2. a Cell Based Smoothed Discrete Shear Gap Method Using Triangular Elements for Static and Free...

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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2012; 91:705741Published online 31 May 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4289

A cell-based smoothed discrete shear gap method using triangularelements for static and free vibration analyses of

ReissnerMindlin plates

T. Nguyen-Thoi1,2,*, , P. Phung-Van3, H. Nguyen-Xuan1,2 and C. Thai-Hoang2

1Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science, Vietnam National

University HCMC, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Vietnam2Division of Computational Mechanics, Ton Duc Thang University, 98 Ngo Tat To St., Ward 19, Binh Thanh Dist.,

Hochiminh City, Vietnam3Faculty of Civil Engineering, Nguyen Tat Thanh University, 300A Nguyen Tat Thanh St, Dist. 4, Hochiminh City,

Vietnam

SUMMARY

The cell-based strain smoothing technique is combined with discrete shear gap method using three-nodetriangular elements to give a so-called cell-based smoothed discrete shear gap method (CS-DSG3) for staticand free vibration analyses of ReissnerMindlin plates. In the process of formulating the system stiffnessmatrix of the CS-DSG3, each triangular element will be divided into three subtriangles, and in each sub-triangle, the stabilized discrete shear gap method is used to compute the strains and to avoid the transverseshear locking. Then the strain smoothing technique on whole the triangular element is used to smooth thestrains on these three subtriangles. The numerical examples demonstrated that the CS-DSG3 is free of shearlocking, passes the patch test, and shows four superior properties such as: (1) being a strong competitor tomany existing three-node triangular plate elements in the static analysis; (2) can give high accurate solutionsfor problems with skew geometries in the static analysis; (3) can give high accurate solutions in free vibra-tion analysis; and (4) can provide accurately the values of high frequencies of plates by using only coarsemeshes. Copyright 2012 John Wiley & Sons, Ltd.

Received 26 July 2011; Revised 3 December 2011; Accepted 5 January 2012

KEY WORDS: ReissnerMindlin plate; shear locking; finite element method (FEM); cell-based smootheddiscrete shear gap technique (CS-DSG3); discrete shear gap method (DSG); strain smooth-ing technique

1. INTRODUCTION

In the past 50 years, many plate bending elements based on the MindlinReissner theory and the

first-order shear deformation theory have been proposed. Such a large amount of elements can be

found in literatures [111], and in recent journal papers [1215]. In formulations of a Mindlin

Reissner plate element using the first-order shear deformation theory, the deflection w and rotations

x , y are independent functions and require at least to be C0-continuous. In practical applications,lower-order displacement-based ReissnerMindlin plate elements are preferred because of their sim-

plicity and efficiency. These elements usually possess high accuracy and fast convergence speed for

displacement solutions. However, for stresses or internal forces, they usually give the low accuracy

[1618] and need a post-process to improve the solution [19, 20]. In addition, the main difficulty

*Correspondence to: T. Nguyen-Thoi, Faculty of Mathematics & Computer Science, University of Science, VietnamNational University HCMC, 227 Nguyen Van Cu, Dist. 5, Hochiminh City, Vietnam.

E-mail: [email protected]; [email protected]

Copyright 2012 John Wiley & Sons, Ltd.


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706 T. NGUYEN-THOI ET AL.

encountered of these elements is the phenomenon of shear locking, which induces over-stiffness as

the plate becomes progressively thinner.

To avoid shear locking, many new numerical techniques and effective modifications have been

proposed and tested, such as the reduced integration and selective reduced integration schemes pro-

posed by Zienkiewicz et al. [21] and Hughes et al. [22, 23], the stabilization procedure proposed

by Belytschko et al. [24, 25], free formulation method proposed by Bergan and Wang [26], the

substitute shear strain method proposed by Hinton and Huang [27], mixed formulation/hybrid ele-ments [2831], etc. However, these elements are still subjected to some drawbacks such as instability

because of rank deficiency or low accuracy or complex performance. To overcome these drawbacks,

Macneal [32] introduced an assumed strain method on the physical elements, and then Park and

Stanley at Stanford [33] first enlarged to the natural elements to give the so-called assumed natural

strain method (ANS), which allows defining the shear strains independently from the approxima-

tion of kinematic variables. In this method, the shear strain field of a 3-node triangular or a 4-node

quadrilateral element is interpolated independently by rational constant shear strains along each

element side, and the shear locking problem will be overcome. It has been proved to be mathe-

matically valid by Bathe and Brezzi [34, 35] and Brezzi et al. [36]. On the basis of this method

and different modifications, many successful models were then presented, including the mixed

interpolated tensorial components (MITC) family proposed by Bathe and colleagues [3440], the

discrete ReissnerMindlin (DRM) family [41,42] and the linked interpolation elements (Q4BL [43]

and T3BL [44]) proposed by various authors, discrete Kirchhoff elements DKT [45] and DKQ

[46] proposed by Batoz and colleagues, the discrete KirchhoffMindlin elements DKMQ [47] and

DKMT [48] proposed by Katili, the refined element RDKQM [49] and DKTM and RDKTM [50,51]

proposed by Wanji and coworkers, etc.

Recently, the discrete shear gap (DSG) method [52], which avoids shear locking, was proposed.

The DSG is somewhat similar to the ANS methods in the terms of modifying the certain strains

within the element, but is different in the aspect of removing collocation points. The DSG method

works for elements of different orders and shapes and has several superior properties [52]. However,

the element stiffness matrix in the DSG still depends on the sequence of node numbers, and hence

the solution of DSG is influenced when the sequence of node numbers changes, especially for the

coarse and distorted meshes.

In the other frontier of developing advanced technologies of nodal integration, Chen et al. [54]

proposed a stabilized conforming nodal integration to eliminate spatial instability in nodal integra-tion in the mesh-free Galerkin methods. In this technique, a constant strain smoothing operation

is incorporated with the compatible strain field in the node-based integration domain, and then

the domain-based integration can be transformed into the boundary-based integration. The method

shows higher efficiency, desired accuracy, convergent properties, and is quite robust in dealing with

irregular nodal spacing in mesh-free discretization. It is also very interesting to see that when

this nodal integration technique is applied to the displacement-based FEM with nodal integra-

tion, a smooth recovered strain fields on the structure domain can be obtained easily without any

post-process operation.

Recently, Liu and Nguyen-Thoi Trung [53] have extended this strain smoothing technique

of mesh-free methods [54] into the conventional FEM using linear interpolations to formu-

late a series of smoothed FEM (S-FEM) models named as the cell-based S-FEM (CS-FEM)

[5562], the node-based S-FEM (NS-FEM) [6367], the edge-based S-FEM (ES-FEM) [6873]

and the face-based S-FEM (FS-FEM) [74, 75]. In these S-FEM models, the finite element

mesh is used similarly as in the FEM models. However, these S-FEM models evaluate the

weak form based on smoothing domains created from the entities of the element mesh such as

cells/elements, or nodes, or edges, or faces. These smoothing domains can be located inside the ele-

ments (CS-FEM) or cover parts of adjacent elements (NS-FEM, ES-FEM, and FS-FEM). These

smoothing domains are linear independent and hence ensure stability and convergence of the

S-FEM models.

For the S-FEM models that have the smoothing domain covering parts of adjacent elements (NS-

FEM, ES-FEM and FS-FEM), the number ofsupporting nodes in smoothing domains is larger than

that in elements. This leads to the bandwidth of stiffness matrix in the S-FEM models to increase

Copyright 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2012; 91:705741

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CS-DSG3 FOR REISSNERMINDLIN PLATES 707

and the computational cost is hence higher than those of the FEM. However, also because of con-

tributing of more supporting nodes in the smoothing domains, the S-FEM models often produce

the solution that is much more accurate than that of the FEM. Therefore in general, when the effi-

ciency of computation (computation time for the same accuracy) in terms of the error estimator

versus computational cost is considered, the S-FEM models are more efficient than the counterpart

FEM models [64, 68, 74]. It is clear that these S-FEM models have the features of both models:

mesh-free [76] and FEM. The element mesh is still used but the smoothed gradients bring the infor-mation beyond the concept of only one element in the FEM: they bring in the information from the

neighboring elements.

For the S-FEM models that have the smoothing domain locating inside the elements such as

CS-FEM [55, 56], the number of supporting nodes in smoothing domains are the same as those in

elements, and hence the bandwidth of stiffness matrix in the CS-FEM is similar to that in the FEM.

However, because of using the strain smoothing technique to increase the accuracy in each element,

the computational cost in the CS-FEM is a little higher than those of the FEM. However, in general,

when the efficiency of computation (computation time for the same accuracy) in terms of the error

estimator versus computational cost is considered, the CS-FEM is still efficient than the counter-

part FEM models. The CS-FEM, however so far, has been developed mainly only for the four-node

quadrilateral elements [5561] and the improvement of accuracy of solutions compared with those

of FEM is still marginal.

This paper hence extends the CS-FEM for triangular elements and for significant improvement of

solutions of plate analysis. The cell-based strain smoothing technique in the CS-FEM is combined

with stabilized discrete shear gap method (DSG3) [52, 77] using three-node triangular elements

to give a so-called cell-based smoothed discrete shear gap method (CS-DSG3) for static and free

vibration analyses of ReissnerMindlin plates. In the process of formulating the system stiffness

matrix of the CS-DSG3, each triangular element will be divided into three subtriangles, and in each

subtriangle the stabilized DSG3 is used to compute the strains and to avoid the transverse shear

locking. Then the strain smoothing technique on whole of the triangular element is used to smooth

the strains on these three subtriangles. The CS-DSG3 hence not only overcomes the drawback of

the DSG3, which depends on the sequence of node numbers of elements, but also improves the

accuracy and the stability of the DSG3. The numerical examples demonstrated that the CS-DSG3

is free of shear locking, passes the patch test, and achieves the high accuracy compared with others

existing elements.

