We consider systems of nonlinear magnetostatics and quasistatics that typically arise in the modeling and simulation of electric machines. The nonlinear problems, eventually obtained after time discretization, are usually solved by employing a vector potential formulation. In the relevant two-dimensional setting, a discretization can be obtained by H1-conforming finite elements. We here consider an alternative formulation based on the H-field which leads to a nonlinear saddlepoint problem. After commenting on the unique solvability, we study the numerical approximation by H(curl)-conforming finite elements and present the main convergence results. A particular focus is put on the efficient solution of the linearized systems arising in every step of the nonlinear Newton solver. Via hybridization, the linearized saddlepoint systems can be transformed into linear elliptic problems, which can be solved with similar computational complexity as those arising in the vector or scalar potential formulation. In summary, we can thus claim that the mixed finite element approach based on the $H$-field can be considered a competitive alternative to the standard vector or scalar potential formulations for the solution of problems in nonlinear magneto-quasistatics.

翻译：本文研究电机建模与仿真中典型出现的非线性静磁与准静态系统。经时间离散后获得的非线性问题通常采用矢量势公式求解。在相关的二维场景中，可通过H1协调有限元实现离散化。本文考虑一种基于H场的新型公式，该公式导致非线性鞍点问题。在讨论解的惟一存在性后，我们研究了H(curl)协调有限元的数值逼近，并给出了主要收敛性结果。特别关注非线性牛顿求解器每一步中线性化系统的高效求解。通过混合化处理，线性化鞍点系统可转化为线性椭圆问题，其计算复杂度与矢量势或标量势公式中的求解复杂度相当。综上所述，基于$H$场的混合有限元方法可作为标准矢量势或标量势公式在非线性磁准静态问题求解中的有力替代方案。