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1. 清华大学电子工程系, 北京 100084;
2. 清华信息科学与技术国家实验室(筹), 北京 100084

Error analysis of equivalence principle algorithm on different equivalence surfaces
DANG Xunwang1, LI Maokun1,2 , YANG Fan1,2, XU Shenheng1,2
1. Department of Electronic Engineering, Tsinghua University, Beijing 100084, China;
2. Tsinghua National Laboratory for Information Science and Technology(TNist), Beijing 100084, China
Abstract:Equivalence principle algorithm (EPA) is a domain decomposition algorithm based on the Huygens' equivalence principle to solve large scale scattering problems. The errors of the equivalence principle operator (EPO) in EPA using cubic, spherical and smooth cubic equivalence surfaces were analyzed to find the source of errors. The sources of the error were investigated and a simple scheme to improve its accuracy was developed. From the numerical examples, it shows that the error of equivalent magnetic current is mainly concentrated on geometrical discontinuity of the equivalence surface, and affected by both the discontinuity of the normal vector on the equivalence surface and the choice of basis functions. Therefore it is suggested to choose smooth equivalence surfaces if possible rather than to use cubic ones directly.
Key words: Huygens equivalence principle     domain decomposition method     error analysis     computational electromagnetics     RWG basis function

1 算法简介 1.1 等效原理简介与符号说明

EPA的基本思路基于等效原理,并可以用电磁场的表面积分形式表达.自由空间中的格林函数为

1.2 等效原理算法简介

 Minc,Jinc—等效面上的等效入射磁流和电流;Msca,Jsca—等效面上的等效散射磁流与电流. 图 1 EPA示意图Fig. 1 An illustration of EPA

1.3 矩量法求解

Minc和Jinc也可以用基函数组{fi1}展开,得到二组向量mcinc和jcinc向量,如式(9)所示.其中脚标c表示对应的分量值是入射场用基函数组的展开系数.

mhincc和mincp以及jcinc和jincp可以通过{fi1}的Gram矩阵[5]U1进行转换,以mcinc和mpinc为例,转换过程如式(10)、式(11)所示.

2 计算结果

 图 2 PEC球的后向RCSFig. 2 Backward RCS of PEC sphere

 图 3 不同算法下不同尺寸立方体等效面的均方根误差Fig. 3 RMSE of cubic equivalence surfaces with different sizes under different methods

 图例 对应的计算方法 方法1 式(16) 方法2 式(17) 方法3 式(18)
 图 4 立方体等效面误差分布Fig. 4 Distribution of error on a cubic equivalence surface

 图 5 不同算法下不同尺寸球形等效面的均方根误差Fig. 5 RMSE of spherical equivalence surface with different sizes under different methods
 图 6 一条棱边上法向向量不连续的示意图Fig. 6 Illustration of discontinuity of normal vector on an edge

 图 7 球形等效面误差分布Fig. 7 Distribution of error on a spherical equivalence surface

 图 8 平滑立方体等效面的误差分布Fig. 8 Distribution of error on a smooth cubic equivalence surface
 图 9 不同算法下不同尺寸平滑立方体等效面的均方根误差Fig. 9 RMSE of a smooth cubic equivalence surface with different sizes under different methods

 图 10 由不同等效面计算RCS的均方根误差Fig. 10 RMSE of RCS calculated from different equivalence surfaces
3 结论

1) 在相同的网格尺寸下,等效面尺寸越大,表面等效磁流的计算误差越小.

2) 等效面的棱边会增加表面等效磁流的误差.对立方体等效面适当的平滑处理可以减少误差,但难以达到球形等效面的误差水平.

3) n×RWG函数不是散度共形函数,不适合用作对等效磁流的展开.

4) 等效面可以通过更为稀疏的网格代表散射体向外辐射作用,同时保证和原先基本相同的精度,适合用作求解多尺度大规模散射问题.

