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.

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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
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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,
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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
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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
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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
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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.
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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
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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:
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?
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.
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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