A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties

Abstract

International audienceWe here propose a multiscale numerical method for the solution of stochastic partial di erential equations with localized uncertainties. It is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales. An efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale, involving a deterministic operator, and local problems at a micro scale for which we have the possibility to use ne approximation spaces. Global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions. Convergence properties of these tensor based methods, which are closely related to spectral decompositions, benefit from the separation of scales. Different types of uncertainties are considered at the micro level. They may be associated with some variability in the operator or source terms, or even with some geometrical variability. In the latter case, specific reformulations of local problems using fictitious domain methods are introduced

Local enrichment of NURBS patches using a non-intrusive coupling strategy: Geometric details, local refinement, inclusion, fracture

“…In this paper, we propose an application of the non-intrusive technique [27,12,11,[28][29][30] to the NURBS context. The idea is to take the NURBS patch to be enriched as the global model.…”

“…An iterative coupling technique is used to perform the substitution in an exact but non-intrusive way: only interface data are transmitted from one model to the other and the global stiffness operator remains unchanged (independently of the shape of the local domain). This strategy has been applied in FEM for the modelling of crack propagation [11], for the modelling of localized uncertainties [28], for 3D-plate coupling [29] and for nonlinear domain decomposition [30]. Let us note that this methodology, involving the coupling of a global model and a local model in an iterative manner, has similarities with some hierarchical global/local methods in FEM: for example, the Chimera method [31], the method of finite element patches [32], numerical zoom [33] or the hp − d method [34][35][36].…”

Non-invasive global–local coupling as a Schwarz domain decomposition method: acceleration and generalization

“…Typically if F was a zone where material coefficients had strong variations, the Fine representation would follow the exact distribution whereas the Auxiliary representation could use a homogenized behavior. An application is the case where the Fine model is stochastic whereas the Auxiliary model is deterministic [10]. The load could also be simplified: l A is associated with body force applied to A and traction applied to ∂ A \ .…”

“…This philosophy was successfully applied in many different contexts like: the introduction of local plasticity and geometrical refinements [1], the computation of the propagation of cracks in a sound model [9], the evaluation of stochastic effects with deterministic computations [10], the taking into account of the exact geometry of connectors in an assembly of plates [11]. In [12] the method was used in order to implement a nonlinear domain decomposition method [13][14][15][16] in a non-invasive manner with Code_aster.…”

Development of a new, more regular, mortar method for the coupling of NURBS subdomains within a NURBS patch: Application to a non-intrusive local enrichment of NURBS patches

“…An iterative coupling technique is used to perform the substitution in an exact but non-intrusive way: only interface data are transmitted from one model to the other and the global stiffness operator remains unchanged (independently from the shape of the local domain). The performance of such a strategy has been highlighted in many applications (see, e.g., [12,32] for the modeling of crack propagation, [33] for the modeling of localized uncertainties, [34] for 3D-plate coupling and [35] for nonlinear domain decomposition).…”