Point Jacobi-type preconditioning and parameter tuning for Calderon-preconditioned Burton-Miller method in transmission problems

TL;DR

This paper introduces a novel Calderon-preconditioned Burton-Miller boundary element method for transmission problems, employing point Jacobi preconditioning and parameter tuning to improve system conditioning and solve high-contrast material scattering efficiently.

Contribution

It proposes a combined preconditioning and parameter tuning strategy based on eigenvalue analysis to enhance the Calderon-preconditioned BM-BEM for transmission problems with high material contrast.

Findings

Improved conditioning for high-contrast transmission problems.

Enhanced efficiency in solving scattering problems with composite materials.

Effective eigenvalue clustering achieved through proposed strategies.

Abstract

It was recently demonstrated that the boundary element method based on the Burton-Miller formulation (BM-BEM), widely used for solving exterior problems, can be adapted to solve transmission problems efficiently. This approach utilises Calderon's identities to improve the spectral properties of the underlying integral operator. Consequently, most eigenvalues of the squared BEM coefficient matrix, i.e. the collocation-discretised version of the operator, cluster at a few points in the complex plane. When these clustering points are closely packed, the resulting linear system is well-conditioned and can be solved efficiently using the generalised minimal residual method with only a few iterations. However, when multiple materials with significantly different material constants are involved, some eigenvalues become separated, deteriorating the conditioning. To address this, we propose an enhanced Calderon-preconditioned BM-BEM with two strategies. First, we apply a preconditioning scheme inspired by the point Jacobi method. Second, we tune the Burton-Miller parameters to minimise the condition number of the coefficient matrix. Both strategies leverage a newly derived analytical expression for the eigenvalue clustering points of the relevant operator. Numerical experiments demonstrate that the proposed method, combining both strategies, is particularly effective for solving scattering problems involving composite penetrable materials with high contrast in material properties.