Higher order x-fem for curved cracks

This paper develops a generalized enriched finite element method to analyze the displacement and stress fields near a blunt crack tip by including the near-field analytic solutions to the shape functions of a conventional isoparametric element using the partition of unity finite element method. In addition, a transitional element is developed by introducing a weighting function to the added near-field solutions to ensure the displacement compatibility between the enriched and conventional elements. The weighting function is constructed by polynomials of the same order as the shape functions of corresponding conventional element. Thus, the enriched, transitional and conventional elements can be expressed in a unified formulation for easy computer implementation. The accuracy of this formulation is demonstrated by numerical examples. The results are in good agreement with theory and conventional finite elements with excessive fine mesh. The examples also show that the newly proposed enriched scheme is more accurate and computational efficient than the conventional isoparametric element with coarse mesh.

International Journal for …, 2002

A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended ÿnite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip ÿelds that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be deÿned by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems. This paper and a companion paper present further developments of the extended ÿnite element method (X-FEM) for modelling cracks and crack growth. The extended ÿnite element method alleviates much of the burden associated with mesh generation for objects with cracks by not requiring the ÿnite elements to conform to the crack surface. Moreover, it provides a convenient way for incorporating near-tip asymptotic ÿelds, so that good accuracy can be obtained for elastic fracture with relatively coarse meshes around the crack.

International Journal for …, 2000

In this paper, the extended finite element method is presented for large elasto-plastic deformation in 3D solid mechanics problems. The X-FEM computational algorithm is presented in the framework of Lagrangian description in order to model the arbitrary discontinuities in large deformations. In X-FEM, the discontinuous surfaces are represented independently of element boundaries and the process is accomplished by partitioning the domain with several tetrahedral sub-elements whose Gauss points are used for integration of the domain of elements. The discontinuity between two bodies is modeled by using the X-FEM technique and applying a modified level set enrichment function. In order to simulate the nonlinear behavior of materials, the Lagrangian plasticity formulation is coupled with the X-FEM technique. Finally, several numerical examples are analyzed to demonstrate the efficiency of the X-FEM technique in large plasticity deformations.

Int. J. Numer. Meth. Engng, 2000

Extensions of a new technique for the ÿnite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended ÿnite element method (X-FEM) allows the representation of crack discontinuities and voids independently of the mesh. A standard displacementbased approximation is enriched by incorporating discontinuous ÿelds through a partition of unity method. A methodology that constructs the enriched approximation based on the interaction of the discontinuous geometric features with the mesh is developed. Computation of the stress intensity factors (SIF) in di erent examples involving branched and intersecting cracks as well as cracks emanating from holes are presented to demonstrate the accuracy and the robustness of the proposed technique. 1742 C. DAUX ET AL.

2012

A new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions. The main novelty consists in modifying the weight function with an intrinsic enrichment which is discontinuous over the finite length of the crack, represented by a segment, but continuous all around the crack tips. An analytical function is used to introduce discontinuities that are incorporated in the kernel in a simple, multiplicative manner.

Journal of Zhejiang University SCIENCE A, 2007

International Journal for Numerical Methods in Engineering, 2013

The aim of this study is the analysis of the convergence rates achieved with domain energy integrals for the computation of the stress intensity factors (SIF) when solving 2-D curved crack problems with the extended finite element method (XFEM). Domain integrals, specially the J-integral and the interaction integral, are widely used for SIF extraction and provide high accurate estimations with finite element methods. The crack description in XFEM is usually realized using level sets. This allows to define a local basis associated with the crack geometry. In this work the effect of the level set local basis definition on the domain integral has been studied. The usual definition of the interaction integral involves hypotheses that are not fulfilled in generic curved crack problems and we introduce some modifications to improve the behavior in curved crack analyses. Despite the good accuracy of domain integrals, convergence rates are not always optimal and convergence to the exact solution cannot be assured for curved cracks. The lack of convergence is associated with the effect of the curvature on the definition of the auxiliary extraction fields. With our modified integral proposal, the optimal convergence rate is achieved by controlling the q-function and the size of the extraction domain.

Computer Methods in Applied Mechanics and Engineering, 2014

The aim of this paper is to provide a-priori error estimates for problems involving curved interfaces and solved with the linear or quadratic extended finite-element method (X-FEM), with particular emphasis on the influence of the geometry representation and the quadrature. We focus on strong discontinuity problems, which covers the case of holes in a material or cracks not subjected to contact as the main applications. The well-known approximation of the curved geometry based on the interpolated level-set function and straight linear or curved quadratic subcells is used, whose accuracy is quantified by means of an appropriate error measure. A priori error estimates are then derived, which depend upon the interpolation order of the displacement, and foremost upon the above error measure and the quadrature scheme in the subcells. The theoretical predictions are successfully compared with numerical experiments.

2013

International Journal of Solids and Structures

The weight function theory for three-dimensional elastic crack analysis received great attention after the work of Rice and . Several applications have been considered since then, particularly in the context of configurational stability, crack path prediction, stress intensity factor expansions, perturbation approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order to formulate the problem in terms of Cauchy Principal Value. In the present note, such hypothesis is further investigated and consequences discussed. A variational statement given in A. Salvadori, 2013 is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to accurate approximation of crack front propagation.