TABLE OF CONTENTS Problems in computational fluid dynamics (CFD) are generally solved by employing the conventional grid-based numerical methods such as the finite difference method (FDM), finite volume method (FVM) and finite element method (FEM) (Anderson, 1995; Hirsch, 1988). These conventional numerical methods have been have dominated for a long time the subject of computational fluid dynamics. An important feature of these methods is that a corresponding Eulerian (for FDM and FVM) or Lagrangian (for FEM) grid is required as the computational frame to provide spatial discretization for the governing equations. However, when simulating some special problems with large distortions, moving material interfaces, deformable boundaries and free surfaces, these methods can encounter some difficulties. FEM cannot well resolve the problems with large mesh element distortion. The Eulerian methods are inefficient in treating moving material interfaces, deformable boundaries, free surfaces, etc. Though a large amount of work on the numerical schemes for solving the fluid dynamic problems have emerged, special difficulties still exist for problems with the above-mentioned features. Attempts have also been made to combine the best features of the FDM and FEM together by using two-grid systems (Lagrangian and Eulerian) like the methods of Coupled Eulerian Lagrangian (CEL) and Arbitrary Lagrange-Eulerian (ALE). These methods have been implemented in some commercial software packages such as MSC/Dytran (MSC/Dytran, 1997), DYNA2D and DYNA3D (Hallquist, 1988; 1986), LS-DYNA (Hallquist, 1998), and AUTODYN (Century dynamics, 1997). In these coupled methods, the computational information is exchanged either by mapping or by special interface treatment between these...
