Extended virtual element method for two-dimensional linear elastic fracture E. Benvenuti a , A. Chiozzi a,* , G. Manzini b , N. Sukumar c a Department of Engineering, University of Ferrara, Via Saragat 1, 44122 Ferrara, Italy b Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Pavia, Italy, c Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA Abstract In this paper, we propose an eXtended Virtual Element Method (X-VEM) for two-dimensional linear elastic fracture. This approach, which is an extension of the standard Virtual Element Method (VEM), facilitates mesh-independent modeling of crack discontinuities and elastic crack-tip singularities on general polygonal meshes. For elastic fracture in the X-VEM, the standard virtual element space is augmented by additional basis functions that are constructed by multiplying standard virtual basis functions by suitable enrichment fields, such as asymp- totic mixed-mode crack-tip solutions. The design of the X-VEM requires an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment fields. An efficient scheme to compute the mixed-mode stress intensity factors using the domain form of the interaction integral is described. The formulation permits integration of weakly singular functions to be performed over the boundary edges of the element. Numerical experiments are conducted on benchmark mixed-mode linear elastic fracture problems that demonstrate the sound accuracy and optimal convergence in energy of the proposed formulation. Keywords: partition-of-unity enrichment; X-VEM; crack discontinuity; crack-tip singularity; mixed-mode fracture; polygonal meshes 1. Introduction Over the past two decades, significant attention has been devoted to the development of numerical techniques to solve problems that admit singular or discontinuous solutions such as fracture propagation in solids. Among these techniques, enriched finite element approxima- tions based on the partition-of-unity framework [1, 2] have received considerable attention. The eXtended Finite Element Method (X-FEM) [3] is one of the most successful methods to anal- yse fracture problems on unstructured triangular and quadrilateral meshes without requiring remeshing. For fracture simulations on polygonal meshes, extended finite element formula- tions have been proposed [4, 5] as well as the scaled boundary element method [6–8]. However, * Corresponding author Email addresses: [email protected] (E. Benvenuti), [email protected] (A. Chiozzi), [email protected] (G. Manzini), [email protected] (N. Sukumar) arXiv:2111.04150v1 [math.NA] 7 Nov 2021
Extended virtual element method for two-dimensional linear elastic fracture
Jun 04, 2023
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