METODI MESHLESS STABILI BASATI SU RADIAL BASIS FUNCTIONS PER LA SIMULAZIONE CFD 3D SU GEOMETRIE COMPLESSE

This work deals with the development and applications of Radial Basis Functions (RBF) to the solution of Partial Differential Equations (PDEs) involving fluid flow and heat transfer. More specifically, the main focus is the study of those stability and accuracy issues arising in presence of Neumann boundary conditions and the consequent development of stabilization techniques. The algorithms presented fall in the broad category of meshless solvers, which are aimed at solving PDEs without relying on the mesh data structure. In order to approximate the solution to a given boundary value problem, these algorithms usually rely on a set of nodes which are scattered on the computational domain with no connectivity information. The two approaches investigated in the present thesis are called Radial Basis Function-Finite Difference (RBF-FD) and Radial Basis Function-Hermite Finite Difference (RBF-HFD). The lack of connectivity information and the reliance on radial functions allow them to provide better geometrical flexibility than traditional mesh-based methods, at the same time, they also enable finite difference-like discretization of differential operators. The interest in meshless methods is motivated by the fact that, as new Computer Aided Engineering (CAE) techniques are developed, it seems that a major obstacle to their greater diffusion is constituted by the intrinsic limitations of the mesh generation. This happens, for instance, in applications requiring automatic shape optimization, where the domain of calculus is subject to extensive deformations requiring frequent remeshing and quality assessments. Similar problems also arise in presence of moving boundaries or multi-phase simulations, furthermore, especially in the field of CFD, highly valuable and experienced operators are often kept busy by activities related to mesh generation for extended periods of time. he starting point of the research activity presented below was the work done at the University of Trieste by Riccardo Zamolo and Enrico Nobile: a clear vision of how to implement the first solver for generic 3D geometries was already established, along with ideas on possible further improvements. However, after some initial success in the solution of heat conduction problems on complex geometries, it soon became clear that some stability issues had to be addressed when dealing with more complex physics. The main topic of research has then become the stability and accuracy of the Radial Basis Function-Finite Difference method in presence of Neumann boundary conditions (BC). This is of critical importance in many cases, for example in the solution of incompressible flows where the projection scheme for pressure correction is adopted and Neumann BC are enforced in the associated elliptic equation.

This work deals with the development and applications of Radial Basis Functions (RBF) to the solution of Partial Differential Equations (PDEs) involving fluid flow and heat transfer. More specifically, the main focus is the study of those stability and accuracy issues arising in presence of Neumann boundary conditions and the consequent development of stabilization techniques. The algorithms presented fall in the broad category of meshless solvers, which are aimed at solving PDEs without relying on the mesh data structure. In order to approximate the solution to a given boundary value problem, these algorithms usually rely on a set of nodes which are scattered on the computational domain with no connectivity information. The two approaches investigated in the present thesis are called Radial Basis Function-Finite Difference (RBF-FD) and Radial Basis Function-Hermite Finite Difference (RBF-HFD). The lack of connectivity information and the reliance on radial functions allow them to provide better geometrical flexibility than traditional mesh-based methods, at the same time, they also enable finite difference-like discretization of differential operators. The interest in meshless methods is motivated by the fact that, as new Computer Aided Engineering (CAE) techniques are developed, it seems that a major obstacle to their greater diffusion is constituted by the intrinsic limitations of the mesh generation. This happens, for instance, in applications requiring automatic shape optimization, where the domain of calculus is subject to extensive deformations requiring frequent remeshing and quality assessments. Similar problems also arise in presence of moving boundaries or multi-phase simulations, furthermore, especially in the field of CFD, highly valuable and experienced operators are often kept busy by activities related to mesh generation for extended periods of time. he starting point of the research activity presented below was the work done at the University of Trieste by Riccardo Zamolo and Enrico Nobile: a clear vision of how to implement the first solver for generic 3D geometries was already established, along with ideas on possible further improvements. However, after some initial success in the solution of heat conduction problems on complex geometries, it soon became clear that some stability issues had to be addressed when dealing with more complex physics. The main topic of research has then become the stability and accuracy of the Radial Basis Function-Finite Difference method in presence of Neumann boundary conditions (BC). This is of critical importance in many cases, for example in the solution of incompressible flows where the projection scheme for pressure correction is adopted and Neumann BC are enforced in the associated elliptic equation.