2. WEAKFORM FOR THE REISSNER-MINDLIN PLATE

Let us now consider a plate under bending deformation. The middle (neutral) surface of the plate is

chosen as the reference plane that occupies a domain R2 as shown in Figure 1. Let w be thetransverse displacement (deflection), and T D

x y

be the vector of rotations, in which x ,

y are the rotations of the middle plane around yaxis and x-axis, respectively, with the positivedirections defined as shown in Figure 1.

Figure 1. ReissnerMindlin thick plate and positive directions of the displacement w and two rotationsx , y .


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uh D

nnXID1

24 NI.x/ 0 00 NI.x/ 0

0 0 NI.x/

35 dI, (10)

where nn is the total number of nodes of problem domain discretized; NI.x/ is shape function atnode I, and dI D wIxIyI

T is the nodal displacement vector associated to node I.The bending and shear strains can be then expressed in the matrix forms as

DXI

BIdI, s D

XI

SIdI , (11)

where

BI D

24 0 NI,x 00 0 NI,y

0 NI,y NI,x

35 , SI D

NI,x NI 0NI,y 0 NI

. (12)

The discretized system of equations of the MindlinReissner plate using the FEM for static

analysis then can be expressed as,

Kd D F, (13)

where

K D

Z

BTDbBd C

Z

STDsSd (14)

is the global stiffness matrix, and the load vector

F D

Z

pN d C fb (15)

in which fb is the remaining part ofF subjected to prescribed boundary loads.

For free vibration analysis, we have K !2M

d D 0, (16)

where ! is the natural frequency and M is the global mass matrix defined by

M D

Z

NTmN

Td. (17)

4. FORMULATION OF A CELL-BASED SMOOTHED DISCRETE SHEAR GAP METHOD

USING TRIANGULAR ELEMENTS

In this section, the cell-based strain smoothing technique [55] is combined with stabilized discrete

shear gap method (DSG3) using three-node triangular elements [52,77] to give a so-called CS-DSG3

for static and free vibration analyses of ReissnerMindlin plates.

4.1. Brief on the discrete shear gap formulation

The formulation of the DSG3 [52, 77] is based on the concept shear gap of displacement along

the sides of the elements. In the DSG3, the shear strain is linear interpolated from the shear gaps

of displacement by using the standard element shape functions. As a result, the operator matrix S

related to the shear part is modified, and its entries are constant and computed from the coordinates

of nodes of elements. The DSG3 element is shear-locking-free and has several superior properties

as presented in [52]. In this paper, we just brief on the DSG3 formulation, which is necessary for

the formulation of the CS-DSG3.


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Figure 2. Three-node triangular element.

Using a mesh of triangular elements, the approximation uh D

w x yT

for a three-

node triangular element e shown in Figure 2 for the ReissnerMindlin plate can be written, at theelement level, as

uhe

D

3

XID1

24

NI.x/ 0 00 NI.x/ 0

0 0 NI.x/

35

NI .x/

deI D

3

XID1NI.x/deI, (18)

where deI D wIxIyIT are the nodal DOFs ofuh

eassociated to node I and NI.x/ are linear

shape functions in a natural coordinate defined by

N1 D 1 , N2 D , N3 D . (19)

The curvatures of the deflection in the element are then obtained by

h D Bde, (20)

wherede D de1 de2 de3

T

is the nodal displacement vector of element,B

contains thederivatives of the shape functions that are constants

B D1

2Ae

26664

0 b c 00 0 d a0 d a b c

B1

0 c 00 0 d0 d c

B2

0 b 00 0 a0 a b

B3

37775 D

1

2Ae

B1 B2 B3

(21)

Figure 3. Three-node triangular element and local coordinates in the DSG3.


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with a D x2x1, b D y2y1, c D y3y1, d D x3x1 as shown in Figure 3, and xi D

xi yiT

,

i D 1, 2, 3, are coordinates of three nodes, respectively; Ae is the area of the triangular element, andBi , i D 1, 2, 3, contains the derivatives of the shape functions of the i th node.

As reported in many literatures on ReissnerMindlin elements, the shear locking often occurs

when the thickness plate becomes small, where the pure bending dominates the plate deformation.

This is because the parasitic transverse shear strains are not eliminated under pure bending condi-

tions. Such a conflict of the bending and shearing strain fields under thin plate situations leads to theshear locking problem. To overcome this conflict, Bletzinger et al. [52] proposed the DSG3 to alter

the shear strain field. The altered shear strains are in the form of

h D Sde, (22)

where

S D1

2Ae

264 b

c Ae 0d a 0 Ae

S1

c ac=2 bc=2d ad=2 bd=2

S3

b bd=2 bc=2a ad=2 ac=2

S3

375

D1

2Ae

S1 S2 S3

(23)

with Si , i D 1, 2, 3, contains the derivatives of the shape functions of the i th node.Substituting Equations (21) and (23) into Equation (14), the global stiffness matrix now becomes

KDSG3 D

neXeD1

KDSG3e , (24)

where the element stiffness matrix, KDSG3e

of the DSG3 element is given by

KDSG3

eD

Ze

BTDbBd C

Ze

STDsSd

D BTDbBAe C STDsSAe

(25)

It was suggested [79] that a stabilization term needs to be added to the original DSG3 element to

further improve the accuracy of approximate solutions and to stabilize shear force oscillations. Sucha modification is achieved by simply replacing Ds in Equation (25) by ODs, as follows:

KDSG3

eD

Ze

BTDbBd C

Ze

ST ODsSd

D BTDbBAe C ST ODsSAe

, (26)

where

ODs Dk t3

t2 C h2e

1 00 1

(27)

in which he is the longest length of the edges of the element and is a positive constant [77].Note that the DSG3 element is formulated based on the idea of shear gap of the one-dimensional

linear Timoshenko beam element [52]. In this idea, the shear gap is computed from two points:

one basis point and one ending point. However, when this idea of shear gap is applied for the two-

dimensional DSG3 triangular plate element, the formulation is not symmetric because only one

element node is chosen to be the basis point of two shear gaps along two element edges, and two

remaining element nodes are the ending points of the shear gaps. As a result, the constant bending

and shear gradient matrices in the DSG3 will be different when different element nodes are chosen

to be the basis point. This can be seen clearly from Equations (21), (23), and (26) in which the

element stiffness matrix in the DSG3 depends on the sequence of node numbers of elements, and

hence the solution of DSG3 is changed when the sequence of node numbers of elements changes,

especially for the coarse and distorted meshes. The CS-DSG3 is hence proposed to overcome this

drawback and also improve the accuracy of the DSG3.


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4.2. Formulation of cell-based smoothed discrete shear gap

In the CS-DSG3, each triangular element is divided into three subtriangles by connecting the central

point of the element to three field nodes, and the displacement vector at the central point is assumed

to be the simple average of three displacement vectors of three field nodes. In each subtriangle, the

stabilized DSG3 is used to compute the strains and also to avoid the transverse shear locking. Then

the strain smoothing technique on the whole triangular element is used to smooth the strains on

these three subtriangles. The formulation of CS-DSG3 is presented in detail as follows.

Consider a typical triangular element e as shown in Figure 4. We first divide the element into

three subtriangles 1, 2, and 3 such as e DS3

iD1i and i \ j D ;, i j , by simplyconnecting the central point O of the triangle with three field nodes as shown in Figure 4. The

coordinates xO D

xO yOT

of the central point O are calculated by

xO D1

3.x1 C x2 C x3/ I yO D

1

3.y1 C y2 C y3/ , (28)

where xi D

xi yiT

, i D 1, 2, 3, are coordinates of three field nodes, respectively.In the CS-DSG3, we assume that the displacement vector deO at the central point O is the simple

average of three displacement vectors de1, de2, and de3 of three field nodes

deO D 13

.de1 C de2 C de3/ . (29)

On the first subtriangle 1(triangle O12), we now construct the linear approximation u1e

Dwe ex ey

Tby

u1e

D N1.x/deO C N2.x/de1 C N3.x/de2 D

3XID1

NI.x/d1I , (30)

where d1 DdeO de1 de2

Tis the vector of nodal DOFs of the subtriangle 1 and NI.x/ is

shape function in a natural coordinate defined by Equations (18) and (19).

The curvatures of the deflection 1 and the altered shear strains 1 in the subtriangle 1 arethen obtained by

1 Dhb11 b

12 b

13

i

b1

24 deOde1

de2

35 D b1d1 (31)

1 Dhs11 s

12 s

13

i

s1

24 deOde1

de2

35 D s1d1 , (32)

sub-triangle

1

23

central point

O1

3

2

Figure 4. Three subtriangles (1, 2, and 3) created from the triangle 123 in the CS-DSG3 byconnecting the central point O with three field nodes 1, 2, and 3.


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where b1 and s1 are, respectively, computed similarly as the matrices B and S of the DSG3

in Equations (21) and (23) but with two following changes: (1) the coordinates of three node

xi D

xi yiT

, i D 1, 2, 3 are replaced by xO , x1 and x2, respectively; and (2) the area Ae isreplaced by the area A1 of subtriangle 1.

