Eigensolutions of the Helmholtz equation for a multiply connected domain with circular boundaries using the multipole Trefftz method

Abstract

In this paper, 2D eigenproblems with the multiply connected domain are studied by using the multipole Trefftz method. We extend the conventional Trefftz method to the multipole Trefftz method by introducing the multipole expansion. The addition theorem is employed to expand the Trefftz bases to the same polar coordinates centered at one circle, where boundary conditions are specified. Owing to the introduction of the addition theorem, collocation techniques are not required to construct the linear algebraic system. Eigenvalues and eigenvectors can be found at the same time by employing the singular value decomposition (SVD). To deal with the eigenproblems, the present method is free of pollution of spurious eigenvalues. Both the eigenvalues and eigenmodes compare well with those obtained by analytical methods and the BEM as shown in illustrative examples.