Analysis of general shaped thin plates by the moving least-squares differential quadrature method

Finite Elements in Analysis and Design 40 (2004) 1453–1474www.elsevier.com/locate/!nel

Analysis of general shaped thin plates by the movingleast-squares di)erential quadrature method

K.M. Liewa;b, Y.Q. Huanga, J.N. Reddyc;∗

aNanyang Center for Supercomputing and Visualization, Nanyang Technological University, Singapore 639798bSchool of Mechanical and Production Engineering, Nanyang Technological University, Singapore 639798

cAdvanced Computational Mechanics Lab, Department of Mechanical Engineering,Texas A&M University, College Station, TX 77843-3123, USA

Received 28 October 2002; received in revised form 30 September 2003; accepted 21 October 2003

Abstract

The mesh-free moving least-squares di)erential quadrature (MLSDQ) method is proposed for solving thefourth-order, partial di)erential equation governing the bending of thin plates according to classical platetheory. The de9ections of an arbitrary shaped plate are expressed in terms of the MLS approximation. Theweighting coe;cients used in the MLSDQ approximation are calculated through a fast computation of theshape functions and their derivatives. The discrete multiple boundary conditions and governing equations aresolved by a least-squares approximation. Numerical examples are presented to illustrate the accuracy, stabilityand convergence of the present method. E)ects of support size, order of completeness and node irregularityon the numerical accuracy are carefully investigated.? 2003 Elsevier B.V. All rights reserved.

Keywords: Di)erential quadrature method; Plates; Moving least-squares approximation; Numerical method;Meshless method; Mesh-free method

1. Introduction

Since the introduction of the di)erential quadrature method (DQM) by Bellman and Casti [1], theDQM has been widely applied to many engineering problems as an alternative numerical techniquefor solving partial di)erential equations in place of the !nite element method (FEM) and the !nitedi)erence method (FDM). Details of the development and application of the DQM can be found ina review paper by Bert and Malik [2].

∗ Corresponding author. Tel.: +1-979-8622417; fax: +1-979-8623989.E-mail address: [email protected] (J.N. Reddy).

0168-874X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.!nel.2003.10.002

mailto:[email protected]

1454 K.M. Liew et al. / Finite Elements in Analysis and Design 40 (2004) 1453–1474

As stated by Civan and Sliepcevich [3], the DQM approximates the partial derivative of a variableat a given discrete point as a linear sum of the function values at all the discrete points in domain ofthat variable. Polynomials and harmonic functions are chosen, for convenience, as the test functionsto calculate the weighting coe;cients in most applications. However, di;culties arise in using theDQM to model engineering problems with discontinuity or interfaces since the approximation isassumed to be continuous in the entire domain. To overcome this drawback, an element-like methodwas developed by combining the DQM with the domain decomposition technique. Chen [4] and Strizet al. [5] proposed the quadrature element method (QEM) and successfully applied it to the staticanalysis of truss and beam structures. Liu and Liew [6,7] proposed the di)erential quadrature elementmethod (DQEM) to study the bending of plates using !rst-order shear deformation plate theory. Chen[8,9] applied the DQEM to torsion and heat conduction problems. Another drawback of the DQMis that a regular discrete node pattern and simple boundary geometry are usually necessary in orderto determine the weighting coe;cients explicitly. Although a domain transformation is possible forirregular shapes, it causes a signi!cant loss of e;ciency and simplicity, especially for problemsinvolving large spatial derivatives. Therefore, the DQM is very similar to a high order FDM schemeand is much limited in its versatility in comparison with the FEM.

In this paper, the MLSDQ method is developed for numerical solution of partial di)erential equa-tions [10]. Usually, to solve this problem with the fourth-order partial di)erential governing equation,the DQM needs a �-type node arrangement in order to satisfy the multiple boundary conditions [2].However, the numerical results obtained by the so-called �-technique may depend on the choiceof �, and a very small distance is always needed to achieve enough numerical accuracy, whichoften results in ill-conditioned matrices. In the present work, a least-squares approximation is usedto treat the multiple boundary conditions. The discrete governing equations and boundary conditionsare approximately satis!ed by minimization of a least-squares norm sum. This method overcomesthe restrictions of the �-point technique; moreover, it provides a symmetric, sparse and banded co-e;cient matrix which may reduce the computation cost and allow the DQM to be easily integratedwith the FEM.

2. Moving least-squares di�erential quadrature formulation

Consider a domain in space � discretized by a set of node points {xi}i=1; :::;N , where N is the totalnumber of nodes, and x= [x y]T in 2D [11] and x= [x y z]T in 3D. The di)erential cubature (DC)method [12,13], or the generalized di)erential quadrature (GDQ) method [14–16], approximates thepartial derivatives of a certain variable u(x) at the point xi as a weighted linear sum of the functionvalues at all the discrete points, i.e.

