DSpace at VNU: A stabilized finite element method for certified solution with bounds in static and frequency analyses of piezoelectric structures

Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

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Comput. Methods Appl. Mech. Engrg.
journal homepage: www.elsevier.com/locate/cma

A stabilized finite element method for certified solution with bounds in static
and frequency analyses of piezoelectric structures
L. Chen a,⇑, Y.W. Zhang a, G.R. Liu b, H. Nguyen-Xuan c, Z.Q. Zhang d
a

Department of Engineering Mechanics, Institute of High Performance Computing, 1 Fusionopolis Way #16-16 Connexis, Singapore 138632, Singapore
School of Aerospace Systems, University of Cincinnati, Cincinnati, OH 45221-0070, USA
c
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University-HCM, Viet Nam
d
Singapore-MIT Alliance (SMA), E4-04-10, 4 Engineering Drive 3, Singapore 117576, Singapore
b

a r t i c l e

i n f o

Article history:
Received 14 January 2012
Received in revised form 21 May 2012
Accepted 22 May 2012
Available online 29 May 2012
Keywords:
Numerical methods

Piezoelectric structures
NS-FEM
Solution bound
Frequency
Stabilization

a b s t r a c t
This paper develops a stabilization procedure in piezoelectric media to ensure the temporal stability of
node-based smoothed finite element method (NS-FEM), and applies it to obtain certified solution with
bounds in both static and frequency analyses of piezoelectric structures using three-node triangular elements. For such stabilized NS-FEM, two stabilization terms corresponding to squared-residuals of two
equilibrium equations, i.e., mechanical stress equilibrium and electric displacement equilibrium, are
added into the smoothed potential energy functional of the original NS-FEM. A gradient smoothing operation is then performed on second-order derivatives of shape functions to achieve the stabilization terms.
Due to the use of divergence theory, the smoothing operation relaxes the requirement of shape functions,
so that the square-residuals can be evaluated using linear elements. The effectiveness of the present stabilized NS-FEM is demonstrated via numerical examples.
Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction
Piezoelectric materials showing an ability of transformation between mechanical energy and electric energy have been widely
used in various applications, where they serve as sensors, actuators, transducers or active damping devices. These applications
range from sub-millimeter length scales in micro-electro-mechanical systems up to large scales in the design of smart electromechanical structures. However, analytical solutions are limited for
solving practical problems of complicated geometry, for which
we have to resort to numerical methods when analyzing and
designing piezoelectric structures, such as the finite element method (FEM) [1,2], the bubble/incompatible displacement method [3],
the mixed and hybrid formulations [4–6] and the piezoelectric finite element with drilling degrees of freedom [7]. Several meshless
methods [8] have also been used to analyze piezoelectric structures such as the meshless point collocation method (PCM) [9],
the point interpolation method (PIM) [10], the radial point interpolation method (RPIM) [11], and the moving Kriging (MK) interpolation-based meshless method [12].
In practical engineering, the upper and lower bound analyses
[13] or the so-called dual analyses [14] have been an important

⇑ Corresponding author. Tel.: +65 64191246.

E-mail address: (L. Chen).
0045-7825/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved.
/>
mean for safety and reliability assessments of piezoelectric structural properties. In order to implement these analyses, two numerical models are usually used: one gives a lower bound and the
other gives an upper bound to the unknown exact solution. The
most popular models giving lower bounds to the exact strain energy and electric energy (or upper bounds to the exact natural frequencies) are the FEM models in which displacement and electric
potential both satisfy fully compatibility conditions, which are
widely used in solving complicated engineering problems. The
models that give upper bounds in energy norm to the exact solutions (or lower bounds to the exact natural frequencies) can be
one of the following models: (1) the stress equilibrium FEM models
[15]; (2) the recovery models using a statically admissible stress
field from displacement FEM solutions [16,17]; (3) the hybrid equilibrium FEM models [18]. These three models, however, are known
to have the following two common disadvantages: (1) the formulation and numerical implementation are complicated and expensive computationally; (2) there exist spurious modes in the hybrid
models or the spurious modes often occur due to the simple fact
that tractions cannot be equilibrated by the stress approximation
field. Due to those drawbacks, these three models are not widely
used in practical applications, and are still very much confined in
the area of academic research.
On another front of computational mechanics, a strain smoothing technique [19,20] was introduced by Chen et al. [19] for spatial
stabilization of nodal integrated meshfree methods, and later


66

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

extended by Yoo and Moran to the natural element method (NEM)
[20]. More recently, Liu [21] has generalized this gradient smoothing technique in order to weaken the consistence requirement for
the field functions, allowing the use of certain types of discontinuous displacement functions. Based on this generalization, a G space
theory and a generalized smoothed Galerkin (GS-Galerkin) weak

form have been developed [22], leading to the so-called weakened
weak ðW2 Þ foundations of a family of numerical methods. Among
them, a cell-based smoothed finite element method (SFEM or CSFEM) [23] was first formulated by introducing the gradient
smoothing technique to (compatible) FEM settings. In such SFEM,
the elements are divided into smoothing domains (SD) over which
the strain is smoothed. In addition, uniquely conceived, an edgebased smoothed finite element method (ES-FEM) constructs
smoothing domains based on the element edges [24]. It is found
that such unique technique gives the ES-FEM remarkable and
superior convergence properties, computational accuracy and efficiency, and spatial and temporal stability. These attractive properties have also led to the applications of ES-FEM to both static and
frequency analyses of piezoelectric structures [25,26]. However,
the ES-FEM usually produces a lower bound to the exact solution
in energy norm as the standard fully compatible FEM [27].
A node-based smoothed finite element (NS-FEM) [28] was also
formulated using smoothing domains associated with nodes in
FEM settings. The most important property of NS-FEM explored
and proven by Liu and coworkers [29,30] is that the NS-FEM is a
general method producing an upper bound solution in energy
norm to the exact solutions of the force-driven elasticity problems,
A simple explanation of the upper-bound property of NS-FEM is
the underestimation of the system stiffness (in a monotonic fashion [31,32]), in contrast to the well known overestimation of the
system stiffness for the displacement-based fully compatible finite
element model. The overestimation behavior of FEM results in the
upper bounds to the exact natural frequencies. In contrast, the
stiffness underestimation behavior of the NS-FEM models can lead
to lower-bound solutions in natural frequencies of free vibrating
solids and structures. However, similar to other nodal integrated
methods [31–33], the NS-FEM suffers from the temporal instability
due to its ‘‘overly soft’’ feature rooted at the use of a relatively
small number of SDs in relation to the nodes. The temporal instability is defined to have spurious non-zero eigen modes. Such models are spatially stable (with positive coercivity constants), and will
not have zero-energy modes. However, when they are excited at

(strictly non-zero) higher energy level, it can behave unphysically.
To eliminate these spurious modes, one possible method is to employ the Lagrangian kernels as Rabczuk et al. [34,35] proposed.
Also, Beissel and Belytschko [36] have developed a scheme to stabilize these nodal integrated methods by the addition to the potential energy functional a stabilization term, which contains the
square of the residual of the equilibrium equation. Further, the
latter has recently been applied to the NS-FEM by adding the stabilization term over the problem domain regulated by a stabilization parameter to the corresponding smoothed potential energy
functional [37]. However, both of these were only limited to
mechanical effects, and did not consider a coupling between
mechanical and electrical variables.
This paper further extends the stabilization technique in [36] to
the piezoelectric media to cure the temporal instability of NS-FEM,
by means of adding to the smoothed potential energy functional of
the original NSFEM two stabilization terms corresponding to
squared-residuals of two equilibrium equations, i.e., mechanical
stress equilibrium and electric displacement equilibrium. These
squared-residual terms can be regarded as an additional constraint
of the system, and be used to cure the ‘‘overly soft’’ behavior of
NS-FEM, for which the spurious non-zero energy modes can be
removed. In order to realize these two stabilization terms, the

gradient smoothing technique is extended to the second-order
derivatives, so that only the first-order derivatives of the shape
function are needed in our formulation. Therefore, the present
squared-residual stabilization technique works very well for linear
elements, such as 3-node triangular elements, and suits ideally in
many ways to the NS-FEM models. Further, the stabilized NSFEM is applied to obtain certified solution with bounds in both
static and frequency analyses of piezoelectric structures. Intensive
benchmark numerical examples are presented to demonstrate the
interesting properties of the proposed method. It is found that
upper bound in energy norm to the exact solutions of static piezoelectric problems and lower bound natural frequencies in vibration
analyses can be achieved using a proper stabilization parameter.

