A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics

Computer Methods in Applied Mechanics and Engineering

Duflot

2007

Self Cite

130

102

The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). In each element, the local estimator is the L2 norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery obtained from the raw nodal XFEM displacements. The XMLS construction is taylored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. The XMLS shape functions are constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two and three dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is always close to 1. and improves upon mesh refinement. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C0. Parametric studies of the general performance of the estimator are also carried out.

Section: Extended Finite Element Methodsmentioning

confidence: 99%

Derivative recovery and a posteriori error estimate for extended finite elements

Computer Methods in Applied Mechanics and Engineering

Duflot

2007

Self Cite

130

102

“…Three different subdomains have to be considered [5] Figure 6d. The [26]. Right: the two cracks are allowed to overlap, this is the formulation adopted in this paper.…”

Section: Tracking the Crack Pathmentioning

confidence: 99%

“…The failure mechanism is petalling. We have performed similar computations in [26] Figure 18 for both discretizations. As can be seen, multiple cracking occurs including crack junctions.…”

Section: Taylor Bar Impactmentioning

confidence: 99%

“…They enrich the moving least squares (MLS) basis functions around the crack tip and improve significantly the convergence behavior. 3 Rabczuk et al proposed the extended element free Galerkin (XEFG) method for cohesive crack initiation, growth and junction in two and three dimensional statics and dynamics, but the closure of the crack along the front is ensured through near-tip enrichment which vanishes along this front [25,26,27] . As noted in [26], the polar coordinate system along a crack front is not well-defined at the kinks.…”

Section: Introductionmentioning

confidence: 99%

“…3 Rabczuk et al proposed the extended element free Galerkin (XEFG) method for cohesive crack initiation, growth and junction in two and three dimensional statics and dynamics, but the closure of the crack along the front is ensured through near-tip enrichment which vanishes along this front [25,26,27] . As noted in [26], the polar coordinate system along a crack front is not well-defined at the kinks. While the fields spanning the space of linear elastic fracture mechanics solutions are known for small strains, this is the case neither in large strain nor for non-linear materials in general.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Three-dimensional non-linear fracture mechanics by enriched meshfree methods without asymptotic enrichment

IUTAM Symposium on Discretization Methods for Evolving Discontinuities

Zi²,

Rabczuk

2007

Self Cite

This paper presents a three-dimensional, extrinsically enriched meshfree method for initiation, branching, growth and coalescence of an arbitrary number of cracks in non-linear solids including large deformations, for statics and dynamics. The novelty of the methodology is that only an extrinsic discontinuous enrichment and no near-tip enrichment is required. Instead, a Lagrange multiplier field is added along the crack front to close the crack. This decreases the computational cost and removes difficulties involved with a branch enrichment. The results are compared to experimental data, and other simulations from the literature to show the robustness and accuracy of the method.

Meshfree Discretization Methods for Solid Mechanics

Rabczuk

Encyclopedia of Aerospace Engineering

Askes

2010

Meshfree methods are used for the spatial discretization of partial differential equations, but in contrast to finite element methods, they do not employ elements in the construction of the interpolants. Instead, a set of nodes is accompanied by a domain of influence for each node, whereby the overlap of domains of influence accounts for the interconnectivity between nodes. In this chapter, we will treat the formulation, implementation, and application to solid mechanics of meshfree methods. The focus will be on the element‐free Galerkin method but we will also provide an overview of other meshfree methods. As regards the applications, we will treat two major classes: firstly, incorporating discontinuities in meshless interpolations and secondly, exploiting the hypercontinuity of meshfree shape functions. Finally, we will discuss some urban myths surrounding meshfree methods, the relation with the new generation of node‐based finite element methods, and the future of meshfree methods.