Proper generalized decomposition solutions within a domain decomposition strategy

Abstract

Domain decomposition strategies and proper generalized decomposition are efficiently combined to obtain a fast evaluation of the solution approximation in parameterized elliptic problems with complex geometries. The classical difficulties associated to the combination of layered domains with arbitrarily oriented midsurfaces, which may require in-plane-out-of-plane techniques, are now dismissed. More generally, solutions on large domains can now be confronted within a domain decomposition approach. This is done with a reduced cost in the offline phase because the proper generalized decomposition gives an explicit description of the solution in each subdomain in terms of the solution at the interface. Thus, the evaluation of the approximation in each subdomain is a simple function evaluation given the interface values (and the other problem parameters). The interface solution can be characterized by any a priori user-defined approximation. Here, for illustration purposes, hierarchical polynomials are used. The repetitiveness of the subdomains is exploited to reduce drastically the offline computational effort. The online phase requires solving a nonlinear problem to determine all the interface solutions. However, this problem only has degrees of freedom on the interfaces and the Jacobian matrix is explicitly determined. Obviously, other parameters characterizing the solution (material constants, external loads, and geometry) can also be incorporated in the explicit description of the solution. KEYWORDS domain decomposition, parameterized solutions, proper generalized decomposition, reduced-order models INTRODUCTIONProper generalized decomposition (PGD 1-4 ) has proven its advantages and applicability in many parameterized problems. PGD reduces the computational complexity induced by a large number of dimensions (the sum of the number of spatial dimensions plus the number of parameters) to the iterative resolution of low dimensional problems, usually one dimensional (1D) or two dimensional (2D). Moreover, the PGD approach provides an explicit expression of the approximated solution. Thus, the online phase, that is, the evaluation of the approximation given the parameters, is very fast and does not require any extra interpolation or another solve (typical, for instance, in Proper Orthogonal Decomposition strategies).However, PGD has never been combined successfully with domain decomposition (DD) strategies. The methodology proposed here makes it possible. The resolution is divided into 2 phases. The first one builds (offline) a PGD model (ie, an

The Coarse Mesh Condensation Multiscale Method for parallel computation of heterogeneous linear structures without scale separation

“…Other multiscale methods or subdomain methods with appropriate reduction have been proposed, such as e.g. in [17,18,19,20]. Another approach which has similarities with the method proposed in this work is the Multiscale Finite Element method [21,22], where shape functions obtained from fine scale calculations in each coarse mesh elements substitute the polynomial shape functions.…”

A parallel noninvasive multiscale strategy for a mixed domain decomposition method with frictional contact

“…Future works also concern the automatic building of macromodes to face highly heterogeneous structures and irregular subdomains inspired by the works of Spillane et al about automatic coarse space for overlapping Schwarz methods and nonoverlapping FETI and balancing domain decomposition approaches. Extensions to model reduction techniques are also considered, either to tackle quasi‐static problem with many time steps or to faster prototype the assemblies by using a simplified representation of the subdomains and interfaces …”

6 Localized model reduction for parameterized problems

“…In the context of the proper generalized decomposition method (Chapter 3 of this volume of Model order reduction) a domain decomposition strategy has been proposed in [55], and in [94] hierarchical model reduction [112,94,92,102] has been combined with an iterative substructuring method.…”