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.

Abstract The aim of this paper is to analyse two-dimensional linear elastic continuum containing multiple interacting cracks. Both the mechanical model and the numerical approach are addressed throughout the text as key concepts for the computational framework, whose main characteristics will be described. The Splitting Method is a decomposition method considered for mechanical modeling of multiple interacting cracks. Accordingly, the original problem is divided into a set of global and local sub-problems. The Generalized Finite Element Method (GFEM) is adopted aiming at finding accurate numerical solutions for local sub-problems. Such problems are conceived so as to consider the stress concentration and the effects of interaction on the cracks. The main findings are related to the effectiveness of the proposed combination between the Splitting Method and the GFEM to provide accurate results, as well as the versatility of the conceived computational framework for analyzing different scenarios, including cracks of multilinear shapes and mixed mode fractures. Finally, it is possible to verify that the GFEM provides precise results using simpler meshes, in comparison with standard FEM used, for example in Franc2D(r).

The Generalized Finite Element Method (GFEM) is a numerical method based on the Finite Element Method (FEM), presenting as its main feature the possibility of improving the solution by means of local enrichment functions. In spite of its advantages, the method demands a complex data structure, which can be especially benefited by the Object-Oriented Programming (OOP). Even though the OOP for the traditional FEM has been extensively described in the technical literature, specific design issues related to the GFEM are yet little discussed and not clearly defined. In the present article it is described an Object-Oriented (OO) class design for the GFEM, aiming to achieve a computational code that presents a flexible class structure, circumventing the difficulties associated to the method characteristics. The proposed design is evaluated by means of some numerical examples, computed using a code implemented in Python programming language.