Development of General Finite Differences for complex geometries using a sharp interface formulation

Abstract

We apply sharp interface boundary conditions to collocated meshfree approximations. Using the sharp interface formulation with the General Finite Difference (GFD) shape functions, we solve several test cases on meshfree grids with uniform and variable resolutions. Numerical results for the 2D Poisson equation indicate that the sharp interface formulation performs as well as the traditional collocated boundary approach with respect to both the convergence rate and accuracy. The analytic lid driven cavity results indicate that our discretization of a semi-implicit approximate projection scheme is indeed consistent and spatially converges at a second order rate under diffusive time stepping and at a first order rate under convective time stepping due to the first order in time splitting error. To resolve sharp gradients, variable resolution meshfree grids are considered for the classic lid driven cavity problem and for the uniform flow over a cylinder problem. Comparison to appropriate data sets indicate that our approach accurately estimates the various flow parameters using small cloud sizes of 13 neighbors while using high clustering ratios.