A robust adaptive model reduction method for damage simulations D. Ryckelynck ⇑ , D. Missoum Benziane, S. Cartel, J. Besson MINES ParisTech, Centre des matériaux, CNRS UMR 7633, BP 87 91003 Evry Cedex, France article info Article history: Received 18 October 2010 Accepted 29 November 2010 Available online 12 January 2011 Keywords: APHR method POD Separated representation Continuum damage Bifurcation 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 sim- ulation 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 pro- posed scheme versus the amount of global linear solutions involved in the classical incremental New- ton–Raphson scheme. The quality of the proposed approximation is compared to the one provided by the classical snapshot Proper Orthogonal Decomposition (POD) method. Ó 2011 Published by Elsevier B.V. 1. Introduction Continuum damage elasto-plastic models are widely used for fa- tigue life prediction or crack growth on metallic components [1–3]. Due to the bifurcation of the solutions, the damage models are very sensitive to any model modification. Nonlocal models have been proposed to avoid the mesh dependency of the solution assuming the element size is small enough. For more details about mesh non- local models, we refer the reader to [4]. Various approaches are available to study the bifurcation modes related to the continuum damage models [5,4,6–8]. In this paper we assume that a numerical simulation aims to estimate one of the admissible solutions of the nonlinear continuous model. Only one prediction must be forecast per simulation. Model perturbations are introduced to obtain vari- ous predictions of the mechanical state. The reduced basis approx- imations introduce errors that can be amplified during the bifurcation of the solution. This paper aims to show that the APHR (A Priori Hyper Reduction) method proposed in [9] enables to build robust reduced approximations for continuum damage models based on Rousselier constitutive law [10,11]. This reduced approx- imation is a time/space separated representation of the displace- ment, which is defined over the whole time interval of the simulation and the entire domain. The development of large FE models increases the need of low order models created by model reduction methods. The availability of reduced-order models (ROMs) can greatly facilitate the solution of series of mechanical problems appearing in optimization prob- lems for instance. The formulation of the reduced-equations differs from the formulation of the detailed equations in the choice of the 0927-0256/$ - see front matter Ó 2011 Published by Elsevier B.V. doi:10.1016/j.commatsci.2010.11.034 ⇑ Corresponding author. E-mail address: david.ryckelynck@mines-paristech.fr (D. Ryckelynck). Computational Materials Science 50 (2011) 1597–1605 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci