时域电磁数值方法统一理论的研究

Wireless technologies that use electromagnetic waves have been more and more applied in our work and daily life, such as remote health monitoring, Terahertz medical imaging, wireless power transfer and underground pipe leakage detection. These applications present new challenges to numerical methods that solve electromagentic problems, requiring them to be hightly accurate and highly efficient as well as capable of solving problems of very low to very high frequencies; this is especially true for time-domain methods. In the past three decades, a large number of time-domain numerical methods of different features have been proposed, developed and applied. Such a large number of the methods appear to have been developed from different mathematical backgrounds and to have their respective advantages and disadvantages. As a result, they have become difficult or challenging not only for beginners or practitioners to understand and to use but also for the methods to be hybridized and interlaced. This research intends, based on the initial work by the applicant and his group, to research the theory that unifies all the time-domain numerical methods for electromagnetic problems, to break the theoretical barriers that appear to disconnnect different numerical methods from each other, and finally to establish a full theory and methodology that embodies current and future numerical methods; this research will provide a single analysis platform that includes all numerical time-domain numerical methods, laying a solid theoretical and technological foundation for a new highly effective electromagnetic modeling and simulation technology.

电磁领域中的无线技术在我们现代生活中的应用已越来越广泛，如远程健康监护，太赫兹医疗成像，无线能量传输和地下管道泄漏传感等，这些应用对电磁数值计算方法产生了新的挑战，要求数值方法可以精确、高效的求解从低频到高频的所有电磁现象，尤其是时域方法。在过去三十年中，大量新型各具优势与特色的时域数值方法相继提出并应用，从表面看，它们是基于各种不同的数学理论和方法推导得出的，因而容易造成初学者和应用者理解上的困难与使用上的混乱甚至错误，同时也使得不同方法的混合交叉与取长补短变得十分困难。本项目拟在团队原有研究的基础上，研究将时域电磁数值计算方法归纳、统一的理论，从而全面突破现有数值方法互不相干、彼此独立的现状，最终建立完整的时域计算电磁学统一方法论和今后发展新时域数值方法的统一数学框架，并由此研制出能够涵盖各种时域方法的统一分析平台，形成高精度高效率有坚实理论基础和平台支撑的新兴电磁仿真技术。

本项目的研究目标是将时域电磁数值计算方法归纳在一个统一的数学框架内，从而突破现有数值方法互不相干、彼此独立的现状，建立时域计算电磁学统一方法论对今后发展新的高精度高效率时域数值方法提供坚实的理论基础和平台支撑。基于此，本项目提出了基于空间节点的无网格方法作为涵盖各种数值方法的统一分析平台，完成了从无网格方法推导出其它数值方法的理论框架与实现手段。本项目对该方法进行了进一步的深入研究与开发，确立了在展开函数能满足无散度特性的情况下，无网格方法的数学框架，形成了节点形状函数的条件和数学计算方案，并用之于多物理多尺度问题的计算。与此同时，我们继续对时域有限差分方法进行了研究与应用，推导出了解析解，新的无时间离散空间差分法，新时间反演的理论和源重构以及失效天线阵元定位的方法，应用电磁计算对热点无线传输和特殊天线系统进行了设计与优化，为将来在相关领域中的进一步研究，特别是各种数值方法的交叉奠定了坚实基础。