Discretisation of the velocity-pressure formulation with integrated radial-basis-function networks

Abstract

This study is concerned with the use of integrated radial-basis-function networks (IRBFNs) for the discretisation of the velocity-pressure formulation in two dimensions on Cartesian grids. In the approximation of the field variables (i.e. velocity components and pressure), instead of using low-order polynomial interpolants, we employ global IRBFNs along grid lines (i.e. one-dimensional IRBFNs). In the imposition of boundary condition for the pressure, we propose two treatments, namely Treatment A and Treatment B. For both treatments, Neumann boundary conditions are transformed into Dirichlet ones. The former is based on values of the pressure at interior nodes along a grid line and first derivative values of the pressure at two extreme nodes of that grid line; while the latter relies on values of the pressure at interior nodes along a grid line together with both first and second derivative values of the pressure at two extreme nodes of that grid line. The proposed method is verified successfully through the simulation of a benchmark test, namely the isothermal lid-driven cavity flow problem.