A compact RBF-FD based meshless method for the incompressible Navier—Stokes equations

Abstract

Meshless methods for solving fluid and fluid-structure problems have become a promising alternative to the finite volume and finite element methods. In this paper, a mesh-free computational method based on radial basis functions in a finite difference mode (RBF-FD) has been developed for the incompressible Navier—Stokes (NS) equations in stream function vorticity form. This compact RBF-FD formulation generates sparse coefficient matrices, and hence advancing solutions will in time be of comparatively lower cost. The spatial discretization of the incompressible NS equations is done using the RBF-FD method and the temporal discretization is achieved by explicit Euler time-stepping and the Crank—Nicholson method. A novel ghost node strategy is used to incorporate the no-slip boundary conditions. The performance of the RBF-FD scheme with the ghost node strategy is validated against a variety of benchmark problems, including a model fluid—structure interaction problem, and is found to be in a good agreement with the existing results. In addition, a higher-order RBF-FD scheme (which uses ideas from Hermite interpolation) is then proposed for solving the NS equations.

A coupled meshfree-mesh-based solution scheme on hybrid grid for flow-induced vibrations

“…As a result, maximum number of nodes and their distribution within the domain 17 is limited by ill-conditioning effect. The limitation was later overcome by the use of local RBFs which 18 compromise on spectral accuracy in bargain of better conditioned problems with improved accuracy 19 [3,55,48,41]. This is done by localizing the influence domain around each particle.…”

“…As a result, sparse 20 and well conditioned coefficient matrices are generated irrespective of total number of data points 21 and their density within the domain. RBF in Finite Difference mode [55,48,41,60] and RBF based 22 differential quadrature methods [3] are the two famous local RBF techniques which are used for the 23 solution of Navier Stokes equations in meshfree domain. However, like other meshfree methods, RBF 24 based methods also suffer from high computational cost.…”

An ALE Based Hybrid Meshfree Local RBF-Cartesian FD scheme for Incompressible flow around moving boundaries

“…Subsequently, RBF-FD weights ( ( ) ( ) ) are calculated , corresponding to all neighbouring points. RBF-FD approximation of differential operator of a field variable then is given by weighted linear sum of values of same variable at the neighbouring nodes [20]:…”

Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

“…Recently, the LRBFCM [30][31][32] and MAPS [33] have been used for numerical solutions of two-dimensional flow fields. Sanyasiraju and Chandhini [30] used the LRBFCM and the fractional step algorithm to solve the two-dimensional primary-variables formulation of the Navier-Stokes equations.…”

“…Bourantas et al [31] adopted the LRBFCM to simulate steady-state equations of the velocity-vorticity formulation; hence, only the steady solutions for forced, natural and mixed convection are efficiently acquired. The streamfunction-vorticity formulation is analyzed by the LRBFCM [32] and the flow fields in a lid-driven square cavity are compared well with other numerical results. Since the definition of streamfunction [1,2] usually appears only in two-dimensional problems, their approach might be difficult to be extended to three-dimensional problems.…”