Abstract：To solve the multi-scale model of geoelectric field induced by geomagnetic storms, the domain decomposition method is put to use in this paper. As for each decomposed domain, modeling and solving can be worked out by means of finite element method (FEM). In order to simplify the iterative process and avoid the processing of the boundary conditions on the interface generated by the non-overlapping domain decomposition, the overlapping domain decomposition method is adopted. The potential function on the virtual boundary is chosen as the unknown variable, based on the linear relationship in computation of FEM, the problem of multi-scale induced geoelectric field is finally transformed into an asymmetric complex linear equation. In order to save computational memory, the GMRES iterative algorithm of domain decomposition is deduced without directly solving the specific coefficient matrix of the equations according to the mathematical principles of Arnoldi's orthogonalization algorithm and generalized minimal residue (GMRES) method. In this paper, a small scale model is derived from the whole model of the large scale induced geoelectric field, and a multi-scale solution model is formed. For 2D and 3D cases, the iterative variable selection and the derivation of the iterative process are described. Taken the results of FEM direct solution as benchmark, the correctness is verified. Besides, the efficiency of the proposed algorithm is verified by comparing with direct iteration and relaxation iteration in 2D case. The proposed algorithm is used to calculate a numerical example of multi-scale geoelectric field, which provides a new method for the calculation of multi-scale alternating electromagnetic field.
王泽忠, 陶瑞祥. 基于区域分解求解磁暴感应地电场的广义极小残量迭代算法[J]. 电工技术学报, 2018, 33(6): 1265-1275.
Wang Zezhong, Tao Ruixiang. Generalized Minimal Residual Iterative Algorithm for Solving Induced Geoelectric Field During Geomagnetic Storms Based on Domain Decomposition. Transactions of China Electrotechnical Society, 2018, 33(6): 1265-1275.
[1] 刘春明, 林晨翔, 王璇, 等. 大地电导率横向变化对地磁暴感应电场H极化及地磁感应电流的影响[J]. 电工技术学报, 2016, 31(24): 113-119. Liu Chunming, Lin Chenxiang, Wang Xuan, et al.Evaluating the influence of lateral change of condu- ctivity on the “H Polarization” of induced geoelectric field and geomagnetically induced currents[J]. Transactions of China Electrotechnical Society, 2016, 31(24): 113-119. [2] 董博, 王泽忠, 刘连光, 等. 大地电导率横向突变处磁暴感应地电场的邻近效应[J]. 地球物理学报, 2015, 58(1): 238-246. Dong Bo, Wang Zezhong, Liu Lianguang, et al.The proximity effect on the induced geoelectric field at the interface of different conductivity structures with lateral variations during geomagnetic storms[J]. 3 Chinese Journal of Geophysics, 2015, 58(1): 238-246. [3] 刘连光, 吴伟丽. 磁暴影响电力系统安全风险评估思路与理论框架[J]. 中国电机工程学报, 2014, 34(10): 1583-1591. Liu Lianguang, Wu Weili.Security risk assessment ideas and theoretical framework for power system considering geomagnetic storm[J]. Proceedings of the CSEE, 2014, 34(10): 1583-1591. [4] Liu C M, Li Y L, Pirjola R.Observations and modeling of GIC in the Chinese large-scale high- voltage power networks[J]. Journal of Space Climate, 2014, 4(A03): 1-7. [5] Marti L, Rezaei-Zare A, Boteler D.Calculation of induced electric field during a geomagnetic storm using recursive convolution[J]. IEEE Transactions on Power Delivery, 2014, 29(2): 802-807. [6] Liu C, Liu L, Pirjola R, et al.Calculation of geomagnetically induced currents in mid-to low- latitude power grids based on the plane wave method: a preliminary case study[J]. Space Weather-the Inter- national Journal of Research & Applications, 2016, 7(4): 1-9. [7] 刘连光, 张鹏飞, 王开让, 等. 