Differential quadrature method in computational

Differential quadrature method in computational mechanics: A review Charles W Bert and Moinuddin Malik School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman OK 73019-0601 The differential quadrature method is a numerical solution technique for initial and/or boundary problems. It was developed by the late Richard Bellman and his associates in the early 70s and, since then, the technique has been successfully employed in a variety o f problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review o f the differential quadrature method, which should be o f general interest to the computational mechanics community.

CONTENTS 1 INTRODUCTION............................................................................ 1 2 THE DIFFERENTIALQUADRATUREMETHOD....................... 2 2. I The quadrature rules................................................................ 2 2.2 Weighting coefficients and sampling points ........................... 4 2.3 Examples of differential quadrature solutions ........................ 5 Example 1: Heat transfer in a triangular fin ............................ 5 Example 2: Torsion of a rectangular-cross-sectionshaft ........ 7 Example 3: A freely vibrating cantilever beam..................... 10 Example 4: Steady-state heat conduction in a slab with temperature-dependent conductivity............................... 1I Example 5: An integro-differential equation......................... 13 Example 6: Cooling/heating by combined convection and radiation........................................................................... 15 Example 7: One-dimensional, time-dependent heat diffusion in a sphere ...................................................................... 16 3 CHRONOLOGICALDEVELOPMENTOF THE DIFFERENTIAL QUADRATUREMETHOD........................................................ 17 4 GENERAL REMARKS................................................................. 22 5 CLOSURE ..................................................................................... 25 ACKNOWLEDGMENTSAND DEDICATION.............................. 25 REFERENCES.................................................................................. 25 1 INTRODUCTION Along with the evergrowing advancement o f faster computing machines, the research into the development of new methods for numerical solution o f problems in engineering and physical sciences also is an ongoing parallel activity. Such research interests, o f course, remain motivated by needs o f modern technology. As an example, simulation o f many dynamic systems often requires very fast numerical solution o f the equations o f the system mathematical models. Another example is the computer aided design process in which the database often requires large computer storage and the interpolative manipulations for the operating design parameters may be less accurate as well as quite time consuming. In such cases fast numerical solution o f the system equations offers the possibility o f more accurate and effi-

cient real-time analysis and design, bypassing fully or partially the need o f a database. This paper focuses on the differential quadrature method (Bellman, 1973; Bellman and Adomian, 1985; Bellman and Roth, 1986) which has a relatively recent origin and is gradually emerging as a distinct numerical solution technique for the initial- and/or boundary-value problems o f physical and engineering sciences. The problem areas in which the applications o f the differential quadrature method (referred to hereafter, for brevity, as the quadrature method or simply as the DQM) may be found in the available literature include biosciences, transport processes, fluid mechanics, static and dynamic structural mechanics, static aeroelasticity, and lubrication mechanics. It has been claimed that the DQM has the capability o f producing highly accurate solutions with minimal computational effort. The method has seemingly a high potential as an alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review o f the differential quadrature method. In the following, first the basic mathematical concepts underlying the DQM are presented. The implementation o f the method for the solution o f actual problems is elaborated through some examples. Due to its rather recent origin, the DQM is possibly not well known to the computational mechanics community. For this reason, the paper also aims to familiarize the readers with the DQM and, therefore, this section is written in a pedagogical manner. In the next section, a review o f the chronological development o f the method is presented. The paper is concluded with remarks on the issues that concern the application and further development o f the DQM. The meanings o f the symbols used in the paper are defined within the text.