Substituting deO in Equation (29) into Equations (31) and (32), and then rearranging we obtain

1 Dh

13b11 C b

12

13b11 C b

13

13b11

i

B1

24 de1de2de3

35 D B1de (33)

1 Dh

13s11 C s

12

13s11 C s

13

13s11

i

S1

24 de1de2

de3

35 D S1de. (34)

Similarly, for the second subtriangle 2 (triangle O23) and third subtriangle 3 (triangleO31), the curvatures of the deflection j , the altered shear strains j and matrices Bj , Sj ,

j D 2, 3, respectively, can be obtained by cyclic permutation.Now, applying the cell-based strain smoothing operation in the CS-FEM [55, 56], the constant

bending strains 1 , 2 , 3 and constant shear strains 1 , 2 , 3 are respectively used to

create a smoothed bending strain Qe and a smoothed shear strain Qe on the triangular element esuch as

Qe D

Ze

h e .x/d D 1

Z1

e .x/d C 2

Z2

e .x/d C 3

Z3

e .x/d (35)

Qe D

Ze

h e .x/d D 1

Z1

e .x/d C 2

Z2

e .x/d C 3

Z3

e .x/d, (36)

Figure 5. Patch test of the element.

Table I. Patch test.

Element w5 x5 y5 mx5 my5 mxy5

MIN3 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033DSG3 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033ES-DSG3 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033MITC4 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033CS-DSG3 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033Exact 0.6422 1.1300 0.6400 0.0111 0.0111 0.0033


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where e .x/ is a given smoothing function that satisfies at least unity propertyRe

e .x/d D 1.

Using the following constant smoothing function:

e .x/ D

1=Ae x 2 e

0 x e, (37)

where Ae is the area of the triangular element, the smoothed bending strain Qe and the smoothedshear strain Qe in Equations (35) and (36) become

Qe DA1

1 C A22 C A3

3

Ae(38)

Qe DA1

1 C A22 C A3

3

Ae. (39)

Substituting j , j D 1, 2, 3, into Equation (38), the smoothed bending strain Qe is expressed by

Qe D QBde, (40)

(a) (b) (c)

Figure 6. Square plate models and their discretizations using triangular elements: (a) clamped plate; (b)simply supported plate; and (c) four discretizations of a quarter of plate using triangular elements.

(a) (b)

Figure 7. Results of various methods in the shear locking test of the clamped square plate subjected touniform load with mesh 4 4: (a) central deflection and (b) central moment.


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where QB is the smoothed bending strain gradient matrix given by

QB DA1B

1 C A2B2 C A3B

3

Ae. (41)

Substituting j

, j D 1, 2, 3, into Equation (39), the smoothed shear strainQ

e is expressed by

Qe D QSde, (42)

where QS is the smoothed shear strain gradient matrix given by

QS DA1S

1 C A2S2 C A3S

3

Ae. (43)

Table II. Convergence of central deflection wc=

qL4=100D

and central moments Mc=

qL2=10

for the

clamped square plate subjected to uniform load q.

MeshAnalytical

t=L Method 4 4 8 8 10 10 12 12 16 16 solutions [44]

wc=

qL4=100D

CS-DSG3 0.1123 0.1227 0.1241 0.1248 0.1256MITC4 (M) 0.1348 0.1289 0.1281 0.1276 0.1271MITC4 0.1211 0.1251 0.1256 0.1259 0.1262RDKTM [50] 0.1550 0.1350 0.1305 0.1289

0.001 DKTM [50] 0.1159 0.1250 0.1266 0.1271 0.1267DSG3 0.0440 0.0873 0.0985 0.1058 0.1141MIN3 0.0754 0.1102 0.1160 0.1192 0.1224ES-DSG3 0.0904 0.1202 0.1234 0.1248 0.1259

CS-DSG3 0.1357 0.1467 0.148 0.1487 0.1495MITC4 (M) 0.1569 0.1524 0.1517 0.1513 0.1509MITC4 0.1431 0.1488 0.1494 0.1497 0.150

0.1 DSG3 0.0927 0.1366 0.1419 0.1447 0.1473 0.1499MIN3 0.0915 0.1246 0.1285 0.1305 0.1324ES-DSG3 0.1306 0.1499 0.1506 0.1508 0.1508

Mc=

qL2=10

CS-DSG3 0.1574 0.2110 0.2176 0.2211 0.2246MITC4 (M) 0.1896 0.2201 0.2234 0.2251 0.2269MITC4 0.189 0.2196 0.223 0.2249 0.2267RDKTM [50] 0.3075 0.2518 0.2397 0.2353

0.001 DKTM [50] 0.2525 0.2362 0.2340 0.2329 0.2291DSG3 0.0767 0.1627 0.1816 0.1938 0.2077MIN3 0.1272 0.1979 0.209 0.2151 0.2212ES-DSG3 0.1444 0.2121 0.2188 0.2223 0.2255

CS-DSG3 0.1618 0.2145 0.2208 0.2242 0.2276MITC4 (M) 0.1901 0.2222 0.2258 0.2277 0.2296

0.1 MITC4 0.1898 0.2219 0.2256 0.2275 0.2295 0.231DSG3 0.1253 0.2070 0.2167 0.2217 0.2264MIN3 0.1416 0.2093 0.2174 0.2215 0.2255ES-DSG3 0.1699 0.2201 0.2246 0.2269 0.2292


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Therefore, the global stiffness matrix of the CS-DSG3 is assembled by

QK D

neXeD1

QKe, (44)

where QKe is the smoothed element stiffness matrix given by

QKe D

Ze

QBTDb QBd C

Ze

QST ODs QSd

D QBTDb QBAe C QST ODs QSAe .

(45)

Note that the constant bending and shear gradient matrices in three subtriangles will be different

because three basis points and shear gaps in three subtriangles are different and these gradient matri-

ces are also different from those of the vertex-defined triangle. From Equations (41), (43), and (45),

it is clearly seen that the element stiffness matrix in the CS-DSG3 does not depend on the sequence

of node numbers, and hence the solution of CS-DSG3 is always stable when the sequence of node

numbers changes.

(a) (b)

Figure 8. Convergence of results of clamped square plate with t =L D 0.001: (a) central deflection and (b)central moment.

(a) (b)

Figure 9. Convergence of results of clamped square plate with t =L D 0.1: (a) central deflection and (b)central moment.


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Note that the introduction of the central points in the triangular elements in the CS-DSG3 is only

for the intermediate formulation of the element stiffness matrix. In the final form of element stiff-

ness matrix as shown in Equation (45), the nodal displacement vectors of the central points will be

replaced by those of three vertex nodes. Hence, no extra DOFs are associated with these central

points. This means that the nodal unknowns in the CS-DSG3 are the same as those in the DSG3 of

the same mesh.

It should be mentioned that similar to the DSG3, the CS-DSG3 also employs the constant bendingand transverse shear strain fields, and only five deformation modes are considered. The formulation

hence still misses a rigid mathematical justification and can suffer from rank deficiency. However,

the CS-DSG3 elements pass the patch test and are free of spurious kinematic modes as shown in

various numerical examples in Section 5.

In addition, similar to the DSG3, the CS-DSG3 also employs the idea of discrete shear gaps,

which introduce two nodal indicators of shear deformation or two discrete Kirchoff constraints at

the element nodes. As a result, the constraint index [2] of the CS-DSG3 is 3/2, which is the ideal

number for the shear strain constraints engendered in the thin plate limit.

Table III. Convergence of central deflection wc=

qL4=100D

and central moments Mc=

qL2=10

for the

simply supported plate (w D s D 0) subjected to uniform load q.

NumberAnalytical

t=L Mesh type 4 4 8 8 10 10 12 12 16 16 solutions [44]

wc=

qL4=100D

CS-DSG3 0.3517 0.3928 0.3977 0.4004 0.4030MITC4(M) 0.4107 0.4075 0.407 0.4068 0.4065MITC4 0.3969 0.4041 0.4049 0.4053 0.4057

0.001 DSG3 0.2019 0.3299 0.3543 0.3689 0.3845 0.4062MIN3 0.2948 0.3767 0.3873 0.3931 0.3989

ES-DSG3 0.3229 0.3906 0.3969 0.4001 0.403

CS-DSG3 0.3748 0.4143 0.419 0.4215 0.4240MITC4 (M) 0.4329 0.4288 0.4282 0.4280 0.4277MITC4 0.4190 0.4255 0.4261 0.4265 0.4268

0.1 DSG3 0.2981 0.4001 0.4108 0.4162 0.4213 0.4273MIN3 0.3199 0.3931 0.4010 0.4050 0.4087ES-DSG3 0.3652 0.4161 0.4205 0.4227 0.4247

Mc=

qL2=10

CS-DSG3 0.3721 0.4527 0.4622 0.4673 0.4724MITC4 (M) 0.4075 0.4612 0.4676 0.4710 0.4745

MITC4 0.4075 0.4612 0.4676 0.4710 0.47450.001 DSG3 0.2495 0.3942 0.4209 0.4369 0.4542 0.4789

MIN3 0.3316 0.4399 0.4539 0.4615 0.4691ES-DSG3 0.3687 0.4559 0.4646 0.4692 0.4735

CS-DSG3 0.3754 0.4533 0.4625 0.4675 0.4725MITC4 (M) 0.4075 0.4612 0.4676 0.4710 0.4745MITC4 0.4075 0.4612 0.4676 0.4710 0.4745

0.1 DSG3 0.3223 0.4452 0.4584 0.4651 0.4714 0.4789MIN3 0.3499 0.4501 0.4611 0.4668 0.4722ES-DSG3 0.3895 0.4591 0.4663 0.4701 0.4740


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5. NUMERICAL RESULTS

In this section, various numerical examples are performed to show the accuracy and stability

of the proposed CS-DSG3 compared with the analysis solutions. For comparison, several other

elements such as DSG3 [52], ES-DSG3 [15], MIN3 [80], MITC4 [37], and MITC4(M) in which

MITC4(M) uses k t3=

t2 C h2

eas the multiplier of the matrix ODs for the transverse shear have

also been implemented in our package. In addition, the stabilized parameter in Equation (27) inthe CS-DSG3 is fixed at 0.1 for both static analysis and free vibration analysis.

