Abstract
Abstract The Stable Generalized Finite Element Method (SGFEM) is essentially an improved version of the Generalized Finite Element Method (GFEM). Besides of retaining the good flexibility for constructing local enriched approximations, the SGFEM has the advantage of presenting much better conditioning than that of the conventional GFEM. Actually, bad conditioning is well known as one of the main drawbacks of the GFEM, while affecting severely the precision of the numerical scheme used for solving the linear system associated to the problem. Despite of its consistent mathematical basis, the numerical experiments so far conducted on using SGFEM are not yet clearly conclusive, especially regarding the robustness of the method. Therefore, the main purpose of the present paper is to give a contribution in this direction, through further investigating the SGFEM accuracy and stability. In particular, the so called Flat-Top SGFEM is a recent proposed version of the method hereby considered. As a flat-top Partition of Unit (PoU) is used for constructing the augmented approximation space with polynomial enrichments this version of the method is called SGFEM with flat-top PoU, or simply FT-SGFEM. Some computational aspects are briefly addressed, as the ones related to the implementation and integration of the flat-top for 2-D analysis. The numerical simulations consist essentially of linear analysis of panels presenting edge cracks and reentrant corners on its boundaries. Our findings from the numerical tests done are highly relevant regarding accuracy of the SGFEM versions, which present order of convergence similar to the conventional GFEM. Moreover, the measure of stability given by the scaled condition number presented in particular by the FT-SGFEM is comparable to the conventional FEM order.
Highlights
The Generalized Finite Element Method GFEM is a Partition of Unity PoU based Galerkin method, according to which the basic approximation space provided by a PoU is enlarged by shape functions constructed trough the product of the PoU by functions with good approximation skills, referred to as enrichment functions
Owing to such special feature and considering that the unity is always taken as the first component of the set of enrichment functions, the GFEM/XFEM can be understood as an extension of the conventional Finite Element Method FEM for which the local approximations provided by the shape functions are enlarged by means of enrichment functions
4 CONCLUSIONS The numerical investigation carried out with the GFEM versions has included the use of a flat-top partition of unity for constructing the augmented or enriched approximation space
Summary
The Generalized Finite Element Method GFEM is a Partition of Unity PoU based Galerkin method, according to which the basic approximation space provided by a PoU is enlarged by shape functions constructed trough the product of the PoU by functions with good approximation skills, referred to as enrichment functions. A mesh of finite elements is used to provide a PoU, which is commonly defined through the piecewise linear Lagrangian shape functions embodied in the elements Owing to such special feature and considering that the unity is always taken as the first component of the set of enrichment functions, the GFEM/XFEM can be understood as an extension of the conventional Finite Element Method FEM for which the local approximations provided by the shape functions are enlarged by means of enrichment functions. The problems considered for the computational experiments hereby reported consist of two-dimensional linear analysis of panels presenting edge cracks and reentrant corners These problems are typically useful to demonstrate the efficacy of the GFEM for exploring special enrichments.
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