Proper Generalized Decomposition solutions within a Domain Decomposition strategy

Author (s): Huerta, A., Nadal, E., and Chinesta F.
Journal: International Journal for Numerical Methods in Engineering

Volume: 113, Issue 13
Pages: 1972 – 1994
Date: 2018

Abstract:
Domain Decomposition strategies and the 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 mid-surfaces, 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 to solve a nonlinear problem to determine all the interface solutions. But 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, geometry) can also be incorporated in the explicit description of the solution.

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Bibtex:

@article {AH-HNC:18,
        Author = {Antonio Huerta and Enrique Nadal and Francisco Chinesta},
        Title = {Proper {G}eneralized {D}ecomposition solutions within a {D}omain {D}ecomposition strategy},
        Fjournal = {International Journal for Numerical Methods in Engineering},
        Journal = {Int. J. Numer. Methods Eng.},
        Volume = {113},
        Number = {13},
        Pages = {1972--1994},
        Year = {2018},
        Doi = {10.1002/nme.5729}
        }