基于JASMIN框架结构的并行多层快速多极子算法及应用研究
|
摘要
三维目标电磁问题的快速分析一直是计算电磁学研究的热点。本文围绕多层快速多极子算法(MLFMA)开展了一系列优化研究及并行处理,并采用优化的并行算法分析求解三维复杂电大目标电磁散射和辐射问题。
本文首先介绍了用于金属目标电磁散射问题分析的表面积分方程方法及其求解方法——矩量法,并详细阐述了基于矩量法的快速求解方法——快速多极子方法和多层快速多极子方法的基本原理和步骤。
其次,本文研究了多层快速多极子方法中的优化策略。通过修正多极子模式数,减少角谱方向数目,从而在保证计算精度的情况下,减小内存存储量和计算时间。利用聚合配置因子的角谱对称性,使得电场积分方程(EFIE)的聚合配置因子内存存储量降为原来四分之一,而混合场积分方程(CFIE)为原来一半。本文还对插值技术做了分析研究,实现了高斯插值和拉格朗日插值技术的优化,并对这两种插值技术在改善插值前后对目标雷达散射分析做误差分析。
再次,本文研究了用于多层快速多极子方法中的预条件技术。本文通过在广义最小残差(GMRES)迭代方法中加入块对角(BD)和稀疏近似逆(SAI)预条件,加速积分方程中的迭代收敛,并比较这两种预条件技术的具体构造代价和加速效果。接着,本文研究了并行的多层快速多极子算法,用于电大复杂目标电磁分析。
首先实现了基于共享式内存的OpenMP标准的并行多层快速多极子,并分析了其并行效率。其次将属于非结构网格的多层快速多极子算法移植到结构网格的JASMIN框架,利用框架可移植性和可扩展性高、负载平衡好的优点,实现了分布式内存结构的并行,并用于电磁特性分析研究,同时做了并行效率分析。并行程序采用C++/Fortran混合编程,双精度数据结构存储,实现高性能计算。
最后,将基于JASMIN框架结构的并行MLFMA用于电磁散射和辐射问题的分析,并将数值结果与解析解、商业软件仿真结果以及基于OpenMP并行的MLFMA计算结果比较,比较结果表明了并行算法的可靠性。
The fast analysis of the scattering characteristic of three-dimension structure has always been an intensive research in computational methods for electromagnetics. This paper focuses on the research of multilevel fast multipole algorithm (MLFMA), which is used to solve the complex three-dimensional electromagnetic scattering problems.
Firstly, the paper introduces the integral equation method used in analysis of electromagnetic scattering problems of metal structures. Method of moment (MoM), which is used to solve the integral equation, is also studied. And, the principles and steps of fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) based on MoM is detailed.
Secondly, some optimization strategies for MLFMA are studied in this paper. By adopting modified multipole number method, the multipole number and number of spectrum direction in integration are decreased, as well as the CPU time and memory requirement in the solution, and the high accuracy is hold. The application in spectrum symmetry of aggregation terms reduces the memory requirement of aggregation and disaggregation terms by a factor of four for electrical field integral equation (EFIE), and two for combined field integral equation (CFIE). Interpolation technique, one of the most important components, is studied in this paper. The optimization of Lagrange interpolation and Gaussian interpolation is achieved and used to analyzes the root mean square (RMS) error of radar cross scattering (RCS).
Thirdly, the study of precondition technique in MLFMA is implied in this paper. The block diagonal precondition (BDP) and sparse approximate inverse (SAI) precondition are studied. And, they are achieved in generalized minimum residual (GMRES) iteration solver to accelerate the convergence of the matrix equation. The iterative effects of the two different preconditions are compared with that the precondition is not applied.
Then, in order to improve the ability of solution of large electrical size structure scattering problems, this paper has studied and developed the two kinds of parallel MLFMA. One is with OpenMP standard based on shared memory architecture. The other is with JASMIN frame based on distributed memory architecture. The JASMIN frame has an advantage of high portability and scalability, and a good load balancing. Although the infrastructure is structured and MLFMA is non-structured, the parallel procedure is accomplished overcoming the contradiction. C++/Fortran language and double precision data type are used to realize the procedure for high performance computing. The parallel efficiency of the two kinds of parallel MLFMA is analyzed in this paper, respectively.
