Convergence analysis of the generalized finite element method with global-local enrichments

Gupta, Varun

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https://hdl.handle.net/2142/17013 Copy

Description

Title

Convergence analysis of the generalized finite element method with global-local enrichments

Author(s)

Gupta, Varun

Issue Date

2010-08-31T20:04:10Z

Director of Research (if dissertation) or Advisor (if thesis)

Duarte, C. Armando

Department of Study

Civil & Environmental Eng

Discipline

Civil Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

M.S.

Degree Level

Thesis

Keyword(s)

• Generalized Finite Element Method (GFEM)
• Extended finite element method
• global-local
• Fracture mechanics
• enrichment function
• multiscale
• convergence analysis
• inexact boundary conditions

Abstract

The global-local analysis procedure in the Finite Element Method is broadly used in industry for the analysis of cracks or localized stress concentrations in large, complex, three-dimensional domains. However, the limitations of this technique are well-known. The global-local FEM (GL-FEM) involves two steps: First, the solution of the given problem is computed on a coarse, global, quasi-uniform mesh, in which the cracks or other local features need not be discretized. The solution of this problem is then used as boundary conditions to solve another Finite Element problem, which is basically a local sub-domain, comprised of localized features (like cracks), extracted from the global domain.The efficacy of the so-called Generalized Finite Element Method (GFEM) in solving such multi-scale problems has been quite well proven in past few years. Therefore, combining the two approaches, going one step further from Global-Local Finite Element Analysis, and using the local solution as an enrichment function for the global problem through the Partition of Unity framework of the Generalized Finite Element Method, gives rise to the Generalized Finite Element Method with global-local enrichments (or GFEMg-l). As these classes of methods are relatively new, there are many issues which need to be addressed to make these methods robust enough for their industrial applicability in a comprehensive manner. One of the issues surrounding this GFEMg-l approach concerns the domain size of the local problem containing the complex localized features of a structural problem, and the focus of this study is to provide guidance to address this issue. This study focuses on coming up with guidelines for selecting the size of the enrichment zone for three-dimensional fracture mechanics problems. A theoretical proof and rigorous convergence studies are presented here to provide the guidelines for selecting the size of enrichment zone for practical problems. The effect of inexact boundary conditions, applied to the local problem, on the solution is also investigated.

Graduation Semester

2010-08

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http://hdl.handle.net/2142/17013

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Copyright 2010 by Varun Gupta. All rights reserved.

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