Absolutely Stable Time Domain Integral Equation Methods in Computational Electromagnetism

尽管计算电磁学（CEM）取得了几十年的进步，但一些问题仍然没有有效和准确的解决方案。在微机电系统分析、电磁兼容性和超宽带天线设计中出现的问题涉及复杂的几何结构、小结构和跨均匀区域的大传播距离。具有这些特征的问题的解（原则上）是通过时域积分方程（TDIE）技术最有效地计算出来的。遗憾的是，虽然TDIE已经研究了三十多年，但仍然没有一种用于CEM的TDIE方案能够在保证稳定性的同时有效地模拟曲线几何。因此，目前的技术要么是低阶收敛，要么是不可靠的稳定性。根据该提案创建的数学误差分析，工作估计和计算机程序将展示一种基于卷积正交的新方法，该方法有望提供超线性精度和无条件稳定性。电磁理论支配着光、无线电波和微波的行为。了解电磁波的行为对国家安全（在雷达和监视系统的设计中并保护它们免受干扰）、能源系统（在电力的收集和分配中）和经济（在手机和电视广播中）至关重要。不幸的是，随着电磁系统变得越来越复杂和强大，它们变得越来越难以模拟和设计。例如，现代屏蔽设备免受电磁干扰的问题涉及宽带、长距离传播和小特征。虽然几十年来人们都知道，使用时域积分方程公式可以减轻这种困难的组合，但没有这样的公式可以准确地模拟任意几何而不产生具有指数级误差的解。本文通过引入一种新的求解时间依赖性的建模方法，产生了这样一种方案。

Despite decades of advances in computational electromagnetics (CEM), some problems still elude efficient and accurate solutions. Problems occurring in the analysis of microelectromechanial systems, electromagnetic compatibility, and ultra wide band antenna design involve complex geometry, small structures, and large propagation distances across homogeneous regions. Solutions to problems with these characteristics are most efficiently computed (in principle) by time-domain integral equation (TDIE) techniques. Unfortunately, while TDIEs have been researched for more than thirty years, there is still no TDIE scheme for CEM that can efficiently model curvilinear geometry while ensuring stability. Thus, current techniques are either of low order of convergence or unreliable stability. Mathematical error analysis, work estimates and computer programs created under this proposal will demonstrate a new method, based on convolution quadrature, that promises to deliver both superlinear accuracy and unconditional stability.Electromagnetic theory governs the behavior of light, radio waves, and microwaves. Understanding the behavior of electromagnetic waves is of the utmost concern to national security (in the design of radar and surveillance systems and protecting them from interference), energy systems (in the collection and distribution of electric power), and the economy (in cell phones and television broadcasts). Unfortunately, as electromagnetic systems become more complicated and powerful, they become harder to simulate and therefore design. For instance, modern problems in shielding devices from electromagnetic interference involve broad bandwidths, long propagation distances, and small features. While it has been known for decades that this combination of difficulties can be mitigated using time domain integral equation formulations, no such formulation has been produced that can accurately model arbitrary geometry without producing a solution with exponentially increasing error. This proposal produces such a scheme by introducing a new method for modeling the time dependence of the solution.