Page 212 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 199

where D, K, and N are evaluated by linearization of the FEM equations around
the solution U=Uo. A is the vector of Lagrange multiplers assuring that the
constraints are satisfied in the eigenvector solution as well. Although FEMLAB
provides an eigensolver only for eigensystems modes in the GUI, femeig
accepts the appropriate arguments to solve the generalized eigensystem (5.14)
for any solution U at any time, or for stationary nonlinear problems and the
parametric solver, for any value of the parameter in the p-list.
For instance,
>> sol2=femeig(’In’,{’D’,D,’K’,K, ‘N’,N), ‘Eigpar’,20);
produces a list of the smallest 20 eigenvalues in magnitude and the associated
eigenvectors (the eigen pairs) for the specified D, K, and N matrices. This is
simpler than expressing a generalized eigenvalue problem as (5.14) in the
appropriate format. Help on eigs gives the syntax as
[V,D] =EIGS (A, B) solves the generalized eigenvalue problem
A*V==B*Vector*D. B must be symmetric (or Hermitian) positive
definite and the same size as A.
EIGS (A, K) and EIGS (A, B, K) return the K largest magnitude
eigenvalues.
K, SIGMA) and EIGS (A, B, K, SIGMA) return K eigenvalues based on
EIGS (A,
S1GMA:‘LM‘ or ISM‘ - Largest or Smallest Magnitude
If SIGMA is a real or complex scalar including 0, EIGS finds the
eigenvalues closest to SIGMA.
Here, D is a the diagonal matrix of eigenvalues, and V are the associated
eigenvectors. Comparison with (5.14) gives the following assignments for
appropriate input to eigs in terms of the block matrices K,N,D produced from
assemble:

B=[ DO J; A=“ K Nt (5.15)
00

so you can produce the above block matrices using
>>A=[K N’; N zeros(size(N, l))];
)
)
>>B= [D zeros (size (N’ ) ; zeros (size (N) zeros (size (N, 1) 1 ;
>>[V,D]=eigs(A,B,’SM’);
Either method (eigs or femeig) produces the list of smallest magnitude
eigenvalues and associated eigenvectors.
0.0000 0.0002 0.0022 0.0057 0.0062 0.0121
0.0200 0.0260 0.0299 0.0417 0.0555 0.0613
0.0713 0.0891 0.0989 0.1009 0.1049 0.1088
0.1108 0.1111