 [1] Andriulli F P, Cools K, Bagci H, et al.A multiplicative Calderon preconditioner for the electric field integral equation[J].IEEE Transactions on Antennas and Propagation, 2008, 56(8):2398-2412. Click to display the text [2] Heldring A, Rius J M, Tamayo J M, et al.Multiscale compressed block decomposition for fast direct solution of method of moments linear system[J].IEEE Transactions on Antennas and Propagation, 2011, 59(2):526-536. Click to display the text [3] Li Y J, Jin J M.A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems[J].IEEE Transactions on Antennas and Propagation, 2007, 55(10):2803-2810. Click to display the text [4] Peng Z, Wang X C, Lee J F.Integral equation based domain decomposition method for solving electromagnetic wave scattering from non-penetrable objects[J].IEEE Transactions on Antennas and Propagation, 2011, 59(9):3328-3338. Click to display the text [5] Li M K, Chew W C, Jiang L J.A domain decomposition scheme based on equivalence theorem[J].Microwave and Optical Technology Letters, 2006, 48(9):1853-1857. Click to display the text [6] Li M K, Chew W C.Wave-field interaction with complex structures using equivalence principle algorithm[J].IEEE Transactions on Antennas and Propagation, 2007, 55(1):130-138. Click to display the text [7] Li M K, Chew W C.Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme[J].IEEE Transactions on Antennas and Propagation, 2008, 56(8):2389-2397. Click to display the text [8] Ylä-Oijala P, Taskinen M.Electromagnetic scattering by large and complex structures with surface equivalence principle algorithm[J].Waves in Random and Complex Media, 2009, 19(1):105-125. Click to display the text [9] Shao H, Hu J, Guo H, et al.Fast simulation of array structures using T-EPA with hierarchical LU decomposition[J].IEEE Antennas and Wireless Propagation Letters, 2012, 11:1560-1563. Click to display the text [10] Sun L E, Chew W C, Jin J M.Augmented equivalence principle algorithm at low frequencies[J].Microwave and Optical Technology Letters, 2010, 52(10):2274-2279. Click to display the text [11] Qian Z G, Chew W C.Fast full-wave surface integral equation solver for multiscale structure modeling[J].IEEE Transactions on Antennas and Propagation, 2009, 57(11):3594-3601. Click to display the text [12] Zhang K, Ouyang J, Yang F, et al.Radiation analysis of large antenna array by using periodic equivalence principle algorithm[J].Progress in Electromagnetics Research, 2013, 136:43-59. Click to display the text [13] Shi Y, Wang J, Liang C H.A time-domain equivalence principle and its marching-on-in-degree solution[J].Microwave and Optical Technology Letters, 2014, 56(10):2415-2422. Click to display the text [14] 杨晨.基于等效原理的区域分解算法[D].南京:南京理工大学, 2013.Yang C.Domain decomposition method based on equivalence principle[D].Nanjing:Nanjing University of Science and Technology, 2013(in Chinese). Cited By in Cnki (1) [15] 张博.等效原理算法及其结合快速算法分析目标的电磁散射[D].西安:西安电子科技大学, 2013.Zhang B.Scattering analysis of target using equivalent principle algorithm and its fast algorithm[D].Xi'an:Xidian University, 2013(in Chinese). Cited By in Cnki [16] Chew W C.Waves and fields in inhomogeneous media[M].Piscataway, NJ:IEEE Press, 1995:29, 186. [17] Rao S M, Wilton D, Glisson A W.Electromagnetic scattering by surfaces of arbitrary shape[J].IEEE Transactions on Antennas and Propagation, 1982, 30(3):409-418. Click to display the text [18] Jin J M.Theory and computation of electromagnetic fields[M].Hoboken, New Jersey:John Wiley & Sons, 2011:439-440.

#### 文章信息

DANG Xunwang, LI Maokun, YANG Fan, XU Shenheng

Error analysis of equivalence principle algorithm on different equivalence surfaces

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(10): 1867-1872.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0214