A robust adaptive model reduction method for damage simulations

Abstract

In our opinion, many of complex numerical models in materials science can be reduced without losing their physical sense. Due to solution bifurcation and strain localization of continuum damage problems, damage predictions are very sensitive to any model modification. Most of the robust numerical algorithms intend to forecast one approximate solution of the continuous model despite there are multiple solutions. Some model perturbations can possibly be added to the finite element model to guide the simulation toward one of the solutions. Doing a model reduction of a finite element damage model is a kind of model perturbation. If no quality control is performed the prediction of the reduced-order model (ROM) can really differ from the prediction of the full finite element model. This can happen using the snapshot Proper Orthogonal Decomposition (POD) model reduction method. Therefore, if the expected purpose of the reduced approximation is to estimate the solution that the finite element simulation should give, an adaptive reduced-order modeling is required when reducing finite element damage models.We propose an adaptive reduced-order modeling method that enables to estimate the effect of loading modifications. The Rousselier continuum damage model is considered. The differences between the finite element prediction and the one provided by the adapted reduced-order model (ROM) remain stable although various loading perturbations are introduced. The adaptive algorithm is based on the APHR (A Priori Hyper Reduction) method. This is an incremental scheme using a ROM to forecast an initial guess solution to the finite element equations. If, at the end of a time increment, this initial prediction is not accurate enough, a finite element correction is added to the ROM prediction. The proposed algorithm can be viewed as a two step Newton–Raphson algorithm. During the first step the prediction belongs to the functional space related to the ROM and during the second step the correction belongs to the classical FE functional space. Moreover the corrections of the ROM predictions enable to expand the basis related to the ROM. Therefore the ROM basis can be improved at each increment of the simulation. The efficiency of the adaptive algorithm is checked comparing the amount of global linear solutions involved in the proposed scheme versus the amount of global linear solutions involved in the classical incremental Newton–Raphson scheme. The quality of the proposed approximation is compared to the one provided by the classical snapshot Proper Orthogonal Decomposition (POD) method.