Inclusion of an Inverse Magnetic Hysteresis Model into the Space-Time Finite Element Method for Magnetoquasistatics

TL;DR

The paper integrates the Pragmatic Algebraic Model (PAM) hysteresis into the eddy-current approximation of Maxwell's equations on a 2D cross-section, resulting in the parabolic PDE $\sigma\partial_t u-\operatorname{div}_x\big[ f(|\nabla_x u|)\nabla_x u+g(|\partial_t\nabla_x u|)\partial_t\nabla_x u\big]=j_s-\operatorname{div}_x M^{\perp}$. It develops two numerical schemes: a classical time-stepping finite element method and a space-time finite element method, the latter reformulated with $p=\partial_t u$ to handle second derivatives and solved as a nonlinear saddle-point-like system via Newton with Armijo damping and the parallel MUMPS solver. Numerical tests on a simple geometry and on TEAM Problem 32 demonstrate that both methods produce consistent results for $\mathbf{B}$ and the BH-curve, validating PAM's integration into magnetoquasistatic simulations and highlighting the space-time approach's potential for parallelism and space-time adaptivity. The work provides a efficient and accurate framework for simulating hysteresis in magnetics, with practical implications for electric machines and transformers where hysteresis plays a critical role.

Abstract

In this note we discuss the numerical solution of the eddy current approximation of the Maxwell equations using the simple Pragmatic Algebraic Model to include hysteresis effects. In addition to the more standard time-stepping approach we propose a space-time finite element method which allows both for parallelization and adaptivity simultaneously in space and time. Numerical experiments confirm both approaches yield the same numerical results.

Figures (5)

• Figure 1: Left: The domain $\Omega = (0,1)^2$ consisting of two materials $\Omega_{cu}$ (in blue) and $\Omega_{fe}$ (in red). Right: The space-time mesh $Q = \Omega \times (0,T)$, which has 100 time slices in temporal direction, 53.530 nodes and 293.400 elements.
• Figure 2: Left: The magnetic flux density $B_x$ over time. Right: The BH-curve indicating the hysteresis effect.
• Figure 3: Left: The domain $\Omega$ of the transformer consisting of the iron core $\Omega_{fe}$ (in gray), the windings $\Omega_{cu}$ (in green and red) and air $\Omega_a$ (in blue). Right: The space-time mesh $Q = \Omega \times (0,T)$, which has 100 time slices in temporal direction, 91.405 nodes and 206856 elements.
• Figure 4: Left: The magnetic flux density $B_x$ over time. Right: The BH-curve indicating the hysteresis effect.
• Figure 5: The current density spline $\widetilde{j}$ (in blue) interpolated with respect to the measured values (in orange) of test case 1A from teamproblem32.