A fast directional BEM for large-scale acoustic problems based on the Burton–Miller formulation

Abstract

In this paper, a highly efficient fast boundary element method (BEM) for solving largescale engineering acoustic problems in a broad frequency range is developed and implemented. The acoustic problems are modeled by the Burton-Miller boundary integral equation (BIE), thus the fictitious frequency issue is completely avoided. The BIE is discretized by using the Nyström method based on the curved quadratic elements, leading to simple numerical implementation (no edge or corner problems) and high accuracy in the BEM analysis. The linear systems are solved iteratively and accelerated by using a newly developed kernel-independent wideband fast directional algorithm (FDA) for fast summation of oscillatory kernels. In addition, the computational efficiency of the FDA is further promoted by exploiting the low-rank features of the translation matrices, resulting in two-to three-fold reduction in the computational time of the multipole-tolocal translations. The high accuracy and nearly linear computational complexity of the present method are clearly demonstrated by typical examples. An acoustic scattering problem with dimensionless wave number kD (where k is the wave number and D is the typical length of the obstacle) up to 1000 and the degrees of freedom up to 4 million is successfully solved within 10 hours on a computer with one core and the memory usage is 24 GB.

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

“…Similar conclusion has been obtained for the CIA; see Theorem 3 in [16]. It turns out that the eigenspace of the moment scheme is the same as the one used in the SS-RI algorithm; see (41), (39) and (40). However the derivation here is directly from Theorem 1 and the interpolation theory; one does not need to refer to the derivation of the SS-RI algorithm in Section 3.2.…”

“…The semi-discretized form of the direct boundary integral equation is H(λ)u(λ) = G(λ)q(λ), where H(λ) and G(λ) are complex square matrices, u and q are vector collections of the nodal displacement and traction components; see, e.g. [39]. The matrix T (λ) of the cooresponding NEP consists of the columns of the matrices H(λ) and G(λ) according to the given boundary conditions.…”

“…In general, the entries of H(λ) and G(λ) are distinct functions of λ whose expressions can not be obtained explicitly. For instance, in the acoustic Nyström BEM in [39], the entries of H(λ) and G(λ) associated with well-separated nodes x i and y j are given by the values of double-and singlelayer integral kernels,…”

“…The other steps can be considered as preprocessing and postprocessing parts for the host software. In this paper, RSRR is implemented into the FEM software Ansys R and our in-house fast BEM code [39], which will be used to solve the large-scale examples in Section 6.…”

Is the Burton–Miller formulation really free of fictitious eigenfrequencies?

“…The pulsating sphere is usually used in the literature as a simple example to show the effect of fictitious eigenfrequencies, see for example Refs. [44][45][46][47][48][49][50][51]. Before the numerical analysis of its fictitious eigenfrequencies using the present boundary element eigensolvers, the sound pressure at a position outside the sphere is also calculated to show the effect of fictitious eigenfrequencies.…”