基于间断petrov有限元的Trefftz方法及其在雷达散射截面中的应用

The Trefftz method has been very popular in solving homogeneous Helmholtz equation and Maxwell’s equations with real wave numbers. However, to meet the needs of acoustic and electromagnetic fields, more and more complex problems should be considered. The kind of problems, described by the characters of the complex wave numbers and high frequency, has become a hot and frontier research topic due to its wide applications in acoustic and electromagnetic fields. This project aims to develop a new Trefftz method based on the idea of discontinuous petrov-galerkin method for defining the different trial space and test space. The developed Trefftz method can not only keep the advantages of original Trefftz method in accurately solving homogeneous equations with real wave numbers and domain decomposition method in parallel solving large-scale equations, but also solve the homogeneous problems efficiently with the complex wave numbers and high frequency. Our research topics are given in the following. (1) Develop the new Trefftz method based on discontinuous petrov-galerkin method for efficiently solving homogeneous equations with complex wave numbers. (2) For solving high-frequency problems, the scalable domain decomposition preconditioner corresponding to the developed Trefftz method will be proposed. (3) The developed Trefftz method and its fast solver will be used to numerically simulate a class of typical high-frequency problems ( radar cross section ).

Trefftz方法已在实波数齐次声波方程和电磁场方程组的求解中得到了广泛应用。然而，随着声波和电磁场应用领域提出的问题越来越复杂，如何实现高效数值求解复波数高频问题已成为一个有着广泛工程应用背景的热点前沿课题。本项目希望在保持Trefftz方法高精度求解实波数齐次方程的特点及区域分解法并行求解大规模方程能力的同时，借鉴间断petrov有限元法定义不同的检验空间和测试空间的思想，实现高效求解复波数高频问题。研究内容如下：1. 利用间断petrov有限元法构造不同检验空间和测试空间的思想，改进Trefftz方法用于高精度求解复波数方程；2. 为改进后的Trefftz方法，开发用于求解高频问题的可拓展区域分解算法；3. 将改进后的Trefftz方法及其相应的快速算法应用于一类典型高频问题（雷达散射截面）的数值模拟。

1）在各向异性矩阵为对角矩阵这一般性假设下，首次构造了一组新的平面波基函数，并严格证明了3维各项异性电磁场方程组的平面波逼近估计这一公开问题。工作得到了审稿人的高度评价“the proposed coordinate transform allows to push existing algorithms and error analyses of a bit further the existing isotropic framework”；2）首次提出了求解非齐次高频散射问题的高精度平面波最小二乘方法，理论上严格证明了新方法具有较高的收敛阶；数值结果表明对非齐次散射问题方法几乎没有波数污染；3）将UWVF方法和VTCR方法推广到复波数(非齐次)Helmholtz方程和电磁场方程组；数值结果表明新方法具有较高的精度。文章得到了审稿人的积极评价“（期刊JCAM）Such a study is theoretically signicant and the reported results are new”；4）将UWVF方法推广到复波数各向异性电磁场方程组；数值结果表明新方法具有较高的精度。文章得到了审稿人的积极评价“（期刊AAMM）The paper contains some very interesting results for researchers in numerical analysis. They are particular interest to those studying numerical methods for time-harmonic wave propagation problems”；5）将PWDG方法推广到非齐次各向异性电磁场方程组；数值结果表明新方法具有较高的精度。文章得到了审稿人的积极评价“（期刊NMTMA）The idea is quite innovative and I reckon that the manuscript reflects a non-negligible amount of work carried out by the author”；6）构造了一类求解高频问题的并行区域分解预条件子；数值结果表明所构造的预条件子具有较好的拓展性。文章发表在期刊IJNAM；7）以独立申请人申请发明专利一项。