Radial Basis Function-Finite Difference Meshless Methods for CFD Problems

The objective of this dissertation is to investigate the numerical properties of the RBF-FD meshless approach when it is employed for the solution of CFD problems, with particular reference to fluid-flow problems defined over complex-shaped domains. This objective has been accomplished by developing a MATLAB code which is composed by several elements characterizing the meshless approach to a CFD problem. The work presented in this thesis has focused on the analysis and development of these characterizing elements which are essential in developing an innovative, robust, accurate and flexible numerical method. The node generation is the first problem that has been tackled and a significant portion of this work is dedicated to this element since it is the foundation of every meshless approach. Different algorithms have been proposed for the generation of node distributions on 2D and 3D complex-shaped domains. Such node distributions then proved to be very suitable for the use with RBF-FD discretizations. The developed node generation algorithms are extremely efficient and are based on very simple principles: this is an important insight since the possibility to easily deal with complex geometries represents the main theoretical advantage of meshless methods over mesh-based methods. The RBF interpolation, which is the key element of the RBF-FD method, is then thoroughly studied by exploring all the variables which influence the construction of an accurate and robust interpolant over scattered nodes. The multiquadric RBF has been chosen and extensive numerical tests are conducted for 2D and 3D cases. On the base of these analysis, a RBF-FD code has been developed for the local approximation of the partial derivatives of an unknown function which is defined only at scattered nodes in 2D/3D. The coupling of a node generation algorithm to a RBF-FD scheme leads to an actual meshless approach which can be used to discretize a given PDE over a domain with possible arbitrary shape. Such meshless approach has been employed to perform several test cases for different 2D/3D model problems which have fundamental importance in CFD applications. The solution phase then follows the RBF-FD discretization, and its role in the simulation chain is just as important as the previous aspects. For this purpose, novel multicloud techniques have been proposed for the acceleration of the convergence in the solution of the system of equations arising from RBF-FD discretizations in the case of a 2D Poisson equation. Such multicloud techniques have proven to bring substantial improvements over the traditional solvers employed with the RBF-FD discretizations. Finally, the RBF-FD approach is employed to solve actual 2D/3D fluid-flow problems in the case of the time-dependent, incompressible Navier-Stokes equations. Important problems are addressed, e.g., the development of stable discretizations when dealing with the pressure-velocity coupling using primitive variables, leading to an efficient and stable RBF-FD approach which can be used for the accurate solution of time-dependent fluid-flow problems over arbitrarily shaped domains in 2D and in 3D.