Optimized Domain Decomposition Methods for Wave Propagation in Complex Media

电磁波或弹性波在介质中的传播被介质不连续的界面所扭曲。这发生在许多具有重大实际意义的环境中，涉及通信、光学、遥感和地球物理勘探设备。要了解波与其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，需要解决波在这样的环境中经历的复杂反射和传输。这反过来需要大规模的数值模拟。研究者开发和分析了这类问题的高性能、高效、准确和快速收敛的算法。他最近与同事的工作导致了一种有效的计算策略的发展，该策略将加窗格林函数结合到边界积分方程法中，用于模拟具有无限延伸的界面的波的相互作用。这种计算框架可以模拟高频周期性介质对波的传输和反射。该项目建立在这些方法的基础上，以实现高保真的波在工程结构中传播的模拟，例如薄膜太阳能电池和超表面。研究生参与了这项研究。研究人员开发了一系列算法，主要关注优化的Schwarz区域分解(DD)方法，结合了精心设计的准最优传输算子，并服从简单而有效的预条件策略。结合了直接求解器和迭代求解器的优点，这类方法已经成为求解复杂介质中高频波传播的主要竞争者。这项工作的计算方法是基于边界积分求解器，只要适用，就可以产生高精度且没有数值色散的偏微分方程组的解。该项目利用研究人员和合作者在无限延伸介质(包括周期介质)和介质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，实现了对复杂光子或电子设备等现实工程结构的模拟。这项工作影响到社会感兴趣的各种领域，包括通信、遥感、地震学和光学。该项目具有广泛应用的一个主要部分是开发快速、高精度的周期超材料求解器，并使用由此产生的数值工具来详细研究和设计光子结构和超材料。研究生参与研究并接受高性能科学计算领域的培训。提供了可用于教学和研究目的的边界积分方程解的软件包。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.