Space-time Multi-scale model reduction techniques for frictional contact - Efficient a posteriori and a priori strategies

Industrial complex problems involving large contact zones may lead to prohibitive time of computations. 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. 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. 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. 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.

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