Applications of level sets in an extended finite element method framework

This thesis presents the following, (1) a structured mesh method combined with level sets for treating arbitrary solids within the extended finite element framework. A paradigm is developed for generating structured finite element models from solid models by means of implicit surface definitions. The implicit surfaces are defined by radial basis functions. Internal features, such as material interfaces, sliding interfaces and cracks are treated by enrichment techniques developed in the extended finite element method (XFEM). Methods for integrating the weak form for such models are proposed. These methods simplify the generation of finite element models. Results presented for several examples show that the accuracy of this method is comparable to standard unstructured finite element methods. (2) a method for topology optimization based on implicit functions (level sets). Topology optimization is formulated in terms of the nodal variables that control an implicit function description of the shape. The implicit function is constrained by upper and lower bounds, so that only a band of nodal variables needs to be considered in each step of the optimization. The weak form of the equilibrium equation is expressed as a Heaviside function of the implicit function; the Heaviside function is regularized to permit the evaluation of sensitivities. It is shown that the method is a dual of the Bendsoe-Kikuchi method. The method is applied both to problems of optimizing single material and multi-material configurations; the latter is made possible by enrichment functions based on the extended finite element method that enable discontinuous derivatives to be accurately treated within an element. The method is remarkably robust and no instances of checkerboarding are found. The method handles topological merging and separation without any apparent difficulties. (3) implementation of vector level sets in a three dimensional crack growth problem. In order to capture the physics of the cracks an extended finite element (XFEM) framework is used. In the extended finite element method, the physics of the crack is captured by enriching the displacement field with various continuous and discontinuous functions which describe the near crack field. This allows the propagation of the crack through finite elements, therefore avoiding re-meshing at each crack growth increment. The crack surface is defined by a set of line segments and the evolution by a set of bilinear surfaces. Some examples of 3D crack propagation are solved.