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% Copyright (C) 2005-2016 Julien Pommier.
%
% 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.
gf_workspace('clear all');
disp('2D scalar wave equation (helmholtz) demonstration');
disp('Helmholtz is not handled (for the moment) by gf_solve');
disp('hence this file contains explicit call to the various');
disp('assembly routines needed by the helmholtz equation');
disp('The result is the wave scattered by a disc, the incoming wave beeing a plane wave coming from the top');
disp(' \delta u + k^2 = 0');
disp(' u = -uinc on the interior boundary');
disp(' \partial_n u + iku = 0 on the exterior boundary');
%PK = 10; gt_order = 6; k = 7; use_hierarchical = 0; load_the_mesh=0;
PK=4; gt_order = 3; k = 3; use_hierarchical = 1; load_the_mesh=0;
if (use_hierarchical) s = 'hierarchical'; else s = 'classical'; end;
disp(sprintf('using %s P%d FEM with geometric transformations of degree %d',s,PK,gt_order));
if (load_the_mesh),
disp('the mesh is loaded from a file, gt_order ignored');
end;
if load_the_mesh == 0,
% a quadrangular mesh is generated, with a high degree geometric transformation
% number of cells for the regular mesh
Nt=10; Nr=8;
m=gf_mesh('empty',2);
dtheta=2*pi*1/Nt; R=1+9*(0:Nr-1)/(Nr-1);
gt=gf_geotrans(sprintf('GT_PRODUCT(GT_PK(1,%d),GT_PK(1,1))',gt_order));
ddtheta=dtheta/gt_order;
for i=1:Nt;
for j=1:Nr-1;
ti=(i-1)*dtheta:ddtheta:i*dtheta;
X = [R(j)*cos(ti) R(j+1)*cos(ti)];
Y = [R(j)*sin(ti) R(j+1)*sin(ti)];
gf_mesh_set(m,'add convex',gt,[X;Y]);
end;
end;
fem_u=gf_fem(sprintf('FEM_QK(2,%d)',PK));
fem_d=gf_fem(sprintf('FEM_QK(2,%d)',PK));
mfu=gf_mesh_fem(m,1);
mfd=gf_mesh_fem(m,1);
gf_mesh_fem_set(mfu,'fem',fem_u);
gf_mesh_fem_set(mfd,'fem',fem_d);
sIM=sprintf('IM_GAUSS_PARALLELEPIPED(2,%d)',gt_order+2*PK);
mim=gf_mesh_im(m, gf_integ(sIM));
else
% the mesh is loaded
m=gf_mesh('import','gid','../meshes/holed_disc_with_quadratic_2D_triangles.msh');
if (use_hierarchical),
% hierarchical basis improve the condition number
% of the final linear system
fem_u=gf_fem(sprintf('FEM_PK_HIERARCHICAL(2,%d)',PK));
else,
fem_u=gf_fem(sprintf('FEM_PK(2,%d)',PK));
end;
fem_d=gf_fem(sprintf('FEM_PK(2,%d)',PK));
mfu=gf_mesh_fem(m,1);
mfd=gf_mesh_fem(m,1);
gf_mesh_fem_set(mfu,'fem',fem_u);
gf_mesh_fem_set(mfd,'fem',fem_d);
mim=gf_mesh_im(m,gf_integ('IM_TRIANGLE(13)'));
end;
nbdu=gf_mesh_fem_get(mfu,'nbdof');
nbdd=gf_mesh_fem_get(mfd,'nbdof');
% identify the inner and outer boundaries
P=gf_mesh_get(m,'pts'); % get list of mesh points coordinates
pidobj=find(sum(P.^2) < 1*1+1e-6);
pidout=find(sum(P.^2) > 10*10-1e-2);
% build the list of faces from the list of points
fobj=gf_mesh_get(m,'faces from pid',pidobj);
fout=gf_mesh_get(m,'faces from pid',pidout);
gf_mesh_set(m,'boundary',1,fobj);
gf_mesh_set(m,'boundary',2,fout);
% expression of the incoming wave
wave_expr=sprintf('cos(%f*y+.2)+1i*sin(%f*y+.2)',k,k);
Uinc=gf_mesh_fem_get(mfd,'eval',{wave_expr});
% currently the toolbox does not handle complex valued arrays,
% hence we have to treat both real and imaginary part
[Hr,Rr] = gf_asm('dirichlet', 1, mim, mfu, mfd, gf_mesh_fem_get(mfd,'eval',1), ...
