Multiple Scale Differential Quadrature Method
Direct DQ method is a highly efficient method, being able to yield very accurate results by using a small number of nodes. But most of its applications are limited to one-and two-dimensional problems. Recent years saw considerable efforts to formulate three-dimensional DQ methods. Two approaches are commonly used in developing a three dimensional DQ method. The direct way is to discretize the physical quantity in all three dimensions in the same manner as that for two-dimensional problems. However, paramount difficulties arise in the direct discretization since DQ methods are global schemes with nearly full resultant matrices. It can be expected that the mesh size must be small if direct solution methods such as Gauss elimination are used. This is evidenced from the mesh size reported in the literature. For example, the maximum meshes used in the three-dimensional DQ calculations by Zhou et al., (2002) and Liew et al., (2001) were 8 × 8 × 6 and 11 × 11 × 11, respectively. The three digits indicate the corresponding mesh nodes adopted in the three dimensions. So 8 × 8 × 6 means that two dimensions are discretized by 8 nodes, respectively, and the other by 6 nodes. The mesh sizes in these applications are small. It is also found that increasing the mesh size to 18 × 18 × 18 will invalidate the computation on a computer of 128 MB memory. The restriction of the mesh size due to the employment of direct solution methods will be demonstrated later in the present chapter.