About the Burton-Miller factor in the low frequency region

TL;DR

This work analyzes the Burton-Miller coupling for exterior Helmholtz boundary element methods, focusing on stability in the low-frequency regime where critical frequencies cluster sparsely. It derives the sphere-based eigenstructure of the BIE/dBIE operators to explain low-frequency instabilities when using $\eta=\frac{i}{k}$ and compares alternative $\eta$ choices (notably the Duhamel96 and BruKun01 factors) through numerical experiments on hard and soft spheres. A cheap, one-step Modified Richardson iteration applied to the BM solution is proposed to improve accuracy for sound-hard problems at low $k$, effectively bridging BM and BIE performance in that regime. The results show that while $\eta=\frac{i}{k}$ is robust at higher frequencies, size- and frequency-dependent choices (and the iterative post-processing) can substantially reduce conditioning and error in the low-frequency region, with the Duhamel96 factor often offering the best practical compromise; for sound-soft problems, relying on the BIE alone remains preferable due to dBIE instability as $k\to0$.

Abstract

The Burton-Miller method is a widely used approach in acoustics to enhance the stability of the boundary element method for exterior Helmholtz problems at so-called critical frequencies. This method depends on a coupling parameter $η$ and it can be shown that as long as $η$ has an imaginary part different from 0, the boundary integral formulation for the Helmholtz equation has a unique solution at all frequencies. A popular choice for this parameter is $η= \frac{\mathrm{i}}{k}$, where $k$ is the wavenumber. It can be shown that this choice is quasi optimal, at least in the high frequency limit. However, especially in the low frequency region, where the critical frequencies are still sparsely distributed, different choices for this factor result in a smaller condition number and a smaller error of the solution. In this work, alternative choices for this factor are compared based on numerical experiments. Additionally, a way to enhance the Burton-Miller solution with $η= \frac{\mathrm{i}}{k}$ for a sound hard scatterer in the low frequency region by an additional step of a modified Richardson iteration is introduced.

Figures (9)

• Figure 1: Relative errors for the problem of plane wave scattering from a) a sound hard and b) a sound soft sphere with and without Burton-Miller. The areas depict the range between maximum and minimum relative error at the collocation nodes for $N = 1280$ constant elements. The lines depict the mean relative error.
• Figure 2: Discretization of the unit sphere with a) $N = 1280$ and b) $N = 5120$ almost regular triangles.
• Figure 3: Condition numbers for different Burton-Miller coupling factors $\eta = 0,\frac{{\mathrm{i}}}{k}, \frac{{\mathrm{i}}}{3}, {\mathrm{i}}$ and $\eta$ chosen according to Duhamel96. For the discretization of the unit sphere a) N = 1280, b) N = 5120 triangular elements were used.
• Figure 4: Condition numbers of the linear system for N = 1280 at a) 170.656 Hz, b) 200 Hz, c) 244.083 Hz and d) 343.12 Hz as a function of the Burton-Miller factor in the interval $[-1,1]$. The (blue) dashed curves depict the condition number based on the 2-norm, the (green) continuous curves the condition number based on the Inf-norm. The dotted lines represent $\eta = \frac{{\mathrm{i}}}{k}$.
• Figure 5: a) Relative errors of the BEM solutions compared to the analytic solution on the sound hard sphere for the different Burton-Miller factors (lines) and the relative errors after a single modified Richardson iteration step (symbols) with $w = 2$ and $w = 1$. b) Residuum generated by inserting the Burton-Miller solutions into the BIE. The number of elements used in the mesh is $N = 1280$.