DSpace at VNU: A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner-Mindlin plates

DOI 10.1007/s00466-010-0509-xO R I G I NA L PA P E R A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin Abstract In

DOI 10.1007/s00466-010-0509-x

O R I G I NA L PA P E R

A node-based smoothed finite element method with stabilized

discrete shear gap technique for analysis of Reissner–Mindlin

Abstract In this paper, a node-based smoothed finite

ele-ment method (NS-FEM) using 3-node triangular eleele-ments is

formulated for static, free vibration and buckling analyses

of Reissner–Mindlin plates The discrete weak form of the

NS-FEM is obtained based on the strain smoothing technique

over smoothing domains associated with the nodes of the

ele-ments The discrete shear gap (DSG) method together with

a stabilization technique is incorporated into the NS-FEM to

eliminate transverse shear locking and to maintain stability

of the present formulation A so-called node-based smoothed

stabilized discrete shear gap method (NS-DSG) is then

pro-posed Several numerical examples are used to illustrate the

accuracy and effectiveness of the present method

Keywords Plate bending· Transverse shear locking ·

Finite element method· Node-based smoothed

H Nguyen-Xuan (B) · T Nguyen-Thoi

Department of Mechanics, Faculty of Mathematics and Computer

Science, University of Science, Vietnam National University,

HCM, 227 Nguyen Van Cu, District 5, Ho Chi Minh City, Vietnam

e-mail: nxhung@hcmuns.edu.vn

URL: http://www.math.hcmuns.edu.vn/ ∼nxhung

H Nguyen-Xuan · T Nguyen-Thoi

Division of Computational Mechanics,

Faculty of Civil Engineering, Ton Duc Thang University,

98 Ngo Tat To, Binh Thanh District, Ho Chi Minh City, Vietnam

T Rabczuk · N Nguyen-Thanh

Institute of Structural Mechanics (ISM),

Bauhaus-University Weimar, Marienstr 15,

99423 Weimar, Germany

S Bordas

School of Engineering, Institute of Theoretical,

Applied and Computational Mechanics,

Cardiff University, Wales, UK

finite element· Discrete shear gap (DSG) ·

During the last three decades, lower-order Mindlin–Reissner plate finite elements have often been preferred due

to their simplicity and efficiency They require only

C0-continuity for the deflection and the normal rotations.However, these low-order plate elements in the thin platelimit often suffer from shear locking phenomenon due toincorrect transverse forces under bending Therefore, manyformulations have been developed to overcome the shearlocking phenomenon and to increase accuracy and stabil-ity of numerical methods such as mixed formulation/hybridelements [6 9], stabilization methods [10,11], the enhancedassumed strain (EAS) methods [12,13], the assumed natu-ral strain (ANS) methods [14–17], etc Recently, the discreteshear gap (DSG) method [18] which can be considered as

an alternative form of the ANS was proposed The DSG issomewhat similar to the ANS methods in the aspect of mod-ifying the course of certain strains within the element, but isdifferent in the aspect of lack of collocation points The DSG

method is therefore independent of the order and form of the

element

On an other front of development of finite element

nology, Liu et al have combined the strain smoothing

tech-nique [19] used in meshfree methods [20–27] into the finite

element method using quadrilateral elements to formulate a

cell/element-based smoothed finite element method (SFEM

or CS-FEM) [28–30] for 2D solids It is well known that

low-order elements for solid problems certain inherent drawbacks

such as overestimation of the stiffness matrix and locking

problems Therefore applying the strain smoothing technique

on smoothing domains to these standard FEM models aims

to soften the stiffness formulation, and hence can improve

significantly the accuracy of solutions in both displacement

and stress In CS-FEM, the smoothing domains are created

based on elements, and each element can be subdivided into

1 or several quadrilateral smoothing domains The

theoreti-cal aspects of CS-FEM were fully studied in [29,31,32] The

SFEM has also been extended to general n-sided polygonal

elements (nSFEM) [33], dynamic analysis [34,35], plate and

shell analysis [36,37], kinematic limit analysis [38] and

cou-pled to partition of unity enrichment [39] A general

frame-work for this strain smoothing technique in FEM can be found

in [40,41]

