Efficient, accurate and rapidly convergent algorithms for solutions of wave propagation problems in configurations complex material and geometrical features

TurcDMS-1008076研究者开发了高效，准确和快速收敛的算法，用于评估电磁场和复杂结构之间的相互作用。具体来说，研究者和他的合作者开发了一系列算法，主要关注(1)在包含几何奇点的域中快速，高阶数值解决波传播问题，以及(2)可穿透电磁周期结构的散射问题，特别强调与光子晶体和负折射率材料（NIM）设计相关的共振问题。研究人员专注于基于积分方程公式的大规模并行计算框架的开发和实现，该框架能够产生具有现实复杂性的软波散射问题的快速和高阶解。该方法包括以下几个主要要素：(a)非光滑域上边界积分方程解奇点的高阶解析；(b)基于伪微分演算的良好条件积分方程的设计和分析，导致广泛的电磁传输问题的少量krylov子空间迭代；(c)使用等效的源，基于fft的加速算法，以及利用新可用的图形处理单元（gpu）计算平台的实现，以显着提高计算时间和能力。作为该项目一部分开发的算法对各种应用具有重要意义，例如电磁干扰和兼容性（电子电路），介电/磁涂层导体和复合材料（光子晶体和负折射率材料）。复杂结构中电磁波传播的模拟产生了许多重要的计算挑战，这些挑战来自于非强制公式、振荡解、几何奇点、共振和高频状态中的病态。研究者和他的合作者最近的努力导致了一种高效计算方法的发展，解决了其中的一些困难，其扩展使实现一个雄心勃勃的计划：在高动态范围内模拟高保真的现实散射环境。

TurcDMS-1008076 The investigator develops efficient, accurate and rapidlyconvergent algorithms for evaluation of the interaction betweenelectromagnetic fields and complex structures. Specifically, theinvestigator and his collaborators develop a family of algorithmsthat focus mainly on (1) Fast, high-order numerical solutions forwave-propagation problems in domains that contain geometricsingularities, and (2) Scattering problems from penetrableelectromagnetic periodic structures, with a particular emphasison resonant problems relevant to the design of photonic crystalsand Negative Index Materials (NIM). The investigatorconcentrates on development and implementation of a massivelyparallel computational framework based on integral equationformulations capable of producing fast and high-order solutionsof wave scattering problems of realistic complexity. Theapproach consists of the following main elements: (a) High-orderresolution of the singularities of the solutions of the boundaryintegral equations in non-smooth domains; (b)Pseudodifferential-calculus-based design and analysis ofwell-conditioned integral equation formulations leading to smallnumbers of Krylov-subspace iterations for a wide range ofelectromagnetic transmission problems; and (c) Use of equivalentsources, FFT-based acceleration algorithms, and implementationsthat take advantage of the newly available Graphic ProcessingUnits (GPUs) computational platforms to dramatically enhancecomputational times and capabilities. The algorithms that are developed as part of this projectare of fundamental significance to diverse applications such aselectromagnetic interference and compatibility (electroniccircuits), dielectric/magnetic coated conductors, and compositemeta-materials (photonic crystals and Negative Index Materials). The simulation of electromagnetic wave propagation in complexstructures gives rise to a host of significant computationalchallenges that result from non-coercive formulations,oscillatory solutions, geometric singularities, resonances, andill-conditioning in the high-frequency regime. The recentefforts of the investigator and his collaborators resulted in thedevelopment of a highly efficient computational methodology thatresolved several of these difficulties and whose extensionenables the fulfillment of an ambitious plan: to simulate withhigh fidelity realistic scattering environments with a highdynamic range.