Interfaces and boundaries in physics based simulation of solids and fluids

Recent developments concerning the numerical simulation of solid objects and fluid flows in the field of computer graphics have opened up a plethora of new possibilities in applications such as special effect productions, animated movies, Virtual Reality (VR) applications, medical simulators, and computer games. Despite various techniques for the simulation of solids and fluid flows exist, the accurate incorporation of complex boundary geometries and interface models still poses a great challenge. Nevertheless, a robust handling of these interface descriptions is inevitable for a wide range of applications. Besides other purposes, interface models are frequently used to represent the boundary surfaces of solid objects, the layer between different materials, the boundary geometry of domains trapping a fluid, or even to represent cuts, tears, or cracks separating the material within an object. In order to be able to simulate even more complex phenomena and to enhance existing approaches, advanced methods and new techniques for the efficient and robust numerical simulation of solids and fluids with complex interfaces have to be developed. The contributions of this thesis are organized into three parts. In the first part, two novel methods based on Finite Element (FE) discretizations for the simulation of brittle fracture and cutting of deformable solids are presented. The first chapter in this part focuses on the physically motivated generation of brittle fractures using an adaptive stress analysis. While this approach captures crack interfaces by explicit remeshing and element duplication, the approach described in the second chapter of the first part captures the interface implicitly by using enrichment functions that are directly embedded into the FE discretization. The enrichment based technique is able to capture even highly complex and finely structured cuts with high accuracy while any form of remeshing is completely avoided. The second part of this thesis is concerned with a novel discretization approach for implicit interface representations. An arbitrary surface in three-dimensional space can be represented as an isosurface of a signed distance function. In the first step, the novel approach discretizes the signed distance function into a grid structure using piecewise polynomials. Subsequently, the initial discretization is refined in order to improve the discretization accuracy. The presented method is the first approach that not only refines the grid cells spatially but also varies the degree of the polynomial basis. With this approach even highly complicated surfaces can be accurately discretized while keeping the memory consumption to a minimum. In the third and final part of this thesis a novel approach for the simulation of incompressible fluids and a method to handle non-penetration boundary conditions using the novel concept of precomputed density maps are presented. Building on the Navier-Stokes equations for isothermal incompressible fluids the partial differential equation is spatially discretized using the Smoothed Particle Hydrodynamics (SPH) formalism. Incompressibility is then ensured using a novel pressure solver that enforces both a constant density field throughout the fluid and a divergence-free velocity field. In order to enforce non-penetration an implicit representation of the boundary interface is constructed and a density map is precomputed. Using the novel concept of density maps, non-penetration boundary conditions can be handled using efficient lookups into the map with constant complexity while the requirement to sample the boundary interface geometry with particles vanishes