real(Uinc));
[Hi,Ri] = gf_asm('dirichlet', 1, mim, mfu, mfd, gf_mesh_fem_get(mfd,'eval',1), ...
imag(Uinc));
[null,udr]=gf_spmat_get(Hr,'dirichlet nullspace', Rr);
[null,udi]=gf_spmat_get(Hi, 'dirichlet nullspace', Ri);
ud = udr + 1i*udi;
Qb2 = gf_asm('boundary qu term', 2, mim, mfu, mfd, ones(1,nbdd));
M = gf_asm('mass matrix',mim, mfu);
L = -gf_asm('laplacian',mim, mfu,mfd,ones(1,nbdd));
% builds the matrix associated to
% (\Delta u + k^2 u) inside the domain, and
% (\partial_n u + ik u) on the exterior boundary
A=L + (k*k) * M + (1i*k)*Qb2;
% eliminate dirichlet conditions and solve the system
RF=null'*(-A*ud(:));
RK=null'*A*null;
U=null*(RK\RF)+ud(:);
Udr=gf_compute(mfu,real(U(:)'),'interpolate on',mfd);
Udi=gf_compute(mfu,imag(U(:)'),'interpolate on',mfd); Ud=Udr+1i*Udi;
%figure(1); gf_plot(mfu,imag(U(:)'),'mesh','on','refine',32,'contour',0); colorbar;
%figure(2); gf_plot(mfd,abs(Ud(:)'),'mesh','on','refine',24,'contour',0.5); colorbar;
% compute the "exact" solution from its developpement
% of bessel functions:
% by \Sum_n c_n H^(1)_n(kr)exp(i n \theta)
N=1000; theta=2*pi*(0:N-1)/N; y=sin(theta);
w = eval(wave_expr);
fw = fft(w); C=fw/N;
S = zeros(size(w)); S(:) = C(1); Nc=20;
for i=2:Nc,
n=i-1;
S = S + C(i)*exp(1i*n*theta) + C(N-(n-1))*exp(-1i*n*theta);
end;
P=gf_mesh_fem_get(mfd,'basic dof nodes');
[T,R]=cart2pol(P(1,:),P(2,:));
Uex=zeros(size(R));
nbes=1;
Uex=besselh(0,nbes,k*R) * C(1)/besselh(0,nbes,k);
for i=2:Nc,
n=i-1;
Uex = Uex + besselh(n,nbes,k*R) * C(i)/besselh(n,nbes,k) .* exp(1i*n*T);
Uex = Uex + besselh(-n,nbes,k*R) * C(N-(n-1))/besselh(-n,nbes,k) .* exp(-1i*n*T);
end;
disp('the error won''t be less than ~1e-2 as long as a first order absorbing boundary condition will be used');
Uex=conj(Uex);
disp(sprintf('rel error ||Uex-U||_inf=%g',max(abs(Ud-Uex))/max(abs(Uex))));
disp(sprintf('rel error ||Uex-U||_L2=%g',...
gf_compute(mfd,Uex-Ud,'L2 norm',mim)/gf_compute(mfd,Uex,'L2 norm',mim)));
disp(sprintf('rel error ||Uex-U||_H1=%g',...
gf_compute(mfd,Uex-Ud,'H1 norm',mim)/gf_compute(mfd,Uex,'H1 norm',mim)));
% adjust the 'refine' parameter to enhance the quality of the picture
gf_plot(mfu,real(U(:)'),'mesh','on','refine',8);
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