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JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 1, pp. 123-139, Warsaw 2008

METHODS BASED ON THE DIFFERENTIAL QUADRATUREIN VIBRATION ANALYSIS OF PLATES

Artur Krowiak

Institute of Computing Science, Cracow University of Technology, Cracow, Poland

e-mail: [email protected]

The paper deals with the methods based on the differential quadrature andtheir application to the free vibration analysis of plates. The spline-based dif-ferential quadrature Method (SDQM) is presented as an alternative to knownmethods based on the interpolation polynomial (PDQM). The SDQM uses apolynomial piecewise function to approximate the wanted solution of a go-verning equation. The way of determining the spline functions as well as theway of computing weighting coefficients for the method are presented in thepaper. Then the SDQM is applied to determine natural frequencies of plates.The influence of the spline degree, number of nodes and grid point distribu-tion on the accuracy, convergence and stability is investigated in an example.

All results are compared with values obtained by the conventional differentialquadrature method (PDQM).

Key words:Differential quadrature method, spline interpolation, free vibration

1. Introduction

Most engineering problems are described by partial differential equations. It

is very difficult, if possible at all, to find closed-form solutions to them. Agreat increase in the computational power in recent years enables one to usenumerical methods for solving very complex physical tasks. It is the reasonthat stimulates the development of known methods and the search for new,more efficient ones. Most of these methods rely on the conversion of a phy-sical model of a system from continuous to discrete one. The approximatesolution is searched at some special points in the domain. Among currentlyused discretisation methods, the most popular are: finite difference method,finite element method and finite volume method. These discretisation tech-niques use a low degree interpolation to determine function values at a node

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124 A. Krowiak

and, therefore, they are numbered among low order methods. They allow oneto achieve high accuracy by using a large number of nodes. In many prac-tical engineering problems, the numerical solution is required only at a fewdiscrete points in the domain. The modal analysis could be an example. By

discretizing a partial differential equation, an algebraic eigenvalue problem canbe obtained. Next, the latter is solved giving approximation values of naturalfrequencies (or/and natural modes) of a continuous system. The number of ob-tained frequencies corresponds to the number of nodes of the imposed mesh.Among these frequencies, only a few first are interesting from the practicalpoint of view. However, in order to achieve high accuracy for these values bylow order methods, one has to use a large number of nodes. It requires muchvirtual storage and computational effort. This drawback can be overcome byusing so-called global methods, which take much more information than onlylocal neighborhood of a node to approximate the function. It makes the rate ofconvergence of these methods much higher than low order methods and allowsone to achieve very accurate results with only a few discrete points.

The differential quadrature method (DQM) falls under this category. The-oretical foundations of the DQM were given by Bellman and Casti (1971). Inrecent years, one has been able to notice significant development of the methodand its application in many fields of mechanics. Some main works are quotedby Bert and Malik (1996). Higher efficiency of the DQM for linear problems

than the finite element and finite difference method was proved by Bert et al.(1988, 1993), Wang and Bert (1993), Malik and Civen (1994). In addition, theDQM is much more effective for nonlinear problems comparing with low ordermethods, which was presented by Bellman and Casti (1971), Bellman et al.(1972), Bert et al. (1989), Feng and Bert (1992), Malik and Civen (1994).

The idea of the DQM is based on the approximation of spatial derivativesof a function at each node by a linear combination of function values at alldiscrete points in the domain along the coordinate lines. Considering a two-dimensional case, where the domain of the function f(x, y) is a rectangular

area, partial derivatives with respect to spatial variables at each point (xi, yi)can be expressed in the method as

nf

xn

x= xiy=yj

=Nxk=1

a(n)k (xi)f(xk, yj) =

Nxk=1

a(n)ik f(xk, yj)

(1.1)

mf

ym

x= xi

y=yj

=

Nyk=1

b(m)k (yj)f(xi, yk) =

Nyk=1

b(m)jk f(xi, yk)

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Methods based on the differential quadrature... 125

for i = 1, . . . , N x, j = 1, . . . , N y, where Nx, Ny are the numbers of nodes in

the x and y directions, and a(n)ik , b

(m)jk are the weighting coefficients for the

nth and mth order derivatives with respect to appropriate variables. In orderto approximate the mixed derivatives, the weighting coefficient matrices for

appropriate derivatives with respect to x and y have to be multiplied. Usingformula (1.1), one has to collocate a governing equation at each grid point andimpose boundary conditions in order to reduce the partial differential equationto a system of algebraic or ordinary differential equations, respectively of thecase under consideration. The main stage of the method is the determinationof the weighting coefficients.

2. Polynomial and Spline-based differential quadrature method

Values of the weighting coefficients depend on the way the solution is ap-proximated (selection of trial functions), and they influence the accuracy, co-nvergence and stability of the method. The interpolation polynomial is themost often used to approximate the solution in the differential quadraturemethod (PDQM). Using Lagrange base functions, the rth order derivativesof the interpolation polynomial at the ith discrete point can be expressed asfollows

P(r)(xi) =N

j=1

l(r)j (xi)fj (2.1)

where lj(x) denotes Lagranges base polynomial of the (N 1)th degree.

It is easy to notice that the derivatives of the appropriate Lagrange basefunctions are the weighting coefficients for the polynomial based differentialquadrature method.

The analytical formulas for the coefficients of the first order derivative weregiven by Quan and Chang (1989) and have following forms

a(1)ij =

1

xj xi

Nk=1,k=i,j

xi xkxj xk

for j =i

(2.2)

a(1)ii =

Nk=1,k=i

1

xi xk

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126 A. Krowiak

Due to difficulties with derivation of explicit formulas for higher order derivati-ves of Lagranges base functions, Shu and Richards (1992) gave recurrence re-lationship (2.3) to overcome the problem, i, j = 1, 2, . . . , N ,r = 2, 3, . . . , N 1

a(r)ij =r

a

(1)ij a

(r1)ii

a(r

1)ij

xi xj

for i=j

(2.3)

a(r)ii =

Nj=1,j=i

a(r)ij

There are also other ways based on the Fourier series expansion or B-splinefunctions to approximate the wanted solution in the DQM. But especially

PDQM allows one to achieve very high accuracy by using only a few nodes.However, this method is very sensitive to the type of imposed mesh and thenumber of nodes. When a uniform grid distribution is imposed and too manysampling points are used, the results are inaccurate or the method is notconvergent. The computational instability of the method is the result fromthe manner of the interpolation polynomial. This polynomial oscillates muchwhen its degree N1 (N number of nodes) is too high, particularly when theuniform grid is imposed. In most problems of computational mechanics, theonly way to estimate the accuracy of results is to carry out the computation

again using larger number of nodes. Therefore, the computational stabilityshould be the important feature of a numerical method. To improve stabilityof the DQM, Zhong (2004) used quintic B-spline as trial functions to determinethe weighting coefficients. It considerably improves stability of the method butthe convergence rate is less comparing to the PDQM.

In the present work, another way of the approximation of the solution isshown. In the presented method, the solution of a partial differential equationis approximated by the nth degree polynomial piecewise function (SDQM).The approximation was first applied in the DQM by Krowiak (2006). Thework, that has been done so far, indicates that using a suitably high spli-ne degree the convergence rate of the method is similar to PDQM and thecomputational stability is much better even when using uniformly spacednodes.

When the degree n of the spline is odd then the function is approximatedin the following way

f(x) {si(x), x [xi, xi+1], i= 1, . . . , N 1} (2.4)

where N is the number of nodes.

