This paper proposes well-conditioned boundary integral equations based on the Burton-Miller method for solving transmission problems. The Burton-Miller method is widely accepted as a highly accurate numerical method based on the boundary integral equation for solving exterior wave problems. While this method can also be applied to solve the transmission problems, a straightforward formulation may yield ill-conditioned integral equations. Consequently, the resulting linear algebraic equations may involve a coefficient matrix with a huge condition number, leading to poor convergence of Krylov-based linear solvers. To address this challenge, our study enhances Burton-Miller-type boundary integral equations tailored for transmission problems by exploiting the Calderon formula. In cases where a single material exists in an unbounded host medium, we demonstrate the formulation of the boundary integral equation such that the underlying integral operator ${\cal A}$ is spectrally well-conditioned. Specifically, ${\cal A}$ can be designed in a simple manner that ensures ${\cal A}^2$ is bounded and has only a single eigenvalue accumulation point. Furthermore, we extend our analysis to the multi-material case, proving that the square of the proposed operator has only a few eigenvalues except for a compact perturbation. Through numerical examples of several benchmark problems, we illustrate that our formulation reduces the iteration number required by iterative linear solvers, even in the presence of material junction points; locations where three or more sub-domains meet on the boundary.

翻译：本文提出基于Burton-Miller方法求解传输问题的良态边界积分方程。Burton-Miller方法被广泛认为是一种基于边界积分方程求解外部波动问题的高精度数值方法。虽然该方法同样适用于求解传输问题，但直接构造的公式可能会产生病态积分方程。因此，所得线性代数方程组系数矩阵的条件数极大，导致基于Krylov子空间的线性求解器收敛缓慢。为应对这一挑战，本研究通过利用Calderón公式，改进了针对传输问题的Burton-Miller型边界积分方程。在无界主介质中存在单一材料的情形下，我们展示了如何构建边界积分方程，使得底层积分算子${\cal A}$在谱意义下良态。具体而言，可通过简单方式设计${\cal A}$，确保${\cal A}^2$有界且仅有一个特征值聚点。此外，我们将分析推广至多材料情形，证明所提出算子平方除紧扰动外仅有少量特征值。通过若干基准问题的数值算例，我们表明即使在边界存在材料交界点（即三个或更多子域相交的位置）时，该公式仍能降低迭代线性求解器所需的迭代次数。