Finite Element Computational Homogenization of Nonlinear Multiscale Materials in Magnetostatics

The increasing use of composite materials in the technological industry (automotive, aerospace, ... ) requires the development of effective models that account for the complexity of the microstructure of these materials and the nonlinear behaviour they can exhibit. In this paper we develop a multiscale computational homogenization method for modelling nonlinear multiscale materials in magnetostatics based on the finite element method. The method solves the macroscale problem by getting data from certain microscale problems around some points of interest. The missing nonlinear constitutive law at the macroscale level is derived through an upscaling from the microscale solutions. The downscaling step consists in imposing a source term and determining proper boundary conditions for microscale problems from the macroscale solution. For a two-dimensional geometry, results are validated by comparison with those obtained with a classical brute force finite element approach and a classical homogenization technique. The method provides a good overall macroscale response and more accurate local data around points of interest.