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When nis even, the auxiliary spline knots are defined at the midpoints ofthe nodes in order to be able to meet the conditions for the determination ofthe spline function

z1 =x1 zi+1=1

2 [xi+xi+1] i= 1, . . . , N 1 zN+1=xN

(2.5)and the interpolation function has the form

f(x) {si(x), x [zi, zi+1], i= 1, . . . , N } (2.6)

In Equations (2.4) and (2.6), the ith spline section is defined as

si(x) =n

j=0

cijxj (2.7)

The coefficients cij in Equation (2.7) are determined using the interpolationconditions, the continuity conditions of the derivatives at the nodes and theso-called natural end conditions at the domain boundaries. In the case of anodd degree, these conditions have the following form

si(xi) =fi si(xi+1) =fi+1 i= 1, . . . , N 1

s(k)

i (xi

+1) =s

(k)

i+1(xi

+1) i= 1, . . . , N 2, k= 1, . . . , n 1

s(k)1 (x1) = 0 s

(k)N1(xN) = 0 k=

n+ 1

2 , . . . , n 1

(2.8)Their number is (n+ 1)(N 1), which corresponds to the number of coeffi-cients cij . In the case of an even spline degree, where the number of unknowncoefficients is (n+ 1)N, the auxiliary knots are also used in interpolationconditions (2.9) and the continuity conditions for derivatives (2.10)

si(xi) =fi i= 1, . . . , N si(zi+1) =si+1(zi+1) i= 1, . . . , N 1

(2.9)

and

s(k)i (zi+1) =s

(k)i+1(zi+1) i= 1, . . . , N 1, k= 1, . . . , n 1 (2.10)

To complete the set of equations, the natural end conditions are introducedat the end points

s(k)1 (x1) = 0 s(k)N (xN) = 0 k= n2

, . . . , n 1 (2.11)

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128 A. Krowiak

It insures that in every case of an even spline degree the number of conditionsmatches the number of coefficients cij. These coefficients depend on the griddistribution and unknown function values according to the formula

cij =Nk=1

Cijk(x1, . . . , xN)fk i= 1, . . . , N, j = 0, . . . , n (2.12)

where N=N 1 when nis odd and N=N when n is even.

Taking advantage of the symbolic computation system, one can easily de-termine the weighting coefficients for the differential quadrature method ba-sed on the approximation given by Eqs. (2.4) and (2.6). By manipulating theexpressions in which the unknown function values f(xi) are defined as sym-

bols fi, it is possible to compute derivatives of an arbitrary degree r (r < n1)for function (2.4) or (2.6) at the ith discrete point

f(r)(xi)

s(r)i (xi) i= 1, . . . , N

s(r)i1(xi) i= N when n is odd

(2.13)

In Eq. (2.13), s(r)i (xi) and s

(r)N1(xN) have the forms

s(r)i (xi) =

nj=r

Nk=1

Cijkfk

xjrij

l=jr+1

l

i= 1, . . . , N

(2.14)

s(r)N1(xN) =

nj=r

Nk=1

CN1jkfk

xjrN

jl=jr+1

l

where the advantage has been taken of Equation (2.12).

Equations (2.14) can be expressed in other forms

s(r)i (xi) =

Nk=1

nj=r

Cijkx

jri

jl=jr+1

l

fk i= 1, . . . , N

(2.15)

s(r)N1(xN) =

Nk=1

nj=r

CN1jkx

jrN

jl=jr+1

l

fk

Comparing Eq. (1.1) to (2.15), it is clear that the expressions in square bracketsin the above formula are the weighting coefficients for the rth order derivative

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in the differential quadrature method based on the spline functions which canbe written as follows

a(r)ik =

n

j=r

Cijkxjri

j

l=jr+1

l i= 1, . . . , N (2.16)

a(r)Nk =

nj=r

CN1jkx

jrN

jl=jr+1

l

whenn is odd

3. Free vibration analysis of rectangular plates

Plates belong to basic structural elements in civil and mechanical engineeringand, therefore, they are often subjects of static and dynamic research. Manynumerical methods have been used in dynamical analysis of plates with variousboundary conditions. The obtained results have been used in real structuresor as the comparison of accuracy and convergence for applied methods. Theconventional differential quadrature method has also been applied to the vibra-tion analysis of plates (Shu and Du, 1997; Wang and Bert, 1993). The resultsshow that the convergence rate of the PDQM is very high. Very accurate re-

sults can be obtained applying a grid with points densely concentrated nearboundaries. The use of an arbitrary grid, for example a uniform one, makesthe results inaccurate or the method not convergent. This drawback can beovercome by using the method presented in this paper.

