基于A-λ混合单元法的静磁场数值求解

Numerical Simulation for Magnetostatic Problems Based on A-λ Mixed Finite Element Method

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。

This paper proposes to adopt mixed finite element method to eliminate spurious solution in simulating magnetostatic problems. By introducing a scalar Lagrange multiplier, Coulomb gauge is incorporated into magnetic vector potential formulation based on constrained variational principle, leading to A-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 strategy is 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 which can 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.