有限元法的一种数据结构

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摘要稀疏矩阵非零元素压缩存储技术是处理大型有限元方程的必要手段,采用合适的数据结构是提高计算效率的关键。本文提出一种基于单元边索引的非零元素压缩存储方案,这种方案利用了单元边与总体矩阵中非零元素之间的一一对应关系,概念清晰。由于每个元素在压缩向量中的位置在网格划分阶段就确定了,因此在单元分析过程中就可以直接完成总体系数阵的压缩存储和边界条件处理,无需繁冗的寻址计算,矩阵的求解运算也非常便捷,最大限度地减少了有限元分析的内存需求,提高了计算效率。文中对相应的数据结构和算法描述做了详细的讨论,并通过一个涡流检测问题的有限元算例验证了本文方法的有效性。该方案可以方便地推广用于三维有限元问题和高阶单元。

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	陈德智
	姜贺
	张哲
	潘瑞敏

关键词 ：电磁场有限元, 总体系数阵组装, 稀疏矩阵, 非零元素存储, 数据结构, 单元边

Abstract：Nonzero elements compressed storage is a key technique for dealing with super-large sparse matrices in finite element(FE) analysis, and a proper data structure helps to improve the computation efficiency significantly. A new algorithm for nonzero elements storage is presented based on the fact that the sides of the FE elements are one-to-one corresponding to the nonzero elements of the global FE matrix, consequently, the side information can be used for indexing the nonzero elements in the compressed vector. Since the position of each element in the compressed vector is predefined in the meshing procedure, the assembly and the compression storage of the global matrix, along with the application of the boundary conditions, can be directly done in the element analysis progress, with no need of cumbersome addressing operations. The sparse matrix-vector multiplication can also be calculated fast and conveniently. Data structure and algorithm is discussed in detail, and the effectiveness of the method is validated through a FEM analysis of an eddy current testing problem. The presented method, promising a significant reduction of the memory as well as the CPU time requirements, is suitable for FE analysis of high-dimensional problems and of high-order elements.

Key words： Finite element method for electromagnetic field assembly of global matrix sparse matrix nonzero elements compression storage data structure element side

收稿日期: 2013-12-27 出版日期: 2015-03-23

PACS:

TM15

基金资助:科技部国际热核聚变实验堆（ITER）计划专项（2011GB113003）和湖北省非动力核技术协同创新中心资助项目

作者简介: 陈德智男,1969年生,博士,教授,研究方向为电磁场理论、数值分析与应用。姜贺男,1990年生,硕士研究生,研究方向为电机与电器,电磁场数值分析。

引用本文:

陈德智, 姜贺, 张哲, 潘瑞敏. 有限元法的一种数据结构[J]. 电工技术学报, 2015, 30(1): 1-7. Chen Dezhi, Jiang He, Zhang Zhe, Pan Ruimin. A Data Structure for Finite Element Method. Transactions of China Electrotechnical Society, 2015, 30(1): 1-7.

链接本文:

http://dgjsxb.ces-transaction.com/CN/Y2015/V30/I1/1

[1] 谢德馨, 杨仕友. 工程电磁场数值分析与综合[M]. 北京: 机械工业出版社, 2009.
[2] 李立毅, 黄旭珍, 寇宝泉, 等. 基于有限元法的圆筒型直线电机温度场数值计算[J]. 电工技术学报, 2013, 28(2): 132-138. Li Liyi, Huang Xuzhen, Kou Baoquan, et al. Numerical calculation of temperature field for tubular linear motor based on finite element method[J]. Transactions of China Electrotechnical Society, 2013, 28(2): 132-138.
[3] 黎卫国, 郝艳捧, 熊国锟, 等. 覆冰复合绝缘子电位分布有限元仿真[J]. 电工技术学报, 2012, 27(12): 29-35. Li Weiguo, Hao Yanpeng, Xiong Guokun, et al. Simulation and analysis of potential distribution of iced composite insulator based on finite element method [J]. Transactions of China Electrotechnical Society, 2012, 27(12): 29-35.
[4] 徐超, 王长龙, 绳慧, 等. 三维磁场计算的有限元神经网络模型[J]. 电工技术学报, 2012, 27(11): 126-132. Xu Chao, Wang Changlong, Sheng Hui, et al. FENN model for 3-D magnetic field calculation[J]. Transac- tions of China Electrotechnical Society, 2012, 27(11): 126-132.
[5] Jin Jianming. 电磁场有限元方法[M]. 王建国译. 西安: 西安电子科技大学出版社, 2001.
[6] 秦太验, 徐春晖, 周喆. 有限元法及其应用[M]. 北京: 中国农业大学出版社, 2011.
[7] 侯新录. 结构分析中的有限元法与程序设计: 用Visual C++实现[M]. 北京: 中国建材工业出版社, 2004.
[8] 徐树方．矩阵计算的理论与方法[M]. 北京: 北京大学出版社, 1995.
[9] Ahamed A K C, Magoules F. Fast sparse matrix- vector multiplication on graphics processing unit for finite element analysis[C]. 2012 IEEE 14th Interna- tional Conference on HPCC, 2012.
[10] Niclas J. Optimizing sparse matrix assembly in finite element solvers with one-sided communication[J]. Lecture Notes in Computer Science, 2013(7851): 128-139.
[11] Kaltofen E, Lobo A. Distributed matrix-free solution of large sparse linear systems over finite fields[J]. Algorithmica(New York), 1999, 24(3): 331-348.
[12] 梯华森 R P. 稀疏矩阵[M]. 朱季讷译. 北京: 科学出版社, 1981.
[13] Rajasankar J, Iyer N R. A procedure to assemble only non-zero coefficients of global matrix in finite element analysis[J]. Computer & Structures, 2000, 77(3): 595- 599.
[14] 梁振光, 唐任远. 有限元系数矩阵的快速存储技术[J]. 电工电能新技术, 2004, 23(4): 24-25, 28. Liang Zhenguang, Tang Renyuan. Rapid storage technique of coefficient matrix in finite element method [J]. Advanced Technology of Electrical Engineering and Energy, 2004, 23(4): 24-25, 28.
[15] 侯建国, 安旭文. 有限单元法程序设计[M]. 2版. 武汉: 武汉大学出版社, 2012.
[16] 姚松, 田红旗. 有限元刚度矩阵的压缩存贮及组集[J]. 中南大学学报(自然科学版), 2006, 37(4): 826- 830. Yao Song, Tian Hongqi. Compressed storage scheme and assembly procedure of global stiffness matrix in finite element analysis[J]. J Central South University (Science and Technology), 2006, 37(4): 826-830.
[17] Bao Y, Du G, Chen W, et al. Storage method of FE equations and its application in numerical simulation of sheet metal forming[C]. Proceedings of the 10th Asia- Pacific Conference, Engineering Plasticity and Its Applications, AEPA 2010: 182-187.
[18] 纪国良, 冯仰德. 大规模有限元刚度矩阵存储及其并行求解算法[J]. 数值计算与计算机应用, 2012, 33(3): 230-240. Ji Guoliang, Feng Yangde. Parallel solving large- scale linear system of FEM based on compressed storage format[J]. J Numerical Methods & Computer Applications, 2012, 33(3): 230-240.
[19] 王忠雷, 赵国群, 马新武. 三维有限元刚度矩阵的压缩存储算法[J]. 材料科学与工艺, 2012, 20(2): 96-100, 107. Wang Zhonglei, Zhao Guoqun, Ma Xinwu. Compressed storage algorithm of 3D FEM stiffness matrix[J]. Materials Science and Technology, 2012, 20(2): 96- 100, 107.