Multiscale method with patches for the solution of non-linear stochastic problems with localized uncertainties and non-linearities

International audienceThe quantification and propagation of localized uncertainties in complex multiscale models have gained an ever increasing interest in the scientific community during the last decades. The sources of uncertainties might affect the material behaviour, the boundary conditions. The numerical simulation of complex stochastic multiscale models by traditional monoscale approaches based on refinement or enrichment techniques is very demanding in terms of computational cost. Conversely, multiscale approaches have been designed to manage the high complexity arising from the solution of multiscale stochastic problems. Some of them allow to efficiently couple numerical models exhibiting critical local phenomena (e.g. localized uncertainties, non-linearities and geometrical defects) at different scales. In particular, the multiscale approach proposed in [1] relies on an overlapping domain decomposition method with patches that exploits the localized side of uncertainties. It leads to the definition of an efficient global-local iterative algorithm that requires successive solutions of simple global problems (with deterministic operator) over a deterministic domain and of complex stochastic local problems (with uncertain operator, load, source term or geometry) over patches of interest.In the present work, the approach is extended to non-linear elliptic stochastic problems with localized non-linearities. Convergence and robustness properties of the proposed algorithm are analyzed. The multiscale coupling approach appears to be flexible and non-intrusive as it allows considering different independent global and local models and solvers. At the local level, the stochastic problems are solved using sampling-based approaches. Sampling-based approaches only require evaluations of the solution of local deterministic models that can be performed in parallel and using deterministic solvers, therefore preserving the non-intrusive character of the multiscale coupling strategy. Furthermore, the sample set and the stochastic approximation space are sequentially enriched in order to control the accuracy of local solutions at the micro scale. Numerical examples illustrate the efficiency of the proposed method.References[1] M. Chevreuil, A. Nouy, and E. Safatly. A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties. Computer Methods in Applied Mechanics and Engineering, 255(0):255–274, 2013