A Variational Mesh-Free Method Based on Discretization via Directional Derivatives

摘要

A new mesh-free method based on variational principles (VP) and finite differencing is proposed. The partial derivatives at every node in the VP corresponding to the original partial differential equations in question are discretized first using the total directional derivative formulae expressed in terms of the surrounding node function values, and then the discrete functional expression is obtained by numerical integration over the solution domain covered by a simple background triangular grid. Thereby it should be especially noted that the background triangular grid is used only for the final domain integration but not used for the numerical differentiation at all, so that it is capable of giving high accuracy and is very simple both theoretically and computationally. The method is tested with a numerical example of fluid flow around airfoils with the natural boundary condition at the airfoil surface, and a reliable numerical solution is obtained with high accuracy and efficiency.