Since the results and conclusions coming from application of the PDQMto vibration analysis of plates are common and well know, the SDQM hasbeen applied to check its versatility in using various grid point distributions.In the paper, the SDQM has been applied to determine natural frequencies ofa thin, isotropic, rectangular plate. The dimensionless governing equation for

free vibration of the plate is as follows

4W

X4 + 22

4W

X2Y2+4

4W

Y4 =2W (3.1)

In the above equation W denotes dimensionless mode shape function,X = x/a and Y = y/b are dimensionless coordinates, a and b are leng-ths of the plate edges, = a/b is the aspect ratio and is the dimensionlessfrequency. Its relation with the dimensional circular frequency is following

=a2

D

(3.2)

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130 A. Krowiak

where is the density of the plate material and D= Eh3/[12(1 2)] is theflexural rigidity (E, , h are Youngs modulus, Poissons ratio and the platethickness, respectively). Calculations have been done for square plates (= 1)with following configurations of boundary conditions:

(a) SS-F-SS-F

for X= 0 and X= 1

W = 0 2W

X2 = 0 (3.3)

for Y= 0 and Y = 1

22W

Y2 +

2W

X2 = 0 2

3W

Y3 + (2 )

3W

X2Y = 0 (3.4)

(b) C-F-SS-F

for X= 0

W = 0 W

X = 0 (3.5)

for X= 1

W = 0 2W

X2 = 0 (3.6)

for Y= 0 and Y = 1

22W

Y2 +

2W

X2 = 0 2

3W

Y3 + (2 )

3W

X2Y = 0 (3.7)

(c) C-C-C-C

for X= 0 and X= 1

W = 0 W

X = 0 (3.8)

for Y= 0 and Y = 1

W = 0 W

Y = 0 (3.9)

whereC denotes the clamped edge, SS simply supported edge andF freeedge.

Chebyshev-Gauss-Lobatto grid (3.10) and a uniform grid have been usedto discretize the area 0 X1, 0 Y 1. In both cases, the same numberof points has been used in the X and Y directions

Xi=Yi=12

1 cos

i 1N 1

i= 1, . . . , N (3.10)

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Equation (3.1), written in a discrete form following from the use of the method,is as follows

N

k=1

a(4)ik Wkj + 2

2N

k1=1

N

k2=1

a(2)ik1

a(2)jk2

Wk1k2+4

N

k=1

a(4)jk Wik =

2Wij (3.11)

where a(r)ik denote the weighting coefficients for the rth order derivative in the

SDQM and Wij are unknown nodal values. Identical coefficients a(r)ik appro-

ximate the derivatives in both directions due to the use of the same grid andnumber of nodes in these directions.

The implementation of boundary conditions is the important stage of themethod. Various approaches to this problem were presented by Shu and Du(1997). In the present study, the boundary conditions have been used to cal-

culate function values at the boundary points and points adjacent to the bo-undaries as a linear combination of the values at the interior points. Detailsare presented for theSS-F-SS-Fplate configuration. According to the DQM,the discrete form of the boundary conditions described by Eq. (3.3) and (3.4)is following

W1j = 0 j = 1, . . . , N

Nk=1

a(2)1kWkj = 0 j = 2, . . . , N 1

(3.12)

WNj = 0 j = 1, . . . , N

Nk=1

a(2)NkWkj = 0 j = 2, . . . , N 1

(3.13)

2Nk=1

a(2)1kWik+

Nk=1

a(2)ik Wk1= 0 i= 2, . . . , N 1

2Nk=1

a(3)1kWik+ (2 )N

k1=1

Nk2=1

a(2)ik1a(1)1k2

Wk1k2 = 0 i= 3, . . . , N 2

(3.14)

2Nk=1

a(2)NkWik+

Nk=1

a(2)ik WkN= 0 i= 2, . . . , N 1

2N

k=1

a(3)NkWik+ (2 )