2. Basic piezoelectric formulations
2.1. Governing equations
Consider a 2D piezoelectric solid governed by the equilibrium
equation in the domain X 2 R2 bounded by CðC ¼ Cu þ Ct ;
Cu \ Ct ¼ 0Þ as

'

div r þ b ¼ 0

on X;

div D þ qs ¼ 0

ð1Þ

where r is the Cauchy stress tensor, b represents the vector of body
force applied in the problem domain, D denotes the electric displacement and qs is the free point charge density.
For dynamics problems of linear electroelastic solids, the strong
form of the governing equation is

€ þ gu_ on X;
div r þ b ¼ qu

ð2Þ

where q is the density of the mass, and g is a set of viscosity
parameters.
The strain e and the electric field E are, respectively, derived
from the displacement u and the electric potential u, and could

be written by the vector form

e ¼ rs u;

ð3Þ

E ¼ Àgrad u;
where rs is the symmetric gradient operator,

"

rs ¼

@
@x

0

@
@y

0

@
@y

@
@x

#T

ð4Þ

:

Writing the stress tensor r as the vector form, the constitutive
equations have the following form

r
D

!
¼

cE

ÀeT

e

jS

!

e
E

!
;

ð5Þ

where cE denotes the elastic matrix measured at constant electric
field, jS is the dielectric matrix at constant mechanical strain, and
e is the piezoelectric matrix. These tensors are known experimentally for various kinds of piezoelectric materials. They are usually
not isotropic. To be specific, Eq. (5) can also be written in a component form for the 2D plane piezoelectric problem

rxx 3 2 c11 c12 c13 Àe11 Àe21 32 exx 3
7 6
76
7
6
6 ryy 7 6 c21 c22 c23 Àe12 Àe22 76 eyy 7
7 6
76
7
6
7 6
76
7
6
6 rxy 7 ¼ 6 c31 c32 c33 Àe13 Àe23 76 cxy 7:
7 6
76
7
6
7
6 D 7 6e
6
j12 7
54 Ex 5

4 x 5 4 11 e12 e13 j11
Dy
Ey
e21 e22 e23 j21
j22
2

ð6Þ

On the other hand, Eq. (7) in the following can be recast into a
matrix form in contrast to Eq. (5) as


67

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

e
E

"

!
¼

sE

T

Àd

#

!
ð7Þ

D

eS

d

r

Taking Hamilton’s variational principle yields to

dpðuÞ ¼ 0 ¼

À1
in which use is made of the relationships d ¼ ecÀ1
E ; sE ¼ ecE , and
T
eS ¼ jS þ ecÀ1
e
,
where,
e
is
the
dielectric

matrix
measured
at conS
E
stant stress, d stands for the piezoelectric strain matrix and sE is the
elastic compliance matrix.
At part Cu , the essential boundary condition is given by

u ¼ uC

on Cu ;

u ¼ uC

ð8Þ

where uC is the vector of the prescribed displacement, and uC denotes the prescribed electrical potential, whereas at the part Ct ,
the natural boundary condition is given by

r Á n ¼ tC

on Ct ;

D Á n ¼ qC

ð9Þ

where tC is the vector of prescribed tractions, qC denotes the surface change on Ct , and nj is the surface outward normal of the
boundary Ct .
2.2. Galerkin weak form and finite element formulation


Z 
Z
1 T
1
qu_ u_ þ eT ðuÞrðuÞ À uT b dX À uT tC dC
2
X 2
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
ð10Þ

electric part

Consider the domain discretized into N e of non-overlapping and
e
non-gap elements and N n nodes, such that X ¼ [Nm¼1
Xem and
e
e
Xi \ Xj ¼ 0; 8i – j, the approximation of displacement field uh
and electric potential for a 2D electroelastic problem is given by:

uðxÞ ¼

X

Ni ðxÞui ;

i2nen

X
uðxÞ ¼ Ni ðxÞui ;

ð11Þ

i2nen

where nen is the set of nodes of the element containing
T
x; ui ¼ ½ uxi uyi Š is the vector of nodal displacements, respectively,
in x axis and y axis, ui is the nodal electric potential, and Ni is a matrix of shape functions

Ni ðxÞ ¼

Ni ðxÞ
0
0
Ni ðxÞ

T

ð16Þ

electric part

and then substituting Eqs. (11)–(13) into Eq. (10), we have a set of
piezoelectric dynamic equations

!&

m 0
0

0

€
u
€
u

"

'
þ

#
kuu & u '

kuu
T

kuu

kuu

u

¼

& '
f
;
q

ð17Þ

where

m¼

Z

qNT NdX;

ð18Þ

ðBu ÞT cE Bu dX;

ð19Þ

ðBu ÞT eT Bu dX;

ð20Þ

X

kuu ¼

Z

X

Z

kuu ¼ À
f¼

Z
X

q¼À

Z

ðBu ÞT jS Bu dX;

X

NT ðxÞbdX þ

Z
Ct

Z

NT ðxÞqs dX À

ð21Þ

NT ðxÞtC dC;
Z

NT ðxÞqC dC:

ð22Þ
ð23Þ

Ct

X

3. NS-FEM for the piezoelectricity

mechanical part


Z 
Z
1 T
D ðuÞEðuÞ À uqs dX þ
À
uqC dC
X 2
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}

mechanical part

Z

À
dD ðuÞEðuÞ À duqs dX þ
duqC dC
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}

X

pðuÞ ¼

Z
À T
Á
€ þ deT ðuÞrðuÞ À duT b dX À
du qu
duT tC dC
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Z 

kuu ¼

In this section, a finite element formulation for piezoelectricity
is established via a variational formulation, in which the following
general energy functional p is used to express a summation of the
following two parts: (1) mechanical contribution including kinetic
energy, strain energy, and mechanical external work, and (2) electrical contribution involving dielectric energy and electric external
work [38]

Z

Detailed formulations of the NS-FEM have been proposed in the
previous work [28]. Here, we mainly focus on the extension of the
NS-FEM to the piezoelectric problem using a basic mesh for 3-node
linear triangular elements.
3.1. Gradient smoothing
The strain smoothing method was proposed by Chen et al. in
[19], and later generalized by Liu to form the basis of G space theory [19,20]. Consider the 2D domain X discretized into N s nonoverlapping smoothing domains as shown in Fig. 1, the smoothing
operation on the gradient of a field / for a point xk in a smoothing
domain Xsk is given as follows

 kÞ ¼
/ðx

Z

Xsk

/ðxÞWðx; x À xk ÞdX;

ð24Þ

!
ð12Þ

in which Ni ðxÞ is the shape function for node i. Substituting the
approximations of Eq. (11) into Eq. (3), we obtain

e ¼ rs u ¼

X u
B i ui ;

ð13Þ

i2nen

X
E ¼ Àgrad u ¼ À Bu
i ui ;

ð14Þ

i2nen

where

2

Ni;x
6

u
Bi ¼ 4 0
N i;y

0

3

7
Ni;y 5;
Ni;x

Bu
i ¼

Ni;x
Ni;y

!
ð15Þ
Fig. 1. Division of problem domain X into non-overlapping smoothing domains Xsk
for xk . The smoothing domain is also used as basis for integration.


68

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

ically for problems with complicated geometry, which is also
employed as the mesh platform in this work.

Consider the domain X discretized into N e non-overlapping and
non-gap triangular elements and N n nodes, the local smoothing domains in the NS-FEM are constructed with respect to the nodes of
n
triangular elements, such that X ¼ [Nk¼1
Xsk and Xsi \ Xsj ¼ 0; 8i – j,
in which N n is the total number of nodes in the element mesh. In
this case, the number of smoothing domains are the same as the
number of nodes:N s ¼ N n . For the triangular elements, the smoothing domain Xsk for node k is created by connecting sequentially the
mid-edge-points and the centroids of the surrounding triangles of
the node as shown in Fig. 2.
Fig. 2. Construction of node-based strain smoothing domains based on 3-node
triangular elements.

3.3. Smoothed Galerkin weak form and discrete equations

where Wðx; x À xk Þ is a smoothing function that generally satisfies
the following properties [19]:

Because a smoothed Galerkin weak form with smoothed gradient over smoothing domains is variationally consistent as proven
in [22], using this smoothed or weakened weak form with displacement field and electrical potential satisfying the essential
boundary conditions, we have

Wðx; x À xk Þ P 0;

and

Z
Xsk

Wðx; x À xk ÞdX ¼ 1:

ð25Þ

The Heaviside-type piecewise constant function is employed in this
research:

(
Wðx; x À xk Þ ¼

1=Ask

x 2 Xsk ;

0

x R Xsk ;

Z
Xsk

1
Ask

/;j ðxÞWðx; x À xk ÞdX ¼

Z

/ðxÞnj dC;

Csk

ð27Þ

ek ðxk Þ ¼
¼

Xsk

1
Ask

ek ðxÞWðx; x À xk ÞdX ¼
Z
Xsk

Z
Xsk

ð28Þ

where Ln is the matrix of unit outward normal which can be expressed as

2

nx

0

6
Ln ¼ 4 0

ny

3

7
ny 5:

ð29Þ

Xsk

EðxÞWðx; x À xk ÞdX

Z

1
¼
Àu;j ðxÞWðx; x À xk ÞdX ¼ À s
Ak
Xsk

ð31Þ

electric part

Employing the strain smoothing operation, the smoothed strain
ek over Xsk from the displacement approximation in Eq. (3) can be
written in the following matrix form

ek ¼

X
Bui ðxk Þui :

ð32Þ

i2nsk

Likewise, the matrix form of the smoothed electric field can be
expressed by

E¼

X

Bu
i ðxk Þui ;

ð33Þ

where nsk is the set of nodes associated with the smoothing domain
Xsk . Bui ðxk Þ is the smoothed strain matrix for the mechanical part, and
Bu
i ðxk Þ corresponds to the electric part, i.e., the smoothed electric
field matrix. Those two matrix operations can be written as follows

2
3
bix ðxk Þ
0

6
iy ðx Þ 7
Bui ðxk Þ ¼ 4 0
b
k 5;


biy ðxk Þ bix ðxk Þ

 ðx Þ ¼ 1
b
ih
k
Ask

Similarly, the smoothed electric field can be expressed by

Eðxk Þ ¼

T

"
Bu
i ðxk Þ ¼

#
 ðx Þ
b
ix
k

 ðx Þ ;
b
iy

ð34Þ

k

 ðx Þ; h ¼ x; y, is computed by
where b
ih
k

nx

Z

mechanical part

À

i2nsk

rs uWðx; x À xk ÞdX

Ld uðxÞdX;

À

Z

Á
À
dD ðuÞEðuÞ À duqs dX þ
duqC dC:
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð26Þ

where Csk is the segments of boundary of the smoothing domain Xsk .
 i;j in Eq. (24) to the followSubstituting the smoothed gradient /
ing smoothing operation of strain vector e in Eq. (3) yields the
smoothed strain as follows

Z

Z
Á
€ þ deT ðuÞr
 ðuÞ À duT b dX À
duT qu
duT tC dC
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Z

where Ask is the area of the smoothing domain Xsk .

