SGFEM: a numerical investigation of the two-scale simulation of 3D elastic fracture
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(2020) 42:305
TECHNICAL PAPER
SGFEM: a numerical investigation of the two‑scale simulation of 3D elastic fracture K. F. Santos1 · F. B. Barros1 Received: 31 January 2020 / Accepted: 5 May 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract The Generalized/eXtended Finite Element Method (G/XFEM) is applied as a very useful tool in the resolution of complex structural models using an effective approach to represent the existence of cracks and other micro-defects. This is an unconventional formulation of the Finite Element Method (FEM), in that there is an expansion of the approximate solution field from the use of enrichment functions associated with the nodes. Enrichment functions can be singular functions derived from analytic deductions, polynomial functions or even functions resulting from other solution processes, such as the global–local strategy. The Stable Generalized Finite Method (SGFEM) is a variation of the G/XFEM with a simple modification in its enrichment functions, reducing the condition number of the stiffness matrix as well as the approximate error in the so-called blending elements. Considering this under the global–local strategy, the solution quality and the conditioning of the stiffness matrix in 3D linear fracture problems are investigated here. The crack surface is described by the Heaviside discontinuous functions and singular/crack front functions on a local scale. The solution of the local problem is used to enrich the approximation in the global domain, which may cause bad conditioning of the resulting system of equations and poor approximation errors in the solution. In order to overcome this problem, its projection into the linear polynomial space, according to the SGFEM strategy, is subtracted from the enrichment. Numerical examples, with different load and crack configurations, of linear elastic fracture mechanics are employed. Differently from other works, the meshes of the two scales are kept constant. Only the number of nodes associated with the local enrichment functions is changing. The impact on the accuracy and conditioning of the analysis is assessed, and the importance of using the SGFEM strategy is highlighted. Keywords Finite Element Method (FEM) · Partition of unity (PUM) · Generalized/eXtended Finite Element Method (G/ XFEM) · Fracture mechanics · Stable Generalized Finite Element Method (SGFEM) · Global–local strategy
1 Introduction The Generalized Finite Element Method (GFEM) was initially proposed by Babuška and other researchers by several denominations, such as “Special Finite Element Method ”, [1], “Cloud Method”, [2], “Finite Element Method of the Partition of Unity”, [3]. In spite of certain particularities, these methods have in common the use of the Partition of Technical Editor: João Marciano Laredo dos Reis. * K. F. Santos [email protected] F. B. Barros [email protected] 1
Escola de Engenharia, Av. Antônio Carlos, Pampulha ‑ Belo Horizonte, MG 6627, Brazil
Unity concept to generate the space of enriched solution [4]. It
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