Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

Abstract

SUMMARYA numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a 'universal approximator' based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of 'finite element-type' discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re= 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of 11×11, Re=400 for the driven cavity flow with a density of 33×33 and Rayleigh number Ra= 1 000 000 for the natural convection flow with a density of 27 ×27.

Solving high order ordinary differential equations with radial basis function networks

“…For the approximation of a function and its derivatives, the numerical example above indicated that the indirect approach performs well for a wide range of β. In solving second order DEs (Mai-Duy and Tran-Cong [16]), experience…”

“…Sharan et al [13]; Zerroukat et al [14]; Mai-Duy and Tran-Cong [15,16]; Fedoseyev et al [17] based on a differential process (Kansa [12]) and the indirect approach (IRBF) based on an integration process (Mai-Duy and Tran-Cong [15,18]). Both approaches were tested on the solution of second order DEs and the indirect approach was found to be superior to the direct approach (Mai-Duy and Tran-Cong [15]).…”

Numerical study of stream-function formulation governing flows in multiply-connected domains by integrated RBFs and Cartesian grids

“…Over the last twenty years, radial-basis-function networks (RBFNs) have been developed to solve different types of differential problems encountered in applied mathematics, science and engineering (e.g. [1,2,3,4,5,6,7,8,9]). …”

“…In [4,5,6,7,8], the RBF approximations are constructed using integration (integrated RBFNs (IRBFNs)) rather than the usual differentiation. This approach has the ability to overcome the problem of reduced convergence rates caused by differentiation and to provide effective ways to implement derivative boundary values.…”

“…IRBFNs have been developed for the simulation of flows in simply-connected domains governed by the stream-function formulation and the stream-function -vorticity formulation (e.g. [6]) as well as natural convection in regions between concentric cylinders governed by the stream-function -vorticity formulation (symmetrical flows) (e.g. [4]).…”