# A smoothed finite element method for plate analysis

**ABSTRACT** A quadrilateral element with smoothed curvatures for Mindlin–Reissner plates is proposed. The curvature at each point is obtained by a non-local approximation via a smoothing function. The bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements (smoothing cells). Numerical results show that the proposed element is robust, computational inexpensive and simultaneously very accurate and free of locking, even for very thin plates. The most promising feature of our elements is their insensitivity to mesh distortion.

**0**Bookmarks

**·**

**77**Views

- Computer Modeling in Engineering & Sciences(CMES). 01/2012; 85(2):129-155.
- SourceAvailable from: Hakim Naceur[Show abstract] [Hide abstract]

**ABSTRACT:**A smoothed finite element method formulation for the resultant eight-node solid-shell element is presented in this paper for geometrical linear analysis. The smoothing process is successfully performed on the Element midsurface to deal with the membrane and bending effects of the stiffness matrix. The strain smoothing process allows replacing the Cartesian derivatives of shape functions by the product of shape functions with normal vectors to the element mid-surface boundaries. The present formulation remains competitive when compared to the classical finite element formulations since no inverse of the Jacobian matrix is calculated. The three dimensional resultant shell theory allows the element kinematics to be defined only with the displacement degrees of freedom. The assumed natural strain method is used not only to eliminate the transverse shear locking problem encountered in thin-walled structures, but also to reduce trapezoidal effects. The efficiency of the present element is presented and compared with that of standard solid-shell elements through various benchmark problems including some with highly distorted meshesComputational Mechanics 10/2014; · 2.43 Impact Factor - SourceAvailable from: Ong Thanh Hai
##### Article: On stability, convergence and accuracy of bES-FEM and bFS-FEM for nearly incompressible elasticity

[Show abstract] [Hide abstract]

**ABSTRACT:**We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bES-FEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W 2) procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf-sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM. Keywords: Edge-based smoothed finite element method (ES-FEM); Face-based smoothed finite element method (FS-FEM); Bubble function; Volumetric locking; Nearly-incompressible elasticity.06/2014;

Page 1

A smoothed finite element method for plate analysis

H. Nguyen-Xuan∗

T. Rabczuk†

St´ ephane Bordas‡§

J.F.Debongnie¶

September 18, 2007

Abstract

A quadrilateral element with smoothed curvatures for Mindlin-Reissner plates is proposed. The curvature at

each point is obtained by a non-local approximation via a smoothing function. The bending stiffness matrix is

calculated by a boundary integral along the boundaries of the smoothing elements (smoothing cells). Numerical

results show that the proposed element is robust, computational inexpensive and simultaneously very accurate and

free of locking, even for very thin plates. The most promising feature of our elements is their insentivity to mesh

distortion.

1Introduction

Plate structures play an important role in Engineering Science. There are two different plate theories, the Kirchhoff plate

and the Mindlin-Reissner plate theory. Kirchhoff plates are only applicable for thin structures where shear stresses in

the plate can be ignored. Moreover, Kirchhoff plate elements require C1continuous shape functions. Mindlin-Reissner

plates take shear effects into account. An advantage of the Mindlin-Reissner model over the biharmonic plate model is

that the energy involves only first derivatives of the unknowns and so conforming finite element approximations require

only the use of C0shape functions instead of the required C1shape functions for the biharmonic model. However,

Mindlin-Reissner plate elements exhibit a phenomenon called shear locking when the thickness of the plate tends to

zero. Shear locking occurs due to incorrect transverse forces under bending. When linear finite element shape functions

are used, the shear angle is linear within an element while the contribution of the displacement is only constant. The

linear contribution of the rotation cannot be “balanced” by a contribution from the displacement. Hence, the Kirchhoff

constraint w,x+βy= 0, w,y+βx= 0 is not fulfilled in the entire element any more. Typically, when shear locking occurs,

there are large oscillating shear/transverse forces and hence a simple smoothing procedure can drastically improve the

results. Early methods tried to overcome the shear locking phenomenon by reduced integration or a selective reduced

integration, see References [63,27,28]. The idea is to split the strain energy into two parts, one due to bending and the

other one due to shear. Commonly, different integration rules for the bending strain and the shear strain energy are

used. For example, for the shear strain energy, reduced integration is used while full integration is used for the bending

energy. Reduced integration leads to an instability due to rank deficiency and results in zero-energy modes that can

be eliminated by an hourglass control, [3,6,26,62].

For a general quadrilateral plate element, the deflection and the two rotations of the four-node element can be

interpolated. Often, approximated fields of high degree are used. However, except for the 16-node isoparametric element

∗Division of Computational Mechanics, Department of Mathematics and Informatics, University of Natural Sciences -VNU-HCM,227

Nguyen Van Cu, Vietnam email: nxhung.hcmuns.edu.vn

†Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand email:timon.

rabczuk@canterbury.ac.nz

‡University of Glasgow, Civil Engineering, Rankine building, G12 8LT email: stephane.bordas@alumni.northwestern.edu, Tel. +44 (0)

141 330 5204, Fax. +44 (0) 141 330 4557 http://www.civil.gla.ac.uk/∼bordas

§corresponding author

¶Division of Manufacturing, University of Li` ege, Bˆ atiment B52/3 Chemin des Chevreuils 1, B-4000 Li` ege 1, Belgium JF.Debongnie@ulg.

ac.be

1

Page 2

of [62], most other elements still exhibit shear locking when the thickness tends to zero. To overcome this phenomenon,

a reduced integration scheme is employed in the shear term. However, most of these elements still exhibit either

locking or exhibit artificial zero-energy modes. Another famous class of plate elements are mixed formulation/hybrid

elements [35,34,45] and equilibrium elements [23]. However, such elements are complex and the computational cost is

high. They are not popular in most commercial Finite Element Method (FEM) codes.

The Assumed Natural Strain (ANS) method was developed to eliminate shear locking for bilinear plate elements, [29].

The basic idea is to compute the shear strains not directly from the derivatives of the displacements but at discrete

collocation points, from the displacements. Afterwards, they are interpolated over the element with specific shape

functions. For the bilinear element for example, the collocation points will be placed at the midpoint of the element

edges since the shear stresses are linear in the element and zero in the middle of the element. This reduces in addition

one of the constraints, since it makes one of the Kirchhoff constraints linear dependent on the other constraints. Bathe

and Dvorkin [4] extended the ANS plate elements to shells. The resulting element is known as the MITC (Mixed

Interpolation of Tensorial Components) or Bathe-Dvorkin element, see also [22,5]. Many ANS versions of plate and

shell elements have been developed. A nice overview can be found e.g in the textbooks by [62] and [3].

An alternative to the ANS method to avoid shear locking is the Discrete-Shear-Gap (DSG) method [10]. The DSG

method is in a way similar to the ANS method since it modifies the course of certain strains within the element. The

main difference is the lack of collocation points that makes the DSG method independent of the order and form of the

element. Instead, the Kirchhoff constraints are imposed directly on the element nodes. The Enhanced-Assumed-Strain

(EAS) method principally can be used to avoid locking phenomena as well. While the EAS method was successfully

applied to eliminate membrane locking in shells, it was not very efficient in removing shear locking in plates. The EAS

element of Simo and Rifai [49] gives satisfying results for rectangular plates.

In this work, we present a new plate elements based on the MITC4 element in which we incorporate stabilized

conforming nodal integration (SCNI). The SCNI approach was originally developed in meshfree methods as a nor-

malization for nodal integration [17,60] of the meshfree Galerkin weak form and recently in the FEM [38]. In this

approach, the strain smoothing stabilization has been introduced in the SCNI to meet the integration constraints and

thus fulfills the linear exactness in the Galerkin approximation of the second order partial differential equations. Wang

et al. [52] have shown that the cause of shear locking in Mindlin–Reissner plate formulations is due to the inability of

the approximation functions to reproduce the Kirchhoff mode, and the incapability of the numerical method to achieve

pure bending exactness (BE) in the Galerkin approximation. In their study, the Kirchhoff mode reproducing condition

(KMRC) is ensured for Mindlin-Reissner plates. The approximation functions for the displacement and the rotations

are constructed to meet the KMRC. Then, they derived the integration constraints for achieving BE, and a curvature

smoothing method (CSM) is proposed to meet the bending integration constraints. A further extension of the SCNI to

analysis of transverse and inplane loading of laminated anisotropic plates with general planar geometry [53] is studied.

Other contributions to remove shear locking in meshfree plate discretizations are given in [50,24,32,33,37], and, very

recently, [53,19,2].

In mesh-free methods with stabilized nodal integration, the entire domain is discretized into cells defined by the

field of nodes, such as the cells of a Vorono¨ ı diagram [17,60]. Integration is performed along the edges of each cell.

Although meshfree methods such as EFG obtain good accuracy and high convergence rate, the non-polynomial or

usually complex approximation space increases the computational cost of numerical integration. Recently, applications

of the SCNI to the FEM so-called the smoothed finite element method (SFEM) for two dimensional problems had

been proposed by Liu et al. [38,39]. It was shown that the SFEM is stable, accurate and effective. Following the idea

of the SFEM, Nguyen et al. [44] has formulated the SFEM with a selective cell-wise smoothing technique in order to

eliminate locking in incompressible cases.

We will show by numerical experiments that our element performs better or slightly better than the

original MITC4 element, at least for all examples tested. Moreover, due to the integration technique,

the element promises to be more accurate especially for distorted meshes. We will also show that our

element is free of shear locking.

The outline of the paper is organized as follows. In the next section, we present the basic equations of the plate

problem and the weak form. The curvature smoothing stabilization and the finite element discretization using the

curvature smoothing method are introduced in Section 3. Several numerical examples are given in Section 4. Finally,

2

Page 3

Section 5 closes this manuscript with some conclusions and future plans.

2Governing equations and weak form

Let Ω be the domain in R2occupied by the mid-plane of the plate and w and β = (βx,βy)Tdenote the transverse

displacement and the rotations in the x−z and y −z planes, see Figure 1, respectively. Assuming that the material is

homogeneous and isotropic with Young’s modulus E and Poisson’s ratio ν, the governing differential equations of the

Mindlin-Reissner plate are

− divDbκ(β) − λtγ(β) = 0inΩ(1)

− λtdiv(γ) = pinΩ(2)

w = ¯ w,β =¯β

on Γ = ∂Ω(3)

where t is the plate thickness, p = p(x,y) is the transverse loading per unit area, λ =

correction factor and Dbis the tensor of bending moduli, κ and γ are the bending and shear strains, respectively,

defined as

kE

2(1+ν), k = 5/6 is the shear

κ =

∂βx

∂x

−∂βy

∂βx

∂y−∂βy

∂y

∂x

, γ =

?

∂w

∂x+ βx

∂w

∂y− βy

?

(4)

The Equations (1) – (3) correspond to the minimization of the total potential function

ΠTPE=1

2

?

Ω

κT: Db: κ dΩ +1

2

?

Ω

γT: Ds: γdΩ −

?

Ω

w p dΩ(5)

The weak form of the equilibrium equation follows from the stationarity of eq. (5):

δΠTPE=

?

Ω

δκT: Db: κ dΩ +

?

Ω

δγT: Ds: γdΩ −

?

Ω

δw p dΩ(6)

where the delta denotes the variation. Let us assume that the bounded domain Ω is discretized into nefinite elements,

Figure 1: Assumption of shear deformations for quadrilateral plate element

3

Page 4

Ω ≈ Ωh=

ne

?

e=1Ωe. The finite element solution uh= [w βxβy]Tof a displacement model for the Mindlin-Reissner plate

is then expressed as

np

?

where np is the total number of element nodes, Ni are the bilinear shape functions associated to node i and qi =

[wiθxiθyi]Tare the nodal degrees of freedom of the variables uh= [w βxβy]Tassociated to node i. Then, the discrete

curvature field is

κh= Bbq

uh=

i=1

Ni

0

0

0

0

0

Ni

0Ni

qi

(7)

(8)

where the matrix Bb, defined below, contains the derivatives of the shape functions. The approximation of the shear

strain is written as

γh= Bsq (9)

with

Bs

i=

?

Ni,x

Ni,y

0Ni

0−Ni

?

(10)

By substituting Equation (7) - Equation (9) into Equation (5) and with the stationarity of (5), we obtain a linear

system of an individual element for the vector of nodal unknowns q,

Kq = g (11)

with the element stiffness matrix

K =

?

Ωe(Bb)TDbBbdΩ +

?

Ωe(Bs)TDsBsdΩ(12)

and the load vector

gi=

?

ΩeNi

p

0

0

dΩ(13)

where

Db=

Et3

12(1 − ν2)

1

ν

0

ν

1

0

0

0

1−ν

2

Ds=

Etk

2(1 + ν)

?

1

0

0

1

?

(14)

The element stiffness matrix K is symmetric and positive semi-definite. As already mentioned in the introduction, for

a low-order1element, shear locking is observed that can be eliminated by different techniques, [3,6,26,62]. The aim of

this paper is to propose a stabilized integration for a quadrilateral plate element. Therefore we will

1. apply the curvature smoothing method which was proposed by Chen et al. [17] in meshfree methods

based on the nodal integration and recently in the SFEM by Liu et al. [38] to the bending strains

and

2. adopt an independent interpolation approximation to the shear strains as the MITC4 element [4].

In mesh-free methods based on nodal integration for Mindlin–Reissner plates, convergence requires fulfilling bending

exactness (BE) and thus requires the following bending integration constraint (IC) to be satisfied (Wang et al. [52])

?

Ω

Bb

i(x)dΩ =

?

Γ

Ei(x)dΓ(15)

1in our case a four-node quadrilateral full-integrated bilinear finite element

4

Page 5

where Biis the standard gradient matrix

Bb

i=

0

0

0

0Ni,x

0

Ni,y

−Ni,y

−Ni,x

,Ei=

0

0

0

0Ninx

0

Niny

−Niny

−Ninx

(16)

The IC criterion comes from the equilibrium of the internal and external forces of the Galerkin approximation

assuming pure bending (Wang et al. [52]). This is similar to the consistency with the pure bending deformation in the

constant moment patch test in FEM.

The basic idea is to couple the MITC element with the curvature smoothing method (CSM). Therefore, smoothing

cells are constructed that do not necessarily have to be coincident with the finite elements. We use a mixed variational

principle based on an assumed strain field [48] and the integration is carried out either on the elements themselves,

or over the smoothing cells that form a partition of the elements. The CSM is employed on each smoothing cell

to normalize the local curvature and to calculate the bending stiffness matrix. The shear strains are obtained with

independent interpolation functions as in the MITC element.

There are several choices for the smoothing function. For constant smoothing functions, after transforming the

volume integral into a surface integral using Gauss’ theorem, the surface integration over each smoothing cell becomes

a line integration along its boundaries, and consequently, it is unnecessary to compute the gradient of the shape functions

to obtain the curvatures and the element bending stiffness matrix. In this paper, we use 1D Gauss integration scheme

on all cell edges. The flexibility of the proposed method allows constructing four-node elements even when the elements

are extremely distorted [38].

3A formulation for four-node plate element

3.1The curvature smoothing method (CSM)

The CSM was proposed by Chen et al. [17] and Wang et al. [52] as normalization of the local curvature in meshfree

methods. A curvature smoothing stabilization is created to compute the nodal curvature by a divergence estimation

via a spatial averaging of the curvature fields. In other words, the domain integrals are transformed into boundary

integrals. This curvature smoothing avoids the evaluation of the derivatives of the mesh-free shape functions at the

nodes2, where they vanish, and thus eliminates defective modes. A curvature smoothing at an arbitrary point is given

by

˜ κh(xC) =

?

Ωhκh(x)Φ(x − xC)dΩ(17)

where Φ is a smoothing function that has to satisfy the following properties [60]

Φ ≥ 0and

?

ΩhΦdΩ = 1(18)

For simplicity, Φ is assumed to be a step function defined by

Φ(x − xC) =

?

1/AC,x ∈ ΩC

0,x / ∈ ΩC

(19)

where ACis the area of the smoothing cell, ΩC⊂ Ωe⊂ Ωh, as shown in Figure 2.

Substituting Equation (19) into Equation (17), and applying the divergence theorem, we obtain

˜ κh

ij(xC) =

1

2AC

?

ΩC

?

∂θh

∂xj

i

+∂θh

∂xi

j

?

dΩ =

1

2AC

?

ΓC

(θh

inj+ θh

jni)dΓ(20)

2this applies only to meshfree methods that are based on a nodal integration

5

Page 6

Figure 2: Example of finite element meshes and smoothing cells

Next, we consider an arbitrary smoothing cell, ΩCillustrated in Figure 2 with boundary ΓC=

nb ?

b=1

Γb

C, where Γb

Cis the

boundary segment of ΩC, and nb is the total number of edges of each smoothing cell. The relationship between the

smoothed curvature field and the nodal displacement is written by

˜ κh=˜Bb

Cq (21)

The smoothed element bending stiffness matrix is obtained by

˜Kb=

?

Ωe(˜Bb

C)TDb˜Bb

CdΩ =

nc

?

C=1

(˜Bb

C)T(xC)Db˜Bb

C(xC)AC

(22)

where nc is the number of smoothing cells of the element, see Figure 3.

Here, the integrands are constant over each ΩCand the non-local curvature displacement matrix reads

˜Bb

Ci(xC) =

1

AC

?

