Radial basis function based meshless methods for fluid flow problems

Abstract

This thesis is concerned with the development of meshless methods using radial basis functions for solving fluid flow problems. The advantage of meshless methods over traditional mesh-based methods is that they make use of a scattered set of collocation points in the physical domain and no connectivity information is required. An important objective of the present research is to develop novel meshless methods for unsteady flow problems. Symmetric/unsymmetric radial basis function collocation schemes are proposed for solving an unsteady convection-diffusion equation for various Peclet numbers. Both global and compactly supported radial basis functions are used and the convergence behaviours of various radial basis functions are studied. The performance of the presented schemes is shown by using both uniform as well as scattered distribution of points. Numerical results suggestthat these schemes are capable of obtaining accurate results for low and medium Peclet numbers. Next, two directions have been explored in this thesis for using radial basis functions to solve large scale problems encountered in fluid flow problems. They are namely, domain decomposition schemesand radial basis functions in finite difference mode. These schemes are shown to be computationally efficient and also aid in circumventing the ill-conditioning problem. The performance of both schemes are evaluated by solving the unsteady convection-diffusion problem. The last part of this thesis isconcerned with the solution of the 2D Navier-Stokes equations. Meshless methods based on radial basis collocation and scattered node finite difference schemes are formulated for solving steady and unsteady incompressible Navier-Stokes equations. A novel ghost node strategy is proposed for incor-porating the no-slip boundary conditions. Optimisation strategies based on residual error objective and leave-one-out statistical criterion are proposed to evaluate the optimal shape parameter value in case of the multiquadric RBF for collocation and scattered finite difference approaches respectively.Standard benchmark problems like the driven cavity flows in square and rectangular domains and backward facing step flow problem are solved to study the performance of the developed schemes. Finally, a higher order radial basis function based scattered node finite difference method is proposed for solving the incompressible Navier-Stokes equations.

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