Multi-scale acceleration techniques for non-linear analysis of structures with frictional contact
Résumé
Industrial complex problems involving large contact zones may lead to prohibitive time of computations (non-linear and non-smooth behavior). Unfortunately, classic incremental solvers (e.g. Newton-Raphson) for contact problems could defeat the deployment of model reduction techniques well known for their efficiency to reduce time of computing. We propose to tackle the contact problem with the non-linear iterative LATIN (Large Time Increment) solver [1]. Then, a non-incremental time-space approach is adopted to solve the problem allowing the use of time- space model reduction technique. From a posteriori analyses using SVD (Singular Value Decomposition) of contact problems solutions, we exemplify the multiscale content of time-space reduced basis. Each of this basis vectors depicts a particular scale of the solution of the problem emphasizing the scale separability of contact problems. Moreover, depending on the complexity of the problem, its solution can be compressed into a small number of vectors as in [2,3,4]. We propose to take advantage of these scale separability by making analogies with multigrid solvers and namely the non-linear FAS (Full Approximation Scheme) multigrid solver [5]. Particular iterative solvers (called smoothers: PCG, GS ...) are able to damp rapidly the high-frequency components of the error through iterations whereas low-frequency components are slowly captured. Thus, coarser grids are used in order to correct low-frequency components. Similarly, we propose to consider as coarse grids a representation of iterated solutions over the first modes of the reduced basis (containing large scale information). The smoother is the non-linear LATIN solver and the overall strategy consists in a combination between the FAS multigrid solver and the LATIN method. Such a method increases convergence rate (in comparison to the LATIN method) and provides a well-suited approach for the parametric study framework [6]. [1] Boucard, P. A., Champaney, L. (2003). A suitable computational strategy for the parametric analysis of problems with multiple contact. IJNME, 57(9), 1259-1281.[2] Kerfriden P, Gosselet P, Adhikari S, Bordas S. Bridging proper orthogonal decomposition methods and augmented Newton-Krylov algorithms: An adaptive model order reduction for highly nonlinear mechanical problems. Computer Methods in Applied Mechanics and Engineering 2011; 200:850 – 866. [3] Chinesta, F., Leygue, A., Bordeu, F., et al. (2013). PGD-based computational vademecum for efficient design, optimization and control. ACME, 20(1), 31-59.[4] Vitse, M., Néron, D., Boucard, P. A. (2014). Virtual charts of solutions for parametrized nonlinear equations. Computational Mechanics, 1-11. [5] Giacoma, A., Dureisseix, D., Gravouil, A., Rochette, M. (2014). A multiscale large time increment/FAS algorithm with time space model reduction for frictional contact problems. IJNME, 97(3), 207-230.[6] Giacoma, A., Dureisseix, D., Gravouil, A., Rochette, M. (2015). Toward an optimal a priori reduced basis strategy for frictional contact problems with LATIN solver. CMAME