analog of the governing differential equation, at any point m of the domain, as. N
Pergamon
Int. J. Mech.
So. Vol. 38, No. 6, pp. 589-606. 1996 Copyright 0 1996 Ekvier Science Ltd Printed in Great Britam. All rights reserved 0020-7403196 $15SKl+ 0.00
0020-7403(95)00079-8
THE DIFFERENTIAL QUADRATURE METHOD FOR IRREGULAR DOMAINS AND APPLICATION TO PLATE VIBRATION CHARLES W. BERT and MOINUDDIN School of Aerospace
and Mechanical (Received
Engineering, 19 December
The University
of Oklahoma,
MALIK Norman,
OK 73019-0601,
U.S.A.
1994; and in reoisedform 24 July 1995)
Abstract-By its very basis, the differential quadrature method may be applied to domains having boundaries oriented along the coordinate axes. In this paper, it is shown that quadrature rules may also be formulated for irregular domains using the natural-to-Cartesian geometric mapping technique. The application of the technique is demonstrated through the vibration analysis of thin isotropic plates of general quadrilateral and sectorial planforms. Key words: differential quadrature method, numerical solution methods, partial tions, irregular domains, free vibration, thin plate theory, geometric mapping.
differential
equa-
INTRODUCTION
The differential quadrature method (DQM), proposed in the early seventies by Bellman and associates [l, 21, is a numerical technique for the solution of initial and boundary value problems. The method has been experimented with and its general versatility has been established in a variety of physical problems, such as transport processes [3], structural mechanics [4-73 and hydrodynamic lubrication [8]; an exhaustive list of the literature on the DQM may be found in a forthcoming survey paper [9]. It has been claimed that the DQM has the capability of yielding accurate results with a minimal computational effort. In a recent comprehensive study [lo], it has been shown that the DQM stands out in numerical accuracy as well as computational efficiency over the well-known finite difference and finite element methods. The general efficacy of the DQM coupled with its analytical simplicity have posed the differential quadrature method as a possible alternative to the conventional numerical solution techniques. By its very basis, the differential quadrature method is generally considered to be limited in its application to the domains having boundaries that are parallel to the coordinate axes. Thus, the field domains considered in the applications of the DQM have been the line domains for one-dimensional and axisymmetric problems and rectangular domains for two-dimensional problems. Also, in a recent work [ll], the quadrature solution to the eigenvalue problem of skewed plates has been obtained using oblique reference axes. Mention may also be made of another work [12] in which an L-shaped domain was analyzed by domain decomposition into three rectangles for the quadrature solution to the pool boiling problem in cavities. Very recently, using the concept of domain decomposition, a general methodology has been presented [13] for the analysis of truss and frame structures by the DQM and the method has been named the quadrature element method (QEM). However, general irregular shaped domains having linear and/or curvilinear boundaries which are not parallel to the coordinates axes, are still outside the confines of the quadrature method. In view of the aforementioned limitation, the present work has been undertaken to extend the differential quadrature method to irregular domains in the form of general curvilinear quadrilaterals. The procedure involves reformulation of the quadrature rules using the geometric natural-to-Cartesian mapping concept as commonly employed in finite element analysis [14]. This mapping technique, using cubic serendipity shape functions, was first employed for the static and free vibration analysis of irregular shaped plates by the spline finite strip method [15, 161. Later, the same mapping technique with cubic serendipity and 589
C. W. Bert and M. Malik
Fig. 1. (a) A curvilinear
quadrilateral
region in Cartesian x-y plane, (b) a square parent natural 5-q plane.
domain
in
quartic-linear Lagrangian shape functions was employed for the analysis of arbitrary shaped plates using the finite strip method in conjunction with orthogonal polynomials [17]. More recently, linear serendipity shape functions have also been employed for the static and dynamic flexure analysis of straight-sided quadrilateral plates [18]. The procedure for the reformulation of quadrature rules for irregular domains is given in the next section. The application of the technique is then demonstrated through the free vibration analysis of thin plates of several planforms. The accuracy of the technique is demonstrated by comparing the calculated results of free vibration frequencies with the published results of analytical and numerical solutions. The curvilinear quadrilateral domains encompass a wide range of irregular geometries. This works develops the very first methodology for the extension of the differential quadrature method to irregular shaped domains. It is believed that this methodology in conjunction with domain decomposition [12] and quadrature element [13] concepts should go a long way in the development of the DQM for its employment in a larger class of problems which presently are considered to be in the territory of the finite element method. QUADRATURE
RULES
FOR
CURVILINEAR
QUADRILATERAL
DOMAINS
Let the field domain of interest, shown in Fig. l(a), be a curvilinear quadrilateral region in the Cartesian x-y plane. The geometric mapping of this domain may be accomplished from a square parent domain, - 1 < 5 < 1, - 1 < rl < 1 in the natural 5-q plane, shown in Fig. l(b), using the coordinate transformation x= :
Sit53
Vlxi,
Y =
z
si(5,
?)Yi
(1)
i=l
i=l
where, xi, yi; i = 1,2, . . . , IV, are the coordinates of N, points (the nodes, in finite element terminology) on the boundary of the quadrilateral region. Also, Si = Si(l, q); i = 1,2, . . . , N, are the interpolation or shape functions. By the definition of the shape functions, i.e. that a Si function has a value equal to unity at the ith node and zero at the remaining (Ns - 1) nodes, the region mapped by Eqn (1) and the given quadrilateral domain match exactly at the nodal points. In the notation of the DQM, the parent region is a regular domain and the partial derivatives of a field variable f with respect to the natural coordinates (5, q) at the pre-specified discrete points of the region may be obtained by the quadrature rules. Consider a set of N, x N,, sampling or grid points in the parent region obtained by taking N, points in - 1 < 4 < 1 and N,, points in - 1 < rl d 1, Fig. 2. Then the first-order derivatives af/at and af/aqat a discrete point ri, Y/jare defined by the quadrature rules as af 2
54 = 1.
I
,zlPikfkj
(2)
The differential quadrature method for irregular domains
591
Fig. 2. A quadrature grid.
and (3) where, Jj =f([i, qj). Here, Pik are the first-order t-derivative weighting coefficients associated with the 4 =
(5)
where 1J 1is the determinant of the Jacobian 8(x, y)/a(& q), that is,
The partial derivatives af/dx and aflay at the point xij = x(ri, qj), yij = y(