5.1. Static analysis

5.1.1. Constant bending patch test. The patch test is introduced to examine the convergence of

finite elements. It is checked to ensure that the element can reproduce a constant distribution

of all quantities for arbitrary meshes. It is modeled by several triangular elements as shown in

Figure 5. The boundary deflection is assumed to be w D

1 C x C 2y C x2 C xy C y2

=2. Theresults shown in Table I confirm that, similar to DSG3, ES-DSG3, MIN3, MITC4 elements, the

CS-DSG3 element also fulfills the patch test within machine precision.

(a) (b)

Figure 10. Convergence of results of simply supported square plate (w D s D 0/ with t =L D 0.001: (a)central deflection and (b) central moment.

(a) (b)

Figure 11. Convergence of results of simply supported square plate (w D s D 0/ with t =L D 0.1: (a)central deflection and (b) central moment.


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5.1.2. Square plate. Figures 6(a) and Figures 6(b) describe the models of a square plate (length L,thickness t/ with clamped (w D n D s D 0/ and simply supported (w D s D 0/ boundaryconditions subjected to a uniform load q D 1, respectively. The material parameters are given byYoungs modulus E D 1.092.000 and Poissons ratio D 0.3. Five uniform discretizations N Nof plate with N D 4, 8, 10, 12, and 16 are used and a quarter of these discretizations are plotted inFigure 6(c). The detailed expressions of analytical solutions can be found in [44].

First, a shear locking test using a coarse mesh (mesh 4 4) for the clamped plate is performedand the central deflection and central moment are shown in Figure 7. The results show that similarly

(a) (b)

Figure 12. (a) A simply supported (w D 0) skew Morleys model. (b) Four discretizations of a quarter of

plate using triangular elements.

Table IV. Convergence of deflection and principal moments at the central point of Morleys plate.L=t D 1000).

Mesh

Methods 4 4 8 8 10 10 12 12 16 16 Morley solution [81]

Central deflection wc=

qL4=1000D

CS-DSG3 0.4383 0.4019 0.3998 0.3992 0.3994

MITC4 0.3588 0.3571 0.3640 0.3717 0.3835

MITC4 (M) 0.4143 0.4088 0.4075 0.4070 0.4071RDKTM [50] 0.4530 0.4240 0.4190 0.408DKTM [50] 0.3570 0.3390 0.3520DSG3 0.2611 0.2655 0.2768 0.2895 0.3149MIN3 0.2726 0.3101 0.3228 0.3325 0.3465ES-DSG3 0.5926 0.4234 0.4176 0.4179 0.4207

Central max principal moment Mmax=

qL2=100

CS-DSG3 1.4773 1.7683 1.8086 1.8313 1.8548

MITC4 1.5097 1.6756 1.7229 1.7608 1.8116MITC4 (M) 1.5950 1.8149 1.8419 1.8578 1.8764 1.91DSG3 1.2413 1.3065 1.3518 1.4070 1.5182MIN3 1.2496 1.5222 1.5854 1.6306 1.6914ES-DSG3 1.6401 1.8333 1.8648 1.8915 1.9237

Central min principal moment Mmin=

qL2=100

CS-DSG3 0.6194 0.8729 0.9297 0.9670 1.0103

MITC4 0.6738 0.8590 0.9052 0.9424 0.9923MITC4 (M) 0.7188 0.9747 1.0179 1.0415 1.0639 1.08DSG3 0.4736 0.5488 0.5845 0.6231 0.7006MIN3 0.4922 0.6928 0.7470 0.7871 0.8428ES-DSG3 0.7384 0.9016 0.953 0.9952 1.0509


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to DSG3, MIN3, ES-DSG3, and MITC4, the CS-DSG3 also provides a locking-free solution when

the plate thickness becomes progressively small.

For the clamped plate, the convergence of the central deflection and the central moment against

the different mesh densities N N for the thin plate (ratio t =L D 0.001) and thick plate (ratiot=L D 0.1) are presented in Table II and plotted in Figures 8 and 9, respectively. In this analysis,besides the results of the five methods DSG3, MIN3, ES-DSG3, MITC4, and CS-DSG3, we also

added the results of the three methods DKTM, RDKTM [50], and MITC4(M). The results showthat with the same DOFs, the CS-DSG3 is worse than the MITC4 and MITC4(M), but better than

the DSG3 and MIN3 for both thin and thick plates. Compared with the ES-DSG3, the CS-DSG3

usually gives better results for thin plates, but a little worse results for thick plates. Compared with

the DKTM and RDKTM [50] for thin plate, the CS-DSG3 is worse than the DKTM but a little better

than the RDKTM.

For the simply supported plate (w D s D 0/, the convergence of the central deflection and thecentral moment against the different mesh densities N N for the thin plate (ratio t=L D 0.001)and thick plate (ratio t =L D 0.1) are presented in Table III and plotted in Figures 10 and 11. Theobtained comments as in the above clamped plate are again confirmed.

Figure 13. Convergence of deflection at the central point of Morley plate (with L=t D 1000) by variousmethods.

(a) (b)

Figure 14. Convergence of results of at the central point of Morley plate (with L=t D 1000): (a) maxprincipal moment and (b) min principal moment.


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(a) (b) (c)

Figure 15. (a) A circular plate with the clamped boundary condition; (b) four discretizations of a quarterof plate using triangular elements (6, 24, 54, 96 elements); and (c) four discretizations of a quarter of plate

using quadrilateral elements (3, 12, 27, 48 elements).

Table V. Convergence of central deflection wc and central moment Mc of the clamped circular platesubjected to uniform load.

Mesh (Q4: quadrilateral elements)

(T3: triangular elements)

3 Q4 12 Q4 27 Q4 48 Q4Analytical

t=R Methods 6 T3 24 T3 54 T3 96 T3 solution [16, 17]

Central deflection wc

CS-DSG3 7896.2 9304.3 9576.1 9668.9MITC4 (M) 11158 10159 9947.8 9874.6

0.02 MITC4 9068.1 9692.6 9738.5 9759.2 9873.48DSG3 3565.1 7184.3 8471.8 9038.5MIN3 5409.8 8433.2 9158.8 9428.5ES-DSG3 5893.9 8907.6 9456.8 9621.8

CS-DSG3 9.6481 11.089 11.354 11.442MITC4 (M) 12.802 11.892 11.702 11.634


Central moment Mc

CS-DSG3 1.3326 1.8517 1.9559 1.9912

MITC4 (M) 1.8305 1.9523 2.0000 2.01330.02 MITC4 1.8818 1.9404 2.0036 2.0127 2.03125DSG3 0.7463 1.5861 1.8337 1.9182MIN3 1.0716 1.7821 1.9200 1.9705ES-DSG3 1.1736 1.8182 1.9523 1.9917

CS-DSG3 1.3459 1.8517 1.9558 1.9913MITC4 (M) 1.8124 1.9551 1.9953 2.0114



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(a) (b)

Figure 16. Convergence of results of clamped circular plate (t =R D 0.02) subjected to uniform load: (a)central deflection and (b) central moment.

(a) (b)

Figure 17. Convergence of results of clamped circular plate (t =R D 0.2) subjected to uniform load: (a)central deflection and (b) central moment.

5 10 15 205.5

6

6.5

7

Mesh index N

Firstmode

Exact s olution

MITC4DSG3

CS-DSG3 with consistent mass matrix

CS-DSG3 with lumped mass matrix

Figure 18. Affection of the lumped mass matrix and consistent mass matrix to the first frequency of theCS-DSG3.


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5.1.3. Skew plate subjected to a uniform load. We now consider a rhombic plate with simply sup-

ported (w D 0) boundary conditions subjected to a uniform load q D 1 as shown in Figure 12(a).This plate was originally studied by Morley [81]. Geometry and material parameters are given by

length L D 100, thickness t D 0.1, Youngs modulus E D 10.92 and Poissons ratio D 0.3. Fiveuniform discretizations N N of plate with N D 4, 8, 10, 12, and 16 are used and a quarter of thesediscretizations are plotted in Figure 12(b).

The convergence of the deflection and principal moments at the central point by various methodsare presented in Table IV and plotted in Figures 13 and 14. In this analysis, besides the results of six

methods DSG3, MIN3, ES-DSG3, MITC4, MITC4(M), and CS-DSG3, we also added the results

of two methods DKTM and RDKTM [50]. It is seen that the CS-DSG3 shows remarkably excellent

performance compared with the DSG3, MIN3, RDKTM and even MITC4, and is a good competitor

to the ES-DSG3, MITC4(M), and DKTM. These results hence imply that for problems with skew

geometries, the CS-DSG3 can give highly accurate solutions and is a strong competitor to many

existing plate elements.

5.1.4. Circle plate subjected to a uniform load. We now consider a circular plate subjected to a

uniform load q D 1 with the clamped boundary condition as shown in Figure 15(a). The materialparameters are given by Youngs modulus E D 10.92, Poissons ratio D 0.3, the radius R D 5.

Four discretizations of a quarter of plate using triangular elements (6, 24, 54, 96 elements) andquadrilateral elements (3, 12, 27, 48 elements) are shown in Figures 15(b) and (c), respectively.

The detailed expressions of the analytical solution can be found in [16, 17]. The convergence of

the central displacement wc and central moment Mc for the thin plate (ratio t =R D 0.02) and thickplate (ratio t =R D 0.2) are presented in Table V and plotted in Figures 16 and 17, respectively.