地磁暴侵害油气管道的管地电位效应[J]. 电工技术学报, 2016, 31(9): 68-74. Liu Lianguang, Zhang Pengfei, Wang Kairang, et al.PSP interference effect of geomagnetic storm on buried pipelines[J]. Transactions of China Electro- technical Society, 2016, 31(9): 68-74. [8] Boteler D.A new versatile method for modelling geomagnetic induction in pipelines[J]. Geophysical Journal International, 2013, 193(1): 98-109. [9] 刘连光, 于永富, 宗伟, 等. 地磁暴干扰轨道电路工作的仿真计算研究[J]. 铁道学报, 2015, 37(8): 48-52. Liu Lianguang, Yu Yongfu, Zong Wei, et al.Simu- lation calculation about interference of geomagnetic storms on track circuit equipment[J]. Journal of the China Railway Society, 2015, 37(8): 48-52. [10] 朱汉青. 区域分解法在电磁问题分析中的应用研究[D]. 四川: 电子科技大学, 2002. [11] 李世琼, 王泽忠. 开域静电场全源积分人工边界法的GMRES迭代算法[J]. 电工技术学报, 2014, 29(10): 206-212. Li Shiqiong, Wang Zezhong.GMRES iteration method of FEM-actual charge method for open boundry electrostatic field problems[J]. Transactions of China Electrotechnical Society, 2014, 29(10): 206-212. [12] 王立鹏, 王欣彦, 战洪仁, 等. 基于区域分解的EFG-FE耦合法及其在电机运动问题的应用[J]. 电工技术学报, 2012, 27(12): 7-12. Wang Lipeng, Wang Xinyan, Zhan Hongren, et al.EFG-FE coupling method based on DDM and its application on motor motion[J]. Transactions of China Electrotechnical Society, 2012, 27(12): 7-12. [13] 厉天威, 阮江军, 吴田. 并行计算高压输电线路周围电场[J]. 电工技术学报, 2009, 24(9): 1-6. Li Tianwei, Ruan Jiangjun, Wu Tian.Parallel computation of electric field intensity nearby high voltage transmission lines[J]. Transactions of China Electrotechnical Society, 2009, 24(9): 1-6. [14] 王泽忠, 董博, 刘春明, 等. 华北地区大地电性结构三维建模及磁暴感应地电场有限元计算[J]. 电工技术学报, 2015, 30(3): 61-66. Wang Zezhong, Dong Bo, Liu Chunming, et al.Three dimensional earth conductivity structure modelling in north China and calculation of geoelectromagnetic fields during geomagnetic disturbances based on finite element method[J]. Transactions of China Electrotechnical Society, 2015, 30(3): 61-66. [15] 刘连光, 马成廉. 基于有限元方法的直流输电接地极多层土壤地电位分布计算[J]. 电力系统保护与控制, 2015, 43(18): 1-5. Liu Lianguang, Ma Chenglian.Calculation of multi- layer soil earth surface potential distribution of HVDC due to finite element method[J]. Power System Protection and Control, 2015, 43(18): 1-5. [16] Saitoh A, Kamitani A.GMRES with new precondi- tioning for solving BEM-type linear system[J]. IEEE Transactions on Magnetics, 2004, 40(2): 1084-1087. [17] Aiello G, Alfonzetti S, Borzì G, et al.GMRES solution of FEM-BEM global systems for elec- trostatic problems without voltaged conductors[J]. IEEE Transactions on Magnetics, 2013, 49(5): 1701-1704. [18] 王泽忠, 董博, 刘春明, 等. 地磁暴感应地电场的大地电性结构建模方法[J]. 科学技术与工程, 2015, 15(13): 72-76. Wang Zezhong, Dong Bo, Liu Chunming, et al.Earth conductivity modelling technique for calculating geoelectric field during geomagnetic storms[J]. Science Technology and Engineering, 2015, 15(13): 72-76. [19] Ke Chen.Matrix preconditioning techniques and applications[M]. New York: Cambridge University Press, 2005. [20] 洪伟, 孙连友, 尹雷, 等. 电磁场边值问题的区域分解算法[M]. 北京: 科学出版社, 2005.
2018, Vol. 33 