R{u(x)}|x=xi =N∑j=1

cj(xi)u(xj) =N∑j=1

cijuj; (1)

where R denotes a linear di)erential operator that can be any orders of partial derivatives or theircombinations, cij the so-called weighting coe;cients and uj the nodal function values.

The weighting coe;cients cij can be computed by solving a set of linear algebraic equations thatcan be obtained by selecting N monomials from a set of polynomial basis functions and substitutingthem into Eq. (1). In this paper, the approximate function uh(x) of u(x) is obtained by the moving

K.M. Liew et al. / Finite Elements in Analysis and Design 40 (2004) 1453–1474 1455

least-squares (MLS) method and the weighting coe;cients cij are obtained by computing the partialderivatives of uh(x) directly. The main features of the MLS approximation [17] are given in thefollowing section.

The MLS approximation provides a smooth approximate function across irregular grids of nodes.It requires no explicit element mesh and the approximation is local for any given point since onlythe neighboring nodes are taken into account. These features make the MLS approximation analternative choice to the development of meshless methods. To approximate a certain variable u(x)in the domain �, a !nite set of basis functions {pj(x)}j=1; :::;m is introduced. The MLS approximationuh(x) of u(x) for any x∈� is given as

uh(x) = pT(x)a(x) =m∑i=1

pi(x)ai(x); (2)

where the basis p(x) and the related coe;cients a(x) are de!ned as

p(x) = [p1(x) p2(x) · · · pm(x)]T; (3)

a(x) = [a1(x) a2(x) · · · am(x)]T: (4)

For convenience, the !nite basis set can be monomials selected from a complete polynomial. Forexample, a complete cubic basis in the 2D case is of the form

p(x) = [1 x y x2 xy y2 x3 x2y xy2 y3]T: (5)

Noting that the coe;cients a(x) are functions of x, the expression for a(x) can be obtained at eachpoint x by minimizing the following weighted residual:

�(a) =n∑i=1

$i(x)(pT(xi)a(x) − ui)2; (6)

where n is the number of nodes in the neighborhood of x and ui is the nodal value of u(x) in xi.The in9uence of x is governed by a non-zero decreasing weight function $i(x) = $(x − xi) thatvanishes outside the domain of in9uence of x.

The stationarity of �(a) with respect to a(x) results in the following linear equations:

A(x)a(x) = B(x)u (7)

from which

a(x) = A−1(x)B(x)u; (8)

where

A(x) =n∑i=1

$i(x)p(xi)pT(xi);

B(x) = [B1(x) B2(x) · · · Bn(x)]

= [$1(x)p(x1) $2(x)p(x2) · · · $n(x)p(xn)];

u = [u1 u2 · · · un]T:


Substituting Eq. (8) into Eq. (2), the approximation uh(x) can then be expressed in terms of theshape functions as

uh(x) =n∑i=1

�i(x)ui; (9)

where the nodal shape function �i(x) is given by

�i(x) = pT(x)A−1(x)Bi(x): (10)

Using Eq. (9), the weighting coe;cients de!ned in the DQ method can be computed directly. Forinstance, we have

u;1(x) =9u(x)9x =

n∑i=1

9�i(x)9x ui =

n∑i=1

c(1)i (x)ui; (11)

where c(1)i (x) is the set of weighting coe;cients of the !rst derivative of u(x) in x direction at

any point x with respect to node xi. The weighting coe;cients for the other derivatives can beobtained similarly. Therefore, for each node xi in the domain �, we have the corresponding weightingcoe;cients of the !rst derivative such as c(1)

j (xi), which are non-zero only for the points xj withinthe domain of in9uence of xi.

3. Determination of weighting coe cients

From Eq. (11), the weighting coe;cients can be directly obtained by calculating the partial deriva-tives of the shape function �i(x). In this section, a procedure proposed by Belytschko et al. [18] isused to reduce the computational cost and alleviate the ill-conditioning of the matrix A(x). Eq. (10)is rewritten as

�i(x) = pT(x)A−1(x)Bi(x) = �T(x)Bi(x); (12)

which leads to the relationship

A(x)�(x) = p(x): (13)

The coe;cients �(x) can be obtained by an LU decomposition and back-substitution, which requiresfewer computations than the inversion of A(x). Moreover, one can compute the partial derivativesof �(x) in any order using the same matrix after LU decomposition of A(x) as in Eq. (13). Forexample, we have

A; I (x)�(x) + A(x)�; I (x) = p; I (x) (14)

and

A(x)�; I (x) = p; I (x) − A; I (x)�(x); (15)

where (); I (I = 1; 2; 3) means the derivative along the I direction in a Cartesian coordinates frame.Only one back-substitution is required to obtain �; I (x). Similarly, higher derivatives can be computed


in the same way. The second derivatives of �(x) can be computed with a back-substitution usingthe new right-hand vector in the following equations:

A(x)�; IJ (x) = p; IJ (x) − A; IJ (x)�(x) − A; I (x)�; J (x) − A; J (x)�; I (x): (16)

The weighting coe;cients in the DQ method can be obtained accordingly. As an example, theweighting coe;cients related to the !rst and second order partial derivatives take the form

c(I)j (xi) = �T

; I (xi)Bj(xi) + �T(xi)Bj; I (xi) (17)

and

c(IJ )j (xi) = �T

; IJ (xi)Bj(xi) + �T(xi)Bj; IJ (xi) + �T; I (xi)Bj; J (xi) + �T

; J (xi)Bj; I (xi): (18)

4. Application of the MLSDQ method to numerical analysis of Kirchho� plates

4.1. Basic governing equations and boundary conditions

The governing equation for a Kirchho) thin plate takes the form of [19,20]

94w9x4 + 2

94w9x29y2 +

94w9y4 =

qD; (19)

where w and D are the transverse de9ection and 9exural rigidity of the plate, and q is the loaddistributed over the upper surface of the plate.

The bending moments (Mx;My) and the twisting moment Mxy are given as

Mx

My

Mxy

= −D

1 � 0

� 1 0

0 0 1 − �

92w9x2

92w9y2

92w9x9y

; (20)

where � is the Poisson ratio.The shearing forces Qx and Qy are determined from

Qx =9Mx

9x +9Mxy

9y = −D(93w9x3 +

93w9x9y2

); (21a)

Qy =9Mxy

9x +9My

9y = −D(93w9y3 +

93w92x9y

): (21b)

The boundary conditions for an arbitrary edge are considered. If the edge is simply supported, wehave the following equations:

w = 0; (22)

PM Pn = Pn2xMx + 2 Pnx PnyMxy + Pn2

yMy = 0; (23a)


where PM Pn denotes the normal bending moment applied at the edge; Pnx, Pny represent the componentsof the boundary unit normal vector Pn along the x and y axes, respectively.

Taking Eq. (20) into (23a), we have the equivalent expression

( Pn2x + � Pn2

y)92w9x2 + 2(1 − �) Pnx Pny

92w9x9y + ( Pn2

y + � Pn2x)92w9y2 = 0: (23b)

If the edge is clamped, the boundary conditions are Eq. (22) and the following one:

9w9n = Pnx

9w9x + Pny

9w9y = 0: (24)

If the edge is free, the boundary conditions are Eq. (23) and the following one:

Q Pn +9M Pn Pt

9Pt= 0; (25a)

where the subscripts Pn and Pt are, respectively, the normal and tangential directions of the edge; Q Pn

and M Pn Pt represent the shearing force and twisting moments on the edge of the plate, which can beexpressed as

Q Pn = PnxQx + PnyQy; (26)

PM Pn Pt = ( Pn2x − Pn2

y)Mxy + Pnx Pny(My −Mx): (27)

Taking Eqs. (26,27) and (20,21) into Eq. (25a), one can represent the boundary condition inEq. (25a) in terms of the transverse de9ection w. For simplicity, we provide the expression of theedge parallel to the y axis as follows:

93w9x3 + (2 − �)

93w9x9y2 = 0: (25b)

4.2. MLSDQ discretization of the governing equations and boundary conditions

To illustrate the present MLSDQ method, the Kirchho) plate is discretized !rstly by a !nitenumber of nodes (xi; yi) associated with a nodal parameter wi for each node. Then, a domain ofin9uence is formed for each node and the MLS approximation is achieved in the domain of in9uenceas described in Section 2:

wh(x; y) =n∑i=1

�i(x; y)wi; (28)

where n is the number of nodes within the domain of in9uence. The nodal shape function �i(x; y)and its partial derivatives can be obtained by the fast computation technique described in Section 3.Thereafter, the weighting coe;cients with respect to any order of partial derivatives of the ap-proximate de9ection wh(x; y) can be obtained directly. The following expressions give some partial


derivatives from the !rst to the fourth order:

wh;x(x; y) =

n∑i=1

�i;x(x; y)wi =n∑i=1

c(x)i (x; y)wi; (29a)

wh;y(x; y) =

n∑i=1

�i;y(x; y)wi =n∑i=1

c(y)i (x; y)wi; (29b)

wh;xx(x; y) =

n∑i=1

�i;xx(x; y)wi =n∑i=1

c(xx)i (x; y)wi; (29c)

wh;yy(x; y) =

n∑i=1

�i;yy(x; y)wi =n∑i=1

c(yy)i (x; y)wi; (29d)

wh;xxx(x; y) =

n∑i=1

�i;xxx(x; y)wi =n∑i=1

c(xxx)i (x; y)wi; (29e)

wh;yyy(x; y) =

n∑i=1

�i;yyy(x; y)wi =n∑i=1

c(yyy)i (x; y)wi; (29f)

wh;xxxx(x; y) =

n∑i=1

�i;xxxx(x; y)wi =n∑i=1

c(xxxx)i (x; y)wi; (29g)

wh;yyyy(x; y) =

n∑i=1

�i;yyyy(x; y)wi =n∑i=1

c(yyyy)i (x; y)wi: (29h)

The other mixed partial derivatives can be approximated in the same way. Here, we denote theweighting coe;cients with respect to the partial derivatives wh

;xy, wh;xxy, wh

;xyy, wh;xxyy as c(xy)

i , c(xxy)i ,

c(xyy)i , c(xxyy)

i , respectively.Taking Eq. (29) into the governing Eq. (19), the discrete equation by the MLSDQ approximation

for any node (xi; yi) in the plate can be obtained:n∑j=1

(c(xxxx)j (xi; yi) + 2c(xxyy)

j (xi; yi) + c(yyyy)j (xi; yi))wj =

q(xi; yi)D

(30a)

or, in a simple form, using c()ij instead of c()

j (xi; yi):n∑j=1

(c(xxxx)ij + 2c(xxyy)

ij + c(yyyy)ij )wj =

q(xi; yi)D

: (30b)

Note that the number of nodes n within the domain of in9uence maybe di)erent for di)erent nodes(xi; yi).

The boundary conditions in Eqs. (22–25) can be discretized in the same way. However, one shouldnote that the nodal parameter wi is not always equal to the transverse de9ection of the node (xi; yi),since the MLS shape functions �i(x; y) do not satisfy the partition-of-unity condition generally,


i.e., �i(xj; yj) �= �ij. Therefore, for the node (xi; yi) on the boundary, we have the following discreteequations for Eqs. (22–25), respectively:

n∑j=1

�j(xi; yi)wj = 0; (31)

n∑j=1

(( Pn2x + � Pn2

y)c(xx)ij + 2(1 − �) Pnx Pnyc

(xy)ij + ( Pn2

y + � Pn2x)c

(yy)ij )wj = 0; (32)

n∑j=1

( Pnxc(x)ij + Pnyc

(y)ij )wj = 0; (33)

n∑j=1

(c(xxx)ij + (2 − �)c(xyy)

ij )wj = 0: (34)

Once the nodal parameter wi is obtained after solving the discrete governing equations and boundaryconditions, the transverse de9ection at any point (x; y) can be computed using Eq. (28). The bendingmoments and shear forces can be obtained by

Mx

My

Mxy

= −D

1 � 0

� 1 0

0 0 1 − �

n∑j=1

c(xx)j (x; y)wj

n∑j=1

cyyj (x; y)wj

n∑j=1

c(xy)j (x; y)wj

(35)

and

Qx = −Dn∑j=1

(c(xxx)j (x; y) + c(xyy)

j (x; y))wj;

Qy = −Dn∑j=1

(c(yyy)j (x; y) + c(xxy)

j (x; y))wj:

(36)

4.3. Treatment of multiple boundary conditions by least-squares approximation

It is noted that the number of discrete equations is two for the nodes on the boundary of the plate(see Eqs. (22)–(25)). Three boundary conditions exist for corner nodes joining with two edges.However, there is only one degree of freedom for each node. Therefore, the multiple boundaryconditions result in more number of discrete algebraic equations than unknowns. To overcome thisdi;culty when solving the discrete equations, a least-squares approximation is adopted in this paperinstead of the �-point technique. Assuming the total number of degrees of freedom is Na and the


number of discrete equations is Nb, the discrete equations can be expressed as follows:Na∑j=1

Kijwj = Fi (i = 1; 2; : : : ; Nb); (37)

where Nb¿Na and the matrix K is sparse because the MLS approximation is local.The set of algebraic equations in Eq. (37) can be solved by minimizing the following residual

sum:

�(wj) =Nb∑i=1

Na∑

j=1

Kijwj − Fi

2

: (38)

The stationarity of �(wj) with respect to wj, i.e., 9�(wj)=9wj = 0, results in the following set ofequations:

Nb∑k=1

Na∑j=1

KkiKkjwj =Nb∑k=1

KkiFk (i = 1; 2; : : : ; Na) (39)

orNa∑j=1

K̂ ijwj = Fi (i = 1; 2; : : : ; Na); (40)

where

˙

Kij =Nb∑k=1

KkiKkj;˙

Fi =Nb∑k=1

KkiFk : (41)

It is found that the number of the above equations is equal to the number of unknowns, andthe global coe;cient matrix

˙

K is symmetric and banded. This may reduce the computation cost byusing an e;cient equation solver. Moreover, the coe;cient matrix and the right-hand side vector

˙

Fcan be assembled after each discrete equation is formed. This would mean that there is no need toform the matrix K and store it in memory.

The present multiple boundary condition treatment eliminates the drawback of the so-called �-typegrid arrangement [2], which may result in ill-conditioned matrices. In contrast to the introduction ofadditional derivative degrees of freedom for the boundary nodes by Chen et al. [22] and the SBCGEmethod by directly substituting boundary constraints into the discrete governing equations [23], thepresent method has no restrictions when applied to complex boundary conditions.

To achieve more accurate numerical results by the present method, it is recommended to normalizeeach discrete equation before assembling the !nal coe;cient matrix

˙

K and right-hand side vector˙

F . Each discrete equation is normalized by its absolute maximum coe;cient. We have the followingnormalized equations:

Na∑j=1

K ′ijwj = F ′

i (i = 1; 2; : : : ; Nb); (42)


where

K ′ij =

Kijmax(abs(Kij))

; F ′i =

Fimax(abs(Kij))

:

Eq. (42) is used instead of Eq. (37) before forming the least-squares residual functional (38).Similarly, we have

˙

Kij =Nb∑k=1

K ′kiK

′kj;

˙

Fi =Nb∑k=1

K ′kiF

′k : (43)

5. Numerical examples

In this section, to investigate the performance of the MLSDQ method, thin plates with variousgeometries, boundary conditions and loads are solved, and the in9uence of support size, convergenceand numerical accuracy are studied.

A weight function of Gaussian type with a circular support is adopted for the MLS approximationbecause its derivatives with respect to the x or y coordinates exist to any desired order. It takes theform

wi(x; y) =

exp(−(di=c)2) − exp(−(r=c)2)1 − exp(−(r=c)2)

; di6 r;

0; di¿ r;

(44)

where di =√

(x − xi)2 + (y − yi)2 is the distance from a discrete node xi to a sampling point x inthe domain of support with radius r, and c is the dilation parameter. c= r=4 is used in computation.

A scaling factor dmax is de!ned by

dmax =rhm; (45)

where hm is the grid size, which can be regarded as the averaged distance between two neighboringnodes. For regular node arrangements, hm is the adjacent nodal spacing.

Since the governing Eq. (19) has a fourth derivative with respect to x and y, the assumed ap-proximation function should be at least the same order of completeness, i.e., the monomial basisshould be selected from a complete fourth-order polynomial or above. In this computation, ordersNc = 4; 5; 6; 7 of completeness with di)erent scaling factors dmax are considered.

5.1. In>uence of the support size and the completeness order

A simply supported isotropic square plate with unit width a and Poisson’s ratio �=0:3 is consideredas the !rst example. The plate is subjected to two loading conditions, namely, uniform load (UL)q=q0 and hydrostatic load (HL) q=q0x=a. The x- and y-axis coincide with two joined edges of theplate. The whole plate is discretized by an 11 × 11 regular pattern (total 121 nodes) and the grid


x

y

a = 1

(a, a)

ν = 0.3

Fig. 1. Discretization and con!guration of the square plate.

size for this discretization is then hm = 0:1. Details of the con!guration and the grid of the plate areshown in Fig. 1.

Numerical results of the central de9ection Pwc and bending moment PMxc are tabulated inTable 1 for di)erent support sizes and orders of completeness. Exact thin plate solutions by Timo-shenko and Woinowsky-Krieger [19] and Reddy [20,21] are also provided for comparison. Note thatthe minimum scaling factor dmax in Table 1 varies with the completeness order Nc. This is becauseeach domain of in9uence should contain enough discrete points to avoid an ill-conditioned matrixA(x) in Eq. (13). Usually, the minimum value required is identical to the number of terms in theassumed complete polynomial basis. However, more discrete points within the domain of in9uenceare suggested to avoid a singularity.

It is found from Table 1 that dmax¿Nc + 0:5 is required for reasonable numerical solutions. Toachieve good numerical accuracy, the lowest complete polynomial assumption with Nc = 4 needsa larger support size in contrast to the higher ones. This is because the fourth-order governingequation is discretized and satis!ed in a constant type when Nc = 4. However, we found no furtherimprovements on the results with the same support size while increasing the order of completenessfor Nc¿ 5 although p-convergence can be veri!ed. It is also shown that the numerical results ofboth loads exhibit the same characteristics.

To investigate the in9uence of the clamped and free boundary conditions, two more square platessubjected to uniform loads are considered. One is a square plate with all edges built in, and anotheris a plate with three edges simply supported and the fourth edge free. The same con!guration andmesh discretization as in the !rst example (see Fig. 1) are used here. Figs. 2 and 3 show the centralde9ection and bending moment of the clamped plate. Figs. 4 and 5 show the maximum de9ectionat the middle of the free edge and the central bending moment of the plate with three edges simplysupported and the fourth free. It is observed again from Figs. 2–5 that the present results of boththe displacement and the bending moment are in excellent agreement with the analytical solutions[19,20] when a support of suitable size is used. It is also shown in Figs. 4 and 5 that the numericalresults of the plate with the mixed boundaries are much dependent on the support size.


Table 1Central de9ection and bending moment of the simply supported plate ( Pwc = 100wcD=(q0a4); PMxc = 10Mxc=(q0a2); � = 0:3)

Nc dmax UL HL

Pwc PMxc Pwc PMxc

4 4.5 0.38670 0.45186 0.19333 0.225905 0.42354 0.49487 0.21175 0.247425.5 0.41624 0.48620 0.20812 0.243106 0.41019 0.48153 0.20509 0.240766.5 0.40495 0.47744 0.20247 0.238727 0.40390 0.47652 0.20195 0.238267.5 0.40426 0.47697 0.20213 0.238488 0.40472 0.47745 0.20236 0.238738.5 0.40518 0.47793 0.20259 0.23896

5 4.5 0.17268 0.25256 0.095422 0.135535 0.24335 0.33008 0.13498 0.176515.5 0.40381 0.47765 0.20171 0.238646 0.40704 0.47939 0.20352 0.239706.5 0.40447 0.47713 0.20223 0.238577 0.40583 0.47839 0.20292 0.239197.5 0.40565 0.47833 0.20283 0.239168 0.40637 0.47903 0.20318 0.239528.5 0.40629 0.47897 0.20314 0.23949

6 5.5 0.31611 0.39740 0.17322 0.212906 0.23846 0.32756 0.14134 0.186296.5 0.40358 0.47658 0.20299 0.239247 0.40932 0.48153 0.20471 0.240827.5 0.40530 0.47806 0.20262 0.239008 0.40743 0.47993 0.20371 0.239968.5 0.40613 0.47880 0.20307 0.23940

7 6.5 0.32706 0.40996 0.17721 0.218167 0.28302 0.38763 0.15444 0.198967.5 0.40282 0.47599 0.20102 0.237708 0.40223 0.47555 0.19964 0.236728.5 0.40565 0.47835 0.20269 0.23905

Exact [19,20] 0.40624 0.47886 0.20312 0.23943

5.2. Convergence study

The simply supported plate and the clamped plate are chosen to study the h-convergence of thepresent MLSDQ method. The plate is subjected to uniform load or hydrostatic load. Only the !fthcomplete polynomial assumption with various support sizes is considered. Tables 2 and 3 listed thecentral de9ection and the bending moment of the simply supported plate and the clamped plate,


4 5 6 7 8 9

0.7

0.8

0.9

1.0

1.1

wc/

wex

act

dmax

Nc = 4Nc = 5Nc = 6Nc = 7 Exact

Fig. 2. Central de9ection of the clamped plate.

4 5 6 7 8 9

Mxc

/Mex

act

dmax


1.10

1.05

1.00

0.90

0.85

0.80

0.75

0.95

Fig. 3. Central bending moment of the simply supported plate.

respectively. The number of discrete nodes on one edge of the plate is de!ned as Nd, such as,Nd = 11 for the discretization in Fig. 1. The grid size hm is then expressed as

hm =a

Nd − 1: (46)

It is found that both the de9ection and the bending moment converge rapidly.

5.3. Discontinuous loads

A simply supported square plate under a patch load as shown in Fig. 6 is examined to consider thecapability of the present MLSDQ method for problems containing discontinuous loads and bound-aries. As we know, this problem cannot be solved by directly applying the DQ method to the entiredomain. Element division is necessary to apply the DQEM. However, the present MLSDQ methodhas no more di;culty in solving this problem than that with a continuous load. There is no need topay any special attention to the load discontinuity when discretizing the whole domain as the local


4 5 6 7 98

0.5

1.0

wc/

wex

act

dmax


0.0

-0.5

Fig. 4. The maximum de9ection at the middle of the free edge of the plate with three edges simply supported and thefourth free.

4 5 6 7 8 90.2

0.4

0.6

0.8

1.0

1.2

Mxc

/Mex

act

dmax


Fig. 5. The central bending moment of the plate with three edges simply supported and the fourth free.

discontinuity can be controlled by the domain of in9uence. The only di)erence between this problemand the !rst example is the discrete governing equations. That is, for nodes outside the patch, thedistributed load q should be zero. Though the discretization can be arbitrary, it is suggested thatthe patch boundaries be located at the middle of two rows or columns of nodes to obtain betternumerical results. Fig. 6 shows a 11×11 regular discretization. Table 4 lists the maximum de9ectionand bending moment of the plate at the center. Exact solutions based on the theoretical formulae[19] are also shown for comparison.