Finally, the parallel MLFMA based on JASMIN frame (JPMLFMA) is applied in electromagnetic scattering and radiation problems analysis in this paper. The numerical results of JPMLFMA are compared to analytical one, simulation ones of other software and MLFMA solutions based on OpenMP, which demonstrate the reliability and stability of the parallel algorithm.
本文首先介绍了用于金属目标电磁散射问题分析的表面积分方程方法及其求解方法——矩量法,并详细阐述了基于矩量法的快速求解方法——快速多极子方法和多层快速多极子方法的基本原理和步骤。
其次,本文研究了多层快速多极子方法中的优化策略。通过修正多极子模式数,减少角谱方向数目,从而在保证计算精度的情况下,减小内存存储量和计算时间。利用聚合配置因子的角谱对称性,使得电场积分方程(EFIE)的聚合配置因子内存存储量降为原来四分之一,而混合场积分方程(CFIE)为原来一半。本文还对插值技术做了分析研究,实现了高斯插值和拉格朗日插值技术的优化,并对这两种插值技术在改善插值前后对目标雷达散射分析做误差分析。
再次,本文研究了用于多层快速多极子方法中的预条件技术。本文通过在广义最小残差(GMRES)迭代方法中加入块对角(BD)和稀疏近似逆(SAI)预条件,加速积分方程中的迭代收敛,并比较这两种预条件技术的具体构造代价和加速效果。接着,本文研究了并行的多层快速多极子算法,用于电大复杂目标电磁分析。
首先实现了基于共享式内存的OpenMP标准的并行多层快速多极子,并分析了其并行效率。其次将属于非结构网格的多层快速多极子算法移植到结构网格的JASMIN框架,利用框架可移植性和可扩展性高、负载平衡好的优点,实现了分布式内存结构的并行,并用于电磁特性分析研究,同时做了并行效率分析。并行程序采用C++/Fortran混合编程,双精度数据结构存储,实现高性能计算。
最后,将基于JASMIN框架结构的并行MLFMA用于电磁散射和辐射问题的分析,并将数值结果与解析解、商业软件仿真结果以及基于OpenMP并行的MLFMA计算结果比较,比较结果表明了并行算法的可靠性。
The fast analysis of the scattering characteristic of three-dimension structure has always been an intensive research in computational methods for electromagnetics. This paper focuses on the research of multilevel fast multipole algorithm (MLFMA), which is used to solve the complex three-dimensional electromagnetic scattering problems.
Firstly, the paper introduces the integral equation method used in analysis of electromagnetic scattering problems of metal structures. Method of moment (MoM), which is used to solve the integral equation, is also studied. And, the principles and steps of fast multipole method (FMM) and multilevel fast multipole algorithm (MLFMA) based on MoM is detailed.
Secondly, some optimization strategies for MLFMA are studied in this paper. By adopting modified multipole number method, the multipole number and number of spectrum direction in integration are decreased, as well as the CPU time and memory requirement in the solution, and the high accuracy is hold. The application in spectrum symmetry of aggregation terms reduces the memory requirement of aggregation and disaggregation terms by a factor of four for electrical field integral equation (EFIE), and two for combined field integral equation (CFIE). Interpolation technique, one of the most important components, is studied in this paper. The optimization of Lagrange interpolation and Gaussian interpolation is achieved and used to analyzes the root mean square (RMS) error of radar cross scattering (RCS).
Thirdly, the study of precondition technique in MLFMA is implied in this paper. The block diagonal precondition (BDP) and sparse approximate inverse (SAI) precondition are studied. And, they are achieved in generalized minimum residual (GMRES) iteration solver to accelerate the convergence of the matrix equation. The iterative effects of the two different preconditions are compared with that the precondition is not applied.
Then, in order to improve the ability of solution of large electrical size structure scattering problems, this paper has studied and developed the two kinds of parallel MLFMA. One is with OpenMP standard based on shared memory architecture. The other is with JASMIN frame based on distributed memory architecture. The JASMIN frame has an advantage of high portability and scalability, and a good load balancing. Although the infrastructure is structured and MLFMA is non-structured, the parallel procedure is accomplished overcoming the contradiction. C++/Fortran language and double precision data type are used to realize the procedure for high performance computing. The parallel efficiency of the two kinds of parallel MLFMA is analyzed in this paper, respectively.