Table VI. Convergence of six lowest nondimensional frequency parameters $ of a SSSS thin square plate(t =L D 0.005).

Mode sequence number

Meshing Methods 1 2 3 4 5 6

DSG3 5.5626 8.8148 11.8281 13.4126 18.1948 19.2897

MIN3 5.0409 8.6812 10.0678 12.9804 17.2128 18.96434 4 ES-DSG3 4.9168 8.1996 9.4593 11.5035 14.2016 15.0164

MITC4 (M) 4.5576 7.8291 7.8291 9.8260 13.1854 13.1854MITC4 4.6009 8.0734 8.0734 10.305 15.0109 15.0109CS-DSG3 4.4965 7.1241 7.2503 9.0931 10.0933 10.1619

DSG3 4.7327 7.4926 8.2237 10.2755 11.6968 12.4915MIN3 4.5804 7.4049 7.6488 9.9064 11.2774 11.4598ES-DSG3 4.5376 7.2981 7.4659 9.6486 10.8937 11.0280

8 8 MITC4 (M) 4.4712 7.2091 7.2091 9.1140 10.6360 10.6360MITC4 4.4812 7.2519 7.2519 9.2004 10.7796 10.7796CS-DSG3 4.4543 7.0536 7.0791 8.9750 10.0418 10.0477

DSG3 4.5131 7.1502 7.3169 9.3628 10.3772 10.4461MIN3 4.4759 7.1174 7.1704 9.1459 10.2473 10.2570

ES-DSG3 4.4641 7.0870 7.1193 9.0582 10.1444 10.148916 16 MITC4 (M) 4.4498 7.0693 7.0693 8.9411 10.0994 10.0994MITC4 4.4522 7.0792 7.0792 8.9611 10.1285 10.1285CS-DSG3 4.4453 7.0310 7.0367 8.9051 9.95900 9.9592

DSG3 4.4781 7.0905 7.1718 9.1455 10.1643 10.1814MIN3 4.4600 7.0732 7.1003 9.0220 10.0972 10.0997ES-DSG3 4.4537 7.0565 7.0729 8.9731 10.0410 10.0422

22 22 MITC4 (M) 4.4464 7.0479 7.0479 8.9142 10.0200 10.0200MITC4 4.4477 7.0531 7.0531 8.9247 10.0349 10.0349CS-DSG3 4.4440 7.0276 7.0306 8.8949 9.94620 9.94630

Exact [82] 4.4430 7.0250 7.0250 8.8860 9.93500 9.93500


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Table VII. Convergence of six lowest nondimensional frequency parameters $ of a SSSS thick squareplate (t =L D 0.1).



DSG3 4.9970 8.1490 9.4311 11.354 14.1290 14.9353MIN3 4.9129 8.4029 9.3892 12.1445 15.6080 16.9309

4 4 ES-DSG3 4.7376 7.6580 8.4524 10.1882 12.1227 12.7533MITC4 (M) 4.4758 7.4403 7.4403 9.1415 11.5180 11.518MITC4 4.5146 7.6192 7.6192 9.4471 12.2574 12.2574CS-DSG3 4.4032 6.7790 6.8435 8.3901 9.07140 9.0889

DSG3 4.4891 7.0697 7.2530 9.1263 10.2195 10.3361MIN3 4.5114 7.2165 7.3557 9.3924 10.6582 10.7313


DSG3 4.3943 6.8227 6.8587 8.5447 9.4557 9.4616MIN3 4.4297 6.9612 6.9892 8.7917 9.8071 9.8107

16 16 ES-DSG3 4.3846 6.7922 6.8196 8.4744 9.3666 9.3698

MITC4 (M) 4.3731 6.7840 6.7840 8.4026 9.3540 9.3540MITC4 4.3753 6.7918 6.7918 8.4166 9.3728 9.3728CS-DSG3 4.3683 6.7470 6.7511 8.3623 9.2256 9.2257

DSG3 4.3809 6.7854 6.8037 8.4543 9.3441 9.3457MIN3 4.4178 6.9231 6.9373 8.7030 9.6856 9.6866ES-DSG3 4.3759 6.7692 6.7834 8.4173 9.2968 9.2976

22 22 MITC4 (M) 4.3694 6.7618 6.7618 8.3757 9.2800 9.2800MITC4 4.3711 6.7692 6.7692 8.3872 9.3009 9.3009CS-DSG3 4.3674 6.7456 6.7478 8.3587 9.2238 9.2238

Exact [82] 4.3700 6.7400 6.7400 8.3500 9.2200 9.2200

(a) (b)

Figure 19. Six lowest frequencies of SSSS square plates discretized by a uniform mesh 4 4: (a) thin plate(t =L D 0.005) and (b) thick plate (t =L D 0.1).

Again, it is seen that the CS-DSG3 is worse than the MITC4 and MITC4(M) but the CS-DSG3

shows remarkably excellent performance compared with the DSG3 and MIN3 for both thin and

thick plates, and is a good competitor to the ES-DSG3.


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(a) (b)

Figure 20. Six normalized lowest frequencies

!h=!exact

of SSSS square plates discretized by a uniform

mesh 22 22: (a) thin plate (t =L D 0.005) and (b) thick plate (t =L D 0.1).

(b)(a)

(d)(c)

(f)(e)

Figure 21. Shape of six lowest eigenmodes of SSSS square plates (mesh 16 16) by the CS-DSG3(t =L D 0.005). (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, and (f) Mode 6.


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5.2. Free vibration analysis of plates

In this section, we investigate the accuracy and efficiency of the CS-DSG3 element for analyz-

ing natural frequencies of plates. The plate may have free (F), simply (S) supported, or clamped

(C) edges. The symbol, CFSF, for instance, represents clamped, free, supported, and free boundary

conditions along the edges of the rectangular plate. A nondimensional frequency parameter $ isoften used to stand for the frequencies and the obtained results use the regular meshes. The results

of the CS-DSG3 are then compared with analytical solutions and other numerical results, which

are available in the literature. Also note that, in dynamic analysis using the CS-DSG3, we can use

the usual consistent mass matrix defined in Equation (17) to compute. However, in this paper for

computational efficiency [78], the well-known lumped mass matrix is used.

5.2.1. Square plates. We now consider square plates of length L and thickness t . The mate-rial parameters are Youngs modulus E D 2.0 1011, Poissons ratio D 0.3 and the density

mass D 8000. A nondimensional frequency parameter $ D

!2a4t =D1=4

is used, where

D D Et 3=

12.1 2/

is the flexural rigidity of the plate.

First, we analyze two thin and thick SSSS plates corresponding to thickness-to-length t =L D0.005 and t =L D 0.1. The geometry of the plate is shown in Figure 6(b), and four uniform dis-cretizations N N of plate with N D 4, 8, 16, and 22 are used and a quarter of these discretizationsare plotted in Figure 6(c).

Figure 18 shows the affection of using the lumped mass matrix and the consistent mass matrix by

Equation (17) to the first frequency of the CS-DSG3. It is seen that the results of the CS-DSG3 using

the lumped mass matrix are much more accurate than those of the CS-DSG3 using the consistent

Table VIII. Convergence of six lowest nondimensional frequency parameters $ of a CCCC thin squareplate (t =L D 0.005).



DSG3 8.4197 12.772 14.965 17.258 21.389 21.7600

MIN3 7.4097 11.763 13.401 16.355 20.979 21.91974 4 ES-DSG3 6.9741 10.193 11.476 13.055 15.404 15.9360

MITC4 (M) 6.3137 10.169 10.169 12.068 15.857 15.8906MITC4 6.5638 11.523 11.523 13.951 62.605 62.6054CS-DSG3 6.1712 8.6783 8.9731 10.38 11.067 11.2107

DSG3 6.7161 9.7867 10.567 12.998 14.531 15.3143MIN3 6.3460 9.3326 9.6896 12.107 13.632 13.8879


DSG3 6.1786 8.8759 9.068 11.2450 12.218 12.2992MIN3 6.0818 8.7504 8.8301 10.8435 11.9680 12.0010


DSG3 6.0889 8.7239 8.8202 10.8567 11.8519 11.8845MIN3 6.0417 8.6621 8.703 10.6330 11.7280 11.7541


Exact [82] 5.9990 8.568 8.568 10.4070 11.4720 11.4980


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mass matrix, and even better than those of the MITC4 (using the consistent mass matrix). In this

paper, we hence use the lumped mass matrix for the CS-DSG3 in all free vibration analyses with

the aim of increasing the accuracy of solutions and computational efficiency.

Tables VI and VII give the convergence of six lowest frequencies of thin plate ( t=L D 0.005) andthick plate (t =L D 0.1), respectively. In addition, Figures 19 and 20 plot the values of the six lowestfrequencies of the thin plate (t =L D 0.005) and thick plate (t=L D 0.1) for two uniform meshes

Table IX. Convergence of six lowest nondimensional frequency parameters $ of a CCCC thick squareplate (t =L D 0.1).



