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Fall 2008 Schedule

Friday, November 7th
Inverse Problems Seminar
Speaker: Wilkins Aquino
School of Civil and Environmental Engineering Cornell University, Ithaca, NY
Title: Proper Orthogonal Decomposition (POD) for Model Reduction and Basis Enrichment in the Generalized Finite Element Method
Location: Amos Eaton 402
Time: 3:00-4:00 pm

The proper orthogonal decomposition technique has received wide attention in recent years in the context of both model reduction and signal processing. This presentation will explore two applications of proper orthogonal decomposition: a) as a means to model reduction for the inverse characterization of viscoelastic material properties using a vibroacoustic framework, and b) as a strategy for basis enrichment in the context of the generalized finite element method. In Part a), a strategy will be given for the efficient sampling of a material parameter space in order to construct a robust reduced-order model. The reduced-order model is then combined with a global optimization technique to identify estimates to the viscoelastic material properties of a fluid immersed solid from vibroacoustic tests. The sampling process is shown to maximize the generalization capabilities of the reduced-order model over the material search space for a minimal number of full-order analyses. In Part b), a methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that can capture the nature of the solution of the governing equations. Proper orthogonal decomposition 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, one expects that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. The potential of this approach is demonstrated through three numerical examples. Best-approximation studies that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis will be shown.



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