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

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

Gupta, Varun ; Duarte ; C. Armando

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

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/17013/Gupta_Varun.pdf?sequence=1&isAllowed=y

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来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The global-local analysis procedure in the Finite Element Method is broadly used in industry for the analysis ofcracks or localized stress concentrations in large, complex, three-dimensional domains. However, the limitations ofthis technique are well-known. The global-local FEM (GL-FEM) involves two steps: First, the solution of the givenproblem is computed on a coarse, global, quasi-uniform mesh, in which the cracks or other local features need notbe discretized. The solution of this problem is then used as boundary conditions to solve another Finite Elementproblem, which is basically a local sub-domain, comprised of localized features (like cracks), extracted from theglobal domain.The efficacy of the so-called Generalized Finite Element Method (GFEM) in solving such multi-scaleproblems has been quite well proven in past few years. Therefore, combining the two approaches, going one stepfurther from Global-Local Finite Element Analysis, and using the local solution as an enrichment function for theglobal problem through the Partition of Unity framework of the Generalized Finite Element Method, gives rise to theGeneralized 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 thesemethods robust enough for their industrial applicability in a comprehensive manner. One of the issues surroundingthis GFEMg-l approach concerns the domain size of the local problem containing the complex localized features of astructural 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-dimensionalfracture mechanics problems. A theoretical proof and rigorous convergence studies are presented here to provide theguidelines 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.