Transmitted by Associate Editor Isaac Elishakoff ASME Reprint No AMR182 $22 Appl Mech Rev vo149, no 1, January 1996

© 1996 Amencan Society of Mechanical Engineers

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Appl Mech Rev vol 49, no 1, January 1996

Bert and Malik: Differential quadrature method in computational mechanics

2 THE DIFFERENTIAL QUADRATURE METHOD 2.1 T h e q u a d r a t u r e

rules

The numerical methods for the solution of initial- and/or boundary-value problems, in general, seek to transform, either through a differential or an integral formulation, the governing differential and/or integro-differentiai equations into an analogous set of first-order or algebraic equations in terms of the discrete values of the field variable (the function) at some prespecified discrete points of the solution domain. In the differential quadrature method, this is accomplished by expressing at each grid point, the calculus operator value of a function with respect to a coordinate direction at any discrete point as the weighted linear sum of the values of the function at all the discrete points chosen in that direction. In order to go into the mathematical basis of the DQM, consider a function W = q'(x,y) having its field on a rectangular domain 0 _ 2 ) Z A (r-l,t.lJ k=l

m=l

k=l

m=l

Now following Eqs (8) and (9), one may easily obtain the following recurrence relationships for the weighting coefficients

[A,,>1= [A,,,][A'r "] = [A

It may be seen that having the matrix [AO)] of first-order derivative weighting coefficients, one can obtain the weighting coefficients of the higher-order derivatives by successive multiplications of the [,4(1)] matrix by itself. Eqs (8) through (10) are given for the x-partial derivatives; the equations for the y-partial derivatives follow in an identical manner. In its usual sense, the term quadrature refers to the approximation of an integral of a function by a linear weighted sum of the function values at some sampling points taken between the limits of integration. It is interesting to mention that the quadrature rule for function derivatives was actually formulated as an analogous extension of integral quadrature by Bellman and Casti (1971). The x-integral of the function W(x,y)on any line y =yj is N,

d

(11)

fxa__oW(x,yj)dx=ZCkqJkj k=l

where {W}j and{Wit)), are the column vectors of the

Nx

J

values each of the function and its rth-order x-partial derivative, respectively, at the sampling points on a line y = yj. Also, [A(r)] is the Nx x Nx matrix of weighting coefficients of the rth-order derivatives. Further, noting the definition of the differential operators

and they-integral on any line x = xi is

~y~ W(xi'y)dy=Z DeWi' =0

Eqs (11) and (12) are the rules of integral quadrature. Here, Ck and D e are the weighting coefficients for integrals in the x and y directions, respectively. Using Eqs (3) through (5) in Eqs (11) and (12), one obtains

Oxr

O

one may also express the quadrature rule from Eq (1) as

N

(12)

f=l

/l

j = It2

Fig 2. Quadrature grid for a parallelogram region

Fig 3. Quadrature grid for a concentric, circular, sectorial region


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Appl Mech Rev vol 49, no 1, January 1996


~(xO-i ) c,=T; a ° o = 1 , 2 ..... Nx

(13)

k=l

and Nr

•

)~-I

=--;

C=l

g=

1,2,...,Ny

(14)

I't

The quadrature rules as given by Eqs (1), (2), (13), and (14) may also be used for linear combinations of the function derivatives and integrals, but with respect to one independent variable only. That is to say, a quadrature rule is not given in the form of Eqs (1) and (2), for example, in the case of a mixed partial derivative of the type ~r+s)~t}l/Oxr~y~. However, following the definition of calculus operators, the differential quadrature analog of a mixed derivative may be obtained as o(r+s)~I/ x,,yi

OxrOyS

or

=0

where

k=l

(xi-xo)'n(xk)=

H

H (x*-xo)"

(18)

u=l,u~k

The off-diagonal terms of a weighting coefficient matrix of the second- and higher-order derivatives may be obtained through the following recurrence relationship

g v

qJ(x'y)dxdy= Z C k ZDe~k'" =0

i,k=l,2,...,N x a n d k # i (17)

(15)

and, similarly, for mixed integration as gx

for

t)=l.u~i

t=l

k=l

I~' f~

I-I(xi)

A'7 =

rl(xi)=

/[

Nx Nv = - %~ ~(r) ~-~ n('~)~u Oxr t ~yS )1 xi'Y' - ~'# ~ik ~ uj£ --kg ( OskI~

(1973). The weighting coefficients for the derivatives may be obtained directly, and most accurately, irrespective of the number and positions of the sampling points, from the explicit formulae (Quan and Chang, 1989a; Shu and Richards, 1992a-b). These extremely useful formulae, taken from Shu and Richards (1992a-b), are given here for the interest of the readers. These are given with respect to the x-coordinate only; the formulae with respect to the y-coordinate would follow in an identical manner. The off-diagonal terms of the weighting coefficient matrix of the first-order derivative are given by

