Meshfree methods and extended finite element methods for arbitrary discontinuities

In the first part, a new vector level set method for modeling propagating cracks in the element free Galerkin method is presented. With this approach only nodal data are used to describe the crack; no geometrical entity is introduced for the crack trajectory, and no partial differential equations need be solved to update the level sets. The nodal description is updated as the crack propagates by geometric equations. In addition, new methods for crack approximations in EFG are introduced, using a jump function to account for the displacement discontinuity along the crack faces and the Westergard's solution enrichment near the crack tip. These enrichments, being extrinsic, can be limited only to the nodes surrounding the crack and are naturally coupled to the level set crack representation.; Another meshfree method with discontinuous radial basis functions and their numerical implementation for elastic problems are presented in the second part. We study the following radial basis functions: the multiquadratic (MQ), the Gaussian basis functions and the thin-plate basis functions. These radial basis functions are combined with step function enrichments directly or with enriched Shepard functions. The formulation is coupled with level set methods and requires no explicit representation of the discontinuity.; Numerical results show the robustness of the method, both in accuracy and convergence.; Finally, a methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial differential equation loses hyperbolicity. The discrete discontinuity is treated by the extended finite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Two models for the simulation of arbitrary evolving dynamic cracks, a continuous crack model and element-wise cracking model, are introduced. In the element-wise cracking model, a cohesive segment is initiated and passes through the center node extends across the entire element once the element cracks by the loss of hyperbolicity criteria, the crack segments can be added at arbitrary position and with arbitrary direction, the model is capable of modelling complex fracture behaviors including the simulation of crack nucleation at multiple locations and crack branching easily. Both methods are applied to several dynamic crack growth problems including the branching of the cracks.