Development of mesh-free methods and their applications in computational fluid dynamics

Abstract

The recent decade has witnessed a research boom on the mesh-free methods. It is well-known that the mesh-free method has a few clear advantages over the mesh-based method such as the requirement of node generation instead of mesh generation and easy deletion/insertion of new nodes. Up to date, a lot of attentions of mesh-free researchers have been devoted to the solution of partial differential equations in the weak form. As a result, many mesh-free methods actually belong to the finite element community. In my study, two mesh-free methods: least square-based finite difference (LSFD) method and local RBF-based differential quadrature (LRBF-DQ) method have been developed. Their abilities of dealing with the problems in fluid mechanics have also been demonstrated by applications to different types of flow problems with dynamic and geometric complexity, such as unsteady flow around two cylinders, natural convection within complex geometry, and compressible flow with shock waves. Both methods belong to the non-integral types of mesh-free methods. In other words, they solve the strong form of partial differential equations (PDEs), and the discretization process consists only of mesh-free derivative approximation. In this regard, they are truly mesh-free since no requirement of additional background meshes is needed.