Sequential Subspace Mode Adaptation for the Reduced-Order Homogenization of Dissipative Microstructures using E3C Hyper-Reduction

Abstract

Three-dimensional inelastic computational homogenization of complex engineering components requires a multitude of nonlinear microstructural simulations, making it computationally expensive. This work investigates a projection-based model order reduction (pMOR) method with 'Sequential Subspace Mode Adaptation', which can be easily integrated into existing codes using linear subspaces. Starting with a 'conventional' linear subspace strain approximation, the dynamic online construction of a second -- lower dimensional -- affine subspace embedded in the linear subspace determined offline leads to a further reduction of the dimensionality. A second novelty is the outline of the E3C hyper-reduction method for non-crystalline dissipative materials with internal variables, introducing a viscous regularization of non-differentiable stress-strain relations. In addition, a theoretical discussion is provided, illustrating that the E3C method aims at satisfaction of a projected and hyper-reduced variant of the classical Hill-Mandel macro-homogeneity condition. The latter theoretically implies equivalence with the high-dimensional model and satisfaction of both the hyper-reduced weak equilibrium and compatibility conditions. The influence of training batch size, material nonlinearity, and microstructure on the performance are evaluated through parameter studies. Three-dimensional elastoplastic two-scale simulations with hundreds of thousands of macroscopic degrees of freedom illustrate the efficiency and accuracy, with computational times approaching those of single scale simulations.