Hybrid Methods for the Time Domain Integral Equations of Computational Electromagnetics

“电磁场时间域积分方程组的混合方法”项目提出了一个由时间域积分方程组(TDIEs)的基本数值分析和双重杂交程序组成的多学科程序。对一类新的基于卷积求积的TDIE方法(CQ-TDIe)的稀疏化、求积误差、误差估计和快速求解器等问题进行了基本的数值分析。杂交工作的第一步是将CQ-TDIE与体积有限元相结合。这种混合不仅可以很容易地模拟非均匀和复杂的介质，而且还提出了一种基于完美时间域积分的边界条件的诱人前景。第二种，也是更具投机性的杂交方法，将CQ-TDIe与基于Galerkin的旧技术相结合，目的是控制扩散和耗散。预计由此产生的混合方法具有CQ方法的稳定性和空间精度，但具有与Galerkin方法相关的更高的效率和更小的色散。特别是，这项技术将非常适合于分析某些类别的现代技术，这些技术涉及通过均匀区域(如电磁干扰分析中的出现)或机械移动部件(如许多纳米技术问题中所必需的)进行长传播距离的分析。物理过程的数值模拟降低了商业原型成本，使危险实验的结果能够安全预测，甚至可以通过照亮内部过程而使科学发现难以在实验室中进行检查。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。“电磁场时间域积分方程组的混合方法”开创了模拟电磁场与物质相互作用的两种新方法。虽然电磁分析的计算机化方法已经存在，但研究人员将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(如纳米技术模拟中所需)的问题。除了这些好处，新方法还代表着一项重要的数学进步：它基于一种三十多年来一直被认为不稳定的技术，这项技术直到最近才被付诸实践。因此，研究人员预计，在这项工作的支持下，在电磁问题分析方面取得的进展将很容易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.