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Proper generalized decomposition solutions within a domain decomposition strategy

Article dans une revue avec comité de lecture
Auteur
HUERTA, Antonio
86136 Departament de Matematica Aplicada III [Barcelona]
NADAL, Enrique
300772 Universitat Politècnica de València = Universitad Politecnica de Valencia = Polytechnic University of Valencia [UPV]
ccCHINESTA SORIA, Francisco
86289 Laboratoire Procédés et Ingénierie en Mécanique et Matériaux [PIMM]
111023 École Centrale de Nantes [ECN]

URI
http://hdl.handle.net/10985/13823
DOI
10.1002/nme.5729
Date
2018
Journal
International Journal for Numerical Methods in Engineering

Résumé

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.

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