Assuming /;j exists (the assumed field / is continuous), and
introducing divergence theorem to Eq. (24), we shall have

 ;j ðxk Þ ¼
/

Z

dps ðuÞ ¼ 0 ¼

Z
Csk

Z
Csk

nh ðxÞNi ðxÞdC:

ð35Þ

Using the Gaussian integration along the segments of boundary

uðxÞnj dC:

ð30Þ

Csk , we have:
"N
#
Nseg X

gau
1X

wm;n Ni ðxm;n Þnh ðxm;n Þ
ðh ¼ x; yÞ;
bih ¼ s
Ak m¼1 n¼1

ð36Þ

3.2. Construction of smoothing domains
Smoothing domain of the NS-FEM is constructed based on the
nodes of elements, as illustrated in Fig. 2, and the elements used
can be 3-node triangular element, 4-node quadrilateral element,
and n-side polygonal element. The only requirement of the
smoothing domain is non-overlap, and not required to be convex.
In order to simplify meshing, the NS-FEM generally relies on the
3-node triangular elements that can be usually generated automat-

where Nseg is the number of segments of the boundary Csk ; Ngau is
the number of Gaussian points used in each segment, wm;n is the
corresponding weight of Gaussian points, nh is the outward unit
normal corresponding to each segment on the smoothing domain
boundary and xm;n is the nth Gaussian point on the mth segment
of the boundary Csk .
Substituting the approximated displacements and electric
potential in Eq. (3), and the smoothed strains and electric field,


L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

respectively, from Eqs. (32) and (33) into the smoothed Galerkin
weak form leads to the following equation

Z
ðBu ÞT cE Bu udX þ duT ðBu ÞT eT Bu udX
X
X
X
Z
Z
u T
u
u T
u
þ du ðB Þ eB udX À du ðB Þ js B udX
X
ZX
Z
Z
À duT
NT bdX À duT
NT tC dC þ du NT qs dX
X
Ct
X
Z
T
þ du
N qC dC ¼ 0:

ð37Þ

duT

Z

qNT Nu€ dX þ duT

Z

Ct

Eliminating du and du yields the following two discrete equilibrium equations

m 0
0

!&

€
u
€
u

0

"

'
þ

 uu
k

kT

uu

#
 uu & u ' & f '
k
¼
;
 uu
u
q
k

ð38Þ

where f and q are computed similarly by Eqs. (22) and (23), respectively, and the mass matrix m adopts a consistent mass matrix, thus
can be calculated in the same way as Eq. (18). The stiffness matrix is
then assembled by

 uu ¼
k

Ns Z
X

Xsk

k¼1

 ¼
k
uu

ðBu ÞT cE Bu dX;

Ns Z
X

u T T

Xsk

k¼1

u

ðB Þ e B dX;

Ns Z
X
 uu ¼ À
k
ðBu ÞT jS Bu dX:

Xsk

k¼1

gard, the gradient smoothing based on divergence theorem has been
proposed to eliminate the spatial instability in the nodal integrated
methods, such as EFG, NEM and NS-FEM. This technique produces
the smoothed derivatives of shape functions using only the shape
function values and does not need to calculate the derivatives of
shape functions. To be specific, the NS-FEM has been proven spatially stable [29]. On the other hand, the ‘‘overly-soft’’ property of
NS-FEM leads to spurious non-zero eigen modes, that is, temporal
instability. This kind of instability does not influence the calculation
of the static problems, but, it affects the time-dependent analyses
(e.g., dynamics problems, transient analyses, and so on).
One approach to cure this temporal instability in the NS-FEM is
to use a scheme of Beissel and Belytschko [36], in which a modified
potential energy functional is constructed by adding a smoothed
squared-residual stabilization term into the smoothed potential
energy functional [37]. In this regard, we extend the stabilization
technique in [36] to the piezoelectric media, by means of adding
two stabilization terms corresponding to squared-residuals of
two equilibrium equations into the smoothed potential energy
functional of the original NSFEM

ps ðuÞ ¼
ð39Þ

 uu ¼
k
 uu ¼
k

ðBu ÞT cE Bu Ask ;

k¼1
Ns
X

ðBu ÞT eT Bu Ask ;

k¼1
Ns
X

 ¼À
k
uu

ðBu ÞT jS Bu Ask :

smoothed potential functionalÀelectric part

Z
al2
þ
ðdiv r þ bÞT ðdiv r þ bÞdX À c ðdiv D þ qs Þ2 dX ;
j X ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
E X
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl

ð41Þ

ð42Þ

al2c

ð45Þ

Eliminating the u yields the following equation

where a is the dimensionless, real, finite and non-negative stabilization parameter; lc is the characteristic length of the elements in the
mesh that is determined by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
areaðXÞ
;
2
Ne

ð49Þ

where areaðXÞ is the area of the problem domain, and N e is the
number of elements. E is the effective Young’s modulus of the material, and the average diagonal value in the elastic matrix is employed for anisotropic material as E ¼ ðc11 þ c22 þ c33 Þ=3. Likewise,
j is the effective dielectric coefficient of the material as
j ¼ ðj11 þ j22 Þ=2.
It is clear from Eq. (48) that the stabilization terms consist of the
squared-residuals of mechanical stress equilibrium and electric
displacement equilibrium in Eq. (1). It is constructed by
considering.

ð46Þ

 uu . Therewhere G denotes the Moore–Penrose pseudoinverse of k
fore the natural frequency x and mode k can be computed by
solving the following eigenvalue problem


h
i
 uu À k
 uu Gk
T
Àx2 ½mŠ þ k
fkg ¼ 0:
uu

ð48Þ

lc ¼
ð44Þ

Z

smoothed squared residualÀmechanical part smoothed squared residualÀelectric part

ð43Þ

In this work, modal analysis of the system is analyzed for
dynamics problems of linear electroelastic solids. Hence, Eq. (38)
reduces to the following equation without damping and forcing

terms.

 uu À k
 uu Gk
 T Šfug ¼ 0;
€ g þ ½k
½mŠfu
uu

smoothed potential functionalÀmechanical part


Z
1 T
À
D ðuÞEðuÞ À uqs dX þ
uqC dC
X 2
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}

ð40Þ

k¼1

 uu Šfug þ ½k
 uu Šfug ¼ 0;
€ g þ ½k
½mŠfu

T


½kuu Šfug þ ½kuu Šfug ¼ 0:


Z 
Z
1 T
1
qu_ u_ þ eT ðuÞr ðuÞ À uT b dX À uT tC dC
2
X 2
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}
Z 

All entries in matrix B in Eq. (39) are constants over each
smoothing domain, the stiffness matrix in Eq. (39) can therefore
be rewritten as
Ns
X

69

ð47Þ

4. Stabilization of NS-FEM
4.1. Governing equations and variational principle

With regard to the nodal integrated methods, direct nodal integration leads to a numerically spatial instability in meshfree settings
(spurious zero-energy modes exist) due to vanishing derivatives of
shape functions at the nodes during integration [29,32]. In this re-

(i) while a ! 0, the functional in Eq. (48) converges to the original smoothed potential energy functional;
(ii) while lc ! 0, the functional in Eq. (48) also converges to
the original smoothed potential energy functional, for
any finite a;
(iii) for a finite model (lc is finite positive constant), the strongform equilibrium system equation is better enforced by
using a larger a, and the weakened weak form [22] is better
enforced using a smaller a. Therefore, adjusting the stabilization parameter a suppresses the ‘‘overly soft’’ effect of
original NS-FEM models, thereby achieving a desired
stability.
In this work, we prefer to use a possible a to obtain desired
number of smallest eigen-modes for a given 2D solids, so that we


70

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

can obtain the upper bounds in energy norm to the exact solution
(or lower bounds to the exact natural frequency). Because of the
known fact that a fully compatible FEM model can give lower
bounds to the exact strain energy and electric energy (or upper
bounds to the exact natural frequency), the use of our stabilized
NS-FEM and FEM can bound the solutions from both sides with
complicated geometry as long as a triangular element mesh can
be generated.
Taking variation and applying stationary condition to Eq. (48)

yields

dps ðuÞ ¼ 0 ¼

Z

Z
Á
 ðuÞ À duT b dX À
du_ T qu_ þ deT ðuÞr
duT tC dC
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
À

