Power iterative multiple reciprocity boundary element method for solving three-dimensional Helmholtz eigenvalue problems

Abstract

The multiple reciprocity boundary element method (MRBEM) has been employed to solve the three-dimensional Helmholtz equation, ▿ 2 gf + k 2 gf = 0. In the present technique, the Helmholtz equation is arranged as ▿ 2 gf + k 0 2 gf + gf/ γ = 0 where k 0 is an estimate of k and λ is equal to ( k 2 − k 0 2) −1. As the term gf/λ is treated as a source, the power iteration technique with Wielandt's spectral shift is used to find the value of λ. The boundary integral equation is formulated with the fundamental solution to ▿ 2 gf + k 0 2 gf + δ i = 0. The domain integral related to the above source is transformed into a series of boundary integrals, with the aid of the higher order fundamental solutions based on the spherical Bessel functions. The eigenvalue k 2 can also be described using only boundary integrals. Test calculations demonstrate that the present technique is efficient for finding k 2 and easier to handle than the conventional determinant search scheme.