| Abstract |
In the past decades, mesh-free particle methods have advanced steadily and amassed a rich body of literature. They have successfully demonstrated their ability to simulate complex physics in various fields of research ranging from applied engineering to biology. Remarkably, the mesh-free representation of curved, dynamic surfaces has received far less attention than the mesh-free approximation of differential operators. The most commonly used, truly mesh-free approach for tracking and quantifying curved surfaces is limited to lower-order accuracy, with its fundamental basis having been proposed in the nineties. In this thesis, we bring high-order geometric computing to mesh-free particle methods. We propose the particle closest-point method, which implicitly tracks and quantifies evolving curved surfaces in a narrow band around the surface without the need of any connected mesh. It represents surfaces as the zero level-set of high-order polynomials that are obtained in local regression problems in a Newton-Lagrange basis on a set of unisolvent nodes, enabling robustness on scattered data. It is a purely geometric approach that computes comprehensive surface information in the form of signed-distance values, closest points, surface normals, and mean and Gaussian curvatures. We demonstrate that the surface quantities converge with the theoretical orders and show its capabilities in simulations of multi-phase hydrodynamics, where we consider oscillating and dividing droplets. We leverage the comprehensive and accurate geometric information of the particle closest-point method to address an issue that is encountered in solving partial differential equations on curved surfaces that are represented explicitly: When the surface undergoes large deformations, the surface point distribution inevitably becomes locally irregular, leading to a loss of accuracy and stability in the approximation of surface differential operators. We propose the self-adaptive implicit surface sampling method, which computes globally adaptive and locally regular point distributions for curved surfaces. It is based on two key mechanisms, the relaxation of the surface point distribution and a routine to find an appropriate number of points. The relaxation is done by minimizing a global potential which consists of local point-point interactions that depend on the local curvature of the surface. We minimize the global potential using a gradient descent accelerated by a line search. For finding an appropriate number of points for the discretization of the surface we propose a local integral support measure to decide if there is a lack or excess of points. We show the potential of the proposed method by computing adaptive and locally regular point discretizations of a variety of parametric and non-parametric surfaces from both synthetic and real-world data. We find robust and rapid convergence to the final point set size based on the integral support measure and low average deviations from the target spacing of the points. To demonstrate the impact of the contributions of this thesis we combine the two methods with the surface variant of discretization-corrected particle strength exchange to solve partial differential equations on deformable surfaces and coupled bulk-surface equations. We propose time-integration schemes that outline how the methods can be combined. We solve the reaction-diffusion equation on the surface of an oscillating droplet and show various Turing patterns. We further simulate a morphogenetic model, where local surface concentrations induce growth of the surface. We reveal the importance of maintaining a regular surface discretization by means of the self-adaptive implicit surface sampling method and find rich, organic shapes governed by the morphogenetic model. |