Because only problems having simple regular domains and under simple external environments can be solved by using DQ, the application of this method is very limited.
The author has proposed the DQEM for solving a generic engineering or scientific problem having an arbitrary domain configuration. Like the FEM, in this method, the analysis domain of a problem is first separated into a certain number of subdomains or elements. Then the DQ, GDQ or EDQ discretization is carried out on an element-basis. The governing differential or partial differential equations defined on the elements, the transition conditions on inter-element boundaries, and the boundary conditions on the analysis domain boundary are in computable algebraic forms after the DQ, GDQ or EDQ discretization. By assembling all discrete fundamental equations an overall algebraic system can be obtained which is used to solve the problem.
The DQFDM has also been proposed by the author. The finite difference operators are derived by DQ. They can be obtained by using the weighting coefficients for DQ discretizations. The derivation is straight and easy. By using different orders or the same order but different grid DQ discretizations for the same derivative or partial derivative, various finite difference operators for the same differential or partial differential operator can be obtained. Finite difference operators for unequally spaced and irregular grids can also be generated through the use of GDQ.
DQEM and DQFDM have been used to develop solution algorithms for computational mechanics. In this paper, numerical results are presented to demonstrate these two discrete analysis methods.