Plenary and Semi-Plenary Lectures
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| Title: |
| Asymptotic a posteriori bounds for the outputs of the Element Free Galerkin method |
| Lecturer: |
| Antonio Huerta |
| Abstract: |
A novel approach for implicit residual-type error estimation in mesh-free methods is presented. Upper and lower bounds of the error in energy norm can be computed with the ultimate goal of obtaining bounds for outputs of interest. It precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting in mesh-free methods.
Assessment of functional outputs of the solution (goal-oriented error estimation) in computational mechanics problems is a real need in standard engineering practice. Techniques providing bounds require using error estimators in the energy norm of the solution. Bounds for quantities of interest (functional outputs) are recovered combining upper and lower bounds of the energy error for both the original problem (primal) and a adjoint problem (associated with the selected functional output).
The need of obtaining reliable upper and lower bounds of the error for quantities of interest has motivated the use of residual error estimators, which are currently the only type of estimators ensuring computable bounds for the error. Classical residual type estimators require flux-equilibration procedures (hybrid-flux techniques) to properly set boundary conditions for local problems. Flux-equilibration requires a domain decomposition, which is natural in finite elements but not in mesh-free methods. And, moreover, it is performed by a complex algorithm, strongly dependent on the finite element type and requiring a particular data structure.
The idea of using flux-free estimators, based on the partition-of-the-unity concept and using local subdomains different than elements, has been already proposed in finite elements. The main advantage of the flux-free approach is the simplicity in the implementation. Obviously, this is especially important in the 3D case. From the mesh-less point of view, another advantage is the fact that the local subdomains where the error equation is solved are the support of the functions characterizing the partition of unity. This is a concept that also exists in mesh-free methods and thus the extension is possible. Moreover, boundary conditions of the local problems are trivial and the usual data structure of a code is directly employed.
To the authors knowledge implicit residual based estimators have not been proposed for mesh-free methods. However, these residual based approaches are now standard in finite elements because they are more mathematically sound, more precise and allow to compute upper and lower bounds for energy norms as well as functional outputs. |
 Antonio Huerta is full Professor of Applied Mathematics at the Universitat Politècnica de Catalunya (Barcelona, Spain) since 1993. He obtained his civil engineering degree in 1983 at the Universitat Politècnica de Catalunya, and a Ph. D. from Northwestern University in 1987 (USA). In 1989, he was awarded the Thomas A. Jaeger Prize by the International Association for Structural Mechanics in Reactor Technology and in 2002 was elected Fellow of the International Association of Computational Mechanics.
His research interests are focused in computational methods in applied sciences and engineering, in particular: finite elements and mesh-free methods, nonlinear computational mechanics, error estimation and adaptivity, convection-dominated flow as well as incompressible flows. He has published over eighty papers and chapters in books; recently, with Professor Jean Donéa, he has published the book entitled “Finite Element Methods for Flow Problems (Wiley, 2003).
He has served in numerous positions at his university (Chairman of the applied mathematics department, Board of trustees, Senate…) and governmental offices.
He is member of several editorial boards of prestigious journals in numerical methods and engineering mechanics. He is the director of the Laboratori de Càcul Numèric (LaCàN), www-lacan.upc.es, of the Universitat Politècnica de Catalunya. |
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