Helmholtz-equation eigenvalues and eigenmodes for arbitrary domains

Abstract

An integral equation technique is employed to obtain eigenvalues and eigenmodes of the homogeneous Holmholtz equation for a two- or three-dimensional closed region of arbitrary shape with arbitrary first-order homogeneous boundary conditions. The method is described for general (i.e., nonseparable) geometries, with a discussion of the simplification introduced by having a separable geometry given in an Appendix. A numerical example is given for a nonseparable geometry, i.e., a two-dimensional right triangle of arbitrary enclosed angle with Neumann boundary conditions. Results for the special case of an isosceles right triangle agree very well with a known analytical solution.

Method of Fundamental Solutions for Plate Vibrations in Multiply Connected Domains

“…On the other hand, it was shown independently by Panich [24] and by Brakhage & Werner [25] that the mix potential method for indirect BIE is also feasible. In simply connected domains, there are no spurious eigenvalues if the complex-valued BIE is employed [26]. However, the spurious eigenvalues always appear when the complexvalued BIE is employed to the multiply connected problem [11,27].…”

The Boundary Element Method in Acoustics

“…For example iterative methods such as the secant method are applied to the problem of finding the roots of the equation det(A k ) = 0 in references [81], [27] and [1]. However, this is not a satisfactory method when the matrix A k is large [84].…”

The Boundary Collocation Method With Meshless Concept for Acoustic Eigenanalysis of Two-Dimensional Cavities Using Radial Basis Function

“…To solve acoustic problems by using the complex-valued BEM, the influence coefficient matrix would be complex arithematics [9,10]. Therefore, Tai and Shaw [11] employed only the real-part kernel to solve the eigenvalue problems and to avoid the complex-valued computation. The computation of the real-part kernel method or the MRM [11,12] has some advantages, but it still faces both the singular and hypersingular integrals.…”

“…Therefore, Tai and Shaw [11] employed only the real-part kernel to solve the eigenvalue problems and to avoid the complex-valued computation. The computation of the real-part kernel method or the MRM [11,12] has some advantages, but it still faces both the singular and hypersingular integrals. To avoid the singular and hypersingular integrals, De Mey [13] used imaginary-part kernel to solve the eigenvalue problems.…”