Development and comparison of three-dimensional h-conforming and b-conforming multiscale formulations for composites accounting for eddy currents

  • Niyonzima, Innocent (Université Grenoble Alpes - G2ELab)
  • Marteau, Antoine (Université Grenoble Alpes - G2ELab)
  • Meunier, Gérard (CNRS - G2ELab)
  • Galopin, Nicolas (Université Grenoble Alpes - G2ELab)
  • Chadebec, Olivier (CNRS - G2ELab)

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Multiscale and homogenization methods have now been popularly used in areas of physics such as mechanics, porous flows, biomedical engineering, electromagnetics etc. Their use leads to the reduction of the computational cost associated to the numerical modeling of problems with composites by replacing the heterogeneous composite material by an equivalent material with local homogeneous properties. Among widely used multiscale methods, the HMM method [1] also known as the FE² method in the mechanics community presents a competitive advantage for modeling multiscale problems with separated scales. The method consists in replacing the composite material by a homogeneous material for which a coarse mesh can be used at the macroscale. The missing constitutive law at the Gauss points of the macroscale are obtained by upscaling fields of interest from the mesoscale solution, such as the homogenized magnetic field h or the homogenized magnetic induction b from the mesoscale solution. The HMM algorithm uses an iterative scheme combining the coupled resolution of the macro-problem and meso-problems until convergence is reached. The authors have recently shown that the upscaling of the fields of interest for electromagnetic problems with eddy currents might require special upscaling techniques instead of the simple volume average usually used in mechanics and in some static electromagnetic problems [2]. The upscaled magnetic field h needed for b-conforming formulations and the upscaled electric field e needed for h-conforming formulations with the macroscale current density should be filtered, e.g., using the Helmholtz decomposition or the boundary average of the concerned fields at the boundary of the mesoscale problem (see [2]). In the present research work, the multiscale methods including the approach developed in [2] for the 2D problems is extended to 3D problems and applied to the homogenization of idealized 3D periodic soft magnetic composites with spherical inclusions. REFERENCES [1] Abdulle, A., Weinan, E., Engquist, B., & Vanden-Eijnden, E. (2012). The heterogeneous multiscale method. Acta Numerica, 21, 1-87. [2] A. Marteau, I. Niyonzima, G. Meunier, J. Ruuskanen, N. Galopin, P. Rasilo and O. Chadebec, (2023). “Magnetic Field Upscaling and B-Conforming Magnetoquasistatic Multiscale Formulation”, in IEEE Transactions on Magnetics, doi: 10.1109/TMAG.2023.3235208.