(16)

f=l

Using the quadrature rules for the various order derivatives, one may write the quadrature analog of a given differential equation at each grid point of its solution domain and, consequently, obtain a set of first-order or algebraic equations in terms of the grid-point function values. One may also form the quadrature analog equations of the boundary conditions. The first-order equations may be integrated in time or the algebraic equations of the differential equations and their boundary conditions be solved simultaneously to obtain the unknown grid-point function values. Obviously, the same procedure also applies to integral or integro-differential equations.

2.2 Weighting coefficients and sampling points Two extensively decisive factors in the accuracy of the differential quadrature solutions are: one, the accuracy of the weighting coefficients and two, the choice of sampling points. 1 In order to obtain the weighting coefficients, one may solve the Vandermonde system of equations, such as Eqs (6), (7), (13), and (14), using the usual linear equation solvers. However, Vandermonde matrices are known to be inherently ill-conditioned (Press et al 1988) and, in fact, it is experienced that the weighting coefficients obtained by a direct solution of the Vandermonde equations become increasingly inaccurate with an increasing number of sampling points. Better accuracy in the weighting coefficients may be obtained using the analytical solution method of Hamming IThese two issueswill be discussedfurtherin the nexttwo sections.

A)[)=r[ Aff=OA)~) x,A~[-Ok for

(19)

i , k = l , 2 ..... Nx a n d k # i

where 2 _
where r I is a dummy variable to ~. The exact solution of Eq (79) subject to the boundary condition f=lat

~=0

(80)

is (Civan and Sliepcevich, 1986)

La;

i = 2,3 ..... N

(87)

j=2

where L/j = d~. I)

-e~iCj

(88)

so that Lu are the weighting coefficients of the integro-differential operator

df Iie~_nf(rl) dq

d~ f({)

= 3 e - 4 + c o s l - sin 1

ea-ri

(e -l)+2-cos

(81)

where in Eqs (79) and (81), "e" is the base of the natural logarithms. For the quadrature formulation of Eq (79), the quadrature rule for the integral term is written as

which may be obtained from the equations N

• I _

-I

j=l

(89) u = l , 2 ..... N.


Appl Mech Rev vo149, no 1, January 1996


The results in Table 5 show the exact values of the function f i t ) at t = 0.1 intervals in 0.1 _ 9, Type II DQ solutions match to eight or more decimal places with the exact solution. It should be noted that the differential quadrature solution of the foregoing example was first considered by Civan and Siiepcevich (1986). However, the present solution is modified in the quadrature formulation of the integral term of the governing equation. Consequently, the results presented in Table 5 are more accurate than those in the cited work.

Example 6: Co®ling/heating by combined convection and radiation Consider a body subjected to heat transfer by combined convection and radiation. Assuming a lumped capacity model of the body, the time rate of change in temperature of the body is given by (Lienhard, ! 987)

'~

(90)

where O = O(T) is the absolute temperature of the body, x is a nondimensional time, and Oo and Os are the absolute temperatures of, respectively, the ambient gas and distant surroundings with which radiation exchange occurs. Also, h is the convection heat transfer coefficient, F is the view factor, and ~ is the Stefan-Boltzmann constant. In the solution of Eq (90), one would be interested in knowing the temperature versus time history of the body for some given value of its initial temperature. A long-term integration of Eq (90) should indicate a steady-state or final temperature ®f which the body would eventually reach. However, the final temperature O r of the body may be obtained directly by applying the physical condition dO O---~Of, d~ ---~0as ~--~oo

(91)

to Eq (90) giving the quartic equation

04'f + "T~h 0 "f -

4

_h._ 0

O s - Fc~

a

= 0

T.f

and substituting in Eq (90), one obtains the governing equation in the normalized time domain 0 _