dfrac{1}{\cn}\left(\utauy\right)\right)\right.& -\left.\dphi{0}{y}\pn\right\rbrace\\
&jw\left\lbrace\mu\left[\dphi{1}{x}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}+\dudy+\dcdy\dfrac{\un}{\cn}\right)+\dphi{1}{y}\left(2\left(\dvdy+\dcdy\dfrac{\vn}{\cn}\right)\right)\right]\right.& +\left.\rho\phi_1\left(\dvdt+\dfrac{\vn}{\cn}\divu+\dfrac{1}{\cn}\left(\utauy\right)\right)\right.& -\left.\dphi{1}{y}\pn\right\rbrace&\qquad\bm{V}\\
&jw\left\lbrace\mu\left[\dphi{2}{x}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}+\dudy+\dcdy\dfrac{\un}{\cn}\right)+\dphi{2}{y}\left(2\left(\dvdy+\dcdy\dfrac{\vn}{\cn}\right)\right)\right]\right.& +\left.\rho\phi_2\left(\dvdt+\dfrac{\vn}{\cn}\divu+\dfrac{1}{\cn}\left(\utauy\right)\right)\right.& -\left.\dphi{2}{y}\pn\right\rbrace\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
&jw\left\lbrace\varepsilon\left(\dphi{0}{x}\dpn{x}+\dphi{0}{y}\dpn{y}\right)+\phi_0\left(\divu+\dcdt\right)\right\rbrace\\
&jw\left\lbrace\varepsilon\left(\dphi{1}{x}\dpn{x}+\dphi{1}{y}\dpn{y}\right)+\phi_1\left(\divu+\dcdt\right)\right\rbrace&&&\qquad\bm{P}\\
&jw\left\lbrace\varepsilon\left(\dphi{2}{x}\dpn{x}+\dphi{2}{y}\dpn{y}\right)+\phi_2\left(\divu+\dcdt\right)\right\rbrace\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\
&jw \left\lbrace\phi_0\left(1-\cn-\qn\right)\right\rbrace\\
&jw \left\lbrace\phi_1\left(1-\cn-\qn\right)\right\rbrace&&&\qquad\bm{C}\\
&jw \left\lbrace\phi_2\left(1-\cn-\qn\right)\right\rbrace
\end{aligned}\right)\]
\[\text{LOCAL\_MATRIX}=\left(\begin{aligned}
&U_0&U_1&U_2&&V_0 &V_1&V_2&&P_0&P_1&P_2&&Q_0&Q_1&Q_2\\
&jw\left\lbrace\mu\left[\dphi{0}{x}\left(2\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right)+\dphi{0}{y}\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_0\left(\dfrac{\phi_0}{\Delta t}+2\dfrac{\phi_0 }{\cn}\dudx+2\dfrac{\un}{\cn}\dphi{0}{x}+\left(\dcdx\dfrac{2\un}{\cn^2}\phi_0\right)\right)\right\rbrace&~\cdots~&~\cdots~& & jw\left\lbrace\mu\left[\dphi{0}{y}\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_0\left(\dfrac{\un }{\cn}\dphi{0}{y}+\dfrac{\phi_0}{\cn}\left(\dudy+\dcdy\dfrac{\un}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& \dphi{0}{x}\phi_0&~\cdots~&~\cdots~&&0&~\cdots~&~\cdots~&\\
&jw\left\lbrace\mu\left[\dphi{1}{x}\left(2\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right)+\dphi{1}{y}\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_1\left(\dfrac{\phi_0}{\Delta t}+2\dfrac{\phi_0 }{\cn}\dudx+2\dfrac{\un}{\cn}\dphi{0}{x}+\left(\dcdx\dfrac{2\un}{\cn^2}\phi_0\right)\right)\right\rbrace&~\cdots~&~\cdots~& & jw\left\lbrace\mu\left[\dphi{1}{y}\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_1\left(\dfrac{\un }{\cn}\dphi{0}{y}+\dfrac{\phi_0}{\cn}\left(\dudy+\dcdy\dfrac{\un}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& \dphi{1}{x}\phi_0&~\cdots~&~\cdots~&&0&~\cdots~&~\cdots~&\\
&jw\left\lbrace\mu\left[\dphi{2}{x}\left(2\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right)+\dphi{2}{y}\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_2\left(\dfrac{\phi_0}{\Delta t}+2\dfrac{\phi_0 }{\cn}\dudx+2\dfrac{\un}{\cn}\dphi{0}{x}+\left(\dcdx\dfrac{2\un}{\cn^2}\phi_0\right)\right)\right\rbrace&~\cdots~&~\cdots~& & jw\left\lbrace\mu\left[\dphi{2}{y}\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0 }{\cn}\right)\right]+\rho\phi_2\left(\dfrac{\un }{\cn}\dphi{0}{y}+\dfrac{\phi_0}{\cn}\left(\dudy+\dcdy\dfrac{\un}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& \dphi{2}{x}\phi_0&~\cdots~&~\cdots~&&0&~\cdots~&~\cdots~&\\
\end{aligned}\right)
\]
\end{document}
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