Hybrid Methods for the Time Domain Integral Equations of Computational Electromagnetics

“电磁学时域积分方程的混合方法”项目提出了一个多学科的方案，包括时域积分方程（TDIEs）的基本数值分析和混合的双重方案。 基本的数值分析致力于稀疏化，正交误差，误差估计和快速求解器的一个新的和令人兴奋的一类TDIE方法的基础上卷积正交（CQ-TDIE）的问题。杂交工作的第一个方面是将CQ-TDIE与体积有限元法相结合。 这种混合不仅可以方便地模拟非均匀和复杂的介质，但它也提出了一个完美的时域积分为基础的边界条件诱人的前景。第二，更投机，杂交方法结合CQ TDIE与旧的基于Galerkin的技术，目的是控制分散和耗散。 可以预期，由此产生的混合方法具有CQ的稳定性和空间精度，但具有更高的效率和减少的色散与Galerkin方法。 特别是，该技术将非常适合于分析某些类别的现代技术，包括通过均匀区域的长传播距离（如电磁干扰分析中出现的）或机械运动部件物理过程的数值模拟降低了商业原型成本，能够安全预测危险实验的结果，甚至可以通过阐明难以在实验室中检查的内部过程来做出科学发现。 在个人通信技术占主导地位的时代，电磁现象的模拟变得尤为关键。“电磁场时域积分方程的混合方法”为电磁场与物质相互作用的模拟创造了两种新的方法。 虽然电磁分析的计算机化方法已经存在，但研究人员创建的方法允许更准确和有效地模拟涉及生物组织，长传播距离（如通信模拟中发生的）和机械移动部件（如纳米技术模拟所需）的问题。 除了这些好处之外，新方法还代表了一个重要的数学进步：它基于一种三十多年来一直被认为不稳定的技术，直到最近才变得实用。 因此，研究人员预计，在这项工作的主持下，电磁问题的分析所取得的进展将很容易地转化为其他科学领域，包括声学，固体力学和流体力学。

The project "Hybrid methods for the time domain integral equations of electromagnetics" advances a multidisciplinary program consisting of basic numerical analysis of time domain integral equations (TDIEs) and a two-fold program of hybridization. The basic numerical analysis is devoted to issues of sparsification, quadrature error, error estimates and fast solvers for a new and exciting class of TDIE methods based on convolution quadrature (CQ-TDIE). The first prong of the hybridization work concerns combining CQ-TDIE with the volume finite element method. Not only would this hybridization allow for the easy simulation of inhomogeneous and complex media, but it also raises the tantalizing prospect of a perfect time domain integral based boundary condition. The second, and more speculative, hybridization approach combines CQ-TDIE with older Galerkin-based techniques with the aim of controlling dispersion and dissipation. It is anticipated that the resulting hybrid methods have the stability and spatial accuracy of CQ, but with the greater efficiency and reduced dispersion associated with Galerkin approaches. In particular, the technique would be ideally suited for the analysis of certain classes of modern technology involving long propagation distances through homogeneous regions (as occur in electromagnetic interference analysis) or mechanically moving parts (as necessary in many nanotechnology problems).Numerical simulation of physical processes reduces business prototyping costs, enables the safe prediction of the outcome of dangerous experiments, and can even make scientific discoveries by illuminating internal processes difficult to examine in the laboratory. In an era dominated by personal communication technology, the simulation of electromagnetic phenomena becomes especially crucial. "Hybrid methods for the time domain integral equations of electromagnetics" creates two new methods for the simulation of the interaction between electromagnetic fields and matter. While computerized approaches to electromagnetic analysis already exist, the methods to be created by the investigators allow for more accurate and efficient simulation of problems involving biological tissue, long propagation distances (such as occur in communications simulations), and mechanically moving parts (as needed in nanotechnology simulations). In addition to these benefits, the new method represents an important mathematical advance: it is based on a technique thought unstable for over three decades, which has only recently been made practical. The investigators therefore expect that the advances made in the analysis of electromagnetic problems under the auspices of this work will be easily translated to other scientific areas including acoustics, solid mechanics, and fluid mechanics.