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.

翻译：近期研究表明，广泛应用于求解外部问题的基于Burton-Miller公式的边界元法（BM-BEM）可有效适配于求解传输问题。该方法利用Calderon恒等式改善基础积分算子的谱特性，使得平方BEM系数矩阵（即算子的配点离散版本）的大部分特征值在复平面上聚集于少数点。当这些聚集点紧密分布时，所得线性系统具有良好的条件数，可通过广义最小残差法以少量迭代高效求解。然而，当涉及材料常数差异显著的多重介质时，部分特征值会发生分离，导致条件数恶化。为此，我们提出一种增强型Calderon预条件BM-BEM，采用两种策略：首先，应用受点Jacobi方法启发的预条件方案；其次，通过调优Burton-Miller参数以最小化系数矩阵的条件数。两种策略均基于新推导的相关算子特征值聚集点解析表达式。数值实验表明，结合两种策略的所提方法能特别有效地求解涉及高材料特性对比的复合可穿透介质的散射问题。