Enforcement of boundary conditions in meshfree methods using interpolating moving least squares

Abstract

A technique for enforcement of boundary conditions in meshfree methods based on interpolating moving least squares (IMLS) via usage of singular weights for the solution of partial differential equations is presented. Due to the specific singular choice of the weight functions, which is needed to guarantee the interpolation, there is a problem to find the inverse of the singular matrix. We extend the perturbation technique applied to transport equations by Kunle [Entwicklung und Untersuchung von Moving Least Square Verfahren zur numerischen Simulation hydrodynamischer Gleichungen, Dissertation, Fakultät für Physik, Eberhard-Karls-Universität zu Tübingen] to allow the correct evaluation of all necessary derivatives for the meshfree collocation solution of boundary value problems in interpolation points at a reasonable cost. The inverse is carried out using the regularized weight function. It turns out that a stable inverse is obtained when the vanishing regularization parameter is considered. The detailed description of this strategy is presented. In contrast to standard kernel functions used in EFGM, RKPM, etc., the singular kernel functions lead to truly interpolating functions which satisfy the Kronecker-delta property and they can therefore be used for enforcement of Dirichlet boundary conditions when solving boundary value problems. As a confirmation of the mathematical derivations we give a solution to a model BVP with a known analytical solution, as well as the experimental convergence study of the method to analytical solutions.