ΓC

0

0

0

0Ninx

0

Niny

−Niny

−Ninx

dΓ(23)

We use Gauss quadrature to evaluate (23) with one integration point over each line segment Γb

C:

˜Bb

Ci(xC) =

1

AC

nb

?

b=1

0

0

0

0Ni(xG

b)nx

0

b)ny

−Ni(xG

−Ni(xG

b)ny

b)nx

Ni(xG

lC

b

(24)

where xG

band lC

bare the midpoint (Gauss point) and the length of ΓC

b, respectively.

6

Page 7

The smoothed curvatures lead to high flexibility such as arbitrary polygonal elements, and a slight reduction in

computational cost.The element is subdivided into nc non-overlapping sub-domains also called smoothing cells.

Figure 3 illustrates different smoothing cells for nc = 1, 2, 3 and 4 corresponding to 1-subcell, 2-subcell, 3-subcell and

4-subcell methods. The curvature is smoothed over each sub-cell. The values of the shape functions are indicated at

the corner nodes in Figure 3 in the format (N1,N2,N3,N4). The values of the shape functions at the integration nodes

are determined based on the linear interpolation of shape functions along boundaries of the element or the smoothing

cells.

Figure 3: Division of an element into smoothing cells (nc) and the value of the shape function along the boundaries of

cells: k-Subcell stands for the shape function of the MISCk element, k = 1, 2, 3, 4

Therefore the element stiffness matrix in (12) can be modified as follows:

˜K =˜Kb+ Ks=

nc

?

C=1

(˜Bb

C)TDb˜Bb

CAC+

?

Ωe(Bs)TDsBsdΩ(25)

It can be seen that a reduced integration on the shear term Ksis necessary to avoid shear locking. We will denote

these elements by SC1Q4, SC2Q4, SC3Q4 and SC4Q4 corresponding to subdivision into nc =1, 2, 3 and 4 smoothing

7

Page 8

cells, Figure 3. However, we will show that these elements fail the patch test and they exhibit an instability due to rank

deficiency. Therefore, we employ a mixed interpolation as in the MITC4 element and use independent interpolation

fields in the natural coordinate system [4] for the approximation of the shear strains:

?

γx

γy

?

= J−1

?

γξ

γη

?

(26)

where

γξ=1

2[(1 − η)γB

ξ+ (1 + η)γD

ξ], γη=1

2[(1 − ξ)γA

η+ (1 + ξ)γC

η] (27)

where J is the Jacobian matrix and the midside nodes A, B, C, D are shown in Figure 1. Presenting γB

based on the discretized fields uh, we obtain the shear matrix:

ξ,γD

ξand γA

η,γC

η

Bs

i= J−1

?

Ni,ξ

Ni,η

−b12

−b22

iNi,ξ

iNi,η

b11

iNi,ξ

b21

iNi,η

?

(28)

where

b11

i = ξixM

,ξ, b12

i = ξiyM

,ξ, b21

i = ηixL

,η, b22

i = ηiyL

,η

(29)

with ξi∈ {−1,1,1,−1},

shear term Ksis still computed by 2 ×2 Gauss quadrature while the element bending stiffness Kbin Equation (12) is

replaced by the smoothed curvature technique on each smoothing cell of the element.

ηi∈ {−1,−1,1,1} and (i,M,L) ∈ {(1,B,A);(2,B,C);(3,D,C);(4,D,A)}. Note that the

3.2Hu–Washizu variational formulation

We use a modified Hu–Washizu variational formulation [54] given for an individual element by

Πe

HW(w, ˜ κ,M) =

1

2

?

Ωe˜ κT: Db: ˜ κdΩ −

?

ΩeM : (˜ κ − κ)dΩ +1

2

?

ΩeγT: Ds: γdΩ −

?

Ωew p dΩ(30)

where M is the moment tensor. Partitioning the element into nc sub-cells such that the sub-cells are not overlapping

nc ?

Πe

HW(w, ˜ κ,M) =

and form a partition of the element Ωe, Ωe=

ic=1Ωe

ic, the functional energy, Πe

HW, can be rewritten as

1

2

nc

?

ic=1

?

Ωe

ic

˜ κic: Db: ˜ κicdΩe

ic−

nc

?

ic=1

?

Ωe

ic

M : (˜ κ − κ)dΩe

ic+1

2

?

Ωeγ : Ds: γdΩ −

?

Ωew pdΩ(31)

where

˜ κic=

1

Aic

?

Ωe

ic

κ(x)dΩe

icand Ae=

nc

?

ic=1

Aic

(32)

with Aicis the area of the smoothing cell, Ωe

To reduce Πe

HWfrom a three-field potential to a two-field potential, we need to find a strict condition on the

smoothing cells ΩCfor the orthogonality condition [48,49]:

ic.

