Improved conditioning and accuracy of a two-scale generalized finite element method for fracture mechanics

Many problems of engineering relevance in computational mechanics involve analysis of structural behavior spanning different spatial scales. Examples of such industrial problems include fracture in engine components, structural members of aircrafts, and pipeline joints. The presence of small cracks can lead to failure of these structures, due to intense thermal and mechanical loadings. Therefore, engineering decisions regarding such structures require accurate response prediction methodologies. The efficacy of the Generalized/eXtended Finite Element Method (GFEM or XFEM) in solving problems involving cracks, material interfaces or localized stress concentrations in large, complex, three-dimensional domains has been well established in the recent past. The superior properties of 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 functions are not always available. This research work focuses on advances of a two-scale GFEM for the accurate and efficient computation of the numerical solution for problems where only limited a priori knowledge about the solution is available. This method, termed as the Generalized FEM with global-local enrichments (GFEM gl ) is based on the solution of interdependent global and local scale problems, and can be applied to a broad class of multiscale problems of relevance to the industry. In this approach, the enrichment functions are obtained from the numerical solution of a fine-scale boundary value problem defined around a localized region of interest. The local problems focus on the resolution of fine-scale features of the solution, while the global problem addresses the macro-scale structural behavior. The local solutions are embedded into the global solution space using the Partition of Unity Method. A rigorous a priori error estimate for the method is presented along with numerical verification of convergence properties predicted by the estimate. The analysis shows optimal convergence of the method on problems with strong singularities and the method can deliver the same accuracy as direct numerical simulations (DNS) while using much fewer degrees of freedom as compared to the DNS. This document further reports on extensions of the method to two-scale fracture problems exhibiting nonlinear material behavior. The nonlinear model problem focuses on structures with plastic deformations at regions that are orders of magnitude smaller than the dimensions of the structural component. It is shown that the GFEMgl can produce accurate nonlinear solutions at a computational cost much lower than available FEMs. The issue of ill-conditioning of the system of equations obtained with the GFEM/XFEM has been 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 a simple local modification of the enrichments employed in the GFEM, which near-orthogonalizes the enrichment space to the finite element approximation space. Another bonus feature of this method is the improved accuracy over the GFEM/XFEM. This work proposes the SGFEM for two- 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 the SGFEM in 2-D and 3-D. It is shown that the SGFEM with these additional enrichments leads to significant 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 available Generalized FEM (GFEM)