On the convergence of higher order finite element methods for nonlinear magnetostatics

TL;DR

This work develops a rigorous finite element framework for nonlinear magnetostatics using a vector potential formulation and higher-order elements. By casting the problem as a convex minimization of $W(a)=\int_\Omega w(\operatorname{curl} a) - h_s \cdot \operatorname{curl} a\,dx$, the authors prove well-posedness and derive order-optimal error estimates under smoothness assumptions, incorporating quadrature and curved-domain mappings. They establish global linear convergence of a damped Newton method with a discretization-parameter–independent rate and prove local quadratic convergence in a mesh-dependent neighborhood, applicable to general inhomogeneous, nonlinear, anisotropic materials including permanent magnets. The theory is extended to curved domains and to 2D, and is supported by numerical tests (smooth solutions, TEAM Problem 13, PMSM) that confirm mesh-independent Newton behavior and expected convergence orders, highlighting the approach’s practical relevance for electric machine simulations. Overall, the paper provides a solid mathematical foundation for reliable, high-order FE simulations of nonlinear magnetostatics in complex geometries and materials.

Abstract

The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of such devices. We study the numerical solution of the vector potential formulation of nonlinear magnetostatics by means of higher-order finite element methods. Numerical quadrature is used for the efficient handling of the nonlinearities and domain mappings are employed for the consideration of curved boundaries. The existence of a unique solution is proven on the continuous and discrete level and a full convergence analysis of the resulting finite element schemes is presented indicating order optimal convergence rates under appropriate smoothness assumptions. For the solution of the nonlinear discretized problems, we consider a Newton method with line search for which we establish global linear convergence with convergence rates that are independent of the discretization parameters. We further prove local quadratic convergence in a mesh-dependent neighborhood of the solution which becomes effective when high accuracy of the nonlinear solver is demanded. The assumptions required for our analysis cover inhomogeneous, nonlinear, and anisotropic materials, which may arise in typical applications, including the presence of permanent magnets. The theoretical results are illustrated by numerical tests for some typical benchmark problems.

Figures (3)

• Figure 1: Geometry sketch (left), the magnitude of the $B$-field (middle) and the magnitude of the $H$-field
• Figure 2: Left: Geometric setup of TEAM problem 13; compare with teamproblem13. Right: Current density $j$ in the coil (black arrows) and magnitude of the magnetic flux $|b|$ in the iron plates (color).
• Figure 3: Left: geometric setup of PMSM with iron (grey), magnet (red), coil (green) and air (cyan). The orientation of the magnets is depicted with red arrows. Middle: magnitude of the magnetic flux $|b| = |\operatorname{curl} a|$. Right: magnitude of the magnetic field $|h| = |\partial_b w(b)|$.

Theorems & Definitions (28)

• Remark 2
• Theorem 3
• proof
• Remark 4
• Remark 6
• Theorem 7
• proof
• Lemma 8
• proof
• Theorem 9