Stable 3D XFEM/vector level sets for non‐planar 3D crack propagation and comparison of enrichment schemes

Abstract

Summary We present a three‐dimensional vector level set method coupled to a recently developed stable extended finite element method (XFEM). We further investigate a new enrichment approach for XFEM adopting discontinuous linear enrichment functions in place of the asymptotic near‐tip functions. Through the vector level set method, level set values for propagating cracks are obtained via simple geometrical operations, eliminating the need for solution of differential evolution equations. The first XFEM variant ensures optimal convergence rates by means of geometrical enrichment, ie, the use of enriched elements in a fixed volume around the crack front, without giving rise to conditioning problems. The linear enrichment approach, significantly simplifies implementation and reduces the computational cost associated with numerical integration, while providing nonoptimal convergence rates similar to standard finite elements. The 2 dicretization schemes are tested for different benchmark problems, and their combination to the vector level set method is verified for nonplanar crack propagation problems.

Discrete and Phase Field Methods for Linear Elastic Fracture Mechanics: A Comparative Study and State-of-the-Art Review

“…As an alternative, aiming at combining advantages of both explicit and implicit representations, Fries and Baydoun [141] proposed a method where level set functions were directly computed from explicit crack representations using linear segments (2D) or triangles (3D). Similarly, in the vector level set method [142][143][144] linear segments (2D) or quadrilaterals (3D) are used to update the level set description of the crack and are subsequently discarded. Another instance of a method combining elements from both types of representations is the method of Sadeghirad et al [145], where an explicit representation is constructed in order to correct the level set representation by removing disconnected parts of the crack.…”

Fracture mechanism simulation of inhomogeneous anisotropic rocks by extended finite element method

“…In order to take into account the anisotropy in computations, Asadpoure et al [16,46,47] introduced a set of enrichment functions inspired by analytical solutions of Sih et al [48] and Viola et al [49] which used the notion of complex numbers. Recently XFEM has been developed to model a 3D crack propagation [50][51][52][53] and multi-crack growth [24][25][26]. In addition, in the field of rock engineering, XFEM is one of the powerful tools that has been successfully applied to simulate hydraulic fracturing in hydrocarbon reservoirs, given the inherent anisotropy of the rock formations [54][55][56].…”

“…In addition, the concept of T-stress is incorporated into the formulation of the stress field near a crack tip in an inhomogeneous anisotropic material in order to introduce a new criterion called "Inhomogeneous Anisotropic Maximum Tangential Stress (IAMTS)" to predict the crack initiation angle and fracture path in such materials. In three dimensions, incorporating such higher order terms require special treatments proposed recently in [50][51][52].…”

An improved crack tracking algorithm with self‐correction ability of the crack path and its application in a continuum damage model

“…The LSM, which was developed by Osher and Sethian, determined the crack propagation direction by solving the Hamilton‐Jacobi equation and has been widely used in practical cracking problems. However, the standard LSM has some disadvantages in the description of the crack propagation process, and so, many studies for improving the LSM have been performed in the XFEM and meshfree method . Oliver et al proposed the GTA based on solving a thermal‐like problem at the first of each loading step for tracking multiple crack paths within the framework of the EFEM; however, it has a high computational cost and involves some programming complexity.…”

“…However, the standard LSM has some disadvantages in the description of the crack propagation process, [4][5][6] and so, many studies for improving the LSM have been performed in the XFEM and meshfree method. [7][8][9] Oliver et al 10,11 proposed the GTA based on solving a thermal-like problem at the first of each loading step for tracking multiple crack paths within the framework of the EFEM; however, it has a high computational cost and involves some programming complexity. Furthermore, the numerical results of the GTA are sensitive to the Dirichlet boundary conditions applied to the crack tracking problem, and so, the improvements of the GTA have also been studied by many researchers to overcome these drawbacks, such as a domain crack TA.…”