À

Á

Z

X1 Z
Ni;m ðxÞnn dCui :
As s
i2ns k Ck

Z
Csk

u;m ðxÞnn dC
ð52Þ

From Eqs. (51) and (52), it is clear that smoothing operation relaxes
the requirement of field function. Consequently, the smoothed second-order derivative only requires C 0 continuity, so that the squareresiduals can be evaluated using linear elements.
Next, the smoothing operation is applied to the divergence of
the Cauchy stress tensor r. In this regard, one can arrive at the
smoothed divergence of the Cauchy stress tensor that can be expressed in the following vector form:

&

'

r xx;x þ r xy;y
:
r yy;y þ r yx;x

ð53Þ

Substituting the constitutive relation of Eq. (6), on defining Csm

smoothed potential functionalÀelectric part

as

2 Z
2 Z
2alc
2alc
þ

ðdiv drÞT ðdiv rÞdX þ
ðdiv drÞT bdX
E
E
X
X
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Csm ¼

smoothed square residualÀmechanical part

2 Z
2 Z
2alc
2alc
À
ðdiv dDÞðdiv DÞdX À
ðdiv dDÞqs dX:
j X
j X
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl
ffl}

1
Ask

u;mn ðxÞWðx; x À xk ÞdX ¼

k

div r ¼

À
dDT ðuÞEðuÞ À duqs dX þ
duqC dC
X
Ct
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Xsk

¼

smoothed potential functionalÀmechanical part

Z

Z

 ;mn ðxk Þ ¼
u

c11

c12

c13

c31

c32

c33

e11

e21

e13

e23

c31

c32

c33

c21

c22

c23

e13

e23

e12

e22

!
;

ð54Þ

we have

ð50Þ
div r ¼

&

smoothed square residualÀelectric part

r xx;x þ r yx;x
r yy;y þ r xy;y

'

¼ Csm Ks ;

ð55Þ

where Ks is defined as

Â

ÃT
xy;x exx;y eyy;y c
xy;y ÀEx;x ÀEy;x ÀEx;y ÀEy;y :
Ks ¼ exx;x eyy;x c

4.2. Discretization
In the proposed variational principle in Eq. (50), the discretization of smoothed total potential functional manifested by the first
two terms can directly use the procedure in Section 3.3. Therefore,
we now construct specifically the discretization on the last two
stabilization terms of Eq. (50), which consist of the square-residuals of mechanical stress equilibrium and electric displacement
equilibrium.
4.2.1. Square-residual of mechanical stress equilibrium
Obviously, the stabilization term regarding to the square-residual of mechanical stress equilibrium includes the second-derivatives of the displacements, whereas the assumed displacement
fields used in this work do not have the second-order derivatives
over the whole problem domain. Further, the use of T3 element
leads to zero second order derivatives of shape function, i.e.,
N ;mn ðxÞ ¼ 0 ðm; n ¼ x; yÞ, hence, the stabilization term will have
no contribution for the stabilization if N ;mn ðxÞ is calculated directly
using the FEM shape functions. In order to realize the stabilization
term, the present work performs the gradient smoothing technique
on the second-order derivatives u;mn (m; n ¼ x; yÞ of the displacement fields (the assumed displacement fields u and its gradient
u;m are continuous). Letting /ðxÞ in Eq. (27) be u;m , i.e.,
/ðxÞ ¼ u;m , we have the smoothed the second-order derivatives
of displacements in a smoothing domain Xsk

 ;mn ðxk Þ ¼
u

¼

Z
Xsk

u;mn ðxÞWðx; x À xk ÞdX ¼
Z

1
Ask

Csk

0
@

1

1
Ask

Z
Csk

Employing the strain–displacement and electric field-potential
relationships in Eq. (5), then
 y;xx u
 y;xy u
 y;yx u
 x;yx þ u
 x;xy u
 y;yy u

 x;yy þ u
 x;xx u
 ;xx u
 ;yx u
 ;xy u
 ;yy ŠT :
Ks ¼ ½ u
ð57Þ
s

Substituting Eqs. (51) and (52) into K leads to the smoothed the
smoothed divergence of the Cauchy stress tensor over the smoothing domain Xsk

div r ¼

&

r xx;x þ r yx;x
r yy;y þ r xy;y

'

¼ Csm Ks ¼ Csm

X
Bsi di ;

ð58Þ

i2nsk

where nsk is the set of nodes associated with the smoothing domain
Xsk ; di ¼ ½ ui ui ŠT , and Bsi is expressed by

"
Bsi

¼

Ni;xx

0

Ni;yx

Ni;xy

0

Ni;yy

0

Ni;yx

Ni;xx

0

Ni;yy

Ni;xy

#T
ð59Þ

;

where

Ni;mn ¼

1
Ask

Z
Csk

Ni;m ðxÞnn dC;

m; n ¼ x; y

ð60Þ

4.2.2. Square-residual of electric displacement equilibrium
Performing the similar smoothing operation as before, the
smoothed divergence of electric displacements can be expressed
in the form

u;m ðxÞnn dC

X
Ni;m ðxÞui Ann dC

(

div D ¼

i2nsk

X1 Z
¼
Ni;m ðxÞnn dCui :
As s
i2ns k Ck

ð56Þ

ð51Þ

k

In the same way, the smoothed second-order derivatives of
 ;ij over the smoothing domain Xsk can be exelectric potential, u
pressed by:

Dx;x
Dy;y

)

¼ Cse Ks ¼ Cse

X
Bsi di ;

ð61Þ

i2nsk

where Cse is defined by

Cse ¼ ½ e11

e12

e13

e21

e22

e23

Àj11

Àj12

Àj21

Àj22 Š:
ð62Þ

Also, it is worth noting that

Bsi

is the same as that given in Eq. (59).


71

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Substituting these equations to Eq. (50) leads to the following
equation

duT

Z

qNT Nu€ dX þ duT

Z

X

Â

Z

ðBu ÞT cE Bu udX þ duT

X

ðBu ÞT eT Bu udX þ du

X

À du

Z

ðBu ÞT eBu udX

X

Z

ðBu ÞT js Bu udX À duT

Z

X

À duT

NT bdX

X

Z

NT tC dC þ du

Ct

Z

NT q s d X þ d u

Z
Ct

X

2

NT q C d C þ

E

T

dd

Z

2
À ÁT

À ÁT
2alc T
ðBs ÞT Csm Csm Bs d þ ðBs ÞT Csm bdX À
dd
j
X
Z
À ÁT
À ÁT
 ðBs ÞT Cse Cse Bs d þ ðBs ÞT Cse qs dX ¼ 0:

Â

2alc

ð63Þ

X

!&

€
u
€
u

"

'
þ

 uu
k

kT

0 0
uu
& '
f
¼
þ ff m g þ ff e g
q

 uu
k

kuu

#

Âs à Âs Ã
þ k
m þ ke

!&

u

u

'
ð64Þ

s ; k
s ; fm
k
m
e

in which
and f e are the newly introduced matrices in the
discretized algebraic equations of system that are then assembled by
2X
Ns Z
 s ¼ 2alc
k
ðBs ÞT ðCsm ÞT Csm Bs dX
m
E k¼1 Xsk
2 Ns
À ÁT
2alc X
ðBs ÞT Csm Csm Bs Ask ;
E k¼1
2X
Ns Z
À ÁT
 s ¼ À 2alc
ðBs ÞT Cse Cse Bs dX

k
e
s
j
X

¼

k¼1

ð65Þ

k

2 Ns
lc X

2a
¼À
j

À ÁT
ðBs ÞT Cse Cse Bs Ask ;

ð66Þ

k¼1

2 Ns Z
À ÁT

2alc X
ðBs ÞT Csm bdX;
fm ¼ À
E k¼1 Xsk
2 Ns Z
À ÁT
2alc X
ðBs ÞT Cse qs dX:
fe ¼
s
j
X
k¼1

(7)

6. Numerical examples

Eliminating du and du yields

m 0

(4)
(5)
(6)

e. compute the smoothed matrix for the divergence of
stress Bs ðxk Þ by Eq. (59), and obtain the stabilized stiff s and k
 s from the square-residuals,
ness matrices k

m
e
respectively, of mechanical stress equilibrium and electric displacement equilibrium, using Eqs. (65) and (66);
f. evaluate the contribution of load vector over the current
smoothing domain;
g. assemble the contribution of the current smoothing
domain to form the global system stiffness matrix and
load vector;
calculate the consistent mass matrix m;
implement essential boundary conditions;
solve the linear system of equations to obtain the nodal displacements and electric potentials (static analysis); and
eigenmodes and frequencies (eigenvalue problems);
post-processing of desired results.

