基于传输线法的非线性静磁场四边形有限元加速求解方法研究
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摘要
有限元方法是目前工程应用中最常用的数值计算方法之一。随着求解模型的日益复杂,以及越来越多的多物理场耦合问题和参数优化设计等需求的出现,有限元的计算耗时过长的问题日益凸显。为了优化有限元求解的时长,实现有限元求解的加速计算,基于传输线迭代法(transmissionline method,TLM),提出一种黑盒电路模型,针对四边形分网下的非线性静磁场有限元问题进行加速求解。首先,介绍TLM方法的原理,并且针对有限元四边形分网模型,应用传统伽辽金法对静磁场问题进行有限元分析,得到相关矩阵;然后,通过导纳矩阵的等效关系,将非线性有限元矩阵建模为等效的黑盒电路网络;最后,应用TLM方法对电路网络进行迭代求解。应用所提出的非线性有限元求解方法,分析某型号含永磁轴对称结构接触器的静磁场分布。通过与传统牛顿迭代法以及商业有限元软件COMSOL进行结果对比,验证计算方法的可行性和正确性。
Finite element method(FEM) is the most commonly used numerical computation method. With the increasing complex solution models and the advent of multi-physics problems, the time cost using the finite element method is one of the headaches for simulation engineers. To solve the overlong solution time of finite element method and accelerate the solving process, this paper proposed a black-box circuit model based on the transmission line method(TLM) to solve the magnetostatic problem using FEM with quadrilateral shape element. Firstly, we introduced the principle of TLM and applied the traditional Galerkin method into the quadrilateral shape finite element to obtain the global stiffness matrix. Then the nonlinear finite element matrix was modelled as nonlinear circuit network according to the equivalence of admittance matrix. Finally, we solved the circuit network using TLM iteration. The distribution of an axisymmetrical contactor's magnetostatic field was obtained by the method proposed.Compared with the traditional Newton-Raphson(NR) iteration and commercial FEM software COMSOL, the feasibility and correctness of the method was verified.
Finite element method(FEM) is the most commonly used numerical computation method. With the increasing complex solution models and the advent of multi-physics problems, the time cost using the finite element method is one of the headaches for simulation engineers. To solve the overlong solution time of finite element method and accelerate the solving process, this paper proposed a black-box circuit model based on the transmission line method(TLM) to solve the magnetostatic problem using FEM with quadrilateral shape element. Firstly, we introduced the principle of TLM and applied the traditional Galerkin method into the quadrilateral shape finite element to obtain the global stiffness matrix. Then the nonlinear finite element matrix was modelled as nonlinear circuit network according to the equivalence of admittance matrix. Finally, we solved the circuit network using TLM iteration. The distribution of an axisymmetrical contactor's magnetostatic field was obtained by the method proposed.Compared with the traditional Newton-Raphson(NR) iteration and commercial FEM software COMSOL, the feasibility and correctness of the method was verified.
引文
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[18]Jin Jianming.The finite element method in electromagnetics[M].2nd ed.New Jersey:Wiley-IEEE,2002:39-40.
[2]孙明灿,唐任远,韩雪岩,等.轴向磁通永磁电机空载气隙磁场建模分析[J].中国电机工程学报,2018,38(5):1525-1533.Sun Mingcan,Tang Renyuan,Han Xueyan,et al.Analysis and modeling for open circuit air gap magnetic field prediction in axial flux permanent magnet machines[J].Proceedings of the CSEE,2018,38(5):1525-1533(in Chinese).
[3]谢裕清,李琳,宋雅吾,等.油浸式电力变压器绕组温升的多物理场耦合计算方法[J].中国电机工程学报,2016,36(21):5957-5965.Xie Yuqing,Li Lin,Song Yawu,et al.Multi-physical field coupled method for temperature rise of winding in oil-immersed power transformer[J].Proceedings of CSEE,2016,36(21):5957-5965(in Chinese).
[4]金亮,邱运涛,杨庆新,等.基于云计算的电磁问题并行计算方法[J].电工技术学报,2016,31(22):5-11.Jin Liang,Qiu Yuntao,Yang Qingxin,et al.A parallel computing method to electromagnetic problems based on cloud computing[J].Transactions of China Electrotechnics Society,2016,31(22):5-11(in Chinese).
[5]金亮,汪东梅,邱运涛,等.基于弹性云计算集群的国际TEAM Problem 7基准问题计算方法[J].电工技术学报,2017,32(8):144-150.Jin Liang,Wang Dongmei,Qiu Yuntao,et al.Acalculation method for the international benchmark problem of TEAM 7 based on elastic cluster in cloud computing[J].2017,32(8):144-150(in Chinese).
[6]谢德馨,朱占新,吴东阳,等.大规模工程涡流场有限元计算的困境与展望[J].中国电机工程学报,2015,35(5):1250-1257.Xie Dexin,Zhu Zhanxin,Wu Dongyang,et al.Plight and perspective of large-scale engineering Eddy current field FEM computation[J].Proceedings of the CSEE,2015,35(5):1250-1257(in Chinese).
[7]Dommel H W.Digital computer solution of electromagnetic transients in single-and multiphase networks[J].IEEE Transactions on Power Apparatus and Systems,1969,PAS-88(4):388-399.
[8]Johns P B,O’Brien M.Use of the transmission-line modelling(t.l.m.)method to solve non-linear lumped networks[J].Radio and Electronic Engineer,1980,50(1/2):59-70.
[9]Hui S Y R,Christopoulos C.Numerical simulation of power circuits using transmission-line modelling[J].IEEProceedings A-Physical Science,Measurement and Instrumentation,Management and Education,1990,137(6):379-384.
[10]Lobry J,Trecat J,Broche C.The transmission line modeling(TLM)method as a new iterative technique in nonlinear 2-D magnetostatics[J].IEEE Transactions on Magnetics,1996,32(2):559-566.
[11]Knight A M.Time-stepped eddy-current analysis of induction machines with transmission line modeling and domain decomposition[J].IEEE Transactions on Magnetics,2003,39(4):2030-2035.
[12]Deblecker D,Lobry J,Broche C.Use of transmission-line modelling method in FEM for solution of nonlinear eddy-current problems[J].IEE Proceedings-Science,Measurement and Technology,1998,145(1):31-38.
[13]Knight A M.Efficient parallel solution of time-stepped multislice eddy-current induction motor models[J].IEEETransactions on Magnetics,2004,40(2):1282-1285.
[14]Asghari B,Dinavahi V.Novel transmission line modeling method for nonlinear permeance network based simulation of induction machines[J].IEEE Transactions on Magnetics,2011,47(8):2100-2108.
[15]Asghari B,Dinavahi V.Real-time nonlinear transient simulation based on optimized transmission line modeling[J].IEEE Transactions on Power Systems,2011,26(2):699-709.
[16]Yang Wenying,Peng Fei,Dinavahi V.Nonlinear axisymmetric magnetostatic analysis for electromagnetic device using TLM-based finite-element method[J].IEEETransactions on Magnetics,2017,53(4):1-9.
[17]颜威利,杨庆新,汪友华.电气工程电磁场数值分析[M].北京:机械工业出版社,2005.Yan Weili,Yang Qingxin,Wang Youhua.Numerical analysis of electromagnetic field in electrical engineering[M].Beijing:China Machine Press,2005(in Chinese).
[18]Jin Jianming.The finite element method in electromagnetics[M].2nd ed.New Jersey:Wiley-IEEE,2002:39-40.