In the effort to overcome shortcomings of low-order

ele-ments, Liu et al have then extended the concept of smoothing

domains to formulate a family of smoothed FEM (S-FEM)

models with different applications such as the node-based

S-FEM (NS-FEM) [42,43], edge-based S-FEM (ES-FEM)

[44–49], face-based S-FEM (FS-FEM) [50,51] Similar to

the standard FEM, these S-FEM models also use a mesh of

elements In these S-FEM models, the discrete weak form

is evaluated using smoothed strains over smoothing domains

instead of using compatible strains over the elements as in

the traditional FEM The smoothed strains are computed by

integrating the weighted (smoothed) compatible strains The

smoothing domains are created based on the features of the

element mesh such as nodes [42], or edges [44] or faces [50]

These smoothing domains are linear independent and hence

stability and convergence of the S-FEM models are ensured

They cover parts of adjacent elements, and therefore the

num-ber of supporting nodes in smoothing domains is larger than

that in elements This leads to bandwidth of the stiffness

matrix in S-FEM models increased and the computational

cost is hence higher than those in the FEM However, due

to contribution of more supporting nodes in the smoothing

domains, S-FEM often produces the solution that is much

more accurate than that of the FEM Therefore in general,

when the efficiency of computation (computation time for the

same accuracy) in terms of the error estimator versus

com-putational cost is considered, the S-FEM models are more

efficient than the counterpart FEM models [43,46,47,51] It

can be argued that these S-FEM models have the features of

both models: meshfree and FEM The element mesh is stillused but the smoothed strains bring the non-local informa-tion from the neighboring elements A general and rigoroustheoretical framework to show properties, accuracy and con-vergence rates of the S-FEM models was given in [32].Among these S-FEM models, the NS-FEM [42,43] showssome interesting properties in the elastic solid mechanicssuch as: 1) it can provide an upper bound to the strain energy;2) it can avoid volumetric locking without any modification

on integration terms; 3) super-accurate and super-convergentproperties of stress solutions are gained; and 4) the stress

at nodes can be computed directly from the displacementsolution without using any post-processing In this paper,

we exploit several interesting properties of the NS-FEM foranalyzing plates The NS-FEM has been already extended toperform adaptive analysis [52], linear elastostatics and vibra-tion 2D solid problems [53] Also, alpha finite element meth-ods (αFEM) have been recently proposed have been recentlyproposed as an alternative to the NS-FEM and shown to sig-nificantly improve the results obtained by conventional andsmoothed FEM techniques, at the cost of the introduction of

a problem-dependent parameterα [54–56].

This paper presents a formulation of the node-basedsmoothed finite element method (NS-FEM) for Reissner–Mindlin plates using only three-node triangular mesheswhich are easily generated for complicated domains Theevaluation of the discrete weak form is performed by using

a strain smoothing technique over smoothing domains ciated with nodes of elements Transverse shear locking can

asso-be avoided through the discrete shear gap (DSG) method.The stability and accuracy of NS-FEM formulation is fur-ther improved by a stabilization technique to give a so-callednode-based smoothed finite element method with a stabi-lized discrete shear gap method (NS-DSG) Several numeri-cal examples are presented to demonstrate the reliability andeffectiveness of the present method

The layout of the paper is as follows Next sectiondescribes the weak form of governing equations and the for-mulation of 3-node plate element In Sect.3, a formulation

of NS-FEM with the stabilized discrete shear technique isintroduced Section4recalls some techniques relevant to thepresent approach Section5presents and discusses numeri-cal results Finally, we close our paper with some concludingremarks

2 The formulation of 3-node plate element

2.1 Discrete governing equations

We consider a domainΩ ⊂ R2occupied by the referencemiddle surface of a plate Letw and β = (βx , βy) T be the

transverse displacement and the rotations about the y and x

Fig 1 a 3-Node triangular

element; b Local coordinates

axes, see Fig.1, respectively We assume that the material

is homogeneous and isotropic with Young’s modulus E and

Poisson’s ratioν, the governing differential equations of the

static and dynamic Mindlin–Reissner plates can be expressed

in the following form [57]

