Optimized Domain Decomposition Methods for Wave Propagation in Complex Media

电磁波或弹性波在介质中的传播因介质不连续的界面而失真。 这发生在许多具有重要实际意义的环境中，涉及通信，光学，遥感和地球物理勘探设备。 为了理解波与它们移动所通过的介质的复杂结构之间的这些相互作用，通常涉及多个散射体以及具有不同材料特性的多个层，需要解决波在这种环境中经历的复杂反射和传输。 这反过来又需要大规模的数值模拟。 研究人员开发和分析高性能，高效，准确，快速收敛的算法，这类问题。 他最近的工作与同事们导致了一个有效的计算策略，将窗口绿色的功能内的边界积分方程方法的相互作用的波与无限延伸的接口模拟的发展。 该计算框架能够模拟高频下周期性介质对波的透射和反射。 该项目建立在这些方法的基础上，能够高保真模拟波在薄膜太阳能电池和超颖表面等工程结构中的传播。 研究生参与了这项研究。研究人员开发了一系列算法，主要集中在优化的施瓦茨域分解（DD）方法，结合精心设计的准最优传输算子，并适合于简单而有效的预处理策略。 结合直接和迭代求解器的优点，这类方法已成为解决复杂介质中高频波传播问题的主要竞争者。 这项工作的计算方法是基于边界积分求解器，只要适用，可以产生高阶精度和无数值色散的偏微分方程的解决方案。 该项目利用了研究人员和合作者在无限延伸介质（包括周期性介质）和介电复合介质的边界积分方程处理方面的最新进展，结合DD方法固有的模块化和并行性，以模拟现实的工程结构，如复杂的光子或电子设备。 这项工作影响了社会利益的各个领域，包括通信，遥感，地震学和光学。 该项目的一个重要组成部分是开发周期性超材料的快速，高精度的求解器，并将所得的数值工具用于光子结构和超材料的详细研究和设计。 研究生参与研究，并在高性能科学计算领域接受培训。 该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The propagation of electromagnetic or elastic waves in a medium is distorted by interfaces where the medium is discontinuous. This happens in many settings of great practical importance, involving devices for communications, optics, remote sensing, and geophysical exploration. To understand these interactions between waves and the complex structures of the media through which they move, which often involve multiple scatterers as well as multiple layers with different material properties, requires resolving the complicated reflections and transmissions that waves undergo in such environments. This in turn requires large-scale numerical simulations. The investigator develops and analyzes high-performance, efficient, accurate, and rapidly convergent algorithms for this class of problems. His recent work with colleagues resulted in the development of an efficient computational strategy that incorporates windowed Green's functions within the boundary integral equation approach for the simulation of interaction of waves with infinitely extending interfaces. This computational framework enables simulation of transmission and reflection of waves by periodic media at high frequencies. This project builds on these methods to enable high-fidelity simulations of waves propagating in engineering structures such as thin film solar cells and metasurfaces. Graduate students participate in the research.The investigator develops a family of algorithms that focus mainly on optimized Schwarz domain decomposition (DD) methods, incorporate carefully designed quasi-optimal transmission operators, and are amenable to simple yet effective preconditioning strategies. Combining the merits of direct and iterative solvers, this class of methods has emerged as a leading contender for solution of high-frequency wave propagation in complex media. The computational methodology underlying this work is based on boundary integral solvers that, whenever applicable, can produce solutions to partial differential equations with high-order accuracy and no numerical dispersion. The project leverages recent advances introduced by the investigator and collaborators in the boundary integral equation treatment of infinitely extending media (including periodic media) and dielectric composite media, combined with the modularity and parallelism inherent to DD methods, to enable simulations of realistic engineering structures such as complex photonic or electronic devices. The work affects a variety of areas of societal interest, including communication, remote sensing, seismology, and optics. A major part of the project with wide applications is the development of fast, highly accurate solvers for periodic metamaterials, and use of the resulting numerical tools in detailed investigation and design of photonic structures and metamaterials. Graduate students participate in the research and are trained in the field of high-performance scientific computing. Software packages for solution of boundary integral equations that can be used for teaching and research purposes are made available.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.