Benchmark problems are examined to demonstrate the validity
of the proposed stabilization scheme within the framework of NSFEM for the piezoelectricity. The strain energy used in this research
is defined as

EðXÞ ¼

1
2

Z

eT DedX:

ð69Þ

X

Numerical errors are then calculated by the following equations

Relative displacement error in L2 norm : eu
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Á2
uPNn À exact
u
u
À unumerical
i
;
¼ t i¼1 Pi N À
Á
n
exact 2
i¼1 ui
Relative electric potential error in L2 norm : eu
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
uPNn À exact
Á2
u
u À unumerical
i
¼ t i¼1 PiN À
Á
n
exact 2
i¼1

ui

ð70Þ

ð71Þ

ð67Þ

where the superscript exact denotes the exact solution (if the exact
solution does not exists, exact denotes the reference solutions), and
numerical denotes the numerical solution obtained using a numerical method.

ð68Þ

6.1. Patch test

k

5. Numerical implementation
The numerical procedure for the stabilized NS-FEM is outlined
as follows:
(1) divide the problem domain into a set of elements and obtain
information on node coordinates and element connectivity;
(2) create the smoothing domains using the rule given in
Section 3.2;
(3) loop over smoothing domains
a. determine the node connecting information of the
smoothing domain Xsk associated with node k;
b. calculate the outward unit normal for each boundary
segment of the smoothing domains Xsk ;

c. compute the smoothed strain matrix Bu ðxk Þ and the
smoothed electric field matrix Bu ðxk Þ by using Eq. (34);
 uu for the
d. evaluate the smoothed stiffness matrix k

 uu for
mechanical field, kuu for the electric field, and k
the mechanical-electric coupling field over the current
smoothing domain by using Eqs. (39)–(41);

A standard patch test is first considered, whose nodal distribution and geometry are presented in Fig. 3. The piezoelectric material PZT-4 as listed in Table 1 is employed in this patch test. The
boundary conditions for the mechanical displacements and the
electric potential are assumed to be [5]

ux ¼ s11 r0 x;

uy ¼ s13 r0 y;

u ¼ g 31 r0 y;

Fig. 3. Nodal setting and geometry of piezoelectric patch test.

ð72Þ


72

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81
Table 1
Piezoelectric material properties of PZT-4 and PVDF.

PZT-4 [6] q ¼ 7500 kg=m3

2

PVDF [9] q ¼ 1800 kg=m3

3

3
2:18 6:33
0
7
6
cE ¼ 4 6:33 2:18
0 5 GPa
0
0
0:775
2

139 74:3
0
7
6
cE ¼ 4 74:3 113
0 5 GPa
0
0
25:6
e¼

jS ¼

!
0
0
13:44
Coulomb=m2
À6:98 13:84
0
!
6:00
0
 10À9 F=m
0
5:47

u
v
/
T xx
T yy
T xy
Dx
Dy

!
0
0
0

Coulomb=m2
0:046 0:046 0

jS ¼

!
0:1062
0
 10À9 F=m
0
0:1062

that the desired results gained by the present method with
stabilization parameter a ¼ 0:05 match the exact solutions (other
parameters are found to match as well, however, the corresponding results are not listed here due to the length limit), and hence
the method successfully passes the patch test.

Table 2
The results of patch test.
Variable

e¼

Results
Exact

sNS-FEM ða ¼ 0:05Þ

2.376547556249814eÀ006
À1.818789953339896eÀ007

À1.066703050081747eÀ009
1.0
0
0
0
0

2.376547526243648eÀ006
À1.818789951339893eÀ007
À1.066712350081748eÀ009
1.0
À6.753880495917718eÀ015
À3.8934368989602721eÀ016
À8.139613253960852eÀ014
4.599293089164516eÀ013

6.2. Single-layer piezoelectric strip
In order to examine the accuracy of the present method, a
piezoelectric strip of L  2h ¼ 1 mm  1 mm undergoing a shear
deformation condition as depicted in Fig. 4 is considered. The piezoelectric material is polarized along the thickness, i.e., along the y
direction, and is assumed to be transversely isotropic. The strip is
subjected to a uniform stress r0 ¼ 1:0 Pa in the y direction on its
top and bottom boundaries and an applied voltage V 0 ¼ 1000 V
to the left and right boundaries as shown in Fig. 4. The piezoelectric
material PZT-5 is taken and its related parameters are provided as
follows.

2

16:4

6
sE ¼ 4 À7:22

À7:22

0

3

7
0 5 Â 10À6 mm2 =N;
0
0
47:5
!
0
0
584
d¼
 10À9 N=V;
À172 374
0
!
1:53105
0
eS ¼
 10À8 N=V2 :
0
15:05

18:8

ð74Þ

Due to the acting compressive stress together with an applied
electric field perpendicular to the direction of the polarization, a
shear strain is consequently generated in the y direction and expanded slightly in the x direction because of the Poisson effect.
The overall deformation is a superposition of the deformation
due to the shear strain and the compressive loading [7]. The
mechanical and electrical boundary conditions are prescribed to
the edges of the strip
Fig. 4. Piezo-strip under a uniform stress and an applied voltage.

where r0 is an arbitrary stress parameter. For the given boundary
conditions, the corresponding analytical solutions for the stresses
r and for the electric displacements D are obtained as

T xx ¼ r0 ;

T xy ¼ T yy ¼ Dx ¼ Dy ¼ 0:

ð73Þ

In this patch test, the mechanical displacements and the electric
potential are prescribed on all boundaries by the given boundary
conditions with linear functions presented in Eq. (72). Satisfaction
of the patch test then requires that the mechanical displacements
and the electric potential of any interior nodes inside the patch follow ‘‘exactly’’ (to machine precision) the same linear function of
the imposed boundary conditions of the patch. It shows in Table 2

/;y ðx; y ¼ ÆhÞ ¼ 0;

T yy ðx; y ¼ ÆhÞ ¼ r0 ;

T xy ðx; y ¼ ÆhÞ ¼ 0;

/ðx ¼ L; yÞ ¼ ÀV 0 ;

/ðx ¼ 0; yÞ ¼ þV 0 ;

ux ðx ¼ 0; yÞ ¼ 0;

T xy ðx ¼ L; yÞ ¼ 0;
T xx ðx ¼ L; yÞ ¼ 0;

uy ðx ¼ 0; y ¼ 0Þ ¼ 0:
ð75Þ

The analytical solutions for this problem are given by Ohs and
Aluru [9]

ux ¼ s13 r0 x;

uy ¼

d15 V 0 x
þ s33 r0 y;
h



x
/ ¼ V0 1 À 2 :
L

ð76Þ

The numerical simulations of the proposed stabilized NS-FEM
are carried out using a regular mesh with nodal distribution of
7 Â 7 as shown in Fig. 5(a). The mechanical displacements and
the electric potential at the central line ðy ¼ 0Þ with stabilization
parameters a ¼ 0:05 as depicted in Figs. 6–8 are compared directly


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L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

(a)

(b)

Fig. 5. (a) Regular; and (b) irregular meshes for piezo-strip under a uniform stress and an applied voltage.

4

x 10

-5

1

x 10

-6

stabilized NS-FEM α=0.05
Exact

0.8

3.5

0.6

Electric potential(GV)

ux displacement (mm)

3

2.5

2

1.5

0.4
0.2
0
-0.2

-0.4
-0.6

1

-0.8
0.5

-1

stabilized NS-FEM α=0.05
Exact

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 6. Variation of horizontal displacement u at the central line ðy ¼ 0Þ of the
single-layer piezoelectric strip.

-3

x 10

uy displacement (mm)

1

0.8

0.6

0.2

0.2
stabilized NS-FEM α=0.05
Exact
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x (mm)
Fig. 7. Variation of vertical displacement
layer piezoelectric strip.

0.4

0.5

0.6

0.7

0.8

0.9

1

with the analytical solutions available in [9]. It is evident that the
computed results show an excellent agreement with those of the
exact solutions. Additionally, different stabilization parameters a
are employed for the numerical simulations. The numerical errors
in displacement and electric potential solutions calculated by Eqs.
(70) and (71) respectively are presented in Table 3, which demonstrates that the stabilized NS-FEM using different stabilization
parameters a in a proper interval can reproduce the linear behavior
of the exact solutions accurately within round-off errors.
In order to illustrate the robustness of the present method, this
shear problem of the piezoelectric strip is also tested using the
mesh with irregular nodal distribution whose coordinates are generated in the following fashion

y0 ¼ y þ Dy Á r c Á air ;

0

0.3

Fig. 8. Variation of electric potential / at the central line ðy ¼ 0Þ of the single-layer
piezoelectric strip.

x0 ¼ x þ Dx Á rc Á air ;

0.4

0

0.1

x (mm)

x (mm)

1.2

0

v at the central line ðy ¼ 0Þ of the single-

ð77Þ

where Dx and Dy are initial regular element sizes in x and y directions, respectively. r c is a computer-generated random number
between À1.0 and 1.0, and air is a prescribed irregularity factor
whose value is chosen between 0.5 in this research (see Fig. 5(b)).
Also, the numerical errors in displacement and electric potential
solutions are listed in Table 3, and it is found that the stabilized
NS-FEM are in excellent agreement with the linear exact solutions
within machine precision, regardless of the element shape.


