A Reduced Order Model for Geometrically Parameterized Two-Scale Simulations

  • Guo, Theron (Eindhoven University of Technology)
  • Rokos, Ondrej (Eindhoven University of Technology)
  • Veroy, Karen (Eindhoven University of Technology)

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Two-scale simulations are often employed to study macroscopic structures with complicated non-linear microstructures [1]. To model such structures in one scale would require a very fine mesh due to the geometry of the microstructure, which is often intractable for practical solutions. For two-scale simulations, the microstructure is instead modelled on a Representative Volume Element (RVE), while the macrostructure can assume a simplified geometry with a coarse mesh. At every macroscopic integration point, a microscopic boundary value problem is defined and solved to return effective quantities such as stress and stiffness. Since the microscopic problem needs to be repeatedly solved, running two-scale simulations is still computationally expensive. One approach to overcome this issue is to replace the microscopic problem by a fast-to-evaluate reduced order model which remains accurate for a wide range of parameters. In this work, we construct such reduced order models by a combination of the Reduced Basis Method [2] and the Empirical Cubature Method (ECM) [3]. In this context, we discuss geometrical parameters and the consistent derivation of the effective quantities required by the macroscopic solver. To test the methodology, two-scale examples with geometrically parameterized microstructures, consisting of history-dependent elasto-plastic materials, are considered under large deformations. In comparison with the full two-scale solution, our reduced order model achieves a high accuracy as well as significant speed ups. [1] Geers, M. G., Kouznetsova, V. G., Brekelmans, W. (2010). Multi-scale computational homogenization: Trends and challenges. Journal of computational and applied mathematics, 234(7), 2175-2182. [2] Quarteroni, A., Manzoni, A., Negri, F. (2015). Reduced basis methods for partial differential equations: an introduction (Vol. 92). Springer. [3] Hernandez, J. A., Caicedo, M. A., Ferrer, A. (2017). Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Computer methods in applied mechanics and engineering, 313, 687-722.