An automated hierarchical eXtended finite element approach for multiphysics problems involving discontinuities

Abstract

In this thesis, a hierarchical eXtended finite element method for the modeling and numerical simulation of multiphysics problems and its implementation into a framework that uses automated code generation is presented. The approach consists of introducing hierarchically ordered level set functions, motivated by the structure of the considered problem, to decompose a given hold-all domain into several subdomains. The decomposition is guaranteed to be geometrically consistent which means that no overlapping regions or voids can arise. Mathematically, the approach decouples the computational mesh from the physical domains and, thereby, allows for large deformations and topological changes, such as the rise of (new) subdomains. At domain boundaries, quantities, or their gradient, may be modeled discontinuously and eXtended approximation spaces are introduced for the (sharp) representation of such features on the discrete level. The enrichment is realized by Heaviside functions which are defined subject to the hierarchical level set functions and, hence, introduce additional basis functions and coefficients locally at the respective (sub)domain boundary. For imposing interface and boundary conditions, the Nitsche method is used. By design, the developed approach is well suited to be implemented using automated code generation. As a result, the hierarchical eXtended finite element method is implemented as toolbox miXFEMfor the FEniCS framework. Therefore, the core components of FEniCS are significantly extended and new methods (e.g. the subdivision of elements and the assembling of tensors) are added. As the evolution of interfaces is often part of the problem, the framework miXFEMis supplemented by a level set toolbox providing maintaining methods such as reinitialization and volume correction as well as methods for computing a non-material velocity field. The method and its implementation is validated against several examples and then used for the modeling and simulation of different real-world applications in 2d and 3d. Since this thesis is motivated by several research projects where melting and solidification processes are of interest, we focus on these kind of problems and present results for a thermal upsetting process and different welding processes. However, due to the generality and flexibility of the developed framework, it can be used to rapidly implement and simulate problems from different areas such as multiphase flow or other problems with evolving geometries.