arXiv subjects

Heinrich Kraus

Publications and source records attributed to Heinrich Kraus.

3 recordsLinked to original sources

arXiv2024

Higher-Order Generalized Finite Differences for Variable Coefficient Diffusion Operators

We present a novel approach of discretizing variable coefficient diffusion operators in the context of meshfree generalized finite difference methods. Our ansatz uses properties of derived operators and combines the discrete Laplace operator with reconstruction functions approximating the diffusion coefficient. Provided that the reconstructions are of a sufficiently high order, we prove that the order of accuracy of the discrete Laplace operator transfers to the derived diffusion operator. We show that the new discrete diffusion operator inherits the diagonal dominance property of the discrete Laplace operator. Finally, we present the possibility of discretizing anisotropic diffusion operators with the help of derived operators. Our numerical results for Poisson's equation and the heat equation show that even low-order reconstructions preserve the order of the underlying discrete Laplace operator for sufficiently smooth diffusion coefficients. In experiments, we demonstrate the applicability of the new discrete diffusion operator to interface problems with point clouds not aligning to the interface and numerically show first-order convergence.

math.NA↗

arXiv2023

Meshfree One-Fluid Modelling of Liquid-Vapor Phase Transitions

We introduce a meshfree collocation framework to model the phase change from liquid to vapor at or above the boiling point. While typical vaporization or boiling simulations focus on the vaporization from the bulk of the fluid, here we include the possibility of vaporization from the free surface, when a moving fluid comes into contact with a superheated surface. We present a continuum, one-fluid approach in which the liquid and vapor phases are modeled with the same constitutive equations, with different material properties. The novelty here is a monolithic approach without explicit modeling of the interface between the phases, neither in a sharp nor diffuse sense. Furthermore, no interface boundary conditions or source terms are needed between the liquid and vapor phases. Instead, the phase transition is modeled only using material properties varying with temperature. Towards this end, we also present an enrichment of strong form meshfree generalized finite difference methods (GFDM) to accurately capture derivatives in the presence of jumps in density, viscosity, and other physical properties. The numerical results show a good agreement with experimental results, and highlight the ability of our proposed framework to model phase changes with large jumps.

physics.flu-dyn↗

arXiv2022

A Meshfree Point Collocation Method for Elliptic Interface Problems

We present a meshfree generalized finite difference method for solving Poisson's equation with a diffusion coefficient that contains jump discontinuities up to several orders of magnitude. To discretize the diffusion operator, we formulate a strong form method that uses a smearing of the discontinuity; and a conservative formulation based on locally computed Voronoi cells. Additionally, we propose a novel conservative formulation for enforcing Neumann boundary conditions that is compatible with the conservative formulation of the diffusion operator. Finally, we introduce a way to switch from the strong form to the conservative formulation to obtain a locally conservative and positivity preserving scheme. The presented numerical methods are benchmarked against four test cases of varying complexity and jump magnitude on point clouds with nodes that are not aligned to the discontinuity. Our results show that the new hybrid method that switches between the two formulations produces better results than the classical generalized finite difference approach for high jumps in diffusivity.

math.NA↗