ABSTRACT
Previous chapters have discussed a number of meshfree methods to solve the problems of solid mechanics. Meshfree methods can also be applied to problems of fluid mechanics because they basically provide a means of discretizing partial differential equations (PDEs) in the spatial domain. This chapter discusses some of the meshfree methods that suit well and have been applied to solve computational fluid mechanics problems. Simulation and analysis of problems of fluid dynamics have been generally performed
using numerical methods such as the finite difference method (FDM), the finite volume method (FVM), and the finite element method (FEM). These numerical methods have been widely applied to practical problems and have dominated the subject of computational fluid dynamics (CFD). An important feature of these methods is that a corresponding Eulerian (for FDM and FVM) grid or a Lagrangian (for some FEM formulations) grid or both are required as a computational frame to solve the governing equations. When simulating some special problems with large distortions and moving material interfaces, deformable boundaries, and free surfaces, these methods can encounter many difficulties. Although many numerical schemes for the solution of fluid dynamics problems have emerged, difficulties still exist for problems with the above-mentioned features. Attempts have also been made to combine the good features of FDM, FVM, and FEM, and the twogrid systems of Lagrangian grid and Eulerian grid have been used [1]. Computational information is exchanged either bymapping or by special interface treatment between these two grids. This approach is rather complicated and can also cause problems related to stability and accuracy. The search for better methods and techniques is still going on. Meshfree methods or techniques can offer some promising alternatives for solving
problems of CFD. The most attractive feature of the meshfree methods is that there is no need for a mesh or that there is less reliance on the quality of the mesh. When mesh is used, a triangular mesh will often suffice. This opens a new opportunity to solve the abovementioned problems and to conduct adaptive analyses, as it is demonstrated in this chapter. There are basically four types of methods that have been explored for CFD problems:
1. Integral representation methods such as the smoothed particle hydrodynamics (SPH) method [2-8] and the reproducing kernel particle methods [9]. The SPH method is a Lagrangian formulation and is one of the best choices for highly nonlinear, fast dynamic, multiphase, and momentum-driven types of problems. It uses particles and no mesh is needed during the computation.
