Many problems of engineering relevance in computational mechanics involve analysis of structuralbehavior spanning different spatial scales. Examples of such industrial problems include fracturein engine components, structural members of aircrafts, and pipeline joints. The presence of smallcracks can lead to failure of these structures, due to intense thermal and mechanical loadings.Therefore, engineering decisions regarding such structures require accurate response predictionmethodologies.The efficacy of the Generalized/eXtended Finite Element Method (GFEM or XFEM) in solvingproblems involving cracks, material interfaces or localized stress concentrations in large, complex,three-dimensional domains has been well established in the recent past. The superior propertiesof the GFEM/XFEM rely on the use of preselected enrichment functions that are known to ap-proximate the solution of a problem well. However, closed-form analytical enrichment functionsare not always available. This research work focuses on advances of a two-scale GFEM for theaccurate and efficient computation of the numerical solution for problems where only limited apriori knowledge about the solution is available. This method, termed as the Generalized FEMwith global-local enrichments (GFEM gl ) is based on the solution of interdependent global andlocal scale problems, and can be applied to a broad class of multiscale problems of relevance tothe industry. In this approach, the enrichment functions are obtained from the numerical solutionof a fine-scale boundary value problem defined around a localized region of interest. The localproblems focus on the resolution of fine-scale features of the solution, while the global problemaddresses the macro-scale structural behavior. The local solutions are embedded into the globalsolution space using the Partition of Unity Method.A rigorous a priori error estimate for the method is presented along with numerical verificationof convergence properties predicted by the estimate. The analysis shows optimal convergence ofthe method on problems with strong singularities and the method can deliver the same accuracy asdirect numerical simulations (DNS) while using much fewer degrees of freedom as compared tothe DNS.This document further reports on extensions of the method to two-scale fracture problemsexhibiting nonlinear material behavior. The nonlinear model problem focuses on structures withplastic deformations at regions that are orders of magnitude smaller than the dimensions of thestructural component. It is shown that the GFEMgl can produce accurate nonlinear solutions at acomputational cost much lower than available FEMs.The issue of ill-conditioning of the system of equations obtained with the GFEM/XFEM hasbeen well known since the inception of these methods more than a decade ago. The Stable GFEM(SGFEM) provides a robust, yet simple solution to this ill-conditioning. The SGFEM involves asimple local modification of the enrichments employed in the GFEM, which near-orthogonalizesthe enrichment space to the finite element approximation space. Another bonus feature of thismethod is the improved accuracy over the GFEM/XFEM. This work proposes the SGFEM fortwo- and three-dimensional fracture mechanics problems. It is shown that the available crack en-richment functions used in the GFEM/XFEM lead to inaccuracies with the SGFEM. Therefore,this work also proposes the use of additional enrichments to attain optimal convergence with theSGFEM in 2-D and 3-D. It is shown that the SGFEM with these additional enrichments leads tosignificant improvements on the numerical conditioning of the method at a negligible computa-tional cost. The accuracy and conditioning obtained with the SGFEM is compared with availableGeneralized FEM (GFEM).
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Improved conditioning and accuracy of a two-scale generalized finite element method for fracture mechanics
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