Re: [Getfem-users] Badly Scaled Bilaplacian

[Yves Renard]

Dear Oladayo Eluyefa,

Your stiffness matrix is ill-conditionned because you use C0 finite elements 
instead of hermite elements. For this problem you do have to use hermite 
element such as argyris one (FEM_ARGYRIS) but for triangles or FVS element 
(FEM_QUADC1_COMPOSITE) on quadrilaterals (but with a specific integration 
method). Additionally, of course, the stiffness matrix will be singular unless 
a Dirichlet condition is prescribed.

Note also that the Kirchhoff-love term is better conditionned that the 
bi-laplacian term and equivalent but with different boundary conditions.

Yves. 

----- Original Message -----
From: "OLADAYO ELUYEFA (RIT Student)" <address@hidden>
To: address@hidden
Sent: Tuesday, March 17, 2015 6:17:22 PM
Subject: [Getfem-users] Badly Scaled Bilaplacian

I am working on a 4th order 2D Inverse Problem using the matlab OO interface in 
getfem++ . As a result, I calculate the bilaplacian, but I keep getting a badly 
scaled matrix with a close to 0 condition number. Below is the code that 
produces this. Is there a way to correct this. 

Thank you 

Code: 

%% Note equation is -(au'')''=f 
N = 20; 
h = 1/(N+1); 


m = gf_mesh('cartesian', 0:h:1, 0:h:1); 
mf = gf_mesh_fem(m,1); 
mfd = gf_mesh_fem(m,1); 
[xs, ys] = meshgrid(0:1*h:1, 0:1*h:1); 


gf_mesh_fem_set(mf, 'fem', gf_fem('FEM_QK(2,4)')); 
gf_mesh_fem_set(mfd, 'fem', gf_fem('FEM_QK(2,4)')); 

mim = gf_mesh_im(m, gf_integ('IM_GAUSS_PARALLELEPIPED(2,10)')); 

% Boundary conditions: 

border = gf_mesh_get(m, 'outer_faces'); 
gf_mesh_set(m, 'boundary', 1, border); 

% Degrees of freedom 

dof = gf_mesh_fem_get(mf, 'nbdof'); 
dofa = gf_mesh_fem_get(mfd, 'nbdof'); 

u = '256.*(x.^2).*(y.^2).*((1-x).^2).*(1-y).^2'; 
Us = gf_mesh_fem_get(mfd, 'eval', {u}); 

a = '1+0.25.*sin(pi.*x).*sin(pi.*y)'; 
avec = gf_mesh_fem_get(mfd, 'eval', {a}); 
a0 = gf_mesh_fem_get(mfd, 'eval', {1.5})'; 

K = @(a) gf_asm('bilaplacian', mim, mf, mfd, a); 
K1 = gf_asm('laplacian', mim, mfd, mfd, ones(dofa,1)); 

L = @(u) gf_asm('volumic',... 
['u=data(#2);', ... 
't=comp(Hess(#2).Hess(#2).Base(#1))(:,i,i,:,i,i,:);',... 
'M(#2,#1)+=t(j,:,:).u(j)'],... 
mim, ... 
mfd, ... 
mf, ... 
u); 

M = gf_asm('mass matrix', mim, mf); 
Ma = gf_asm('mass matrix', mim, mfd); 



Oladayo Eluyefa, 

Rochester Institute of Technology '15, 
B.S., Computational Mathematics, 
M.S., Applied and Computational Mathematics. 

( address@hidden , address@hidden ) 
(585)-287-1955 



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