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L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Table 3

Single-layer piezoelectric strip: numerical errors in displacement and electric potential solutions.
Stabilization parameter a

0.0

0.01

0.03

0.05

0.1

0.3

1.0

eu
Regular mesh
Irregular mesh

1.36EÀ12
1.30EÀ12

7.56EÀ13
5.86EÀ13

4.41EÀ13
1.15EÀ12

4.54EÀ13
5.39EÀ13

3.36EÀ13
2.33EÀ12

2.33EÀ12
4.03EÀ12

3.61EÀ12
4.02EÀ12

eu
Regular mesh
Irregular mesh

1.15EÀ14
1.46EÀ14

8.12EÀ15
1.23EÀ14

6.65EÀ15
3.04EÀ14

1.29EÀ14
8.37EÀ15

1.38EÀ14
3.76EÀ14

4.39EÀ14
1.71EÀ14

8.04EÀ14
4.36EÀ14

x 10

-8

2.6
2.4

Electric potential

2.2
2
1.8
NS-FEM-T3
sNS-FEM-T3 α=0.01
sNS-FEM-T3 α=0.03
sNS-FEM-T3 α=0.05
sNS-FEM-T3 α=0.1
sNS-FEM-T3 α=0.3
sNS-FEM-T3 α=1.0
FEM-T6
FEM-Q4
FEM-T3
Reference solu.

1.6
1.4
1.2
1
0.8
0.6
0

500

1000

1500

2000

2500

3000

DOF
Fig. 11. Comparison of electric potential at point A of Cook’s membrane.
Fig. 9. Geometry and boundary conditions of Cook’s membrane.

9.5

x 10

2.2

8.5

2.1

8

2
1.9

NS-FEM-T3
sNS-FEM-T3 α=0.01
sNS-FEM-T3 α=0.03
sNS-FEM-T3 α=0.05
sNS-FEM-T3 α=0.1
sNS-FEM-T3 α=0.3
sNS-FEM-T3 α=1.0
FEM-T6
FEM-Q4
FEM-T3
Reference solu.

1.8
1.7
1.6
1.5
1.4

-5

9

0

500

1000

1500

2000

2500

3000

DOF
Fig. 10. Comparison of vertical displacement at point A of Cook’s membrane. Upper
bound solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3
(a 2 ½0:0; 0:1ŠÞ. The lower bound solution is obtained using the FEM-T3 and the
FEM-Q4.

6.3. Cook’s membrane
A benchmark problem, shown in Fig. 9, a clamped tapered panel
subjected to a distributed tip load F ¼ 1 N, resulting in deformation

Strain energy

Vertical displacement v

2.3

x 10

-4

7.5
NS-FEM-T3
sNS-FEM-T3 α=0.01
sNS-FEM-T3 α=0.03
sNS-FEM-T3 α=0.05
sNS-FEM-T3 α=0.1
sNS-FEM-T3 α=0.3
sNS-FEM-T3 α=1.0
FEM-T6
FEM-Q4
FEM-T3
Reference solu.

7
6.5
6
5.5
5

0

500

1000

1500

2000

2500

3000

DOF
Fig. 12. Comparison of strain energy of Cook’s membrane.

dominated by a bending response, is then analyzed. The piezoelectric material PZT-4 whose parameters listed in Table 1 is employed.
The mechanical boundary conditions are similar to the popular
Cook’s membrane [38]. The electric boundary condition of the
lower surface is prescribed by zero voltage (0 V). The geometry,
loading and boundary conditions can be referred to Fig. 9.
Four discretizations (3-node triangular elements) with uniform
nodal distribution: (5 Â 5; 9 Â 9; 17 Â 17, and 33 Â 33Þ, are used for
the present stabilized NS-FEM (sNS-FEM-T3). For comparison, such


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L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Table 4
Cook’s membrane: comparisons of vertical displacement and electric potential at point A, and strain energy in the whole problem domain using different numerical methods.

À4

Vertical displacement 2:109ð10

mm)

Electric potential 1:732ð10À8 GV)

Strain energy 8:5405ð10À5 J=mm2 Þ

Mesh (nodes)

5 Â 5 (% error)

9 Â 9 (% error)

17 Â 17 (% error)

33 Â 33 (% error)

FEM-T3
FEM-Q4
FEM-T6
NS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3

a ¼ 0:01
a ¼ 0:03
a ¼ 0:05
a ¼ 0:10
a ¼ 0:30
a ¼ 1:00

1.0356
1.6617
1.9917
2.2630
2.2012
2.1700
2.1564
2.1406
2.0528
2.0011

(À50.8)
(À21.2)
(À5.6)
(7.3)
(4.4)
(2.9)
(2.2)
(1.5)
(À2.7)
(À5.1)

1.5383

2.0179
2.0739
2.1688
2.1597
2.1544
2.1508
2.1452
2.1349
2.1230

(À27.1)
(À4.3)
(À1.7)
(2.8)
(2.4)
(2.2)
(2.0)
(1.7)
(1.2)
(0.7)

1.8990
2.0897
2.1057
2.1227
2.1225
2.1217
2.1211
2.1201
2.1184

2.1166

(À10.0)
(À0.9)
(À0.2)
(0.6)
(0.6)
(0.6)
(0.6)
(0.5)
(0.4)
(0.4)

2.0470
2.1034
2.1079
2.1133
2.1126
2.1122
2.1120
2.1116
2.1111
2.1105

(À2.9)
(À0.3)
(À0.1)
(0.2)
(0.2)
(0.2)

(0.1)
(0.1)
(0.1)
(0.1)

a ¼ 0:01
a ¼ 0:03
a ¼ 0:05
a ¼ 0:10
a ¼ 0:30
a ¼ 1:00

0.6427
0.8348
1.2221
2.6646
1.9751
1.9077
1.8630
1.8240
1.7007
1.5849

(À62.9)
(À51.8)
(À29.4)
(53.8)
(14.0)
(10.1)
(7.6)

(5.3)
(À1.8)
(À8.5)

1.1629
1.4555
1.5301
1.9597
1.9337
1.9001
1.8817
1.8554
1.8169
1.7893

(À32.9)
(À16.0)
(À11.7)
(13.1)
(11.6)
(9.7)
(8.6)
(7.1)
(4.9)
(3.3)

1.5186
1.6608
1.6808
1.8177

1.7864
1.7801
1.7774
1.7743
1.7703
1.7672

(À12.3)
(À4.1)
(À2.9)
(4.9)
(3.1)
(2.8)
(2.6)
(2.4)
(2.2)
(2.0)

1.6608
1.7037
1.7087
1.7511
1.7434
1.7396
1.7380
1.7363
1.7349
1.7346

(À4.1)

(À1.6)
(À1.3)
(1.1)
(0.7)
(0.4)
(0.3)
(0.3)
(0.2)
(0.1)

a ¼ 0:01
a ¼ 0:03
a ¼ 0:05
a ¼ 0:10
a ¼ 0:30
a ¼ 1:00

4.5180
7.1399
7.8401
9.4103
9.2359
9.0455
8.9201
8.6980
8.1011
6.3611

(À47.1)
(À16.4)

(À8.2)
(10.2)
(8.1)
(5.9)
(4.4)
(1.8)
(À5.1)
(À25.5)

6.6113
8.3230
8.4232
8.8599
8.8333
8.8029
8.7792
8.7289
8.5599
8.0346

(À22.6)
(À2.5)
(À1.4)
(3.7)
(3.4)
(3.1)
(2.8)
(2.2)
(0.2)
(À5.9)

7.8917
8.5148
8.5191
8.6383
8.6345
8.6290
8.6237
8.6114
8.5663
8.4167

(À7.6)
(À0.3)
(À0.3)
(1.1)
(1.1)
(1.0)
(1.0)
(0.8)
(0.3)
(À1.5)

8.3607
8.5392
8.5395
8.5643
8.5653
8.5648
8.5638

8.5610
8.5490
8.5072

(À2.1)
(À0.02)
(À0.02)
(0.3)
(0.3)
(0.3)
(0.3)
(0.2)
(0.1)
(À0.4)

FEM-T3
FEM-Q4
FEM-T6
NS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
FEM-T3
FEM-Q4
FEM-T6
NS-FEM-T3
sNS-FEM-T3

sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3
sNS-FEM-T3

four models are also computed using the standard FEM-T3 and the
standard NS-FEM-T3. In addition, the 4-node quadrilateral elements with the same number of nodes are employed in the standard FEM-Q4 using 2 Â 2 Gaussian points. Also, the six-node
triangular meshes are used by the FEM for comparison, and the
DOFs of these quadratic meshes are the same as those of the linear
meshes that other methods use for fairness. The reference value of
the vertical displacement and the electric potential at the center
tip (A) are 2:109 Â 10À4 mm and 1:732 Â 10À8 GV [7]. It is noted
that the analytical solutions for this problem is unknown, therefore, the FEM-Q4 with a very fine mesh (10,505 nodes) is employed
to provide the reference solution of strain energy in the whole
domain.
Figs. 10 and 11, respectively, show the convergence status of
vertical displacement and electric potential at point A (see Fig. 9)
against the increase of Degree of Freedom (DOF) using different
numerical methods mentioned above. Also, we compare the variation of strain energy in the whole domain, as mesh refines, and the
results are plotted in Fig. 12. Additionally, the corresponding data
of Figs. 10–12 are tabulated in Table 4 in detail for reference.
From the table and figures, it can be first observed that varying
the stabilization parameter a changes the properties of sNS-FEM.
(1) Either decreasing the value of a or reducing the element size
lc makes the solutions calculated by the sNS-FEM converge
to the original NS-FEM ones (sNS-FEM: a ¼ 0:0Þ;
(2) with a relatively small value of a ða 2 ½0:0; 0:1ŠÞ, the sNSFEM is found ‘‘overly-soft’’ and obtains upper bound solutions, whereas the FEM give lower bound solutions. Thus
we can bound the exact solution from both sides with complicated geometry as long as an element mesh could be
generated;

