Implementation of boundary conditions for meshless methods

Abstract

Boundaries and boundary conditions are an aspect of the numerical solution of partial differential equations where meshless methods have had to surmount many initial difficulties due to the lack of a finite-element-like Kronecker delta condition. Furthermore, it is frequently desirable, especially in fluid mechanics, to impose general, nonlinear boundary and interface constraints. This paper describes a computationally efficient algorithm based on d'Alembert's principle that can be used for general constraints both in meshless methods and finite elements. First, a method of partitioning meshless shape functions suitable for imposing linear boundary conditions is developed. Subsequently, an analogous method is developed for nonlinear constraints. Special attention is given to imposing general boundary and fluid-structure interface conditions on the Navier-Stokes equations in terms of conservative variables. Numerical results using d'Alembert's principle with the Reproducing Kernel Particle Method (RKPM), including viscous, supersonic flow past a NACA 7012 airfoil, are shown.