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% Copyright (C) 2011-2016 Mariama Ndiaye, Yves Renard.
%
% This file is a part of GetFEM++
%
% GetFEM++ is free software; you can redistribute it and/or modify it
% under the terms of the GNU Lesser General Public License as published
% by the Free Software Foundation; either version 3 of the License, or
% (at your option) any later version along with the GCC Runtime Library
% Exception either version 3.1 or (at your option) any later version.
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
% or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
% License and GCC Runtime Library Exception for more details.
% You should have received a copy of the GNU Lesser General Public License
% along with this program; if not, write to the Free Software Foundation,
% Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA.
%
% Transient Navier-Stokes equation on a driven cavity with two numerical
% scheme : a projection and a semi-implicite scheme. Without using the
% bricks.
%
% This program is used to check that matlab-getfem is working. This is
% also a good example of use of GetFEM++.
%
clear all;
gf_workspace('clear all');
NX=20; % Space resoltuion
Dt=0.02; % Time step
T = 10; % Time interval
nu=0.005; % Viscosity
v = 30; % Driven velocity
scheme = 1; % 1 : Projection scheme.
% 2 : Semi-implicit scheme
rho = 2; % density
g = 9.81; % gravity constant
%m=gf_mesh('cartesian', [0:1/NX:1],[0:1/NX:1]);
m = gf_mesh('triangles grid',[0:1/NX:1],[0:1/NX:1]);
border = gf_mesh_get(m,'outer faces');
% mark it as boundary #1
gf_mesh_set(m, 'boundary', 1, border);
gf_plot_mesh(m, 'regions', [1]); % the boundary edges appears in red
% create mesh_fem objects
mf_u = gfMeshFem(m,2); % For the velocity.
mf_p = gfMeshFem(m,1); % For the pressure.
mf_f = gfMeshFem(m,1); % For the external forces.
%mim=gf_mesh_im(m, gf_integ('IM_GAUSS_PARALLELEPIPED(2, 2)'));
mim = gf_mesh_im(m, gf_integ('IM_HCT_COMPOSITE(IM_TRIANGLE(13))'));
% assign the fems to all convexes of the mesh_fems
if (scheme == 1)
gf_mesh_fem_set(mf_u,'fem',gf_fem('FEM_PK(2,1)'));
gf_mesh_fem_set(mf_p,'fem',gf_fem('FEM_PK(2,1)'));
gf_mesh_fem_set(mf_f,'fem',gf_fem('FEM_PK(2,1)'));
elseif (scheme == 2)
gf_mesh_fem_set(mf_u,'fem',gf_fem('FEM_PK(2,2)'));
gf_mesh_fem_set(mf_p,'fem',gf_fem('FEM_PK(2,1)'));
gf_mesh_fem_set(mf_f,'fem',gf_fem('FEM_PK(2,1)'));
end;
% Assembly
Fd = [gf_mesh_fem_get(mf_f, 'eval', { 0}); gf_mesh_fem_get(mf_f, 'eval', { rho*g })];
FD =Fd;
U = zeros(gf_mesh_fem_get(mf_u, 'nbdof'), 1); % initial condition
M=rho*gf_asm('mass matrix',mim,mf_u) / Dt;
K=nu*gf_asm('volumic','M(#1,#1)+=comp(vGrad(#1).vGrad(#1))(:,i,j,:,i,j)', mim, mf_u);
F=gf_asm('volumic source',mim,mf_u,mf_f,Fd); %#1 methode d'elmt fini 1, vBase vecteur de base de methode d'EF 1, vGrad grad vect
Kp=gf_asm('volumic','M(#1,#1)+=comp(Grad(#1).Grad(#1))(:,i,:,i)', mim, mf_p);
D=gf_asm('volumic','M(#1,#2)+=comp(Base(#1).vGrad(#2))(:,:,i,i)', mim, mf_p, mf_u);
B=gf_asm('volumic','M(#1,#2)+=comp(vBase(#1).Grad(#2))(:,i,:,i)', mim, mf_u, mf_p);
% for the vorticity computation
Mo=gf_asm('mass matrix', mim, mf_f);
MVo=gf_asm('volumic','t=comp(Base(#1).vGrad(#2));M(#1,#2)+=t(:,:,1,2)-t(:,:,2,1)', mim, mf_f, mf_u);
UBOUND = gf_mesh_fem_get(mf_u, 'dof on region', 1);% fournit le numero des dof sur la frontiere
UNODES = gf_mesh_fem_get(mf_u, 'basic dof nodes');
Kp(1, :) = 0; % In order to fix the pressure on a node for scheme 1.
Kp(1, 1) = 1;
Ndofu = size(D,2); % Dof number for the velocity
Ndofp = size(D,1); % Dof number for the pressure
for t=0:Dt:T
if (scheme == 1)
C=rho*gf_asm('volumic','a=data(#1);M(#1,#1)+=comp(vBase(#1).vGrad(#1).vBase(#1))(i,j,:,k,j,:,k).a(i)', mim,mf_u, U);
A = M + K + C;
L = F + M * U;
for i=UBOUND % Boundary conditions
A(i, :) = 0; A(i,i) = 1; L(i) = 0;
if (mod(i, 2) == 1)
node = UNODES(:, i);
if (abs(node(2)-1) < 1e-10 && abs(node(1)-0.5) < 0.499)
L(i) = v * node(1) * (1-node(1));
end
end
end;
U1_2 = A \ L;
L2 = -D * U1_2/Dt;
L2(1) = 0;
P = Kp \ L2 ;
U = M \ (M * U1_2 - B * P);
elseif (scheme == 2)
C=rho*gf_asm('volumic','a=data(#1);M(#1,#1)+=comp(vBase(#1).vGrad(#1).vBase(#1))(i,j,:,k,j,:,k).a(i)', mim,mf_u, U);
C=C+rho*gf_asm('volumic','a=data(#1);M(#1,#1)+=comp(vBase(#1).vGrad(#1).vBase(#1))(:,i,j,k,k,:,i).a(j)', mim,mf_u, U)/2;
A = [M+K+C, -D'; -D, zeros(Ndofp)];
L = F + M * U;
for i=UBOUND % Boundary conditions
A(i, :) = 0; A(i,i) = 1; L(i) = 0;
if (mod(i, 2) == 1)
node = UNODES(:, i);
if (abs(node(2)-1) < 1e-10 && abs(node(1)-0.5) < 0.499)
L(i) = v * node(1) * (1-node(1));
end
end
end;
A(Ndofu+1, :) = 0; % In order to fix the pressure on a node.
A(Ndofu+1, Ndofu+1) = 1;
UP = A \ [L; zeros(Ndofp,1)];
U = UP(1:Ndofu);
P = UP(Ndofu+1:Ndofu+Ndofp);
end;
Vo = Mo \ (MVo * U); % Vorticity projected on mf_f.
gf_plot(mf_u, U','mesh','off', 'quiver_density', 15, 'quiver_scale', 4);
axis([0 1 0 1]);
hold on;
gf_plot(mf_p,P','refine',1);
gf_plot(mf_f,Vo','refine',1,'contour',[-40,-20,-10,10,20,40,80], 'pcolor', 'off');
hold off;
colorbar; title(sprintf('Quiver plot of U, with color plot of the pressure and vorticity contour lines, t=%g', t));
pause(1);
end;
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