Mesh Generation and Fracture for Deformable Bodies

Numerical simulation complements both theory and experimentation as an important component of today’s research. This thesis focuses on the tightly-woven relationship between the simulation of a Lagrangian solid and its discretization. We introduce a mesh generation technique as well as a new method for simulating meshes and materials with changing topology. The tetrahedral mesh generation algorithm is designed for the Lagrangian simulation of deformable bodies. The algorithm’s input is a level set. First a bounding box of the object is covered with a uniform lattice of subdivision-invariant tetrahedra. The level set is then used to guide a red green adaptive subdivision procedure that is based on both the local curvature and the proximity to the object boundary. The final topology is carefully chosen so that the connectivity is suitable for large deformation and the mesh approximates the desired shape. Finally, this candidate mesh is compressed to match the object boundary. To maintain element quality during this compression phase we relax the positions of the nodes using finite elements, masses and springs, or an optimization procedure. We also propose a virtual node algorithm that allows material to separate along arbitrary paths through a mesh. The material within an element is fragmented by creating several replicas of the element and assigning a portion of real material to each replica. This results in elements that contain both some real material and empty regions. Our new virtual node algorithm automatically determines the number of replicas and the assignment of material to each. Moreover, it provides the degrees of freedom required to simulate the partially or fully fragmented material. This approach provides for the efficient simulation of complex geometry with a simple mesh, i.e. the geometry need not align itself with element boundaries. It also alleviates many shortcomings of traditional Lagrangian simulation techniques for meshes with changing topology. For example, slivers do not require small time step restrictions since they are embedded in well shaped larger elements.