An Efficient Galerkin BEM to Compute High Acoustic Eigenfrequencies

Abstract

An efficient numerical method, using integral equations, is developed to calculate precisely the acoustic eigenfrequencies and their associated eigenvectors, located in a given high frequency interval. It is currently known that the real symmetric matrices are well adapted to numerical treatment. However, we show that this is not the case when using integral representations to determine with high accuracy the spectrum of elliptic, and other related operators. Functions are evaluated only in the boundary of the domain, so very fine discretizations may be chosen to obtain high eigenfrequencies. We discuss the stability and convergence of the proposed method. Finally we show some examples.

A boundary integral equation method for mode elimination and vibration confinement in thin plates with clamped points

“…The main goal of this paper is to introduce a novel high-order boundary integral equation method for the numerical solution of (1) in the presence of a finite collection of punctures (1c). High-order methods for computing eigenvalues of the Laplacian and Helmholtz equations in two and three dimensions have been developed with domain decomposition methods [9,17,18], radial basis functions [43], boundary integral equations [6,45,20], the method of particular solution [7,25,33], the Dirichlet to Neumann map [8], and chebfun [19]. The method of fundamental solutions has also been used to compute eigenvalues of the biharmonic equation [40,3].…”

Fast Computation of High‐Frequency Dirichlet Eigenmodes via Spectral Flow of the Interior Neumann‐to‐Dirichlet Map

“…We need some standard results from potential theory [18]. Let S(k) and D(k) be the single-and double-layer boundary integral operators formed by restricting (23) and (24) respectively to the boundary,…”

“…Such methods often have spectral (i.e. super-algebraic) error convergence, although most BIE implementations remain low-order [35,6,24,59]. They can easily reach j = 10 4 , with relative errors as small as 10 −14 [12], and variants have reached j > 10 6 [60,57].…”

Convergence Analysis of a Galerkin Boundary Element Method for the Dirichlet Laplacian Eigenvalue Problem

“…Different to finite element approaches which require a discretization of the computational domain Ω, the use of boundary integral formulations and boundary element methods to solve the eigenvalue problem needs only a discretization of the boundary Γ. For the discretization of the boundary integral eigenvalue problem collocation schemes [4,5,14,16,17,22,27,29] and Galerkin methods [6,7,31,34] are considered. Both methods yield algebraic nonlinear eigenvalue problems where the matrix entries are transcendental functions with respect to the eigenparameter.…”

“…To our knowledge, a rigorous and comprehensive numerical analysis for the discretization of boundary integral operator eigenvalue problems has not be done so far. Only in few works [6,7,31] the issue of the numerical analysis is addressed. In [31] we analyzed a Galerkin discretization of a Newton scheme for the approximation of a boundary integral operator eigenvalue problem and derived error estimates for eigenpairs for simple eigenvalues.…”