A Generalized Coordinate Differential Quadrature Element Method
Chang-New Chen
Department of Naval Architecture and Marine Engineering
National Cheng Kung University, Tainan, Taiwan
Abstract
The method of differential quadrature approximates a partial derivative of
a variable function with respect to a coordinate at a discrete
point as a weighted linear sum of the function values at all discrete
points along that coordinate direction.
The differential quadrature can be generalized.
The calculation of weighting coefficients is generic. The weighting
coefficients for a grid model defined by a coordinate system having
arbitrary dimensions can also be generated. The configuration of a
grid model can be arbitrary.
The generalized coordinate differential quadrature also has these
characters. In the generalized coordinate differential quadrature,
the variable function is approximated as the linear combination of
one set of appropriate analytical functions and one set of
generalized coordinates. Then,
a partial derivative of a variable function with respect to a
coordinate at a discrete point is expressed as a weighted linear
sum of the generalized coordinates.
In this paper, a generalized coordinate differential quadrature element method is
proposed. In this method, the analysis domain of a problem is
separated into a certain number of subdomains or elements. Then the
differential quadrature 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
differential quadrature discretization. In order to solve the problem,
all discretized governing equations, transition conditions and
boundary conditions have to be assembled to obtain a global algebraic
system. Since all relations governing a continuous problem are
satisfied, the essence of this method is to find a numerically rigorous solution.
Numerical procedures for beam bending vibration and two-dimensional
steady-state heat conduction problems are illustrated.
Various techniques
can be used to calculate the direction cosines of the outward unit normal
vector on the element boundary in the two-dimensional simulation. The
technique of secant approximation is illustrated and used in the sample analyses.