Numerical experiments with the Generalized Finite Element Method based on a flat-top Partition of Unity

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.

On the numerical integration in generalized/extended finite element method analysis for crack propagation problems

“…Additional modifications need to be performed for two-dimensional fracture mechanic problems, aiming to yield an optimal rate order to the condition number. These modifications have been pursued over the last past years in works such as Gupta et al (2013), Zhang et al (2014), Wu and Li (2015), Zhang et al (2016), Zhang et al (2018), Sato et al (2018), Sanchez-Rivadeneira and Duarte (2019) and Oliveira et al (2019).…”

“…The Stable Generalized Finite Element Method (SGFEM), of Babuška and Banerjee (2012), indicates a procedure to modify the enrichment, overcoming the ill-conditioning issue for one-dimensional problems and addressing the blending element problem too. Extensions of this procedure and other strategies have been proposed by Gupta et al (2013), Zhang et al (2014), Gupta et al (2015), Wu and Li (2015), Zhang et al (2016, 2018), Sato et al (2018) and Sanchez-Rivadeneira and Duarte (2019).…”

Preface to the special issue of the International Symposium on Solid Mechanics (MecSol 2017)

“…• Composite Materials: distortions in composite wing structures (Makinde et al, 2018), failure in composite pressure vessels (Vignoli and Savi, 2018), and dynamic analysis (Marques et al, 2018). • X-FEM, G-FEM, and BEM: crack propagation and X-FEM modelling (Angelo et al, 2018); G-FEM modelling (Sato et al, 2018) and dynamic analysis (Weinhardt et al, 2018); BEM analysis of buckling of anisotropic plates (Monteiro and Daros, 2018), and 3D frictional contact (Ubessi and Marczak, 2018). • Structural Reliability Methods and Reliability-Based Design Optimization: Multi-objective optimization (Passos and Luersen, 2018) and experimental crack identification (Oliveira Filho et al, 2018).…”

SGFEM: a numerical investigation of the two-scale simulation of 3D elastic fracture

“…The combined analysis of SGFEM with the global-local technique for fracture mechanics problems is investigated in Malekan and Barros [29] for plane stress and Reissner Mindlin plate problems. Different approaches to manage the numerical problems related to the presence of blending elements and the increasing of the condition number of the system of equations have also been proposed and can be found in Zhang et al [30], Wu and Li [31], Zhang et al [32], Sato et al [33], Zhang et al [34], Sanchez-Rivadeneira and Duarte [35].…”