Modeling Fracture within Local Max-Ent Meshfree Approximation Schemes

The numerical simulation of complex physical phenomena calls for efficient and robust algorithms which can be used in concurrent calculations. For standard static and dynamic problems, the priority of the finite element method is out of discussion, and alternative approaches are far from threatening the finite element leadership. This supremacy may not be true, however, when rather difficult problems are tackled. An example of excruciating problems is the propagation of fractures. In such cases, major complications derive from the necessity to track the evolution of boundaries. In view of subsiding numerical difficulties related to the spatial discretization of volumes, alternative algorithms, in some cases based on brilliant concepts, have been proposed. Meshfree methods represent a particular class of numerical
algorithms that do not rely on the definition of a grid, but, in contrast, use the actual geometry of the simulated object for calculations.

I will present the static version of a meshfree method, recently developed for the analysis of the dynamics of solids and fluids, in view of the simulation of explicit crack propagation in brittle materials. The method relies on the discretization of the system into a finite number of particles, representative of the surrounding material. In the spirit of element erosion procedures, the propagation of cracks is modelled by removing material particles upon the attainment of a fracture criterion. The failure criterion adopted here is based on energetic arguments and relies on a small crack-neighbourhood construction for capturing the local degeneration of the material. The study has been developed in collaboration with Michael Ortiz and Bo Li, Caltech (USA).