Hierarchical and Calderon Regularization of Time Domain Integral Equations for Electromagnetics

基于时间推进(MOT)的时域积分方程组(TDIe)解算器为分析大型复杂结构的瞬变电磁相互作用提供了一条有效的途径。不幸的是，这些解算器在用于分析几何复杂和多尺度结构上的低频到中频电磁暂态时，往往会遇到时间(低频)和空间(密集网格)击穿现象，从而阻碍了它们在许多重要的科学和工程问题中的应用。该项目寻求开发新的平面波时间域加速、正则化MOT-TDIe解算器，该解算器不受时间和空间击穿现象的影响。所提出的技术通过分别利用分层基函数和新的时域Calderon恒等式来消除MOT-TDIe解算器中的时间和空间击穿现象。此外，通过利用Calderon正则化的频谱特性，它们保证了MOT-TDIe求解器的低频稳定性。由这项工作产生的MOT-TDIe求解器将允许对几何复杂和混合尺度结构上的低到中频电磁瞬变进行快速分析。该项目的教育目标是开发一套全面的教材，支持涵盖快速MOT-TDIe技术的所有方面的课程，并将其用于教学和外展。开发快速和高阶准确的MOT-TDIe解算器，稳健并无缝地应用于空间和时间尺度，将允许分析广泛的科学和现实世界的电磁工程问题，这些问题是使用当今方法难以解决的。由于其网格健壮的特性，这项工作产生的电磁模拟器将允许光学科学家严格分析非线性和无序超材料和纳米结构中的瞬变效应，而不需要求助于均匀化近似，从而为控制和引导光纤和芯片上的流动和电磁波和光创造了一种新的途径。此外，它们将允许电子工程师设计数字集成电路和射频无线传感器，而无需诉诸特别的空间分解方法，从而大大节省开发时间和成本。最后，它们将允许汽车和航空航天工程师适当地保护他们的设计免受不必要的电磁干扰，从而通过减少远程产生的系统中断的威胁来提高安全性并影响国家安全。

Marching on in time (MOT)-based time domain integral equation (TDIE) solvers provide an appealing avenue for analyzing transient electromagnetic interactions with large and complex structures. Unfortunately, these solvers often suffer from temporal (low-frequency) and spatial (dense-mesh) breakdown phenomena when applied to the analysis of low- to medium-frequency electromagnetic transients on geometrically intricate and multiscale structures, thereby preventing their application to many important scientific and engineering problems. This project seeks the development of new plane wave time domain-accelerated, regularized MOT-TDIE solvers that are immune to temporal and spatial breakdown phenomena. The proposed techniques eliminate temporal and spatial breakdown phenomena in MOT-TDIE solvers by leveraging hierarchical basis functions and new time domain Calderon identities, respectively. In addition, they guaranty low-frequency stability in MOT-TDIE solvers by exploiting the spectral properties of the Calderon regularizer. MOT-TDIE solvers resulting from this effort will permit the fast analysis of low- to medium-frequency electromagnetic transients on geometrically intricate and mixed-scale structures. The educational objective of this project is to develop a comprehensive set of educational materials supporting a course covering all aspects of fast MOT-TDIE technology and to use them in teaching and outreach.The development of fast and higher-order accurate MOT-TDIE solvers that robustly and seamlessly apply across spatial and temporal scales will permit the analysis of a wide class of scientific and real-world electromagnetic engineering problems that are intractable using present day methods. Given their grid-robust nature, the electromagnetic simulators resulting from this effort will permit optical scientists to rigorously analyze transient effects in nonlinear and disordered metamaterials and nanostructures without resorting to homogenization approximations, thereby creating a new avenue for controlling and directing the flow and electromagnetic waves and light in fibers and on chips. In addition, they will allow electronic engineers to design digital integrated circuits and RF wireless sensors without resorting to ad-hoc spatial decomposition methods, thereby resulting in significant savings in development time and costs. And finally, they will permit automotive and aerospace engineers to appropriately protect their designs from unwanted electromagnetic interference, thereby improving safety and impacting national security by reducing the threat of remotely generated system upsets.