∇ · Db κ(β) + ktγ + ρt3

where t is the plate thickness, ρ is the mass density, ω is the

natural frequency, p = p(x, y) is the transverse loading per

unit area, k = μE/2(1 + ν), μ = 5/6 is the shear correction

factor and Dbis the tensor of bending modulus given by

where∇ = (∂/∂x, ∂/∂y) T is the gradient vector

LetV and V0be defined as

V = {(w, β) : w ∈ H1(Ω), β ∈ H1(Ω)2} ∩ B (6)

V0= {(v, η) : v ∈ H1(Ω), η ∈ H1(Ω)2: v = 0,

withB denotes a set of the essential boundary conditions and

the L2inner products are given as

b2(β, η) =



Ω {(∇β x ) T ˆσ0∇η x + (∇β y ) T ˆσ0∇η y } dΩ

Let us assume that the bounded domainΩ is discretized into

N efinite elements such thatΩ ≈ N e

e=1Ω eandΩ i ∩ Ω j =

∅ , i = j The finite element solution of the static problem of

a low-order1element model for the Mindlin–Reissner plate

is to find(w h , β h ) ∈ V hsuch that

where the finite element spaces,V handV h

where P1(Ω e ) stands for the set of polynomials of degree 1

for each variable

The finite element problem of the free vibration modes is

to find the natural frequencyω h ∈ R+and 0= (w h , β h ) ∈

Since only linear triangular elements are used to obtain

stiff-ness matrices, the finite Reissner–Mindlin plate-bending

ele-ment approximation is simply interpolated using the linear

basis functions for both deflection and rotations without any

additional variables (C0-continuity for the transverse

dis-placement and the normal rotations) Hence, the bending

and geometric strains are constant and unchanged from the

standard finite elements while the transverse shear strains

contain linear interpolated functions It is known that these

low-order plate elements in the thin plate limit often suffer

from shear locking In order to avoid shear locking, the

dis-crete shear gaps (DSG) [18] which were proposed to reform

the shear strains are adopted As a result of using the

three-node triangular elements, the shear strainsγ D SG become

then constant

Fig 2 Triangular elements and smoothing domains associated with

nodes

2.2 Brief on the DSG3 element

In the linear triangular DSG3 element [18], the finite ment approximation (w h , β h ) is simply interpolated using

ele-the linear basis functions for both deflection and rotationswithout any additional variables The bending strains used inthe standard finite elements are unchanged while the trans-verse shear strains are reformulated by the interpolated sheargaps The shear strains can be expressed as a reduced oper-

ator Rh : H1(Ω e ) → Γ h (Ω e ), where Γ h (Ω e ) is defined

Table 1 Summary elements

MITC4 Four node mixed interpolation of tensorial component [14]

DSG3 Discrete shear gap triangle element [18]

ES-DSG3 Edge-based smoothed discrete shear gap triangular element [46]

DKMQ Discrete Kirchhoff Mindlin quadrilateral [70]

RPIM Radial point interpolation method [73]

Pb-2 Ritz Two-dimensional polynomial function Rayleigh–Ritz method [74]

Fig 3 Patch test of the element(E = 100,000; ν = 0.25; t = 0.01)

Table 2 Patch test

derivatives of the shape functions (N1 = 1 − ξ − η, N2 =

ξ, N3 = η) that are only constant (cf Bletzinger al [18]

for more detail)

γ ς (i = 1, 2, 3) are the discrete shear gaps at the

triangular element nodes related to the ς-local coordinate

axis (ς = 1, 2) that are reported as

and(wi , βxi , βyi), i = 1, 2, 3 are the degree of freedoms at node i of the element In order to further improve the accuracy

of approximate solutions and to stabilize shear force lations appearing in the triangular element, the stabilizationtechnique [59] can be used here The idea for the stabilization

oscil-of the original DSG3 element was also introduced in [60].With this remedy, the DSG3 element problem to the staticproblem is to find(w h , β h ) ∈ V hsuch that