Finally, the parallel MLFMA based on JASMIN frame (JPMLFMA) is applied in electromagnetic scattering and radiation problems analysis in this paper. The numerical results of JPMLFMA are compared to analytical one, simulation ones of other software and MLFMA solutions based on OpenMP, which demonstrate the reliability and stability of the parallel algorithm.
引文
[1] Weng Cho Chew, Jian-Ming Jin, Cai-Cheng Lu, Eric Michielssen and Ji-Ming Song. Fast Solution Methods in Electromagnetics. IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 533-543, March, 2007.
[2]盛新庆.计算电磁学要论.科学出版社,2004.
[3] K. S. Yee, J. S. Chen, and A. H. Chang. Conformal finite-difference time-domain(FDTD) with overlapping grids. IEEE Transactions on Antennas and Propagation, 1992, Vol. 40, No.9, pp. 1068-1075, 1992.
[4] A. Taflve. Computational electrodynamics: the finite-difference time-domain Method. Norwood, MA: Artech House, 1995.
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998.
[6] M. A. Morgan and K. K. Mei, Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. IEEE Transactions on Antennas and Propagation, 1979, AP-27, No.2, pp. 202-214, Mar. 1979.
[7] R. F. Harrington. Field Computation by Moment Methods. Krieger Publishing Co., Inc. Melbourne, FL, USA, 1982.
[8]李世智.电磁辐射与散射中的矩量法.电子工业出版社,1985年.
[9] V. Rokhlin. Rapid solution of integral equation of scattering theory in two dimensions. Journal of Computational physics, pp. 414-439. 1990。
[10] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995.
[11] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997.
[12] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code(FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997
[13] S. Velamparambil, J. M. Song, W. C. Chew and K. Gallivan. ScaleME: a portable scaleable multipole engine for electromagnetic and acoustic integral equation. IEEE AntennasPropagation Symposium, 1998.
[14] S. Velamparambil, W. C. Chew, and Jiming Song. 10 Million Unknowns: Is It That Big? IEEE Transaction on Antennas Propagation Magazine, Vol. 45, pp. 43-58, April 2003.
[15] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005.
[16]聂在平,方大纲等.目标与环境电磁散射特性建模——理论、方法与实现.国防工业出版社,2009年.
[17] M. R. Hestenes and E. Stiefel. Method of conjugate gradienis for solving linear Systems. J. Res. Nat. Bur. Standards,Vol.49,pp.409-436, Dec. 1952.
[18] Y. Saad and M. Schultz,“Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems,”SIAM J. Sci. Statist. Comput., Vol.7, No.3, pp.856-869, 1986.
[19] M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, Vol.123, pp.261-292, 2000.
[20]莫则尧,张爱清,刘青凯,等.并行自适应结构网格应用支撑软件框架(JASMIN1.4版)用户指南.高性能计算中心-北京应用物理与计算数学研究所出版.2008年9月.
[21] ?zgur Ergül and Levent Gürel. On the Lagrange Interpolation in Multilevel Fast Multipole Algorithm. IEEE Antennas Propagation Symposium. pp 1891-1894, 2006.
[22] ?zgur Ergül and Levent Gürel. Enhancing the Accuracy of the Interpolations and Anterpolations in MLFMA. IEEE Antennas and Wireless Propagation Letters. Vol.5, pp.467-470, 2006.
[23] Levent Gürel and ?zgur Ergül. Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns. Eletronics Letters. Vol. 43, No. 9. pp. 499-500, April 2007,
[24] Tahir Malas, ?zgur Ergül and Levent Gürel. Sequential and Parallel Preconditioners for Large-Scale Integral-Equation Problems. IEEE Antennas Propagation Symposium. pp. 35-43, 2007.
[25] Tahir Malas, ?zgur Ergül and Levent Gürel. Approximate MLFMA as an Efficient Preconditioner. IEEE Antennas Propagation Symposium. pp. 1289-1292, 2007.
[26] ?zgur Ergül, Tahir Malas, Levent Gürel. Solution of Extremely Large Integral-Equation Problems. IEEE Antennas Propagation Symposium. pp. 166-169, 2007.
[27] Levent Gürel and ?zgür Ergül. Efficient Solution of the Combined-Field Integral Equation with the Parallel Multilevel Fast Multipole Algorithm. Computational Electromagnetics Workshop 2007. 30-31 Aug.
