chapter  3
26 Pages

Triangular Differential Quadrature Method

In applying direct DQ method (DQM) to multi-dimensional problems, transformation (Lam, 1993; Bert and Malik, 1996d) is usually needed to map a non-rectangular physical domain into a normalized computational domain. As a result, a simple governing equation may turn into an awkward one, depending on the irregularity of the physical domain. For straight-edged quadrangles and even curvilinear quadrangles, the transformation does not impose significant difficulty to the solution as long as a proper transformation is available. However, for problems on a triangular domain which are often encountered in practice, singularity arises which has to be eliminated in the implementation of DQM, as pointed out by Zhong (1998). In the implementation of DQM on a triangular domain, one edge of the mesh has to degenerate into a single point, resulting in over-dense grid near the point and therefore the unnecessarily high computational cost. Additionally, the DQM is further hindered in this situation if high-order differential equation is to be solved. To overcome the above obstacles, a triangular differential quadrature (TDQ) method (TDQM) was proposed by Zhong (1998, 2000, 2001). In the present method, not only transformation is rendered unnecessary but also singularity does not appear as well.