In contrast to traditional mesh-based methods for the numerical solution of boundary value problems, e.g., Finite Element (FEM) and Finite Volume (FVM), in the recent period many meshfree approaches have been proposed in order to avoid those typical issues due to the mesh. For example, the quality of the mesh greatly affects the reliability of the final solution in the case of CFD problems and the human intervention of a professional is often still needed when dealing with complex-shaped domains. This in turn increases both cost and time required for the reliable simulation of problems of engineering relevance. Meshless methods, on the other side, usually rely on a simpler distribution of nodes and do not require the storage of connectivity information.
Among others, one of the most promising meshless methods in terms of accuracy and flexibility is the one based on the Radial Basis Function – Finite Difference (RBF-FD) scheme. RBF-FD methods, however, are usually affected by severe ill conditioning issues when Neumann boundary conditions are employed. This fact is the main responsible for the appearance of large discretization errors near the boundary and for the lack of stability of traditional time integration schemes. In order to address this issue, some new algorithms for the robust treatment of boundary conditions have been developed and successfully employed to solve fluid flow problems with heat transfer. Furthermore, it is well acknowledged that the efficient resolution of boundary layers arising in this class of problems requires an adequate spatial discretization in the neighbourhood of the boundary, i.e., increased node/mesh density along the direction of large gradients only. In the context of the RBF-FD method, this result is achieved through an anisotropic node generation algorithm, thus enabling the efficient simulation of a greater variety of CFD problems. The method described above is successfully employed for the accurate solution of a representative 3D heat transfer problem with incompressible fluid flow.