?

ΩeM : ˆ κdΩ =

?

ΩeM : (˜ κ − κ)dΩ = 0(33)

is satisfied. By substituting M through the constitutive relation M = Db˜ κ, we rewrite the orthogonality condition:

8

Page 9

?

nc

?

ΩeM : ˆ κdΩ =

nc

?

1

AC

ic=1

?

Ωe

ic

Db˜ κ : (˜ κ − κ)dΩe

ic=

nc

?

nc

?

ic=1

?

Ωe

ic

Db˜ κ :

?

1

AC

?

ΩC

κdΩC− κ

?

dΩe

ic

=

ic=1

Db˜ κ :

?

Ωe

ic

?

?

ΩC

κdΩC− κ

?

?

dΩe

ic=

ic=1

Db˜ κ :

??

Ωe

icdΩe

AC

ic

?

ΩC

κdΩC−

?

Ωe

ic

κdΩe

ic

?

=

nc

?

ic=1

Db˜ κ :

Aic

AC

?

ΩC

κdΩC−

?

Ωe

ic

κdΩe

ic

?

(34)

where ΩC ⊂ Ωeis a smoothed curvature field defined for every ˜ κ =

does not depend on the integration after processing a smoothed operator, i.e,?

1

AC

?

ΩCκ(x)dΩ and the smoothed curvature ˜ κ

icDb˜ κdΩe

Ωe

ie= Db˜ κAicwith

AC= Aic and ΩC≡ Ωe

ic

(35)

If ΩCcoincides with Ωe

potential:

ic, the orthogonality condition (33) is met and the three-field potential is reduced to a two-field

Πe

HW(w, ˜ κ) =1

2

nc

?

ic=1

?

Ωe

ic

˜ κic: Db: ˜ κicdΩe

ic+1

2

?

Ωeγ : Ds: γdΩ −

?

Ωew pdΩ(36)

Now we show that the proposed total energy approaches the total potential energy variational principle (TPE)

when nc tends to infinity. Based on the definition of the double integral formula, when nc → ∞, Aic→ dAic – an

infinitesimal area containing point xic, applying the mean value theorem for the smoothed strain,

˜ κic=

?

Ωe

ic

κ(x)

Aic

dΩe

ic−→ κ(xic)(37)

where κ(x) is assumed to be a continuous function. Equation (37) states that the average value of κ(x) over a domain

Ωe

icapproaches its value at the converged point xic.

Taking the limit of Πe

HWwhen the number of subcells tends to infinity,

lim

nc→∞Πe

?

HW(w, ˜ κ) =

1

2

lim

nc→∞

nc

?

ic=1

?

Ωe

ic

Db:

??

Ωe

ic

κ(x)

Aic

dΩe

ic

:

??

Ωe

ic

κ(x)

Aic

dΩe

ic

?

dΩe

ic+1

2

?

Ωeγ : Ds: γdΩ −

?

Ωew pdΩ

=1

2

lim

nc→∞

nc

?

ic=1

Db: κ(xic) : κ(xic)dAic+1

2

?

Ωeγ : Ds: γdΩ −

?

Ωew pdΩ

=1

2

?

ΩeDb: κ(x) : κ(x)dΩ +1

2

?

Ωeγ : Ds: γdΩ −

?

Ωew pdΩ = Πe

TPE(w,β)(38)

The above proves that the total potential energy variational principle is recovered from the proposed variational

formulation as nc tends to infinity.

4Numerical results

We will test our new element for different numbers of smoothing cells and call our element MISCk (Mixed Interpolation

and Smoothed Curvatures) with k ∈ {1,2,3,4} smoothing cells for the bending terms. For instance, the MISC1 element

is the element with only one smoothing cell to integrate the bending part of the element stiffness matrix. We will

compare our results to the results obtained with the reduced/selective integrated quadrilateral element (Q4-R), the

MITC4 element and with several other elements in the literatures.

9

Page 10

4.1Patch test

The patch test was introduced by Bruce Irons and Bazeley [7] to check the convergence of finite elements. It is checked

if the element is able to reproduce a constant distribution of all quantities for arbitrary meshes. It is important that

one element is completely surrounded by neighboring elements in order to test if a rigid body motion is modelled

correctly, Figure 4. The boundary deflection is assumed to be w =1

shown in Table 1. While the MITC4 element and the MISCk elements pass the patch test, the Q4-R element and the

SC1Q4, SC2Q4, SC3Q4, SC4Q4 elements fail the patch test. Note that also the fully integrated Q4 element (on both

the bending and the shear terms) does not pass the patch test.