∀(v, η) ∈ V h

0,

a (β h , η h ) + kt(γ D SG (w h , β h ), γ D SG (v, η)) = (p, v)

(21)where

(22)

where heis the longest length of the edges of the element,

α is a positive constant and A e is the area of the triangularelement It is evident that the original DSG3 element is recov-ered whenα = 0 For free vibration and buckling problems,

the second terms ((∇wh − β h , ∇v − η)) in the left hand side

of Eq (15) and Eq (16) are replaced by the terms in Eq (22)

In what follows, we utilize these constant strains toestablish a formulation of a node-based smoothed triangular

Fig 4 Square plate model:

a simply supported plate;

b full clamped plate

Fig 5 Simply supported plate (t /L =0.01): a normalized central

deflection; b relative error under log–log scale

element with the stabilized discrete shear gap technique

(NS-DSG3) for Reissner–Mindlin plates

3 A formulation of NS-FEM with stabilized discrete

shear technique

In the NS-FEM [42,43], the domain discretization is the same

as that of the standard FEM using Ne triangular elements,

but the integration required in the weak form of the FEM

is now performed based on the nodes, and strain

smooth-ing technique [19] is utilized In such a nodal integration

process, the problem domainΩ is again divided into a set

of smoothing domainsΩ (k) such asΩ ≈ N n

k=1Ω (k) and

Ω (i) ∩ Ω ( j) = ∅, i = j in which N n is the total number of

nodes of the problem domain For triangular elements, the

smoothing domainΩ (k) associated with the node k is

0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

Number of elements per edge

Exact solu MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3

(b)

Fig 6 Simply supported plate (t /L =0.01): a normalized central

moment; b relative error under log–log scale

ated by connecting sequentially the mid-edge-points to the

centroids of the surrounding triangular elements of node k as

shown in Fig.2The average curvatures ¯κ, and the average shear strains

¯γ over the cell Ω (k)are defined by

Number of elements per edge

Exact sol.

MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3

The average gradients related to geometric strains ¯∇w, ¯∇β x ,

¯∇β yover the smoothing domainΩ (k)are given by

Number of elements per edge

Exact sol MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3

(b)

Fig 8 Convergence in the deflection of clamped plate (t /L = 0.001):

a normalized central deflection; b relative error under log–log scale

in which A e i is the area of the i th element attached to node

k and N e k is the number of elements associated with node k

illustrated in Fig.2

With the above definitions, a solution of the NS-FEM withthe stabilized discrete shear gap technique using three—nodetriangular elements (NS-DSG3) is now established The NS-DSG3 solution to the static problem is to find(w h , β h ) ∈ V h

such that

Number of elements per edge

Exact sol.

MITC4 DSG3 MIN3 ES−DSG3 NS−DSG3

(b)

Fig 9 Clamped plate (t /L = 0.001): a Normalized central moment;

b relative error under log–log scale

∀(v, η) ∈ V h

0, ¯a(β h , η h )+kt( ¯γ (w h , β h ), ¯γ (v, η))=(p, v)

(27)

The NS-DSG3 solution of the free vibration modes is to

find the natural frequencyω h∈ R+and 0= (w h , β h ) ∈ V h

loadλ h

cr ∈ R+and 0= (w h , β h ) ∈ V hsuch that

0.8 0.85 0.9 0.95 1 1.05

Number of elements per edge

Normalized strain energy Exact solu.

MITC4 MIN3 DSG3 ES−DSG3 NS−DSG3

(a)

0.8 1 1.2 1.4 1.6 1.8 2

and the geometric terms related to the gradients ( ¯∇w, ¯∇β x,

¯∇β y) over the smoothing domain Ω (k)are given by

Fig 11 Performance of NS-DSG3 with varying L /t ratios: a Simply

supported plate; b Clamped plate

Fig 12 A simply supported skew Morley’s model

and modified shear terms are now obtained by performing

the smoothing operation via the smoothing domainΩ (k):

0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

Number of element per edge

Fig 13 Convergence of central deflection for skew Morley’s plate

in which hk =√A (k) is considered as the character length

of the smoothing domainΩ (k).

