Dear users,
1 . I am trying to understand the null-space method for imposing constraints
HU=R
on a system KU=F. I find that the convention used in the documentation is
different under the section for [H,R] =
gf_asm('dirichlet',bnum,mim,mf_u,mf_d,Hd,Rd)
at http://download.gna.org/getfem/doc/getfem_matlab/gfm_50.html and
the example at http://download.gna.org/getfem/doc/getfem_matlab/gfm_13.html.
Specifically,
the matrix N and N' seem to be interchanged.
Could someone please direct me to a reference where I can understand the
theory behind
the null-space technique?
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2. In order to figure out more, I tried a simple problem to solve \frac{d^2
u}{dx^2} = 1 on [0,1]
with boundary conditions u(0) = u(1) =2. However, the solution I obtain is
wrong. The dirichlet conditions
are not imposed correctly. The code is as under
clear all;
gf_workspace('clear');
M = gf_mesh('cartesian',[0:0.025:1]);
MFU = gf_mesh_fem(M,1);
gf_mesh_fem_set(MFU,'fem',gf_fem('FEM_PK(1,1)'));
MIM = gf_mesh_im(M,gf_integ('IM_NC(1,1)'));
nbd = gf_mesh_fem_get(MFU,'nbdof');
A = gf_mesh_fem_get(MFU, 'eval', {-1}) ;
K = gf_asm('laplacian',MIM,MFU,MFU,A) ;
F = gf_asm('volumic source', MIM, MFU, MFU, -A) ;
border = gf_mesh_get(M,'outer faces');
gf_mesh_set(M,'boundary',100,border);
[H,R] = gf_asm('dirichlet',100,MIM,MFU,MFU,ones(1,nbd),2*ones(1,nbd));
[N,U0]=gf_spmat_get(H,'dirichlet_nullspace',R)
KK = N'*K*N;
FF = N'*F;
UU = KK \ FF;
U = (N * UU)' + U0;
x = linspace(0,1,100);
plot(x,0.5*x.^2 - 0.5*x + 2,'b');
hold on;
plot([0:0.025:1],U,'r+');
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Thanks for any help!
Regards
Arvind
PhD Student, Indian Institute of Technology Madras