Commit b0708cb3 authored by Matthieu Constant's avatar Matthieu Constant
Browse files

new version of equations

parent f82c37a4
......@@ -2,7 +2,7 @@
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{bm}
\usepackage[a1paper,landscape,total={32in,20in}]{geometry}
\usepackage[a0paper,landscape,total={45in,30in}]{geometry}
\usepackage[dvipsnames]{xcolor}
\makeatletter
......@@ -124,9 +124,14 @@
\[\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~&\\
&jw\left\lbrace\mu\left[\dphi{0}{x}\left(2\left(\phi{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}+\dfrac{\phi_0}{\cn}\left(2\dudx+\dvdy\right)+2\dfrac{\un}{\cn}\dphi{0}{x}+\dfrac{1}{\cn}\left(\phi_0\left(2\dcdx\dfrac{\un}{\cn}\right)+\vn\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\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(\phi{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}+\dfrac{\phi_0}{\cn}\left(2\dudx+\dvdy\right)+2\dfrac{\un}{\cn}\dphi{0}{x}+\dfrac{1}{\cn}\left(\phi_0\left(2\dcdx\dfrac{\un}{\cn}\right)+\vn\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\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(\phi{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}+\dfrac{\phi_0}{\cn}\left(2\dudx+\dvdy\right)+2\dfrac{\un}{\cn}\dphi{0}{x}+\dfrac{1}{\cn}\left(\phi_0\left(2\dcdx\dfrac{\un}{\cn}\right)+\vn\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\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~&\\
\\
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
&jw\left\lbrace\mu\left[\dphi{0}{x}\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right]+\rho\phi_0\left(\dfrac{\vn}{\cn}\dphi{0}{x}+\dfrac{\phi_0}{\cn}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& jw\left\lbrace\mu\left[\dphi{0}{x}\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\dphi{0}{y}\left(2\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right)\right]+\rho\phi_0\left(\dfrac{\phi_0}{\Delta t}+\dfrac{\phi_0}{\cn}\left(\dudx+2\dvdy\right)+2\dfrac{\vn}{\cn}\dphi{0}{y}+\dfrac{1}{\cn}\left(\un\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\phi_0\left(2\dcdy\dfrac{\vn}{\cn}\right)\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(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right]+\rho\phi_1\left(\dfrac{\vn}{\cn}\dphi{0}{x}+\dfrac{\phi_0}{\cn}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& jw\left\lbrace\mu\left[\dphi{0}{x}\left(\dphi{1}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\dphi{1}{y}\left(2\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right)\right]+\rho\phi_1\left(\dfrac{\phi_0}{\Delta t}+\dfrac{\phi_0}{\cn}\left(\dudx+2\dvdy\right)+2\dfrac{\vn}{\cn}\dphi{0}{y}+\dfrac{1}{\cn}\left(\un\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\phi_0\left(2\dcdy\dfrac{\vn}{\cn}\right)\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(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right]+\rho\phi_2\left(\dfrac{\vn}{\cn}\dphi{0}{x}+\dfrac{\phi_0}{\cn}\left(\dvdx+\dcdx\dfrac{\vn}{\cn}\right)\right)\right\rbrace&~\cdots~&~\cdots~&& jw\left\lbrace\mu\left[\dphi{2}{x}\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\dphi{2}{y}\left(2\left(\dphi{0}{y}+\dcdy\dfrac{\phi_0}{\cn}\right)\right)\right]+\rho\phi_2\left(\dfrac{\phi_0}{\Delta t}+\dfrac{\phi_0}{\cn}\left(\dudx+2\dvdy\right)+2\dfrac{\vn}{\cn}\dphi{0}{y}+\dfrac{1}{\cn}\left(\un\left(\dphi{0}{x}+\dcdx\dfrac{\phi_0}{\cn}\right)+\phi_0\left(2\dcdy\dfrac{\vn}{\cn}\right)\right)\right)\right\rbrace&~\cdots~&~\cdots~&& \dphi{2}{x}\phi_0&~\cdots~&~\cdots~&&0&~\cdots~&~\cdots~\\
\end{aligned}\right)
\]
......
......@@ -169,3 +169,5 @@ while t < tEnd :
fluid.export_vtk(outputdir, t, ioutput)
ii += 1
print("%i : %.2g/%.2g (cpu %.6g)" % (ii, t, tEnd, time.clock() - tic))
if ii==2:
break
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