2(1 + x + 2y + x2+ xy + y2) [18]. The results are

Figure 4: Patch test of elements

Table 1: Patch test

Elementw5

θx5

θy5

mx5

my5

mxy5

Q4-R

SC1Q4

SC2Q4

SC3Q4

SC4Q4

MITC4

MISC1

MISC2

MISC3

MISC4

Exact

0.5440

0.5431

0.5439

0.5440

0.5439

0.5414

0.5414

0.5414

0.5414

0.5414

0.5414

1.0358

1.0568

1.0404

1.0396

1.0390

1.04

1.04

1.04

1.04

1.04

1.04

— no constant moments

-0.676

-0.7314

-0.6767

-0.6784

-0.6804

-0.55

-0.55

-0.55

-0.55

-0.55

-0.55

—

—

—

—

—

—

—

—

—

—

—

—

—

—

—

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.01111

-0.00333

-0.00333

-0.00333

-0.00333

-0.00333

-0.00333

10

Page 11

4.2A sensitivity test of mesh distortion

Consider a clamped square plate subjected to a center point F or uniform load p shown in Figure 5. The geometry

parameters and the Poisson’s ratio are: length L, thickness t, and ν = 0.3. Due to its symmetry, only a quarter (lower

– left) of the plate is modelled with a mesh of 8 × 8 elements. To study the effect of mesh distortion on the results,

interior nodes are moved by an irregularity factor s. The coordinates of interior nodes is perturbed as follows [38]:

x′= x + rcs∆x

y′= y + rcs∆y

(39)

where rcis a generated random number given values between -1.0 and 1.0, s ∈ [0,0.5] is used to control the shapes of

the distorted elements and ∆x,∆y are initial regular element sizes in the x–and y–directions, respectively.

For the concentrated center point load F, the influence of the mesh distortion on the center deflection is given in

Figure 6 for a thickness ratio of (t/L = 0.01 and 0.001). The results of our presented method are more accurate than

those of the Q4-R element and the MITC4 element, especially for extremely distorted meshes. Here, the MISC1 element

gives the best result. However, this element contains two zero-energy modes. In simple problems, these hourglass modes

can be automatically eliminated by the boundary conditions. However, this is not in general the case. Otherwise, the

MISC2, MISC3 and MISC4 elements retain a sufficient rank of the element stiffness matrix and give excellent results.

Let us consider a thin plate with (t/L = 0.001) under uniform load as shown Figure 5a. The numerical results of

the central deflections are shown in Table 2 and Figure 7 and compared to other elements. For the case s = 0, it

can be seen that the MISCk elements are the same or slightly more accurate results than the other

elements.Moreover, all proposed elements are better than the comparison elements especially for

distorted meshes, s > 0.

Table 2: The central deflection wc/(pL4/100D),D = Et3/12(1 − ν2) with mesh distortion for thin clamped plate

subjected to uniform load p

s-1.249-1.00-0.50.000.51.001.249

CRB1 [55]

CRB2 [55]

S1 [55]

S4R [1]

DKQ [31]

ANS-EC [30]

ANS-2ND [30]

Q4BL [61]

NCQ [40]

MITC4

MISC1

MISC2

MISC3

MISC4

Exact solu.

0.1381

0.2423

0.1105

0.1337

0.1694

—

—

—

—

0.0973

0.1187

0.1151

0.1126

0.1113

0.1265

0.1390

0.1935

0.1160

0.1369

0.1658

—

—

—

—

0.1032

0.1198

0.1164

0.1144

0.1130

0.1265

0.1247

0.1284

0.1209

0.1354

0.1543

—

—

—

—

0.1133

0.1241

0.1207

0.1189

0.1174

0.1265

0.1212

0.1212

0.1211

0.1295

0.1460

0.1303

0.1240

0.1113

0.1278

0.1211

0.1302

0.1266

0.1249

0.1233

0.1265

0.1347

0.1331

0.1165

0.1234

0.1418

—

—

—

—

0.1245

0.1361

0.1323

0.1305

0.1287

0.1265

0.1347

0.1647

0.1059

0.1192

0.1427

—

—

—

—

0.1189

0.1377

0.1331

0.1309

0.1288

0.1265

0.1249

0.1947

0.0975

0.1180

0.1398

—

—

—

—

0.1087

0.1347

0.1287

0.1260

0.1227

0.1265

4.3Square plate subjected to a uniform load or a point load

Figure 5a and Figure 13 are the model of a square plate with clamped and simply supported boundary conditions,

respectively, subjected to a uniform load p = 1 or a central load F = 16.3527. The material parameters are given

11

Page 12

(a)

(b)

(c)

(d)

Figure 5: Effect of mesh distortion for a clamped square plate: (a) clamped plate model; (b) s = 0.3; (c) s = 0.4; and

(d) s = 0.5

12

Page 13

00.10.20.3 0.40.5

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

Distorsion s

Normalized deflection wc

Exact solu.