[28] ?zgur Ergül and Levent Gürel. A Hierarchical Partitioning Strategy for an Efficient Parallelization of the Multilevel Fast Multipole Algorithm. IEEE Transaction on Antennas Propagation, Vol. 57, No. 6, June 2009.
[29] Taboada, J. M., Landesa, L., Obelleiro, F., Rodriguez, J.L., Bertolo, J.M., Mourino, J.C., Gomez, A. Parallel FMM-FFT solver for the analysis of hundreds of millions of unknowns . Computational Electromagnetics International Workshop, 2009. CEM 2009.
[30] Taboada, J. M., Landesa, L., Bertolo, J. M., Obelleiro, F., Rodriguez, J.L., Mourino, J.C., Gomez, A. High Scalability Multipole Method for the Analysis of Hundreds of Millions of Unknowns. EuCAP 2009. 3rd European Conference, pp.2753-2756, 2009.
[31]胡俊.复杂目标矢量电磁散射的高效方法-快速多极子方法及其应用.电子科技大学博士学位论文.2000年.
[32] Jun. Hu, Zaiping. Nie and Xiaodong Gong. Sovling electromagnetic scattering and radiation by FMM with curvilinear RWG basis. Chinese Journal of Electronics, Vol. 12, No. 3, pp. 457-460, July, 2003.
[33] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003.
[34]胡俊,聂在平,王军,等.三维电大目标散射求解的多层快速多极子方法.电波科学学报, Vo. 19, No.5, pp. 509-514, 2004.
[35] Jun Hu, Zaiping Nie, Lin Lei, and Li Jun Tian. Fast Solution of Scattering from Conducting Strctures by Local MLFMA Based on Improved Electric Field Integral Equation. IEEE Transactions on Electromagnetic Compatibility, Vol. 50, No. 4, pp.940-945, Nov. 2008.
[36]赵华鹏.多层快速多极子方法优化及其并行化.电子科技大学博士学位论文. 2006年5月.
[37]胡俊,聂在平,温剑,余鹏,王军,赵华鹏,张俊.一种精确通用高效的电磁仿真工具——AUEST_E软件.舰船电子工程电磁兼容性研究专辑. 2004年第12卷.
[38] Xiao-Min Pan and Xin-Qing Sheng. A Sophisticated Parallel MLFMA for Scattering by Extremely Large Targets. IEEE Antennas and Propagation Magazine, Vol. 50, No. 3, June 2008.
[39] Xiao-Min Pan and Xin-Qing Sheng. A high performance parallel MLFMA for scattering by extremely large targets. Asian Pacific Microwave Conference, APMC 2008, Dec. 2008
[40] R. S. Chen, K. F. Tsang, L. Mo,“Wavelet based sparse approximate inverse preconditioned CG algorithm for fast analysis of microstrip circuits,”Microwave and Optical Technology Letters, Vol.35, No.5, pp.383-389, Dec. 2002.
[41]芮平亮.电磁散射分析中的快速迭代求解技术.南京理工大学博士学位论文.2007年9月.
[42] Wen Bin, Hu Jun, Hu Fangjing, Nie Zaiping. Fast Computation of RCS of 3D Electrically Large Structure by Modified Multilevel Fast Multipole Algorithm. IET International Radar Conference 2009. Apr., 2009.
[43] Van den Bosch, Marc Acheroy, and Jean-Paul Marcel. Design, implementation, and optimization of a highly efficient multilevel fast multipole algorithm. Computational Electromagnetics Workshop 2007. 30-31 Aug., 2007.
[44]阙肖峰.导体介质组合目标电磁辐射与散射分析的精确建模和快速算法研究.电子科技大学博士学位论文,2008.
[45] M. Bollhǒfer and M. Volker. Algebraic multilevel methods and sparse approximate inverses. SIAM J. Matrix Anal. Appl., Vol. 24, No.1, pp.191–218, 2002.
[46] J. Lee, J. Zhang, and C. C. Lu. Sparse Inverse Preconditioning of Multilevel Fast Multipole Algorithm for Hybrid Integral Equations in Electromagnetics. IEEE Transaction on Antennas Propagation. Vol.52, No.9, pp.2277-2287, Sep. 2004.
[47] Spécialité. PHD THESIS. Sparse Preconditioners For Dense Linear Systems From Electromagnetic Applications. Institute National Polytechnique de Toulouse.