It is found that the numerical results are quite consistent with the analytical ones and convergerapidly. Also, it is observed that increasing the support size, i.e. dmax, from 5.5 to 7 causes nosigni!cant e)ects on the results. However, it is not preferable to use a larger support size because itincreases the computational cost and results in a reduction of the local accuracy with a discontinuityin the load. Comparing with the DQEM that needs element division, the present element-free MLSDQmethod o)ers a great advantage of analyzing problems with discontinuous loads.


Table 2Central de9ection and bending moment of the simply supported plate ( Pwc = 100wcD=(q0a4); PMxc = 10Mxc=(q0a2); � = 0:3)

dmax Nd UL HL

Pwc PMxc Pwc PMxc

5.5 6 0.40231 0.51505 0.20116 0.2575211 0.40381 0.47765 0.20171 0.2386416 0.40687 0.47919 0.20378 0.2398121 0.40485 0.47772 0.20217 0.23868

6 6 0.41826 0.52010 0.20913 0.2600511 0.40704 0.47939 0.20352 0.2397016 0.40398 0.47677 0.20159 0.2381021 0.40583 0.47856 0.20264 0.23906

6.5 6 0.41237 0.50929 0.20618 0.2546411 0.40447 0.47713 0.20223 0.2385716 0.40573 0.47844 0.20284 0.2392121 0.40639 0.47892 0.20326 0.23951

7 6 0.40325 0.49647 0.20162 0.2482411 0.40583 0.47839 0.20292 0.2391916 0.40603 0.47868 0.20303 0.2393521 0.40510 0.47784 0.20220 0.23863

Exact [19,20] 0.40624 0.47886 0.20312 0.23943

Table 3Central de9ection and bending moment of the clamped supported plate ( Pwc = 100wcD=(q0a4); PMxc = 10Mxc=(q0a2); �= 0:3)

dmax Nd UL HL

Pwc PMxc Pwc PMxc

5.5 6 0.15508 0.31658 0.077538 0.1582911 0.12788 0.23154 0.063950 0.1158016 0.12098 0.21920 0.060032 0.1089121 0.11429 0.20482 0.057150 0.10237

6 6 0.14841 0.29729 0.074205 0.1486411 0.12640 0.22812 0.063206 0.1140716 0.12549 0.22711 0.062814 0.1136921 0.12257 0.22159 0.061205 0.11078

6.5 6 0.14062 0.27986 0.070308 0.1399311 0.12606 0.22811 0.063032 0.1140616 0.12635 0.22877 0.063175 0.1143821 0.12442 0.22492 0.062358 0.11277

7 6 0.13440 0.26603 0.067200 0.1330111 0.12639 0.22860 0.063195 0.1143016 0.12648 0.22895 0.063238 0.1144721 0.12621 0.22849 0.063096 0.11425

Exact [19–21] 0.12653 0.22905 0.063 0.115


0.5a

x

y

a = 1ν = 0.3

(a, a)

0.5a

Fig. 6. Discretization and con!guration of the simply supported plate with a patch load.

Table 4Central de9ection and bending moment of the simply supported plate under a patch load ( Pwc = 100wcD=(q0a4); PMxc =10Mxc=(q0a2); � = 0:3)

dmax Nd Pwc PMxc

5.5 11 0.20457 0.2794315 0.21215 0.2923219 0.21032 0.2907423 0.21149 0.29234

6 11 0.21033 0.2891615 0.21386 0.2951319 0.21282 0.2940623 0.21240 0.29355

6.5 11 0.20806 0.2881715 0.21730 0.2985619 0.21284 0.2941023 0.21398 0.29509

7 11 0.20729 0.2873715 0.21927 0.3003819 0.21186 0.2931023 0.21395 0.29504

Exact [19–21] 0.21322 0.29436

5.4. In>uence of irregular discretization

In this section, the in9uence of irregular nodal arrangements is investigated. The simply sup-ported square plate is considered again as the !rst example. The discrete nodes inside the plate


Fig. 7. Random generated pattern with 121 nodes of the simply supported square plate.

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.00.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1


dmax

wc/w

exac

t

5.0

Fig. 8. Central de9ection of the simply supported plate with irregular discretization.

are distributed randomly. However, the boundary nodes are equally spaced and the total numberof discrete nodes is kept the same as for the regular pattern used in Section 5.1. Fig. 7 showsthe random discretization with a total of 121 nodes which is consistent with the 11 × 11 reg-ular discretization. For convenience, hm = 0:1 is used as in the regular pattern. The maximumde9ection and the bending moment of the plate at the center are plotted in Figs. 8 and 9, re-spectively, in which di)erent support sizes and di)erent orders of assumed basis functions areexamined. It is found that the assumed low-order basis functions (Nc = 4; 5) tend to incorrectvalues even for large support sizes. However, no signi!cant di)erence is noticed between reg-ular and irregular discretizations for high-order basis functions (Nc = 6; 7). More attractively, asmaller support size than for the regular patterns can be used in computation as the singularity isreduced.

