XLME interpolants, a seamless bridge between XFEM and enriched meshless methods

Abstract

In this paper. we develop a method based on local maximum entropy shape functions together with enrichment functions used in partition of unity methcx:ls to discretize problems in linear elastic fracture mechanics. We obtain improved accuracy relative to the standard extended finite element method at a comparable computational cost. In addition. we keep the advantages of the LME shape functions. such as smootlmess and non-negativity. We show numerically that optimal convergence (same as in FEM) for energy norm and stress intensity factors can be obtained through the use of geometric (fixed area) errriclnnent with no special treatment of the nodes near the crack such as blending or shifting.Keywords Local maximum entropy· Convex approximation· Meshless methods· Extrinsic enrichment S. P. A. Bordas's ORCID ID is 0000-0001-7622-2193.

Phase-field modeling of fracture in linear thin shells

“…Since analytical solutions provide limited information, there has been a keen interest in numerically simulating fracture in thin shells in recent years. However, despite the advances made in modeling fracture for solid bodies [1,2,3,4,5], fracture in thin bodies remains a challenge due to the complex interplay between cracks and the shell kinematics and geometry.…”

“…Recently, nonlinear manifold learning techniques have been exploited to parametrize 2D sub-domains of a point-set surface, which are then used as parametric patches and glued together with a partition of unity [38,21]. Here, we combine this methodology with local maximum-entropy (LME) meshfree approximants [39,40,5] because of their smoothness, robustness, and relative ease of quadrature compared with other meshfree approximants.…”

Efficient implementation of Galerkin meshfree methods for large-scale problems with an emphasis on maximum entropy approximants

“…Because the LME basis functions do not satisfy the Kronecker-delta property at nodes, these schemes are referred to as approximants instead of interpolants. The capabilities of LME approximants have been examined in a variety of computational mechanics applications, such as linear and nonlinear elasticity [25,26], plate [27] and thin-shell analysis [28,29], convection-di↵usion problems [30,31], and phase-field models of biomembranes [32,33] and fracture mechanics [34,35,36].…”

XFEM fracture analysis by applying smoothed weighted functions with compact support

“…The idea of XFEM was extended into the meshfree method for linear elastic crack problems (Rabczuk & Belytschko, 2004;Ventura et al, 2002;Amiri et al, 2014) and the extended element-free Galerkin (XEFG) was also developed for cohesive cracks (Rabczuk & Zi, 2007). Meanwhile, Fries (2008) developed a new XFEG for removing the blending region and thereafter this method was applied for three-dimensional modeling of crack growth (Timon et al, 2010).…”

“…The advantage of this XEFG as compared with XFEM is that it has the highly smoothed property, the non-local interpolation property and higher-order continuity, being capable of improving the solution accuracy near the crack tip even though it may have some difficulties in applying Dirichlet boundary condition. Amiri et al (2014) provided an approach based on local maximum entropy (LME) shape functions along with other functions implemented in partition of unity techniques to discretize problems in linear elastic fracture mechanics. The implementation of LME shape functions does not create any troubles in using the boundary conditions since they meet the Kronecker-Delta property.…”