基于曲四面体单元剖分的CN E-H TDFEM方法

计算物理 ›› 2016, Vol. 33 ›› Issue (6): 652-660.

基于曲四面体单元剖分的CN E-H TDFEM方法

叶珍宝¹, 朱剑², 周海京¹

1. 北京应用物理与计算数学研究所, 北京 100094;
2. 南京邮电大学通信与信息工程学院, 南京 210006

收稿日期:2015-09-25 修回日期:2016-03-09 出版日期:2016-11-25 发布日期:2016-11-25
通讯作者: 周海京,E-mail:zhou_haijing@iapcm.ac.cn
作者简介:叶珍宝(1982-),女,博士,助理研究员,主要研究电磁场时域数值计算方法和区域分解方法及其应用,E-mail:ye_zhenbao@iapcm.ac.cn
基金资助:
国家重点基础研究发展计划（2013CB328904）及国家自然科学基金重点项目（61431014）资助

Crank-Nicolson E-H Time-Domain Finite-Element Method Based on Curvilinear Tetrahedral Elements

YE Zhenbao¹, ZHU Jian², ZHOU Haijing¹

1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China;
2. College of Communications and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, 210006, China

Received:2015-09-25 Revised:2016-03-09 Online:2016-11-25 Published:2016-11-25

摘要/Abstract

摘要： 从采用Crank-Nicolson差分格式的基于麦克斯韦旋度方程的E-H时域有限元方法出发，将展开电场和磁场的叠层矢量基函数与曲四面体单元相结合.对金属球谐振腔及介质填充的圆柱形谐振腔的数值模拟表明：相较于规则四面体单元，在剖分单元数目相同的情况下，曲四面体单元离散表面弯曲结构可以获得更高的计算精度.同时，与0.5阶基函数结合曲四面体单元相比，1.0阶基函数与曲四面体单元结合可以用更少的单元数及未知量数目来获得更高的计算精度.

关键词: E-H时域有限元方法, Crank-Nicolson差分, 叠层矢量基函数, 曲四面体单元

Abstract: Based on E-H TDFEM method derived directly from Maxwell's curl equations, Crank-Nicolson difference scheme is implemented for time-partial differential equation to obtain an unconditionally stable algorithm. Curvilinear tetrahedral elements are applied to discretize computational domain and electric and magnetic fields are expanded with same hierarchical vector basis functions. A sphere cavity and a cylindrical cavity partially filled with dielectric rod are simulated. It shows that curvilinear tetrahedral elements can reach higher accuracy with same mesh numbers, compared with tetrahedral elements. Better results can be obtained by curvilinear tetrahedral elements combined with 1.0 order hierarchical basis functions with fewer unknowns than that combined with 0.5 order hierarchical basis functions.

Key words: E-H time-domain finite-element method, Crank-Nicolson scheme, hierarchical vector basis functions, curvilinear tetrahedral elements

中图分类号:

叶珍宝, 朱剑, 周海京. 基于曲四面体单元剖分的CN E-H TDFEM方法[J]. 计算物理, 2016, 33(6): 652-660.

YE Zhenbao, ZHU Jian, ZHOU Haijing. Crank-Nicolson E-H Time-Domain Finite-Element Method Based on Curvilinear Tetrahedral Elements[J]. CHINESE JOURNAL OF COMPUTATIONAL PHYSICS, 2016, 33(6): 652-660.

参考文献

[1] JIN J M. The finite element method in electromagnetics[M]. 3rd ed. New York:Wiley, 2014.
[2] LEE J F. WETD-A finite element time-domain approach for solving Maxwell's equations[J]. IEEE Microwave and Guided Wave Letters, 1994, 4:11-13.
[3] YEE K S. Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[J]. IEEE Trans. Antennas Propagat, 1966, 14:302-307.
[4] KUANG X J, WANG D P, ZHANG L, et al. Finite difference time-domain method based on high order compact scheme[J]. Chinese J Comput Phys, 2014, 31(1):91-95.
[5] GEDNEY S D, NAVSARIWALA U. An unconditionally stable finite element time-domain solution of the vector wave equation[J]. IEEE Microwave and Guided Wave Letters, 1995, 5:332-334.
[6] YE Z B, ZHOU H J. High-order discontinuous Galerkin time-domain finite-element method for three-dimensional cavities[J]. Chinese J Comput Phys, 2015, 32(4):449-454.
[7] ZHOU H J, LIU Y, LI H Y, et al. Computational electromagnetic and applications in numerical simulation of electromagnetic environmental effects and development tendency[J]. Chinese J Comput Phys, 2014, 31(4):379-389.
[8] WU Jo Yu, LEE Robert. The advantages of triangular and tetrahedral edge elements for electromagnetic modelling with the finite element method[J]. IEEE Trans, Microwave Theory Tech, 1997, 45(9):1431-1437.
[9] DONEPUDI K C, KANG G, et al. Higher-order MOM implementation to solve integral equations[C]//IEEE Antennas and Propagation Symposium, 1999, 3:1716-1719.
[10] SUN G, TRUEMAN C W. Unconditionally stable Crank-Nicolson scheme for solving the two-dimensional Maxwell's equations[J]. Electon Lett, 2003, 39:595-597.
[11] DONDERICI Burkay, TEIXEIRA F L. Mixed finite-element time-domain method for transient Maxwell equations in doubly dispersive media[J]. IEEE Trans Microwave Theory Tech, 2008, 56(1):113-120.
[12] CHEN R S, DU L, YE Z B, YANG Y. An efficient algorithm for implementing the Crank-Nicolson scheme in the mixed finite-element time-domain method[J]. IEEE Transaction on Antennas and Propagation, 2009, 57(10):3216-3222.
[13] WEBB J P. Hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements[J]. IEEE Transactions on Antennas and Propagation, 1999, 47(8):1244-1253.
[14] ANDERSEN L S, VOLAKIS J L. Hierarchical tangential vector finite elements for tetrahedral[J]. IEEE Microwave Guided Wave Lett, 1998, 8:127-129.
[15] ZHU Y. Methodologies and algorithms for fast finite-element analysis of electromagnetic devices and systems[D]. PhD Thesis:University of Illinois at Urbana-Champaign, 2002.
[16] 朱剑. 快速算法在电磁场有限元仿真中的应用[D]. 南京:南京理工大学,2010.
[17] CHATTERJEE A, JIN J M, VOLAKIS L. Computation of cavity resonances using edge-based finite elements[J]. IEEE Trans Microwave Theory Tech, 1992, 40(11):2106-2108.
[18] CARPES W P, PICHON Jn L, RAZEK A. Efficient analysis of resonant cavities by finite element method in the time domain[J]. IEE Proc-Microw Antennas Propagation, 2000, 147(1):53-56.