N

k1=1

N

k2=1

a(2)ik1

a(1)Nk2

Wk1k2 = 0 i= 3, . . . , N 2

(3.15)

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References

1. Bellman R., Casti J., 1971, Differential quadrature and long term integra-tion, Journal of Mathematical Analysis and Application, 34, 235-238

2. Bellman R., Kashef B.G., Casti J., 1972, Differential quadrature: A tech-nique for the rapid solution of nonlinear partial differential equations, Journalof Computational Physics, 10, 40-52

3. Bert C.W., Jang S.K., Striz A.G., 1988, Two new methods for analyzingfree vibration of structure components, AIAA Journal, 26, 612-6182

4. Bert C.W., Malik M., 1996, Differential quadrature method in computatio-nal mechanics, Applied Mechanics Review, 49, 1-28

5. Bert C.W., Striz A.G., Jang S.K., 1989, Nonlinear bending analysis of

orthotropic rectangular plates by the method of differential quadrature, Com-putational Mechanics, 5, 217-226

6. Bert C.W., Wang X., Striz A.G., 1993, Differential quadrature for staticand free vibration analysis of anisotropic plates,International Journal of SolidStructures,30, 1737-1744

7. Feng Y., Bert C.W., 1992, Application of the quadrature method to flexuralvibration analysis of a geometrically nonlinear beam, Nonlinear Dynamics, 3,13-18

8. Krowiak A.

, 2006, Symbolic computing in spline-based differential quadraturemethod,Communications in Numerical Methods in Engineering,22, 1097-1107

9. Leissa A.W., 1973, The free vibration of rectangular plates,Journal of Soundand Vibration, 31, 257293

10. Malik M., Civan F., 1994, A comparative study of differential quadratu-re and cubature method vis-a-vis some conventional techniques in contextof convection-diffusion-reaction problems, Chemical Engineering Science, 50,531-547

11. Quan J.R., Chang C.T., 1989, New insight in solving distributed system

equations by the quadrature methods I, Computer and Chemical Engineering,13, 779-788

12. Shu C., 2000, Differential Quadrature and its Application in Engineering,Springer-Verlag, London

13. Shu C., Du H., 1997, Implementation of clamped and simply supported bo-undary conditions in GDQ free vibration analysis of beams and plates, Inter-national Journal of Solids and Structures,34, 819-835

14. Shu C., Richards B.E., 1992, Application of generalized differential quadra-

ture to solve two-dimensional incompressible Nanier-Stokes equations,Interna-tional Journal for Numerical Methods in Fluids, 15, 791-798

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15. Wang X., Bert C.W., 1993, A new approach in applying differential quadra-ture to static and free vibration analysis of beams and plates,Journal of Soundand Vibration, 162, 566-572

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rential equations and its application to Kirchhoff plates, Applied MathematicalModelling, 28, 353-366

Metody oparte na kwadraturach rniczkowych w zastosowaniudo drga pyt

Streszczenie

Praca dotyczy metod opartych na kwadraturach rniczkowych i ich aplikacjido zagadnienia drga wasnych pyt. W pracy, jako alternatyw do znanych me-tod kwadratur rniczkowych, opartych na wielomianie interpolacyjnym (PDQM),przedstawiono metod bazujc na funkcjach sklejanych (SDQM). W SDQM poszu-kiwane rozwizanie przybliane jest funkcj wielomianow, przedziaami zmienn. Wpracy przedstawiono sposb wyznaczenia takiej funkcji interpolacyjnej, jak rwniesposb obliczenia wspczynnikw wagowych, uywanych w metodzie kwadratur r-niczkowych. Nastpnie SDQM uyto do wyznaczenia czstoci drga wasnych pyt,gdzie analizowano wpyw stopnia wielomianu, liczby wzw i ich rozmieszczenia nazbieno, dokadno i stabilno metody. Otrzymane rezultaty porwnano z wy-nikami uzyskanymi przy pomocy konwencjonalnej metody kwadratur rniczkowych(PDQM).

Manuscript received February 21, 2007; accepted for print April 4, 2007

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