DSG3 6.8748 9.8938 11.085 12.636 15.103 15.6402MIN3 6.9924 10.789 12.079 14.539 18.102 19.1320


DSG3 5.9547 8.3618 8.6293 10.299 11.342 11.5397MIN3 6.1140 8.8097 9.0456 11.011 12.355 12.51178 8 ES-DSG3 5.8068 8.0861 8.2701 9.8397 10.760 10.8960

MITC4 (M) 5.7700 8.1376 8.1376 9.6084 10.816 10.8706MITC4 5.8079 8.2257 8.2257 9.7310 10.992 11.0457CS-DSG3 5.7180 7.8901 7.9111 9.3563 10.1380 10.1875

DSG3 5.7616 7.9935 8.0525 9.5772 10.415 10.4697MIN3 5.9281 8.3778 8.4288 10.142 11.150 11.1930ES-DSG3 5.7250 7.9211 7.9627 9.4499 10.263 10.3126

16 16 MITC4 (M) 5.7197 7.9404 7.9404 9.3950 10.290 10.3399MITC4 5.7288 7.9601 7.9601 9.4230 10.326 10.3752CS-DSG3 5.7117 7.8846 7.8965 9.3442 10.1319 10.1803

DSG3 5.7337 7.9381 7.9686 9.4589 10.2760 10.3246MIN3 5.9011 8.314 8.3403 10.011 10.9780 11.0164

ES-DSG3 5.7141 7.8990 7.9206 9.3896 10.1935 10.241122 22 MITC4 (M) 5.7118 7.9102 7.9102 9.3618 10.2099 10.2590

MITC4 5.7166 7.9204 7.9204 9.3764 10.2280 10.2771CS-DSG3 5.7077 7.8810 7.8873 9.3357 10.1273 10.1754

Exact [82] 5.7100 7.8800 7.8800 9.3300 10.1300 10.1800

(a) (b)

Figure 22. Six lowest frequencies of CCCC square plates discretized by mesh 4 4: (a) thin plate(t =L D 0.005) and (b) thick plate (t =L D 0.1).


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(a) (b)

Figure 23. Six normalized lowest frequencies (!h=!exact/ of CCCC square plates discretized by uniformmesh 22 22: (a) thin plate (t =L D 0.005) and (b) thick plate (t =L D 0.1).

Table X. Four lowest nondimensional frequency parameters $ D !L2p

t=D of a thin square plate(t =L D 0.005) discretized by a uniform mesh 4 4 and with various boundary conditions.


Plate type Methods 1 2 3 4

DSG3 13.4384 36.3942 52.2192 83.6418MIN3 13.2647 37.0802 64.1956 94.8655

SSSF ES-DSG3 11.7117 31.415 40.2198 67.8325MITC4 12.4874 30.0113 57.0427 75.9205CS-DSG3 11.6021 26.2662 41.0162 56.2077Exact [7] 11.6850 27.7560 41.1970 59.0660

DSG3 10.3265 20.5677 44.3569 53.5887MIN3 10.5581 19.0247 49.8732 57.4529

SFSF ES-DSG3 9.93190 16.6019 37.7735 45.1925MITC4 10.4426 17.0034 39.5516 54.9420CS-DSG3 9.64290 15.2648 31.5905 39.1789

Exact [7] 9.63100 16.1350 36.7260 38.9450

DSG3 30.7363 58.3350 76.39110 103.0276MIN3 30.7680 62.5272 119.9260 144.4048

CCCF ES-DSG3 23.8495 44.2875 55.80230 73.79370MITC4 29.1083 46.5769 112.7871 124.9989CS-DSG3 24.4269 38.2293 62.98950 70.12510Exact [7] 24.0200 40.0390 63.49300 76.76100

DSG3 26.0322 39.9980 63.4639 81.85270MIN3 26.8688 36.3721 68.6022 110.1215

CFCF ES-DSG3 22.4222 29.1449 47.1992 59.72200MITC4 27.5944 31.3195 49.1215 101.9821CS-DSG3 22.9417 26.0201 38.4104 62.31410Exact [7] 22.2720 26.5290 43.6640 64.46600

DSG3 17.0178 28.3590 52.7620 65.0051MIN3 17.4636 26.2990 57.9568 80.3767

CFSF ES-DSG3 15.5957 21.8959 42.2915 51.3048MITC4 17.5082 22.7384 43.4724 80.8395CS-DSG3 15.4214 19.8854 34.5940 50.1941Exact [7] 15.2850 20.6730 39.8820 49.5000

4 4 and 22 22, respectively. It is observed that the results of CS-DSG3 agree excellently with theanalytical results [82] and are much more accurate than those of the other elements for both thin and

thick plates, and for both coarse and fine meshes. In particular, the CS-DSG3 can provide accurately


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the values of high frequencies of plates by using only coarse meshes. Figure 21 plots the shape of

six lowest eigenmodes of SSSS square plates with mesh 16 16 by the CS-DSG3 (t=L D 0.005). Itis seen that the shapes of eigenmodes express exactly the real physical modes of the plate.

Next, we analyze two thin and thick CCCC square plates shown in Figure 6(a). Four uniform

meshes and two ratios of thickness-to-length t =L are similar to those of the SSSS plate case.Tables VIII and IX give the convergence of six lowest frequencies of thin plate (t=L D 0.005)

and thick plate (t =L D 0.1), respectively. In addition, Figures 22 and 23 plot the values of thesix lowest frequencies of the thin plate (t/L= 0.005) and thick plate (t =L D 0.1) for two uniform

Table XI. Four lowest nondimensional frequency parameters $ D !L2p

t=D of a thin square plate(t =L D 0.005) discretized by a uniform mesh 22 22 and with various boundary conditions.


Plate type Methods 1 2 3 4

DSG3 11.7553 28.258 41.8252 61.1274MIN3 11.7305 28.021 41.7375 60.3199

SSSF ES-DSG3 11.6817 27.8143 41.3866 59.5521

MITC4 11.7085 27.8259 41.5907 59.4952CS-DSG3 11.7117 27.9075 41.6054 59.8642Exact [7] 11.6850 27.7560 41.1970 59.0660

DSG3 9.6608 16.3096 37.5011 39.4050MIN3 9.6591 16.2185 37.1384 39.4010

SFSF ES-DSG3 9.6402 16.1214 36.8606 39.1664MITC4 9.6560 16.1594 36.8250 39.3439CS-DSG3 9.6523 16.1663 36.9537 39.3145Exact [7] 9.6310 16.1350 36.7260 38.9450

DSG3 24.2149 41.4350 64.6795 80.2128MIN3 24.1257 40.6903 64.4940 78.8210

CCCF ES-DSG3 23.8927 40.1428 63.4463 77.6415MITC4 24.0559 40.1776 64.2683 77.5923CS-DSG3 24.0220 40.3201 64.0603 77.9933Exact [7] 24.0200 40.0390 63.4930 76.7610

DSG3 22.3132 27.0330 45.4552 62.2851MIN3 22.3122 26.7138 44.4375 62.2847

CFCF ES-DSG3 22.1684 26.4128 43.8441 61.4711MITC4 22.3107 26.5333 43.7558 62.2403CS-DSG3 22.2521 26.5322 43.9719 61.9548Exact [7] 22.2720 26.5290 43.6640 64.4660

DSG3 15.2635 20.9362 40.9260 50.1777MIN3 15.2635 20.7563 40.3242 50.1788

CFSF ES-DSG3 15.2002 20.5789 39.9116 49.7129MITC4 15.2590 20.6440 39.8569 50.1204CS-DSG3 15.2385 20.6539 40.0256 49.9966Exact [7] 15.2850 20.6730 39.8820 49.5000

(a) (b)

Figure 24. (a) A parallelogram rhombic plate with boundary condition CFFF. (b) Four discretizations of aquarter of plate using triangular elements.


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Table XII. Convergence of six lowest nondimensional frequency parameters $ D

! a2=2p

t=D of a

thin CFFF parallelogram plate (t =L D 0.001).



DSG3 0.4269 1.2390 2.9338 4.3925 5.3783 8.45270

MIN3 0.4169 1.1419 3.2032 4.0874 5.6748 11.49034 4 ES-DSG3 0.3996 0.9774 2.4570 3.2772 4.2868 5.85030

MITC4 0.4034 0.9918 2.9239 3.0466 4.9993 6.35480CS-DSG3 0.3872 0.9086 2.2523 2.3075 3.9334 4.41040

DSG3 0.4134 1.1040 2.7821 3.6708 4.8637 8.6563MIN3 0.4053 1.0191 2.7732 3.0296 4.6516 6.7978

8 8 ES-DSG3 0.3998 0.9617 2.5869 2.8383 4.3300 6.0750MITC4 0.3998 0.9649 2.6694 2.7213 4.3982 5.4487CS-DSG3 0.3968 0.9462 2.5102 2.5438 4.1619 4.9632

DSG3 0.4043 1.0179 2.6736 3.0109 4.4559 6.5803MIN3 0.4003 0.974 2.6286 2.7377 4.3207 5.5610

16 16 ES-DSG3 0.3984 0.9541 2.5744 2.6746 4.2228 5.3329MITC4 0.3986 0.9567 2.5907 2.6504 4.2414 5.2105CS-DSG3 0.3979 0.9520 2.5541 2.6090 4.1827 5.0972

DSG3 0.4019 0.9949 2.6392 2.8569 4.3554 6.0079MIN3 0.3994 0.9652 2.6001 2.6872 4.2608 5.3604ES-DSG3 0.3981 0.9532 2.5692 2.6508 4.2030 5.2283

22 22 MITC4 0.3984 0.9552 2.5776 2.6395 4.2163 5.1728CS-DSG3 0.3980 0.9525 2.5588 2.6175 4.1836 5.1124

Ref [83] 0.3980 0.9540 2.5640 2.6270 4.1890 5.1310

Table XIII. Convergence of six lowest nondimensional frequency parameters $ D

! a2=2p

t=D of a

thick CFFF parallelogram plate (t =L D 0.2).

