发表机构
Universidad Pública de Navarra; New Jersey Institute of Technology（纳瓦拉公立大学; 新泽理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对复杂介质二维时谐弹性透射问题，构建边界积分方程框架，通过重写纳维牵引力算子构造预条件子，大幅减少GMRES迭代次数，提升求解效率。
AI 中文摘要
我们提出了一种用于二维时谐弹性透射问题的边界积分方程（BIE）框架。利用亥姆霍兹分解，弹性动力学位移场可表示为标量压力势和剪切势，因此仅需四个经典亥姆霍兹边界积分算子。关键步骤是在界面的局部法向-切向坐标系中重写纳维牵引力算子，其中二阶法向导数被替换为切向导数和曲率项。这形成了依赖于耦合常数α≠0的单参数族间接组合场积分方程，我们证明其具有唯一可解性。然而，所得算子的主部分存在缺陷，离散系统严重病态。利用亥姆霍兹边界积分算子（BIO）的伪微分演算，我们构造了显式左右解析预条件子，并证明预条件后的算子是单位算子的紧扰动。基于高阶奈斯特龙离散化的数值实验证实，该预条件子可使谱聚类：GMRES迭代次数与离散化无关，且减少一个数量级或更多，即使在高频 regime 和非凸散射体情况下亦然。
英文摘要
We present a boundary integral equation (BIE) framework for two-dimensional time-harmonic elastic transmission problems. Using the Helmholtz decomposition, the elastodynamic displacement fields are expressed in terms of scalar pressure and shear potentials, so that only the four classical Helmholtz boundary integral operators are needed. The key step is a rewriting of the Navier traction operator in the local normal--tangential frame of the interface, in which second-order normal derivatives are eliminated in favour of tangential derivatives and curvature terms. This leads to a one-parameter family of indirect combined field integral equations, depending on a coupling constant $α\ne 0$, which we prove to be uniquely solvable. The principal part of the resulting operator is, however, defective, and the discretized systems are severely ill-conditioned. Using the pseudodifferential calculus of the Helmholtz BIOs, we construct explicit left and right analytical preconditioners and prove that the preconditioned operator is a compact perturbation of the identity. Numerical experiments based on high-order Nyström discretizations confirm that the preconditioner clusters the spectrum: the number of GMRES iterations becomes independent of the discretization and is reduced by one order of magnitude or more, also in the high-frequency regime and for non-convex scatterers.
Comments37 pages, 7 figures
