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Analysis and applications of a generalized finite element method with global-local enrichment functions. (English) Zbl 1169.74597

Summary: This paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the $hp$ cloud method. The procedure combines classical global-local finite element method concepts with the partition of unity approach. It involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization. The local solutions are in turn used to enrich the global space using the partition of unity framework. The computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently.

The effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper. We also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution.

Key aspects of the computational implementation, in particular, the numerical integration of generalized FEM approximations built with global-local enrichment functions, are presented.

The method is applied to fracture mechanics problems with multiple cracks in the domain. Our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems. Our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global-local enrichment functions.

##### MSC:
 74S05 Finite element methods in solid mechanics 74R10 Brittle fracture