The necessity of stabilization for lower order plate ments in bending was shown in [59,60] Stabilization signif-icantly improves the accuracy in the case of very thin platesand distorted meshes and to reduce oscillations of transverseshear forces It is found from numerical experiments that thestabilization parameterα fixed at 0.1 can produce the rea-

ele-sonable accuracy for all cases tested The stiffness matrix ofNS-DSG3 becomes too flexible, ifα is chosen too large On

the contrary, the accuracy of the solution will reduce due tothe oscillation of shear forces, ifα is chosen too small So

far, how to obtain an optimal value of parameterα is an open

question

4 Relationship to similar techniques

The present method can be considered as an alternative form

of nodally integrated techniques in finite element tions [61–68] The crucial idea of these methods is toformulate a nodal deformation gradient via a weightedaverage of the surrounding element values The major con-tribution to the nodal-integral method has been pointed out

formula-by Bonet and Burton [61] In their approach, the node-basedformulation is applied to the volumetric component of thestrain energy in order to eliminate volumetric locking of the

Number of elements per edge

Number of elements per edge

(b)

Fig 14 Skew Morley’s plate: a central max principle moment;

b central min principle moment

standard tetrahedral element Subsequently, Dohrmann et al

[62] proposed a nodally averaged formulation for entire

ponents of the strain energy (i.e including deviatoric

com-ponent) This approach is simple while the method possesses

very interesting properties such as 1) the upper bound

prop-erty in strain energy; 2) free of volumetric locking; 3)

super-accurate and super-convergent properties of stress solutions;

4) the stress at nodes computed directly from the

displace-ment solution without using any post-processing Further

improvements on the stability condition of the nodally

aver-aged formulation have also been devised in Bonet et al

3 3.2 3.4 3.6 3.8 4 4.2

Number of elements per edge

Reference DKMQ MITC4 Q4BL DSG3 ES−DSG3 NS−DSG3

Fig 15 Strain energy of a simply supported skew Morley’s plate

[63], Puso and Solberg [65] and Gee et al [66] Recently,

a weighted-residual method that is used to weakly imposeboth the equilibrium equation and the kinematic equationwas also introduced to create the average nodal strain formu-lation [67,68] for a variety of solid and plate elements.The NS-FEM approach originates from the computation

of smoothed strains via the smoothing domains associatedwith nodes of elements In the NS-FEM, the way to cre-ate smoothing domains is similar to nodal backgrounds inDohrmann et al [62] Furthermore, the NS-FEM works for

arbitrary n-sided polygonal elements When only linear

trian-gular or tetrahedral elements are used, the NS-FEM producesthe same results as the method proposed by Dohrmann et al.[62]

The extension of the nodally integrated techniques toReissner–Mindlin plate elements has been proposed veryrecently in [68] Numerical results show some advantages ofthe nodally integrated formulation compared to the standardFEM However, it was proved in [68] that these formula-tions can not fulfill sufficiently, in general, constant bendingstrain patch test with an arbitrary mesh While only staticplate problems were studied in [68], we dealt with static, freevibration and buckling solutions of Reissner–Mindlin platesusing the NS-FEM Moreover, we show numerically the fulf-ilment of the constant bending strain patch test Although

in several cases the numerical results using the NS-FEMare slightly less accurate than those using the ES-FEM,2itsperformance is much better than several other existing

2 This method for analysis of plates has already been investigated in [46].

Fig 16 Supported and

methods In addition, nodally integrated techniques for

Reissner–Mindlin plate formulations found in the literature

are very limited This may be due to the various

difficul-ties generated from plate models Therefore, the motivation

of this work is to complement the nascent body of

litera-ture on nodally integrated formulations for static, free tion and buckling analyses of Reissner–Mindlin plates.Further developments of the present technique for plateswith complicated behaviors or shell problems will be inves-tigated in forthcoming papers