Ted Belytschko

A finite element method for crack growth without remeshing

SUMMARY An improvement of a new technique for modelling cracks in the nite element framework is presented. A standard displacement-based approximation is enriched near a crack by incorporating both discontinuous elds and the near tip asymptotic elds through a partition of unity method. A methodology that constructs the enriched approximation from the interaction of the crack geometry with the mesh is developed. This technique allows the entire crack to be represented independently of the mesh, and so remeshing is not necessary to model crack growth. Numerical experiments are provided to demonstrate the utility and robustness of the proposed technique. Copyright ? 1999 John Wiley & Sons, Ltd.

Elastic crack growth in finite elements with minimal remeshing

A minimal remeshing finite element method for crack growth is presented. Discontinuous enrichment functions are added to the finite element approximation to account for the presence of the crack. This method allows the crack to be arbitrarily aligned within the mesh. For severely curved cracks, remeshing may be needed but only away from the crack tip where remeshing is much easier. Results are presented for a wide range of two-dimensional crack problems showing excellent accuracy. Copyright

Meshless methods: An overview and recent developments

Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.

Extended finite element method for cohesive crack growth

The extended finite element method allows one to model displacement discontinuities which do not conform to interelement surfaces. This method is applied to modeling growth of arbitrary cohesive cracks. The growth of the cohesive zone is governed by requiring the stress intensity factors at the tip of the cohesive zone to vanish. This energetic approach avoids the evaluation of stresses at the mathematical tip of the crack. The effectiveness of the proposed approach is demonstrated by simulations of cohesive crack growth in concrete.

Arbitrary discontinuities in finite elements

A technique for modelling arbitrary discontinuities in finite elements is presented. Both discontinuities in the function and its derivatives are considered. Methods for intersecting and branching discontinuities are given. In all cases, the discontinuous approximation is constructed in terms of a signed distance functions, so level sets can be used to update the position of the discontinuities. A standard displacement Galerkin method is used for developing the discrete equations. Examples of the following applications are given: crack growth, a journal bearing, a non-bonded circular inclusion and a jointed rock mass. Copyright

MODELING HOLES AND INCLUSIONS BY LEVEL SETS IN THE EXTENDED FINITE-ELEMENT METHOD

A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (S. Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the extended finite-element method (X-FEM) (N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition, the level set function is used to develop the local enrichment for material interfaces. Numerical examples in two-dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique.

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X-FEM) for three-dimensional crack modelling is described. A discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three-dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright

Atomistic simulations of nanotube fracture

The fracture of carbon nanotubes is studied by atomistic simulations. The fracture behavior is found to be almost independent of the separation energy and to depend primarily on the inflection point in the interatomic potential. The rangle of fracture strians compares well with experimental results, but predicted range of fracture stresses is marketly higher than observed. Various plausible small-scale defects do not suffice to bring the failure stresses into agreement with available experimental results. As in the experiments, the fracture of carbon nanotubes is predicted to be brittle. The results show moderate dependence of fracture strength on chirality.

Arbitrary branched and intersecting cracks with the extended finite element method

SUMMARY Extensions of a new technique for the nite element modelling of cracks with multiple branches, multiple holes and cracks emanating from holes are presented. This extended nite 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 elds 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 dierent 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. Copyright ? 2000 John Wiley & Sons, Ltd.

A new implementation of the element free Galerkin method

Abstract Element free Galerkin methods (EFG) are methods for solving partial differential equations with moving least squares interpolants. EFG methods require only nodal data; no element connectivity is needed. In a previous implementation of the EFG method, Lagrange multipliers were used to enforce the essential boundary condition. However, the use of Lagrange multipliers increases the cost of solving the linear algebraic equations. A new implementation is developed based on a modified variational principle in which the Lagrange multipliers are replaced at the outset by their physical meaning so that the discrete equations are banded. In addition, weighted orthogonal basis functions are constructed so the need for solving equations at each quadrature point is eliminated. Numerical examples show that the present implementation effectively computes stress concentrations and stress intensity factors at cracks with very irregular arrangements of nodes; the latter makes it very advantageous for modelling progressive cracking.