SUMMARYWe present a method for simulating quasistatic crack propagation in 2-D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns.

We utilize the shape derivative of the classical Griffith's energy in a level set method for the simulation of dynamic ductile fracture. The level set is defined in the undeformed configuration of the object, and its evolution is designed to represent a transition from undamaged to failed material. No re-meshing is needed since the resulting topological changes are handled naturally by the level set method. We provide a new mechanism for the generation of fragments of material during the progression of the level set in the Griffith's energy minimization. Collisions between different material pieces are resolved with impulses derived from the material point method over a background Eulerian grid. This provides a stable means for colliding with embedded interfaces. Simulation of corotational elasticity is based on an implicit finite element discretization.

We introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities. For this purpose, we use an output least squares approach with level set and augmented Lagrangian methods. Our formulation incorporates the inherent nature of the piecewise constant coefficients, which eliminates the need for a complicated nonlinear solve at every iteration. Instead, we obtain an explicit update formula and therefore vastly speed up computation. We employ our approach to the example problems of Poisson's equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface.