Efficient integral equation solvers for large-scale frequency domain electromagnetic scattering problems

这位研究人员建议开发高度并行、高效、准确和快速收敛的算法来评估波浪与几何复杂结构之间的相互作用。将由研究者开发的算法系列将主要集中于(1)基于Dirichlet-to-Neumann算子的近似的通用方法的设计，该方法能够为具有所有类型的边界条件的广泛的一组散射问题产生本质良好的边界积分方程公式，(2)在计算工具箱中实现这些公式，该计算工具箱提供用于散射问题的快速、高阶解算器，该计算工具箱还可以处理由所有主要CAD引擎产生的CAD模型所描述的几何，(3)基于准最佳传输条件的麦克斯韦方程非重叠区域分解方法(DDM)的设计、有限元和边界元实现。所提出的方法包括以下主要元素：(A)依赖于接受商业CAD格式以及三角剖分和甚至点云作为输入的高阶表面表示的高阶积分方程解；(B)基于伪微分的Dirichlet-to-Neumann算子强制逼近的设计，一方面导致对广泛的电磁问题需要少量Krylov子空间迭代的良好积分方程公式，另一方面导致传输条件优化迭代算子的范数，该迭代算子位于求解Maxwell方程的非重叠区域分解方法的核心；和(C)使用等效源，基于FFT的加速算法来实现快速积分求解器。这些算法将作为拟议工作的一部分而开发，对诸如雷达、电子电路、天线、通信设备、光子学等各种应用具有基本意义。电磁波在复杂结构中传播的模拟带来了一系列重大的计算挑战，这些挑战来自于振荡解、复杂几何的低保真表示以及低频和高频区域的病态。研究人员和他的合作者最近的努力导致了一种高效的计算方法的开发和分析，该方法解决了其中的一些困难，并在此提出的扩展将使研究人员能够实现一个雄心勃勃的计划：模拟高保真的真实散射环境。

The investigator proposes to develop highly-parallel, efficient, accurate and rapidly-convergent algorithms for evaluation of the interaction between waves and geometrically complex structures. The family of algorithms to be developed by the investigator will focus mainly on (1) The design of a general methodology based on approximations of Dirichlet-to-Neumann operators capable of producing intrinsically well-conditioned boundary integral equation formulations for a wide suite of scattering problems with all types of boundary conditions, (2) The implementation of these formulations within a computational toolbox that delivers fast, high-order solvers for scattering problems which can also handle geometries described by CAD models produced by all of the major CAD engines, and (3) The design and Finite Element and Boundary Element implementation of non-overlapping Domain Decomposition Methods (DDM) for the solution of Maxwell's equations based on quasi-optimal transmission conditions. The proposed approach consists of the following main elements: (a) High-order integral equation solvers that rely on high-order surface representations that accept as input commercial CAD formats as well as triangulations and even point clouds; (b) Pseudodifferential calculus-based design of coercive approximations of Dirichlet-to-Neumann operators that on one hand leads to well-conditioned integral equation formulations that require small numbers of Krylov-subspace iterations for a wide range of electromagnetic problems and on the other hand leads to transmission conditions that optimize the norm of the iteration operators that lie at the heart of non-overlapping Domain Decomposition Methods for the solution of Maxwell's equations; and (c) Use of equivalent sources, FFT-based acceleration algorithms to achieve fast integral solvers. The algorithms that are to be developed as part of the proposed work are of fundamental significance to diverse applications such as radar, electronic circuits, antennas, communication devices, photonics. The simulation of electromagnetic wave propagation in complex structures gives rise to a host of significant computational challenges that result from oscillatory solutions, low-fidelity representations of complex geometries, and ill-conditioning in the low and high-frequency regimes. The recent efforts of the investigator and his collaborators resulted in the development and analysis of a highly efficient computational methodology which resolved several of these difficulties and whose extension, proposed hereby, will enable the investigator to fulfill an ambitious plan: to simulate with high fidelity realistic scattering environments.