三维弹性散射问题的正则化积分方程方法
Regularized integral equation methods for elastic scattering problems in three dimensions
摘要：
本文提出了三维空间中封闭和开放表面对弹性波散射的数值模拟新方法。所提出的方法利用新的积分公式，以及有效的高阶奇异积分方法~ \cite{BG18}最近推出的声学情况下的弹性范围的扩展。为了获得配方导致迭代求解器（GMRES）收敛在少量的迭代，我们调查，理论上和计算上，与弹性波卡尔德龙关系的各种运营商的频谱的字符-包括他们的一些可能的组成和组合。特别是，通过依赖于复合算子$NS$的特征值有界远离零和无穷大的事实，新的可解的，低GMRES迭代积分公式的封闭表面的情况下。引入相应的低GMRES迭代方程的开放式表面方程还需要，频谱质量以及精度和效率，使用加权版本的经典积分算子，以匹配奇异的未知密度的边缘。几个数值例子证明了所提出的方法的准确性和效率。
英文摘要：
This paper presents novel methodologies for the numerical simulation of scattering of elastic waves by both closed and open surfaces in three-dimensional space. The proposed approach utilizes new integral formulations as well as an extension to the elastic context of the efficient high-order singular-integration methods~\cite{BG18} introduced recently for the acoustic case. In order to obtain formulations leading to iterative solvers (GMRES) which converge in small numbers of iterations we investigate, theoretically and computationally, the character of the spectra of various operators associated with the elastic-wave Calderón relation---including some of their possible compositions and combinations. In particular, by relying on the fact that the eigenvalues of the composite operator $NS$ are bounded away from zero and infinity, new uniquely-solvable, low-GMRES-iteration integral formulation for the closed-surface case are presented. The introduction of corresponding low-GMRES-iteration equations for the open-surface equations additionally requires, for both spectral quality as well as accuracy and efficiency, use of weighted versions of the classical integral operators to match the singularity of the unknown density at edges. Several numerical examples demonstrate the accuracy and efficiency of the proposed methodology.
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