Generalized finite element method using proper orthogonal decomposition. (English) Zbl 1171.76416
Summary: A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low-order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low-order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best-approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76Q05 | Hydro- and aero-acoustics |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
POD; partition of unity; enrichment; enriched finite elements; generalized finite element methodReferences:
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