Q4−R

MITC4

MISC1

MISC2

MISC3

MISC4

(a)

00.1 0.20.3 0.40.5

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Distorsion s

Normalized deflection wc

Exact solu.

Q4−R

MITC4

MISC1

MISC2

MISC3

MISC4

(b)

Figure 6: The normalized center deflection with influence of mesh distortion for a clamped square plate subjected to

a concentrated load: a) t/L=0.01, b) t/L=0.001

by Young’s modulus E = 1092000 and Poisson’s ratio ν = 0.3. Uniform meshes with N = 2,4,8,16,32 are used and

symmetry conditions are exploited.

For a clamped case, Figure 8 illustrates the convergence of the normalized deflection and the normalized moment

at the center versus the mesh density N for a relation t/L = 0.01. Even for very coarse meshes, the deflection tends to

the exact solution. For the finest mesh, the displacement slightly (.06%) exceeds the value of the exact solution. The

bending moment converges to the analytical value. The rate of convergence in the energy norm is presented in Figure 9

and is for all elements equal to 1.1 but the MISCk elements are more accurate than the MITC4 element in energy.

Tables 3–4 show the performance of the plate element for different thickness ratios, t/L = 10−1∼ 10−5. No shear

locking is observed. In addition, it is observed that the MISCk elements improve the solutions with coarse

meshes while for fine meshes all elements have almost marginal differences.

Next we consider a sequence of distorted meshes with 25, 81, 289 and 1089 nodes as shown in Figure 10. The

numerical results in terms of the error in the central displacement and the strain energy are illustrated in Figure 11.

All proposed elements give stable and accurate results. Especially for coarse meshes, the MISCk elements are more

accurate than the MITC4 element; a reason for this may be that for our finest meshes, fewer elements are distorted in

comparison to coarse meshes.

Now we will test the computing time for the clamped plate analyzed above. The program is compiled by a personal

computer with Pentium(R)4, CPU-3.2GHz and RAM-512MB. The computational cost to set up the global stiffness

matrix and to solve the algebraic equations is illustrated in Figure 12. The MISCk elements and the MITC4 element

give nearly the same CPU time for coarse meshes where the MISCk elements are more accurate. From the plots, we

can conjecture that for finer meshes, the MITC4 element is computationally more expensive than the MISCk element,

and the MISCk elements are generally more accurate. The lower computational cost comes from the fact that no

computation of the Jacobian matrix is necessary for the MISCk elements while the MITC4 element needs to determine

the Jacobian determinant, the inverse of the Jacobian matrix (transformation of two coordinates; global coordinate

and local coordinate) and then the stiffness matrix is calculated by 2x2 Gauss points. Previously, the same tendency

was observed for the standard (Q4 element), see [38] for details.

For a simply supported plate subjected to central concentrate load, the same tendencies as described above are

13

Page 14

−1.5−1−0.500.511.5

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

Distortion parameter

Central deflection wc/(pL4/100D)

CRB1

CRB2

S1

S4R

DKQ

MITC4

MISC1

MISC2

MISC3

MISC4

Exact

Figure 7: Comparison of other elements through the center deflection with mesh distortion

11.522.533.544.55

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

Mesh NxN

Normalized deflection wc

Exact solu.

MITC4

MISC1

MISC2

MISC3

MISC4

(a)

1 1.522.53 3.54 4.55

0.8

0.85

0.9

0.95

1

Mesh NxN

Normalized central moment

Exact solu.

MITC4

MISC1

MISC2

MISC3

MISC4

(b)

Figure 8: Normalized deflection and moment at center of clamped square plate subjected to uniform load

14

Page 15

0.70.80.911.1 1.21.31.41.51.6

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

log10(Number of nodes)1/2

log10(Error in energy norm)

MITC4

MISC1

MISC2

MISC3

MISC4

Figure 9: Rate of convergence in energy norm versus with number of nodes for clamped square plate subjected to

uniform load

15