Mode sequence numberMeshing Methods 1 2 3 4 5 6

DSG3 0.3837 0.8946 2.1154 2.9305 3.5762 5.2992MIN3 0.4047 1.0183 2.6937 3.3188 4.5314 7.9527

4 4 ES-DSG3 0.3717 0.8388 1.9201 2.6050 3.1885 4.3993MITC4 0.3842 0.8590 2.2472 2.4099 3.5907 4.4683CS-DSG3 0.3693 0.8092 1.8323 1.9613 3.0806 3.4338

DSG3 0.3812 0.8427 2.0195 2.3931 3.3217 4.5322MIN3 0.3943 0.9197 2.3563 2.5886 3.7986 5.1173

8 8 ES-DSG3 0.3766 0.8147 1.9372 2.2787 3.1321 4.1642MITC4 0.3800 0.8307 2.0529 2.2332 3.2417 3.9506CS-DSG3 0.3780 0.8251 1.9653 2.1296 3.1531 3.7323

DSG3 0.3790 0.8221 1.9788 2.2261 3.1680 3.9645MIN3 0.3901 0.8874 2.2505 2.4021 3.5630 4.4526

16 16 ES-DSG3 0.3772 0.8121 1.9500 2.1910 3.1023 3.8494MITC4 0.3781 0.8208 1.9999 2.1831 3.1395 3.8069CS-DSG3 0.3785 0.8213 1.9832 2.1616 3.1280 3.7670

DSG3 0.3783 0.8187 1.9738 2.1982 3.1374 3.8689MIN3 0.3891 0.8807 2.2304 2.3704 3.5172 4.3461ES-DSG3 0.3772 0.8129 1.9573 2.1786 3.0999 3.8050

22 22 MITC4 0.3777 0.8190 1.9911 2.1748 3.1224 3.7835CS-DSG3 0.3782 0.8197 1.9834 2.1642 3.1182 3.7648

Ref [83] 0.3770 0.8170 1.9810 2.1660 3.1040 3.7600


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(a) (b)

Figure 25. Six lowest frequencies of CFFF rhombic square plates discretized by mesh 4 4: (a) thin plate(t =L D 0.001) and (b) thick plate (t =L D 0.2).

(a) (b)

Figure 26. Six normalized lowest frequencies (!h=!exact/ of CFFF rhombic square plates discretized bymesh 22 22: (a) thin plate (t =L D 0.001) and (b) thick plate (t =L D 0.2).

Table XIV. Fourteen lowest parameterized natural frequencies $ D

!R2p

t=D of a clamped circular

plate with t=.2R/ D 0.01.

Mode DSG3 MIN3 ES-DSG3 CS-DSG3 ANS4 [85] ANS9 [86] Exact [7] Exact [84]

1 10.4182 10.4082 10.3109 10.2478 10.2572 10.2129 10.2158 10.2162 22.3362 22.2198 21.6702 21.3092 21.4981 21.2311 21.2600 21.260

3 22.3555 22.2444 21.6900 21.3246 21.4981 21.2311 21.2600 21.2604 38.0399 37.7461 36.3124 35.0852 35.3941 34.7816 34.8800 34.8775 38.1428 37.7816 36.3816 35.1137 35.5173 34.7915 34.8800 34.8776 42.9671 43.0344 41.3801 40.3370 40.8975 39.6766 39.7710 39.7717 58.4283 57.8881 54.7796 51.6052 52.2054 50.8348 51.0400 51.0308 58.7024 58.0836 54.8922 51.7926 52.2054 50.8348 51.0400 51.0309 69.0108 68.726 64.6300 61.7079 63.2397 60.6761 60.8200 60.82910 69.5195 69.2354 65.1330 62.0897 63.2397 60.6761 60.8200 60.82911 85.3533 84.3603 77.9311 71.0569 71.7426 69.3028 69.6659 69.66612 85.5382 84.4507 78.1412 71.3304 72.0375 69.3379 69.6659 69.66613 102.975 101.443 93.2686 86.0894 88.1498 84.2999 84.5800 84.58314 103.674 101.758 93.6276 86.5370 89.3007 84.3835 84.5800 84.583


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meshes 4 4 and 22 22, respectively. Again it is seen that the obtained comments in the SSSSplates are confirmed for the CCCC plates.

We further study the five sets of various boundary conditions in this example: SSSF, SFSF, CCCF,

CFCF, CFSF. In this case, one coarse mesh 4 4 and one fine mesh 22 22 are utilized for a thinsquare plate (t =L D 0.005) with various boundary conditions and the first four lowest frequenciesare presented in Tables X and XI, respectively. It is again confirmed that the results of CS-DSG3

agree excellently with the analytical results [7], and are much more accurate than those DSG3,MIN3, and MITC4 for various boundary conditions and for both coarse and fine meshes, and is a

good competitor of the ES-DSG3. In particular, the CS-DSG3 can provide accurately the values of

high frequencies of plates (4th frequency) by using only coarse meshes.

5.2.2. Parallelogram plates. We now consider the thin (t=L= 0.001) and thick (t=L D 0.2) can-tilever rhombic (CFFF) plates. The geometry of the plate is illustrated in Figure 24(a) with skew

angle D 60. The material parameters are Youngs modulus E D 2.0 1011, Poissons ratio D 0.3 and the density mass D 8000. A nondimensional frequency parameter $ is used. Four

(a) (b)

Figure 27. Comparison of fourteen normalized lowest frequencies (!h=!exact/ of a clamped circularplate discretized by a mesh of 394 triangular elements and 222 nodes by various methods. (a) Thin plate

(t=.2R/ D 0.01) and (b) thick plate (t=.2R/ D 0.1).

Table XV. Fourteen lowest parameterized natural frequencies $ D

!R2p

t=D of a clamped circular

plate with t=.2R/ D 0.1.

Mode DSG3 MIN3 ES-DSG3 CS-DSG3 Exact [84] ANS4 [85] ANS4 [85]

1 9.40710 9.96820 9.32620 9.27680 9.2400 9.26050 9.227702 18.3363 20.3060 18.0461 17.8022 17.834 17.9469 17.80103 18.3544 20.3256 18.0673 17.8101 17.834 17.9469 17.80104 28.5094 32.7800 27.8438 27.0648 27.214 27.0345 26.68015 28.5388 32.7905 27.8856 27.0795 27.214 27.6566 27.22466 31.9501 37.1591 31.1280 30.4370 30.211 30.3221 29.85627 39.9949 47.5437 38.6936 36.7727 37.109 37.2579 36.39668 40.0120 47.5993 38.7042 36.8531 37.109 37.2579 36.39669 45.6439 55.3530 44.1130 42.4101 42.409 43.2702 42.108910 45.8554 55.6608 44.3286 42.5429 42.409 43.2702 42.108911 52.8749 64.9340 50.5469 46.6672 47.340 47.7074 46.059612 52.9039 64.9946 50.5964 46.7856 47.340 47.8028 46.098513 60.3690 75.7085 57.7532 54.0439 54.557 56.0625 53.933214 60.4325 75.8198 57.8604 54.1967 54.557 57.1311 54.7720

Note: The alternative form of MITC4 [85] using a consistent mass; the alternative form of MITC4 [85] using a

lumped mass.


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(b)(a)

(d)(c)

(f)(e)

(h)(g)

Figure 28. Shape of eight lowest eigenmodes of clamped circular plate by the CS-DSG3 (t=.2R/ D 0.1).(a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, (f) Mode 6, (g) Mode 7, and (h) Mode 8.

(a) (b)

Figure 29. (a) A square triangular plate and (b) a rhombic triangular plate.


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uniform discretizations N N of plate with N D 4, 8, 16, and 22 are used and a quarter of thesediscretizations are plotted in Figure 24(b). Tables XII and XIII show the convergence of the six low-

est frequencies of a CFFF rhombic plate corresponding to the thin plate ( t=L D 0.001) and thickplate (t =L D 0.2). In addition, Figures 25 and 26 plot the values of the six lowest frequencies of thethin plate (t=L D 0.001) and thick plate (t =L D 0.2) for two uniform meshes 4 4 and 22 22,respectively. Again, it is confirmed that the results of CS-DSG3 agree excellently with those of

the semi-analytical pb-2 Ritz method [83], and are much more accurate than those DSG3, MIN3,and MITC4 for both coarse and fine meshes, and is a good competitor of the ES-DSG3. In partic-

ular, the CS-DSG3 can provide accurately the values of high frequencies of plates by using only

coarse meshes.

5.2.3. Circle plates. In this example, the circular plate with the clamped boundary as shown in

Figure 15(a) is again studied. The material parameters are Youngs modulus E D 2.0 1011,

Table XVI. Six lowest parameterized natural frequencies $ D

!L2=2p

t=D of thin triangular plates(t =L D 0.001).