(3) it is also worth noting that increasing the stabilization
parameter a in this interval with small values can improve
the numerical accuracy of the method, because the least-

-1

Log 10(displacement error)

Ref. solution

-1.5

-2

-2.5

-3

-3.5
-1

NS-FEM-T3
sNS-FEM-T3 α=0.01
sNS-FEM-T3 α=0.03
sNS-FEM-T3 α=0.05
sNS-FEM-T3 α=0.1
sNS-FEM-T3 α=0.3
sNS-FEM-T3 α=1.0
FEM-T6

-0.5

0

0.5

1

1.5

2

Log10(t)
Fig. 13. Comparison of computational efficiency in term of displacement error for
Cook’s membrane.

squared stabilization term introduces ‘‘stiffening’’ effects
into the ‘‘overly soft’’ NS-FEM-T3, which makes the sNSFEM model more close to the exact models. At the same
time, such stiffening effects in sNS-FEM eliminate temporal
instability existed in the standard NS-FEM, which will be
demonstrated later;
(4) by gradually increasing a, the additional constraint of the
squared-residual of the equilibrium equation takes more
effects, leading to a stiffer system. However, the solution
bound property is numerically found uncertain. Consequently, the a in this range is not recommended;


76

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

(5) while a is larger than a certain value, sNS-FEM can find the
lower bound solution similarly to ‘‘overly stiff’’ fully compatible FEM. Nonetheless, the accuracy with this range of
a values is not guaranteed;
(6) in order to find the upper bound solution in energy norm (or
lower bound solution in natural frequency) and to stabilize
the system, the value of a can be selected as 0.03–0.1.
In addition to the above, the computational efficiency in terms
of the displacement error against computational time(s) is compared for the present sNS-FEM-T3 and the FEM-T6, and the results
are plotted in Fig. 13. It can be observed that the sNS-FEM-T3 is
obviously less efficient compared to the FEM-T6. Two factors associated with computational cost lead to this performance for our
proposed sNS-FEM-T3: (1) the ‘‘overhead’’ cost for all operations
until the stiffness matrix is formed (including mapping, Gauss integration and etc.), and (2) the solver time to solve the resultant system equations. Firstly, in our sNS-FEM models, the overhead time
includes the smoothing operations that need the construction of
smoothing domains and thus relatively complicated data storage
structures. Also, the forming penalty, i.e., two stabilization terms,
requires computations that the FEM does not. Therefore, the overhead cost of the present sNS-FEM models is apparently increased.
The second factor is attributed to the bandwidth of the global stiffness matrix. As we know, each element of FEM-T6 definitely requires 6 nodes to assemble the global stiffness matrix. On the
other hand, each integration (node-based smoothing) domain in
the sNS-FEM models needs information from all the nodes belonging to the elements that include the current node (called as influence nodal set here) for assembling. For a specific smoothing
domain, this influence nodal set arises from the surrounding discretization of the current node, and generally, about 7 nodes are
included in this influence nodal set as shown in Fig. 2. Consequently, this difference leads to a little larger bandwidth, and the
sNS-FEM-T3 is, accordingly, a little more expensive for solving
the resultant system equations. Based on these two factors, the
‘‘overall’’ computational cost of the present sNS-FEM is thus
increased compared to the FEM-T6.
However, the important feature of the sNS-FEM-T3 lies in that
an upper bound to the exact solutions in energy norm (or a lower
bound in natural frequencies) can be achieved with a proper
parameter, while the FEM-T3 produces the lower bound in energy

norm (or the upper bound in natural frequencies). Therefore, we
now can use our stabilized NS-FEM together with the FEM to

Fig. 15. Construction of strain smoothing domains for the nodes falling on the bimaterial interface.

numerically obtain both upper and lower bounds of solutions for
piezoelectric problems as long as a triangular element mesh can
be generated.
6.4. Bimorph MEMS device
The purpose of this problem is to simulate the linear tilt angle of
the reflected light through a mirror of a MEMS device. The device is
constructed from two parallel bimorphs made of PVDF material
connected by a mirror as shown in Fig. 14(a). Each bimorph with
length L ¼ 10 lm and height H ¼ 1 lm is assumed. The bimorph
beam is divided into the top and bottom layers as show in
Fig. 14(b).
The following boundary conditions are applied to layer 1 of
bimorph beam

/ð1Þ ðx; y ¼ 0Þ ¼ V;

T ð1Þ
yy ðx; y ¼ 0Þ ¼ 0;

ð1Þ
T xy
ðx; y ¼ 0Þ ¼ 0;

/ð1Þ ðx; y ¼ hÞ ¼ 0;

ð2Þ
T ð1Þ
yy ðx; y ¼ hÞ ¼ T yy ðx; y ¼ hÞ;

ð2Þ
T ð1Þ
xy ðx; y ¼ hÞ ¼ T xy ðx; y ¼ hÞ;

/;xð1Þ ðx
/;xð1Þ ðx

ð78Þ

¼ 0; yÞ ¼ 0;

u ðx ¼ 0; yÞ ¼ 0;

v

¼ L; yÞ ¼ 0;

T ð1Þ
xx ðx

T ð1Þ
xy ðx

ð1Þ

(a)

(b)
Fig. 14. (a) Bimorph MEMS device; (b) parallel bimorph geometry.

¼ L; yÞ ¼ 0;

ð1Þ

ðx ¼ 0; yÞ ¼ 0;
¼ L; yÞ ¼ 0


77

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

7.5

x 10

-3

NS-FEM-T3
sNS-FEM-T3 α=0.1
FEM-T6
FEM-Q4
FEM-T3
Reference solu.

Tip deflections (um)

7

6.5

6

5.5

5

electroded
surfaces

4.5

4

0

1000

2000

3000

4000

5000

6000

7000

8000

DOF
Fig. 16. Convergence of bimorph tip deflection in the MEMS device. Upper bound
solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3, whereas lower
bound solution is obtained using the FEM-T3, the FEM-Q4 and the FEM-T6.

Fig. 17. Schematic of a transducer.

Boundary conditions for layer 2 are

/ð2Þ ðx; y ¼ hÞ ¼ /ð1Þ ðx; y ¼ hÞ;

v

ð2Þ

ðx; y ¼ hÞ ¼ v

ð1Þ

T ð2Þ
yy ðx; y ¼ 2hÞ ¼ 0;

ðx; y ¼ hÞ;

uð2Þ ðx; y ¼ hÞ ¼ uð1Þ ðx; y ¼ hÞ;
ð2Þ

/ ðx; y ¼ 2hÞ ¼ V;

ð2Þ
T xy
ðx; y ¼ 2hÞ ¼ 0;

/ð2Þ
;x ðx ¼ 0; yÞ ¼ 0;

uð2Þ ðx ¼ 0; yÞ ¼ 0;

v ð2Þ ðx ¼ 0; yÞ ¼ 0;

/ð2Þ
;x ðx ¼ L; yÞ ¼ 0;

ð2Þ
T xx
ðx ¼ L; yÞ ¼ 0;

ð2Þ
T xy
ðx ¼ L; yÞ ¼ 0:

ð79Þ
The centers of bimorphs are connected by a 1 lm long mirror.
Linear elasticity is assumed to the mirror. When a voltage is applied across the thickness, the bimorphs vertically displace in

opposite directions and rotate the mirror with the tilt angle. As a
result, the direction of the reflected light can change when the various voltages are applied.
In the simulation, the bimorph is discretized into three triangular meshes with uniform nodal distribution of 41 Â 5; 81 Â 9, and
161 Â 17. Two smoothing domains are constructed for the nodes
falling on the bi-material interface as shown in Fig. 15, in order
to avoid the discontinuity within one integration domain as the
standard FEM does. Since the analytical solutions of this problem

are not available, the reference solutions are calculated using the
FEM-Q4 with a very fine mesh (92,345 nodes) in this study.
In Fig. 16, the convergence process of the bimorph tip deflection
with the nodal distribution refined using different numerical
methods are compared. Note that the stabilization parameter a is
selected as 0.05 that falls in the suggested interval in Section 6.3.
From the figure, it can be found again that the tip deflection of
the FEM-T3, Q4 and T6 are no-more than the exact solutions and
converge from below, the corresponding values of the stabilized
sNS-FEM-T3 and the NS-FEM-T3 are no-less than the exact ones
and converge from above. Thus, again, we also can bound the exact
or reference solutions from two directions. Further, obviously, the
stabilized NS-FEM-T3 with a proper parameter improves significantly the accuracy in comparison with the standard NS-FEM-T3.
Meanwhile, the proposed sNS-FEM-T3 is again found slightly less
accurate than the quadratic FEM-T6 with the same DOFs.
We also varied the voltage linearly, and Table 5 lists the tip
deflection for different applied voltages with nodal distribution
of 161 Â 17. Further, the tilt angle of mirror could be calculated
by normal deflection of beam at point connection between mirror
and beam in which the horizontal displacement of this point is

