基于<em>A-λ</em>混合单元法的静磁场数值求解

doi:10.19595/j.cnki.1000-6753.tces.162025

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摘要提出采用混合单元法解决静磁场有限元计算中的伪解问题。针对静磁场磁矢势方程,基于约束变分原理,引入Lagrange标量乘子施加Coulomb规范,得到A-λ混合列式,并进一步地识别出Lagrange乘子的梯度为激励电流的不协调部分。基于Newton-Raphson法,对材料非线性问题建立相应的迭代解法。混合单元中的磁矢势A采用棱边元离散,Lagrange乘子λ采用节点元离散。对混合单元法离散得到的鞍点问题,可以通过增广Lagrange乘子技术,将其转换为一个等价问题,并采用Uzawa法进行迭代求解。与传统的节点元和棱边元相比,混合单元可以有效地消除伪解,获得较高的数值精度。

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关键词 ：静磁学混合有限元法, Coulomb规范, Lagrange乘子法, Uzawa法

Abstract：This paper proposes to adopt mixed finite element method to eliminate spurious solution in simulatingmagnetostatic problems. By introducing a scalar Lagrange multiplier, Coulomb gauge is incorporated into magnetic vector potential formulation based on constrained variational principle, leading toA-λ mixed formulation. Furthermore, the gradient of the scalar Lagrange multiplier is identified as the incompatible component of exciting current source. By means of Newton-Raphson method, iterative strategyis established for material nonlinearity problems. Edge elements are employed to discretize magnetic vector potential, and nodal elements are employed to discretize Lagrange multiplier. By augmented Lagrange multiplier technique, the saddle point problem arising from mixed finite element discretization can be transferred to an equivalent problem whichcan be iteratively solved by Uzawa method. Compared with conventional nodal element and edge element, the mixed element can suppress potential spurious solutions, and obtain a more accurate solution.

Key words： Magnetostatics mixed finite element method Coulomb gauge Lagrange multiplier method Uzawa method

收稿日期: 2016-12-31 出版日期: 2018-03-14

PACS:

TM153+.1

基金资助:国家自然科学基金面上项目（11272074）和国家重大科技专项（2011ZX02403-004）联合资助项目

通讯作者: 关振群男,1965年生,教授,博士生导师,研究方向为计算力学和工程科学计算。E-mail：guanzhq@dlut.edu.cn

作者简介: 江鹏男,1987年生,博士研究生,研究方向为电磁及多场耦合。E-mail：jiangpeng87@mail.dlut.edu.cn

引用本文:

江鹏, 李敬, 张群, 罗林山, 关振群. 基于A-λ混合单元法的静磁场数值求解[J]. 电工技术学报, 2018, 33(5): 1167-1176. Jiang Peng, Li Jing, Zhang Qun, Luo Linshan, Guan Zhenqun. Numerical Simulation for Magnetostatic Problems Based on A-λMixed Finite Element Method. Transactions of China Electrotechnical Society, 2018, 33(5): 1167-1176.

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