Computationally Efficient Boundary Element Methods for High-Frequency Helmholtz Problems in Unbounded Domains

Abstract

This chapter presents the application of the boundary element method to high-frequency Helmholtz problems in unbounded domains. Based on a standard combined integral equation approach for sound-hard scattering problems we discuss the discretization, preconditioning and fast evaluation of the involved operators. As engineering problem, the propagation of high-intensity focused ultrasound fields into the human rib cage will be considered. Throughout this chapter we present code snippets using the open-source Python boundary element software BEM++ to demonstrate the implementation.

The Wave-Matching Boundary Integral Equation — An energy approach to Galerkin BEM for acoustic wave propagation problems

“…It has also been shown that different values of the coupling parameter can be chosen on different sections of the obstacle so long as its imaginary part is one-signed [33]; the ideas presented in this paper can equivalently be thought of as applying different values for the coupling parameter to different basis functions. [10,39]. The connection between these approaches and the Burton-Miller formulation is wellestablished, and it will be seen that the BIE proposed herein has strong similarities to some of them too.…”

“…We remark that multiplication by implements a Dirichlet-to-Neumann map, since , and note that expressions like that for appear in the OSRC literature e.g. following (10) in [36].In the case where , becomes imaginary and the wave becomes evanescent, obeying a non-oscillatory exponential taper in the direction . The choice of whether follows the positive or negative imaginary branch of the square root is in some senses arbitrary, but we choose the positive branch because this makes physical sense.…”

“…Taking the surface-normal derivative of (6) gives: (9) It will be assumed that the obstacle has a locally-reacting admittance boundary condition, so and are related by: (10) Here is the value of locally-reacting specific admittance, which for simplicity is considered to be uniform over the obstacle.…”

“…In this formulation the incoming wave is still used for 'testing' purposes, but the discretisation of surface quantities is performed using the basis functions from section 2.2, with related to by the admittance boundary condition (10). The testing functions in (31) will again be the set of incoming basis functions .…”

“…BEM produces full interaction matrices, linking every basis function to every other basis function, leading to computational cost and storage requirements that scale . Matrix compression techniques such as fast-multipole [9] and adaptive-crossapproximation [10] are capable of significantly reducing these requirements, but the scaling of computation cost and storage with frequency is still sufficiently unfavourable so as to preclude full audible-bandwidth simulation for most realistic sized room acoustics problems of interest.…”