Mode sequence number Methods 1 2 3 4 5 6

DSG3 0.6288 2.4399 3.4796 5.9913 8.3737 11.2081MIN3 0.6272 2.4242 3.4344 5.9458 8.2460 11.0178ES-DSG3 0.6241 2.3970 3.3631 5.7976 7.9696 10.5356

0 CS-DSG3 0.6223 2.3417 3.2130 5.5353 7.5546 9.50930Rayleigh-Ritz [87] 0.6240 2.3770 3.3080 5.6890 7.7430 Pb2 Rayleigh-Ritz [88] 0.6250 2.3770 3.3100 5.6890 7.7430 Experimental [89] 0.5880 2.3180 3.2390 5.5400 7.5180 ANS4 [85] 0.6240 2.3790 3.3170 5.7240 7.7940 10.2000

DSG3 0.5915 2.2360 3.6467 5.5686 7.8875 11.1572MIN3 0.5886 2.2229 3.5717 5.5286 7.7447 10.9544ES-DSG3 0.5843 2.1979 3.4937 5.3893 7.4745 10.4532

15 CS-DSG3 0.5824 2.1513 3.3332 5.1162 7.1483 9.43820Rayleigh-Ritz [87] 0.5840 2.1810 3.4090 5.2800 7.2640 Pb2 Rayleigh-Ritz [88] 0.5860 2.1820 3.4120 5.2790 7.2630 ANS4 [85] 0.5830 2.1810 3.4130 5.3030 7.2890 10.0950

DSG3 0.5928 2.2293 3.9729 5.8523 7.8772 11.9930MIN3 0.5866 2.2164 3.8656 5.8059 7.6697 11.5791ES-DSG3 0.5795 2.1881 3.7620 5.6464 7.3608 10.9609


DSG3 0.6265 2.4298 4.5709 6.9888 8.7055 13.4984MIN3 0.6125 2.4038 4.4237 6.9194 8.3447 12.7848ES-DSG3 0.6001 2.3557 4.2652 6.6442 7.9210 11.8923


DSG3 0.6969 2.9282 6.0785 9.3379 11.8340 17.8590MIN3 0.6631 2.8106 5.8343 8.9727 11.3197 16.3597ES-DSG3 0.6371 2.6804 5.4909 8.1353 10.6516 14.5225



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Poissons ratio D 0.3, the radius R D 5 and the density mass D 8000. The plate is discretizedby the mesh of 394 triangular elements and 222 nodes. Two thicknessspan ratios h=.2R/ D 0.01and 0.1 are considered. As shown in Table XIV and Figure 27(a), in case of the thicknessspan ratio

h=.2R/ D 0.01, the frequencies obtained from the CS-DSG3 element are much closer to analyticalsolutions [84] than DSG3 and ES-DSG3, and is a good competitor to quadrilateral plate elements

such as the ANS solutions (ANS4) [85] and the higher order ANS solutions (ANS9) [86]. As shown

in Table XV and Figure 27(b), in case of the thicknessspan ratio h=.2

R/ D 0.1, the CS-DSG3 areagain closer to exact solutions [84] than DSG3 and ES-DSG3, and is a good competitor to quadrilat-

eral plate elements such as the ANS4 element that used 432 quadrilateral elements (or 864 triangular

elements). Figure 28 plots the shape of the eight lowest eigenmodes of clamped thick circular plate

(t=.2R/ D 0.1) by the CS-DSG3. It is seen that the shapes of eigenmodes express exactly the realphysical modes of the plate.

5.2.4. Triangular plates. We now consider cantilever (CFF) triangular plates with various shape

geometries, see Figures 29(a) and (b). The material parameters are Youngs modulus E D 2.01011,

Table XVII. Six lowest parameterized natural frequencies $ D

!L2=2p

t=D of thick triangular

plates (t =L D 0.2).


Methods 1 2 3 4 5 6

DSG3 0.5872 1.9529 2.4678 4.1305 5.2243 6.2886MIN3 0.6042 2.1109 2.7737 4.6581 6.1311 7.3208ES-DSG3 0.5856 1.9376 2.4411 4.0678 5.1289 6.1359

0 CS-DSG3 0.5820 1.8779 2.3706 3.8693 4.8877 5.6733Pb2 Rayleigh-Ritz [88] 0.5820 1.9000 2.4080 3.9360 FEM [90] 0.5810 1.9010 2.4100 ANS4 [85] 0.5820 1.9150 2.4280 3.9840 5.0180 5.9440

DSG3 0.5478 1.8160 2.4429 3.8111 5.0253 6.0033MIN3 0.5656 1.9555 2.7772 4.2851 5.8857 6.9977ES-DSG3 0.5457 1.8018 2.4121 3.7529 4.9317 5.8327


DSG3 0.5365 1.8126 2.4954 3.7549 4.9796 5.9219MIN3 0.5582 1.9551 2.8595 4.2605 5.8273 6.9945ES-DSG3 0.5329 1.7963 2.4552 3.6902 4.8717 5.6942


DSG3 0.5448 1.9269 2.6246 3.9196 5.1474 5.9838MIN3 0.5724 2.1044 3.0165 4.5535 6.0525 7.2736ES-DSG3 0.5380 1.9002 2.5682 3.8239 4.9806 5.6000


DSG3 0.5691 2.1281 2.7093 4.3427 5.3437 6.5278MIN3 0.6017 2.3742 3.2377 5.1762 6.5578 8.1003ES-DSG3 0.5569 2.0692 2.5955 4.1109 4.7967 6.0192



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Poissons ratio D 0.3 and the density mass D 8000. A nondimensional frequency parameter$ D !L2.t=D/1=2=2 of triangular square plates with the aspect ratio t =L D 0.001 and 0.2 arecalculated. The mesh of 170 triangular elements and 108 nodes is used to analyze the convergence

for modes via various skew angles such as D 0, 15, 30, 45, 60. Tables XVI and XVII give theconvergence of the six lowest modes of the thin triangular plate (t=L D 0.001) and thick triangularplate (t=L D 0.2), respectively. In these two tables, the CS-DSG3 element is also compared with the

alternative MITC4 finite element formulation [85] (the ANS method (ANS4) using a mesh of 398four-node quadrilateral elements or 796 triangular elements) and two other well-known numerical

methods such as the RayleighRitz method [87] and the pb-2 Ritz method [88]. In addition, the six

lowest frequencies of a thin plate (t=L D 0.001) and a thick plate (t =L D 0.2) with the skew angles D 60 and D 15 by various methods are plotted in Figures 30 and 31, respectively. From theresults given in Tables XVI and XVII, Figures 30 and 31, it is again verified that the CS-DSG3 are

much closer to reference solutions [85] than DSG3, MIN3, and ES-DSG3, and is a good competitor

to two well-known quadrilateral plate elements such as the RayleighRitz method [87] and the pb-2

Ritz method [88].

(a) (b)

Figure 30. Comparison of six lowest frequencies of a thin triangular plate with the skew angle D 60 byvarious methods (mesh of 170 triangular elements and 108 nodes): (a) thin plate (t =L D 0.001) and (b) thick

plate (t =L D 0.2).

Figure 31. Comparison of six lowest frequencies of a thin triangular plate with the triangle angle D 15by various methods (mesh of 170 triangular elements and 108 nodes): (a) thin plate (t =L D 0.001) and (b)

thick plate (t =L D 0.2).


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(b)(a)

(d)(c)

(f)(e)

Figure 32. Shape of six lowest eigenmodes of thin triangular plate (t =L D 0.001) with the skew angle D 60 by CS-DSG3: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, and (f) Mode 6.

Figure 32 plots the shape of the six lowest eigenmodes of thin triangular plate (t =L D 0.001) withthe skew angle D 60 by the CS-DSG3. It is seen that the shapes of eigenmodes express exactlythe real physical modes of the plate.

6. CONCLUSIONS

A cell-based smoothed finite element method with the stabilized DSG technique using triangu-

lar elements (CS-DSG3) is formulated for static and free vibration analyses of ReissnerMindlinplates. Through the formulations and numerical examples, some concluding remarks can be drawn

as follows:

The CS-DSG3 is formulated from three different techniques including DSG, cell-based

smoothed strain and using k t3=t 2 C h2e

as the multiplier of the matrix ODs for the transverseshear with the aim of resolving simultaneously the shear locking and improving the accuracy

of the solutions for the plate analysis.

In the CS-DSG3, the smoothing domains and the resulting smoothed strains are element-based

(or cell-based). The formulation of the CS-DSG3 is hence only based on the elements that are

similar to that of the original DSG3.

The CS-DSG3 uses only three DOFs at each vertex node without additional DOFs. The

CS-DSG3 is free of shear locking and passes the patch test.

Compared with the DSG3, the CS-DSG3 not only overcomes the drawback of the DSG3 whose

results depend on the sequence of node numbers of elements, but also improves the accuracy

and the stability of the DSG3.

For static analysis, the results of the CS-DSG3 agree well with analytical solutions and results

of several other published elements in the literature. The CS-DSG3 is much more accurate than

the DSG3, MIN3 and is a good competitor to the ES-DSG3, DKTM, and RDKTM [50].

For free vibration analysis, the CS-DSG3 is stable temporally, agrees well with analytical solu-

tions, and shows many superior properties. The CS-DSG3 gives much more accurate results

than the DSG3, MIN3, MITC4, and ES-DSG3 and shows to be a strong competitor to exist-

ing complicated quadrilateral plate elements such as the RayleighRitz method, the pb-2 Ritz


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method. In particular, the CS-DSG3 can provide accurately the values of high frequencies of

plates by using only coarse meshes.

For the problems with skew geometries, the CS-DSG3 shows to be a strong competitor to others

methods.

In conclusion, the CS-DSG3 is based on the formulation of three-node triangular elements with-

out adding any additional DOFs. The CS-DSG3 is free of shear locking and passes the patch test.

The CS-DSG3 is very promising to provide an effective tool for static and free vibration analyses.

The CS-DSG3 shows four superior properties such as: (1) being a strong competitor to many exist-

ing three-node triangular plate elements in the static analysis; (2) can give high accurate solutions

for problems with skew geometries in the static analysis; (3) can give high accurate solutions in free

vibration analysis; and (4) can provide accurately the values of high frequencies of plates by using

only coarse meshes. In addition, the performance of the CS-DSG3 is also simple and only based on

the elements; hence, it will be easy to extend to the flat shell element. This extension will highlight

the advantage of the CS-DSG3, which uses only triangular elements, because the geometry of shell

structures is often much more complicated than that of the plate structures.

ACKNOWLEDGEMENTS

This work was supported by Vietnam National Foundation for Science & Technology Development(NAFOSTED), Ministry of Science & Technology, under the basic research program (Project No.:107.02.2010.01).

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