[48]吴建平,王正华,李晓梅.块对角占优性与对称矩阵的块对角预条件.数值计算与计算机应用.2003年12月.
[49] K. Sertel and J. L. Volakis,“Incomplete LU preconditioner for fmm implementation,”Microwave Opt. Technol. Lett., Vol.26, No.7, pp.265-267, 2000.
[50] Miguel Hermanns. Parallel Programming in Fortran 95 using OpenMP. 19 April, 2002.
[51] http://www.openmp.org.
[52] A. Buchau, S. M. Tsafak, W. Hafla, and W. M. Rucker.“Parallelization of a Fast Multipole Boundary Element Method with Cluster OpenMP”. IEEE Transaction on Magnetics, Vol. 44, No. 6, June 2008.
[53]莫则尧,袁国兴编著.消息传递并行编程环境MPI.科学出版社,北京,2001.
[54] Cao Xiaolin, Mo Zeyao, Liu Xu, Xu Xiaowen, Zhang Aiqing. Parallel Implementation of Fast Multipole Method Based on JASMIN. Science in China Series F: Information Sciences. Vol. 52, No.1, Jan., 2009.
[55] Andrew F. Peterson, Scott L. Ray, Raj Mittra. Computational methods for electromagnetics. Wiley-IEEE Press, Dec. 1997
[56] W. C. Chew, C. P Davis, K. F. Warnick, Z. P. Nie, J. Hu, S. Yan. L. Gürel. EFIE and MFIE, Why the Difference? IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, APSURSI, 2008.
[57] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982.
[2]盛新庆.计算电磁学要论.科学出版社,2004.
[3] K. S. Yee, J. S. Chen, and A. H. Chang. Conformal finite-difference time-domain(FDTD) with overlapping grids. IEEE Transactions on Antennas and Propagation, 1992, Vol. 40, No.9, pp. 1068-1075, 1992.
[4] A. Taflve. Computational electrodynamics: the finite-difference time-domain Method. Norwood, MA: Artech House, 1995.
[5]金建铭.电磁场有限元方法.西安电子科技大学出版社,1998.
[6] M. A. Morgan and K. K. Mei, Finite-element computation of scattering by inhomogeneous penetrable bodies of revolution. IEEE Transactions on Antennas and Propagation, 1979, AP-27, No.2, pp. 202-214, Mar. 1979.
[7] R. F. Harrington. Field Computation by Moment Methods. Krieger Publishing Co., Inc. Melbourne, FL, USA, 1982.
[8]李世智.电磁辐射与散射中的矩量法.电子工业出版社,1985年.
[9] V. Rokhlin. Rapid solution of integral equation of scattering theory in two dimensions. Journal of Computational physics, pp. 414-439. 1990。
[10] J. M. Song and W. C. Chew. Fast multipole solution of three dimensional integral equation. IEEE Antennas Propagation Symposium, pp.1528-1531, 1995.
[11] J.M.Song, C.C.Lu, and W.C.Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Transactions on Antennas and Propagation. Vol. 45, No. 10, pp. 1488-1493, 1997.
[12] J. M. Song, C. C. Lu, W. C. Chew and S.W. Lee. Introduction to Fast Illinois Solver Code(FISC). IEEE Antennas Propagation Symposium. pp. 48-51, 1997
[13] S. Velamparambil, J. M. Song, W. C. Chew and K. Gallivan. ScaleME: a portable scaleable multipole engine for electromagnetic and acoustic integral equation. IEEE AntennasPropagation Symposium, 1998.
[14] S. Velamparambil, W. C. Chew, and Jiming Song. 10 Million Unknowns: Is It That Big? IEEE Transaction on Antennas Propagation Magazine, Vol. 45, pp. 43-58, April 2003.
[15] S. Velamparambil, W. C. Chew. Analysis and Performance of a Distributed Memory Multilevel Fast Multipole Algorithm. IEEE Transactions on Antennas and Propagation, Vol.53, No. 8 II, pp. 2719-2727, August 2005.
[16]聂在平,方大纲等.目标与环境电磁散射特性建模——理论、方法与实现.国防工业出版社,2009年.
[17] M. R. Hestenes and E. Stiefel. Method of conjugate gradienis for solving linear Systems. J. Res. Nat. Bur. Standards,Vol.49,pp.409-436, Dec. 1952.
