Complex Differential Quadrature Method
Theory of complex variables is the branch of mathematics investigating functions of complex variables and is extremely important in complex analysis. It is hard to find another mathematical branch in which complex variable has not been mentioned. This is a very special mathematical phenomenon because most of mathematical branches are irrelevant. It is often an important mathematical event if two branches of mathematics are linked through a subtle way, but complex analysis comes into other mathematical branches in a natural way. Without the aid of complex analysis, we are even unable to prove the fundamental theorem of algebra. Theory of complex variable is very important not only as a tool for math-
ematical analysis, but also as an effective technique for solving some tough problems in areas as diverse as electromagnetism, thermodynamics, fluid mechanics, thermal mechanics, elastic mechanics and acoustics. It is even able to provide a surprisingly concise solution to singular problems, which are hard to obtain by using an alternative method. In mechanics of elasticity, the stress field of a crack tip in a plate is singular, indicating that the stress there becomes larger and larger as we get closer and closer to the crack tip. The solution obtained from complex variable theory reveals that the singular behavior of such a tip is described by the reciprocal of distance r from the crack tip. Another striking application of complex variable theory is that it provides a complete solution to the lifting problem of a two-dimensional wing, which singularity at the trailing edge is removed through smart introduction of Kutta condition. Therefore, theory of complex variables remains important in the future. In view of this, it is necessary to generalize DQ method from the real domain to the complex domain, termed as complex DQ method. In this chapter, complex DQ method (Zong, 2003a; Zhang et al., 2007) is
introduced. First, extension of Lagrange interpolation to the complex plane is presented. It turns out to be composed of two parts: polynomial and rational interpolations of an analytic function. Based on Lagrange interpolation in the complex plane, complex differential quadrature method is constructed suited to solving the potential problem (Laplace’s equation) and the plane linear elastic problem (the so-called bi-harmonic equation). Three important issues in the method are:
(2) Use of analytic function or Airy’s stress function in the form of complex variable; and
(3) Application of boundary conditions to determination of the unknowns.