As an alternative example, a simply supported circular plate of radius a under uniform load isstudied to consider the in9uence of both irregular discretization and plate geometry. The discrete


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

Mxc

/M Nc = 4Nc = 5Nc = 6Nc = 7 Exact

d

exac

t

5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.05.0

max

Fig. 9. Central moment of the simply supported plate with irregular discretization.

Radius a =13.0=

x

y

ν

Fig. 10. Random discretization of a simply supported circular plate.

pattern of the plate is shown in Fig. 10 where the nodes inside the circle are randomly generatedand the boundary nodes are equi-spaced. A total of 265 nodes is used in computation. Table 5lists the central de9ection and bending moments of the plate when di)erent support sizes r anddi)erent orders of assumed polynomials are used. It is found that a fourth-order complete polynomialassumption is su;cient to furnish excellent numerical results with irregular grids. This di)ers fromthe simply supported square plate in the !rst example where a higher order of the assumed basis isneeded to ensure convergence. The reason for this is the fact that the theoretical de9ection of the


Table 5Central de9ection and bending moment of the simply supported circular plate under uniform load ( Pwc=10wcD=(q0a4); PMxc=Mxc=(q0a2); PMyc = Myc=(q0a2); � = 0:3)

Nc r Pwc PMxc PMyc

4 0.4 0.63692 0.20623 0.206240.45 0.63628 0.20603 0.206020.5 0.63703 0.20625 0.206260.55 0.63701 0.20625 0.20625

5 0.55 0.62246 0.20190 0.202500.6 0.63671 0.20616 0.206140.65 0.63694 0.20622 0.206230.7 0.63702 0.20625 0.20625

6 0.65 0.62628 0.20336 0.203140.7 0.63638 0.20606 0.206070.75 0.63694 0.20622 0.206230.8 0.63704 0.20625 0.20626

7 0.7 0.64548 0.20779 0.209300.75 0.64002 0.20672 0.207460.8 0.63695 0.20627 0.206200.85 0.63728 0.20632 0.20632

Exact [19–21] 0.63702 0.20625 0.20625

plate is just a fourth-order polynomial distribution given by [19–21]

w =q0(a2 − x2 − y2)

64D

(5 + �1 + �

a2 − x2 − y2

): (47)

Also, it is observed from Table 5 that, for higher order assumed bases, a larger support size providesbetter results.

5.5. Skew plate

As a !nal example, a simply supported skew rhombic plate with edge length a under uniformloading is studied to consider the in9uence of the skew grid. Fig. 11 shows the 11×11 discretizationof a skew rhombic plate with skew angle ' = 45◦. The !fth-order complete polynomial assumptionwith Nc = 5 and support radius r = 0:65a and 0:85a are used in the computation. Values for thede9ection and the principal bending moments at the center of the plate are shown in Table 6. TheFEM solutions by Sengupta [24], Zienkiewicz and Lefebvre [25] and the di)erential quadrature (DQ)solutions by Liew and Han [26] are also listed for comparison where the thickness to span ratio istaken to be 0.01 in computation. It is found that the present results are in close agreement with theFEM and DQ results for all skew angles. Again, it is shown that a larger support size results inbetter values.


a x

y

α = 45°

Fig. 11. Skew rhombic plate with 11 × 11 discretization.

Table 6The central de9ection and principal bending moments of the simply supported rhombic skew plate under uniform load( Pwc = 1600wcD=(q0a4); PMc = 40Mc=(q0a2); � = 0:3)

Skew angle Reference Pwc PM 1c PM 2c

15◦ Present (r = 0:65a) 0.0669 0.2600 0.1254Present (r = 0:85a) 0.0640 0.2562 0.1176Sengupta [24] 0.0605 0.2461 0.1030Zienkiewicz and Lefebvre [25] 0.0652 0.2580 0.1126Liew and Han [26] 0.0640 0.2566 0.1170






6. Conclusions

In this paper, the MLSDQ approximation is successfully applied to analyze Kirchho) plate prob-lems. It is found that the MLSDQ method o)ers advantage in solving problems with discontinuitiesin loads and boundaries. A least-squares method is adopted to deal with the multiple boundary con-ditions that avoids the introduction of �-arrangements of boundary nodes or C1 continuous weightingcoe;cients. Moreover, the present multiple boundary treatment yields a symmetric global momentmatrix, which may reduce computational cost, and it can be easily coupled with the !nite elementmethod.

It is concluded from the numerical results that the !fth order or above complete polynomialbasis is suitable for regular nodal patterns while the sixth order or above is suggested for irregulardiscretizations. The support size is dependent on the order of the completeness and the nodal spacing.However, use of a larger support size is recommended for higher assumed bases, especially forirregular patterns.

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Documents

Analysis of general shaped thin plates by the moving least-squares differential quadrature method