[18] Y. Saad and M. Schultz,“Gmres: a generalized minimal residual algorithm for solving non-symmetric linear systems,”SIAM J. Sci. Statist. Comput., Vol.7, No.3, pp.856-869, 1986.
[19] M. Eiermann, O. G. Ernst, and O. Schneider. Analysis of acceleration strategies for restarted minimum residual methods. Journal of Computational and Applied Mathematics, Vol.123, pp.261-292, 2000.
[20]莫则尧,张爱清,刘青凯,等.并行自适应结构网格应用支撑软件框架(JASMIN1.4版)用户指南.高性能计算中心-北京应用物理与计算数学研究所出版.2008年9月.
[21] ?zgur Ergül and Levent Gürel. On the Lagrange Interpolation in Multilevel Fast Multipole Algorithm. IEEE Antennas Propagation Symposium. pp 1891-1894, 2006.
[22] ?zgur Ergül and Levent Gürel. Enhancing the Accuracy of the Interpolations and Anterpolations in MLFMA. IEEE Antennas and Wireless Propagation Letters. Vol.5, pp.467-470, 2006.
[23] Levent Gürel and ?zgur Ergül. Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns. Eletronics Letters. Vol. 43, No. 9. pp. 499-500, April 2007,
[24] Tahir Malas, ?zgur Ergül and Levent Gürel. Sequential and Parallel Preconditioners for Large-Scale Integral-Equation Problems. IEEE Antennas Propagation Symposium. pp. 35-43, 2007.
[25] Tahir Malas, ?zgur Ergül and Levent Gürel. Approximate MLFMA as an Efficient Preconditioner. IEEE Antennas Propagation Symposium. pp. 1289-1292, 2007.
[26] ?zgur Ergül, Tahir Malas, Levent Gürel. Solution of Extremely Large Integral-Equation Problems. IEEE Antennas Propagation Symposium. pp. 166-169, 2007.
[27] Levent Gürel and ?zgür Ergül. Efficient Solution of the Combined-Field Integral Equation with the Parallel Multilevel Fast Multipole Algorithm. Computational Electromagnetics Workshop 2007. 30-31 Aug.
[28] ?zgur Ergül and Levent Gürel. A Hierarchical Partitioning Strategy for an Efficient Parallelization of the Multilevel Fast Multipole Algorithm. IEEE Transaction on Antennas Propagation, Vol. 57, No. 6, June 2009.
[29] Taboada, J. M., Landesa, L., Obelleiro, F., Rodriguez, J.L., Bertolo, J.M., Mourino, J.C., Gomez, A. Parallel FMM-FFT solver for the analysis of hundreds of millions of unknowns . Computational Electromagnetics International Workshop, 2009. CEM 2009.
[30] Taboada, J. M., Landesa, L., Bertolo, J. M., Obelleiro, F., Rodriguez, J.L., Mourino, J.C., Gomez, A. High Scalability Multipole Method for the Analysis of Hundreds of Millions of Unknowns. EuCAP 2009. 3rd European Conference, pp.2753-2756, 2009.
[31]胡俊.复杂目标矢量电磁散射的高效方法-快速多极子方法及其应用.电子科技大学博士学位论文.2000年.
[32] Jun. Hu, Zaiping. Nie and Xiaodong Gong. Sovling electromagnetic scattering and radiation by FMM with curvilinear RWG basis. Chinese Journal of Electronics, Vol. 12, No. 3, pp. 457-460, July, 2003.
[33] Jun Hu, Zaiping Nie and Lei Lin. Solving 3-D electromagnetic scattering from conducting object by MLFMA with curvilinear RWG basis. IEEE Antennas and Propagation Society International Symposium, pp: 460-463, 2003.
[34]胡俊,聂在平,王军,等.三维电大目标散射求解的多层快速多极子方法.电波科学学报, Vo. 19, No.5, pp. 509-514, 2004.
[35] Jun Hu, Zaiping Nie, Lin Lei, and Li Jun Tian. Fast Solution of Scattering from Conducting Strctures by Local MLFMA Based on Improved Electric Field Integral Equation. IEEE Transactions on Electromagnetic Compatibility, Vol. 50, No. 4, pp.940-945, Nov. 2008.
