Solution of Eigenvalue Problems for Open Domains with Boundary Element Methods


Boundary element methods have been applied to eigenvalue problems since their early days of developments. Their success in these problems, however, has been limited so far because of lack of good numerical methods. As a matter of fact, even standard linear eigenfrequency problems in mechanics reduce to non-linear eigenvalue problems with boundary integral equations because the kernel functions used in BEM depend on the frequency non-linearly. In addition, one had so far few choices as solvers of these non-linear eigenvalue problems except for the direct search of zeros of determinants of BEM matrices.

With recent developments of non-linear eigenvalue solvers based on contour integrals, however, one is now able to apply BEM to eigenvalue problems easily. Indeed, with the Sakurai-Sugiura method, one takes a contour in the complex plane in the interior of which one finds non-linear eigenvalues of the given integral equations. These eigenvalues are obtained as standard eigenvalues of certain small matrices computed from the relevant contour integrals. The computation of the integrand includes inversions of BEM matrices which are accelerated with fast BEMs such as fast multipole methods.

This approach is particularly interesting in eigenfrequency problems for open domains such as infinite wave guides. Although eigenfrequencies are usually related to closed domains, open domains do have eigenfrequencies which are complex valued. These complex eigenfrequencies are closely related to the physics of the problem. For example, they correspond to the peaks of the scattering amplitude vs frequency curves in scattering problems and their imaginary parts are related to the sharpness of these peaks. The computation of these eigenfrequencies is not very easy with domain type methods such as FDM or FEM because the corresponding eigenfunctions blow up in the far fields and a naive domain truncation may seriously deteriorate the quality of the numerical results. This is, however, not the case with BEM in which the behaviour of the solution is built in the fundamental solution.

However, care has to be taken in applying this approach because one may otherwise pick up spurious eigenvalues called fictitious eigenvalues. Fictitious eigenvalues have been recognised as a source of inaccuracy in BEM solutions for time harmonic wave problems and many "resonance free" boundary integral equations have been proposed. Although these equations are without real valued fictitious eigenvalues, they do have complex fictitious eigenvalues. It is very important to make distinction between true and fictitious eigenvalues when the true ones are complex valued. Fortunately, however, we can modify BEMs slightly to make this distinction simple.

The planned talk will explain issues mentioned above with illustrative numerical examples. Also, it will show that the distribution of eigenvalues of integral operators gives insights into the accuracy of frequency domain BEMs as well as the stability of time domain BEMs.