Table 5

Bimorph MEMS device: comparisons of bimorph tip deflection and mirror tilt angle using different numerical methods.
Applied voltage (V)
Tip deflection (lmÞ
1.00
2.00
5.00
10.00
20.00
50.00
Tilt angle (°)
1.00
2.00
5.00
10.00
20.00
50.00

FEM-T3
0.004821
0.009643
0.024107
0.048214
0.096427
0.241068

0.5525
1.1050
2.7635
5.5335
11.1194

28.8250

FEM-T6
0.004890
0.009780
0.024449
0.048898
0.097797
0.244492

0.5603
1.1207
2.8028
5.6123
11.2794
29.2738

FEM-Q4
0.004868
0.009737
0.024342
0.048684
0.097369
0.243422

0.5579
1.1158
2.7905
5.5877
11.2294

29.1333

NS-FEM-T3
0.005283
0.010566
0.026416
0.052831
0.105663
0.264157

0.6054
1.2109
3.0284
6.0653
12.2000
31.8916

Stabilized NS-FEM-T3 (a ¼ 0:05Þ
0.004914
0.009829
0.024572
0.049144
0.098289
0.245722

0.5632
1.1264
2.8169
5.6406
11.3369

29.4356

Ref. solution
0.004901
0.009802
0.024505
0.049010
0.098020
0.245050

0.5616
1.1233
2.8092
5.6252
11.3055
29.3472


78

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Model 1

Model 1

Model 1

Model 2

Model 2

Model 2

Model 3

Model 3

Model 4

Model 4

Model 3

Model 4

Fig. 18. Mode shapes of piezoelectric transducer solved by (a) the NS-FEM-T3 with 369 nodes; (b) the stabilized NS-FEM-T3 with parameter a ¼ 0:05 using 369 nodes; and (c)
the FEM-Q4 with 8241 nodes.

supposed to be zero, and the corresponding results are tabulated in
Table 5 as well. It is observed that the tip displacements of the
bimorphs and the tilt angle of the mirror for both FEM and NSFEM models vary linearly with applied voltages, which is very similar to the finding by Ohs and Aluru [9] using the meshless point
collocation method (PCM). More importantly, the results of the
proposed sNS-FEM are found in good agreement with the reference
value with a fraction of percent error. All of these indicate the proposed sNS-FEM can solve the piezoelectric problems effectively.
6.5. Eigenvalue analysis of a piezoelectric transducer
A cylindrical piezoelectric transducer is considered as the final
example for the eigenvalue analysis using a piezoelectric ceramic
wall of the material PZT-4 of which the material parameters are
given in Table 1. Brass end caps are placed on the top and the bot-

tom of the transducer as depicted in Fig. 17. The material parameters of the brass are as follows: E = 105 GPa, v = 0.37 and
q = 8500 kg/m3. In this model, the inner and outer covering surfaces of the structure are designed with electrodes. The transducer
is particularly modeled as an axisymmetric structure [39]. All the
models are constrained under open-circuit voltages and in all
cases the potentials on the inner surface are restrained to zero,
and therefore, the frequencies correspond to those for antiresonance.
Eigenvalue analysis of the piezoelectric transducer is performed
using the sNS-FEM-T3 to demonstrate the proposed stabilization
scheme on curing the temporal instability presented in the standard NS-FEM-T3. Fig. 18 illustrates the mode shapes solved by
the sNS-FEM (a ¼ 0:05Þ and the NS-FEM-T3 with one regular mesh
setting of 369 nodes. In addition, the FEM-Q4 with a very fine mesh
(8,241 nodes) is employed to provide the reference solutions as is


79

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Model 5

Model 5

Model 5

Model 8

Model 8

Model 8

Model 13

Model 13

Model 13

(b) Model 19

(c) Model 19

(a) Model 19

Fig. 18 (continued)

Table 6
Piezoelectric transducer: comparisons of natural frequencies (kHz) using different numerical methods.a

a

Methods

FEM-T3 (% error)

FEM-T6 (% error)

NS-FEM T3 (% error)

sNS-FEM-T3 ða ¼ 0:05Þ (% error)

Ref. solution

Experiment [39]

Mode
Mode
Mode
Mode
Mode
Mode
Mode
Mode

18.93 (2.0)
40.7 (0.9)
59.19 (2.8)
64.94 (1.2)
90.62 (3.5)
161.18 (2.9)
277.74 (2.4)
408.54 (1.2)

18.71
40.52
58.24
64.40
89.02
158.80
274.55
406.21

18.19
39.33
55.24
61.07
83.48
148.18
260.52
391.37

18.39
39.87
56.24
62.38
85.10
153.70
266.79
400.32

18.56
40.32
57.55
64.14
87.53
156.61
271.29
403.79

18.6
35.4

54.2
63.3
88.8

1
2
3
4
5
8
13
19

(0.8)
(0.5)
(1.2)
(0.4)
(1.7)
(1.4)
(1.2)
(0.6)

(À2.0)
(À2.5)
(À4.0)
(À4.8)
(À4.6)
(À5.4)
(À4.0)
(À3.1)

(À0.9)
(À1.1)
(À2.3)
(À2.8)
(À2.8)
(À1.9)
(À1.7)
(À0.9)

Upper bound of natural frequencies is obtained using the FEM-T3 and the FEM-T6, whereas lower bound solution is obtained using the NS-FEM-T3 and the sNS-FEM-T3.

done before. One can clearly find spurious non-zero energy modes
(e.g., mode 13 and mode 19) in Fig. 18(a) if no stabilization is used,
which indicates the temporal instability of the original NS-FEM

(a ¼ 0:00Þ. Employing the stabilization scheme with a proper a
value yields the changes of the natural modes, and the vanish of
spurious modes as plotted in Fig. 18(b).


80

L. Chen et al. / Comput. Methods Appl. Mech. Engrg. 241–244 (2012) 65–81

Additionally, the corresponding natural frequency solutions
solved by the numerical methods mentioned above are listed in
Table 6. The relative error of natural frequency is given by

e¼

xnum À xref
 100%:
xref

ð80Þ

From Eq. (80), it is clear that the negative relative error means that
the numerical solution is smaller than the reference value, and vice
versa. It is observed that all the results by the stabilized NS-FEM-T3
give the negative relative errors in natural frequency with the
corresponding reference solutions, which contrast with results by
the FEM-T3. It means again that the stabilized NS-FEM-T3 produces
a lower bound solution in natural frequency, while the FEM-T3 produces the upper bound. In that case, we can still bound the exact
solution from both sides, although the direction is opposite with regard to the solution in energy norm. It is also noted that the remarkable improvement of accuracy can be achieved by the present
stabilized sNS-FEM-T3 comparing to the standard NS-FEM-T3, with
the relative errors being less than 3 percent, although it is again
found less accurate compared to the FEM-T6 under the same Dofs.
Noted that this problem was also investigated by the experimental
work described in Mercer et al. [39] and the experimental result is
thus given in Table 6 for comparison. Also, the solutions of our sNSFEM-T3 are found in excellent agreement with the corresponding
experimental values.

7. Conclusions
In this work, a stabilization technique stemming from [36] is
extended to piezoelectric media to cure temporal instability of
node-based smoothed finite element method (NS-FEM). In this stabilization scheme, two stabilization terms corresponding to
squared-residuals of two equilibrium equations, i.e., mechanical
stress equilibrium and electric displacement equilibrium, are
added to the smoothed potential energy functional of the original

NSFEM. Through the formulations and numerical examples, the
following conclusions can be drawn:
1. The gradient smoothing technique is performed on the
second-order derivatives to realize two formulated stabilization
terms, and only the first order derivatives of the shape function
are needed. Therefore, the present squared-residual stabilization technique works very well for 3-node triangular linear
elements, and suits ideally in many ways to the NS-FEM
models.
2. The sNS-FEM can always pass the standard patch test and thus
converge to the exact solution with the mesh refinement.
3. With a small stabilization parameter, the stabilized NSFEM
(sNS-FEM) behaves ‘‘overly soft’’ as the original NS-FEM does,
which, however, is found more accurate than the original NSFEM with a properly small a. In that case, the sNS-FEM can be
used to find an upper bound to the exact solutions in energy
norm (or a lower bound in natural frequencies), which contracts
with the bound properties of the FEM solutions. Therefore, we
now can use our stabilized NS-FEM together with the FEM to
numerically obtain both upper and lower bounds of solutions
for piezoelectric problems as long as a triangular element mesh
can be generated.
4. Temporal instability and spurious modes in the original NS-FEM
can be eliminated by the added two stabilization terms. This is
because the least-squared stabilization term introduces stiffening effects into the ‘‘overly soft’’ NS-FEM models, which makes
the sNS-FEM models temporally stable.

5. Increasing a to a certain value, both the accuracy and bound
property can not be guaranteed for sNS-FEM. Numerical examples indicate that a sNS-FEM-T3 model with a stabilization
parameter a 2 ½0:03; 0:1Š can be made to achieve both temporal
stability and certified solution with bounds.

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