[36]赵华鹏.多层快速多极子方法优化及其并行化.电子科技大学博士学位论文. 2006年5月.
[37]胡俊,聂在平,温剑,余鹏,王军,赵华鹏,张俊.一种精确通用高效的电磁仿真工具——AUEST_E软件.舰船电子工程电磁兼容性研究专辑. 2004年第12卷.
[38] Xiao-Min Pan and Xin-Qing Sheng. A Sophisticated Parallel MLFMA for Scattering by Extremely Large Targets. IEEE Antennas and Propagation Magazine, Vol. 50, No. 3, June 2008.
[39] Xiao-Min Pan and Xin-Qing Sheng. A high performance parallel MLFMA for scattering by extremely large targets. Asian Pacific Microwave Conference, APMC 2008, Dec. 2008
[40] R. S. Chen, K. F. Tsang, L. Mo,“Wavelet based sparse approximate inverse preconditioned CG algorithm for fast analysis of microstrip circuits,”Microwave and Optical Technology Letters, Vol.35, No.5, pp.383-389, Dec. 2002.
[41]芮平亮.电磁散射分析中的快速迭代求解技术.南京理工大学博士学位论文.2007年9月.
[42] Wen Bin, Hu Jun, Hu Fangjing, Nie Zaiping. Fast Computation of RCS of 3D Electrically Large Structure by Modified Multilevel Fast Multipole Algorithm. IET International Radar Conference 2009. Apr., 2009.
[43] Van den Bosch, Marc Acheroy, and Jean-Paul Marcel. Design, implementation, and optimization of a highly efficient multilevel fast multipole algorithm. Computational Electromagnetics Workshop 2007. 30-31 Aug., 2007.
[44]阙肖峰.导体介质组合目标电磁辐射与散射分析的精确建模和快速算法研究.电子科技大学博士学位论文,2008.
[45] M. Bollhǒfer and M. Volker. Algebraic multilevel methods and sparse approximate inverses. SIAM J. Matrix Anal. Appl., Vol. 24, No.1, pp.191–218, 2002.
[46] J. Lee, J. Zhang, and C. C. Lu. Sparse Inverse Preconditioning of Multilevel Fast Multipole Algorithm for Hybrid Integral Equations in Electromagnetics. IEEE Transaction on Antennas Propagation. Vol.52, No.9, pp.2277-2287, Sep. 2004.
[47] Spécialité. PHD THESIS. Sparse Preconditioners For Dense Linear Systems From Electromagnetic Applications. Institute National Polytechnique de Toulouse.
[48]吴建平,王正华,李晓梅.块对角占优性与对称矩阵的块对角预条件.数值计算与计算机应用.2003年12月.
[49] K. Sertel and J. L. Volakis,“Incomplete LU preconditioner for fmm implementation,”Microwave Opt. Technol. Lett., Vol.26, No.7, pp.265-267, 2000.
[50] Miguel Hermanns. Parallel Programming in Fortran 95 using OpenMP. 19 April, 2002.
[51] http://www.openmp.org.
[52] A. Buchau, S. M. Tsafak, W. Hafla, and W. M. Rucker.“Parallelization of a Fast Multipole Boundary Element Method with Cluster OpenMP”. IEEE Transaction on Magnetics, Vol. 44, No. 6, June 2008.
[53]莫则尧,袁国兴编著.消息传递并行编程环境MPI.科学出版社,北京,2001.
[54] Cao Xiaolin, Mo Zeyao, Liu Xu, Xu Xiaowen, Zhang Aiqing. Parallel Implementation of Fast Multipole Method Based on JASMIN. Science in China Series F: Information Sciences. Vol. 52, No.1, Jan., 2009.
[55] Andrew F. Peterson, Scott L. Ray, Raj Mittra. Computational methods for electromagnetics. Wiley-IEEE Press, Dec. 1997
[56] W. C. Chew, C. P Davis, K. F. Warnick, Z. P. Nie, J. Hu, S. Yan. L. Gürel. EFIE and MFIE, Why the Difference? IEEE International Symposium on Antennas and Propagation and USNC/URSI National Radio Science Meeting, APSURSI, 2008.
[57] S. Rao, D. Wilton, A. Glisson. Electromagnetic scattering by surfaces of arbitrary shape. IEEE Transactions on Antennas and Propagation, Vol. 30, pp. 409-418, 1982.