Multi-Domain Differential Quadrature Method
There arise difficulties when we apply DQ method to the elastic problems where discontinuities are present. If an elastic structure under consideration is made of two or more materials, at least one of stress components at the interface of the two different materials is not continuous, leading to a finite jump at the interface. It is mathematically clear that a finite discontinuity can only be described by a first-order numerical scheme (Sod, 1978; Zong et al., 2005). Any high-order numerical scheme must change to first-order scheme at the discontinuity (but remains a high-order scheme elsewhere). DQ method is equivalent to Lagrange interpolation and differentiation using
Lagrange polynomial, and thus it is a high-order numerical scheme, being differentiable to many orders. However, it fails at the discontinuity where a first-order scheme is required. In these cases, direct application of DQ method would yield very bad results due to the nature of the method. A multi-domain DQ method is formulated in this chapter to solve plane
elastic problems in the presence of material discontinuity. By putting the boundary of each sub-domain on the interface of two different materials, discontinuity is transformed to proper imposition of compatibility conditions at the interface. It turns out that the compatibility conditions at the interfaces have dominant influence on the numerical accuracy. Numerical examples show that the proposed method, which is first-order accurate at the interfaces and high-order accurate elsewhere, can properly capture the material discontinuity and yield good results when compared with those given by FEM.