Absolutely Stable Time Domain Integral Equation Methods in Computational Electromagnetism

尽管在计算电磁学（CEM）的几十年的进步，一些问题仍然逃避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形状、小结构和跨越均匀区域的大传播距离。具有这些特征的问题的解决方案是最有效地计算（原则上）的时域积分方程（TDIE）技术。不幸的是，虽然TDIE已经研究了三十多年，但仍然没有用于CEM的TDIE方案，其可以有效地建模曲线几何形状，同时确保稳定性。因此，目前的技术要么是低阶收敛或不可靠的稳定性。 数学误差分析、工作估计和计算机程序将证明一种基于卷积求积的新方法，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（设计雷达和监视系统并保护它们免受干扰），能源系统（收集和分配电力）和经济（手机和电视广播）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，因此也越来越难以设计。 例如，屏蔽设备免受电磁干扰的现代问题涉及宽带宽、长传播距离和小特征。 虽然几十年来已经知道，可以使用时域积分方程公式来减轻这种困难的组合，但是还没有产生这样的公式，其可以精确地对任意几何形状进行建模而不产生具有指数增加的误差的解。 该建议通过引入一种新的方法来模拟解决方案的时间依赖性，从而产生这样一个方案。

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.