Computational Homogenization for Laminated Ferromagnetic Cores in Magnetodynamics

Abstract

In this paper, we investigate the modeling of ferromagnetic multiscale materials. We propose a computational homogenization technique based on the heterogeneous multiscale method (HMM) that includes both eddy-current and hysteretic losses at the mesoscale. The HMM comprises: 1) a macroscale problem that captures the slow variations of the overall solution; 2) many mesoscale problems that allow to determine the constitutive law at the macroscale. As application example, a laminated iron core is considered.

Error estimates for finite element approximations of nonlinear monotone elliptic problems with application to numerical homogenization

“…We refer to [40,41] for multiscale simulations of the magnetostatics and magnetodynamics of such iron cores. Setting.…”

“…Those are the most general hypotheses for the maps A ε under which homogenization for (1) can be established, see [15,18,42]. Many physical processes can be modeled by parabolic partial differential equations (PDEs) of the form (1), e.g., non-Newtonian fluids, ferromagnetic materials or composites with nonlinear materials, see [12,41]. Using standard numerical methods, like the finite element method (FEM), to discretize the problem (1) in space leads to high computational cost as the small scale ε of the spatial heterogeneities of A ε has to be resolved.…”

Linearized Numerical Homogenization Method for Nonlinear Monotone Parabolic Multiscale Problems

“…K is the numerically homogenized map defined in (35). In particular, in the a priori error estimates of Theorems 4.3 and 4.4 the upscaling error is quantified by e HM M given by…”

“…where the numerically homogenized nonlinear map A 0,h k is given by (35). We note that the terms (51a) and (51b) can be bounded using Lemma 5.3.…”

“…We assume throughout this article that the equation (1) is of monotone type to guarantee the well-posedness of the problem, i.e., the maps A ε (x, ξ) = a ε (x, ξ)ξ are strongly monotone and Lipschitz continuous with respect to ξ ∈ R d , with constants independent of ε. This class of problems arises in many applications, e.g., material laws in elasticity or constitutive relations in magnetodynamics [34,35]. The nonlinearity of the tensor a ε in (1) with respect to the solution gradient ∇u ε makes the problem challenging both computationally and for the analysis because it combines the difficulties of the finescale structure of the data (with small variations at the microscopic scale) and the nonlinearity of the problem.…”