Frequency extraction for BEM-matrices arising from the 3D scalar Helmholtz equation

TL;DR

This paper introduces a data-sparse representation of BEM matrices for the 3D Helmholtz equation that efficiently captures frequency dependence using phase extraction, enabling rapid sampling across frequencies for complex geometries.

Contribution

It presents a novel frequency-dependent BEM matrix representation based on phase extraction and adaptive approximation techniques, improving efficiency for complex 3D scattering problems.

Findings

The representation accurately captures frequency dependence with few samples.

It enables fast matrix sampling at arbitrary frequencies.

The method is effective for complex 3D geometries.

Abstract

The discretisation of boundary integral equations for the scalar Helmholtz equation leads to large dense linear systems. Efficient boundary element methods (BEM), such as the fast multipole method (FMM) and $\Hmat$ based methods, focus on structured low-rank approximations of subblocks in these systems. It is known that the ranks of these subblocks increase linearly with the wavenumber. We explore a data-sparse representation of BEM-matrices valid for a range of frequencies, based on extracting the known phase of the Green's function. Algebraically, this leads to a Hadamard product of a frequency matrix with an $\Hmat$. We show that the frequency dependency of this $\Hmat$ can be determined using a small number of frequency samples, even for geometrically complex three-dimensional scattering obstacles. We describe an efficient construction of the representation by combining adaptive cross approximation with adaptive rational approximation in the continuous frequency dimension. We show that our data-sparse representation allows to efficiently sample the full BEM-matrix at any given frequency, and as such it may be useful as part of an efficient sweeping routine.