Reliable Computation of Microscopic Solutions in Numerical Homogenization Context for Static Heat Conduction with Heterogeneous Heat Generation

  • Belgrand, Louis (CEA, DES, IRESNE, DEC, SESC, LMCP)
  • Ramière, Isabelle (CEA, DES, IRESNE, DEC, SESC, LMCP)
  • Largenton, Rodrigue (EDF R&D, Dpt MMC)
  • Lebon, Frédéric (Aix-Marseille Université, CNRS, Centrale Mars)

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Numerical homogenization is mainly used to determine effective properties of an equivalent macroscopic media of an heterogeneous material. However, in a multiscale context, there is a growing interest in the knowledge of the local behavior [1]. In this perspective, we would like to obtain the effective properties but also reliable microstructural solutions of steady-state heat conduction problems with heterogeneous conductivity and heat generation. The aim of this work is to discuss different numerical strategies to accurately take into account the heterogeneity of heat generation in a multiscale approach. Asymptotic developments indicate that a second-order homogenization is required [3]. However, the authors suggest that satisfactory results with a first-order like homogenization technique could be obtained considering the averaged source term in the microscale problem. Some more recent works [4] propose another first-order FE2 homogenization by including the volumetric heat generation rate in the homogenized macroscopic problem. In this case, the microscopic problem remains unchanged. A comparison of both approaches will be conducted, especially in regards to microscale fields. Moreover microscopic solutions, required at each integration point of the macroscale problem, must be computed in reasonable computation times to make this numerical coupling approach efficient. It is well known that periodic boundary conditions (BC) are much more precise than Dirichlet BC on the vicinity of the Representative Volume Element boundary, at the expense of computational time. In order to obtain the best efficiency, defined by the ratio between precision and CPU cost, it has been recently proposed [2] improvement of Dirichlet-like solutions. Such strategies will be tested against heterogeneous heat generation problem. [1] L. Belgrand, I. Ramière, R. Largenton, F. Lebon. Mathematics, 10, 4437, 2022. [2] L. Belgrand, I. Ramière, M. Josien, F. Lebon In preparation, 2023 [3] V. Blanc. Phd Thesis, Aix-Marseille University, 2009. [4] G.R. Ramos, T. dos Santos, R. Rossi. International Journal for Numerical Methods in Engineering, 111(6):553–580, 2017.