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

研究者建议开发高度并行、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。研究者开发的算法家族将主要集中在(1)基于Dirichlet-to-Neumann算子近似的一般方法设计，该方法能够为具有所有类型边界条件的广泛散射问题生成本质上条件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，该计算工具箱可提供快速，(3)基于准最优传输条件的求解Maxwell方程组的非重叠域分解方法（DDM）的设计和有限元及边界元实现。建议的方法包括以下主要元素：(a)高阶积分方程求解器，它依赖于高阶曲面表示，接受作为输入的商业CAD格式以及三角测量甚至点云；(b)伪微分Dirichlet-to-Neumann运营商的微积分强制近似设计,一方面导致状态良好的积分方程公式,需要少量Krylov-subspace迭代范围广泛的电磁问题,另一方面会导致传输条件,优化迭代的规范运营商的核心非重叠区域分解方法麦克斯韦方程的解决方案;(c)使用等效源，基于fft的加速算法来实现快速积分求解。作为提议工作的一部分，将要开发的算法对雷达、电子电路、天线、通信设备、光子学等各种应用具有根本意义。复杂结构中电磁波传播的模拟带来了一系列重大的计算挑战，这些挑战来自振荡解、复杂几何形状的低保真表示以及低频和高频状态下的不良调节。研究者和他的合作者最近的努力导致了一种高效的计算方法的发展和分析，解决了其中的一些困难，并且在此提出的扩展，将使研究者能够实现一个雄心勃勃的计划：以高保真度模拟现实散射环境。

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.