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

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

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

L Chena,⇑, Y.W Zhanga, G.R Liub, H Nguyen-Xuanc, Z.Q Zhangd

a

Department of Engineering Mechanics, Institute of High Performance Computing, 1 Fusionopolis Way #16-16 Connexis, Singapore 138632, Singapore

b

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

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 ele-ments 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 oper-ation is then performed on second-order derivatives of shape functions to achieve the stabilizoper-ation 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 sta-bilized NS-FEM is demonstrated via numerical examples

Ó 2012 Elsevier B.V All rights reserved

1 Introduction

Piezoelectric materials showing an ability of transformation

be-tween mechanical energy and electric energy have been widely

used in various applications, where they serve as sensors,

actua-tors, transducers or active damping devices These applications

range from sub-millimeter length scales in

micro-electro-mechan-ical systems up to large scales in the design of smart

electrome-chanical 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

meth-od (FEM)[1,2], the bubble/incompatible displacement method[3],

the mixed and hybrid formulations[4–6]and the piezoelectric

fi-nite element with drilling degrees of freedom[7] Several meshless

methods [8]have also been used to analyze piezoelectric

struc-tures such as the meshless point collocation method (PCM)[9],

the point interpolation method (PIM)[10], the radial point

interpo-lation method (RPIM)[11], and the moving Kriging (MK)

interpola-tion-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

mean for safety and reliability assessments of piezoelectric struc-tural properties In order to implement these analyses, two numer-ical 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 en-ergy and electric enen-ergy (or upper bounds to the exact natural fre-quencies) 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 solu-tions (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 equi-librium FEM models[18] These three models, however, are known

to have the following two common disadvantages: (1) the formu-lation and numerical implementation are complicated and expen-sive 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 smooth-ing technique[19,20]was introduced by Chen et al.[19]for spatial stabilization of nodal integrated meshfree methods, and later

0045-7825/$ - see front matter Ó 2012 Elsevier B.V All rights reserved.

⇑ Corresponding author Tel.: +65 64191246.

E-mail address: chenl@ihpc.a-star.edu.sg (L Chen).

Contents lists available atSciVerse ScienceDirect

Comput Methods Appl Mech Engrg.

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c m a

extended by Yoo and Moran to the natural element method (NEM)

[20] More recently, Liu[21]has generalized this gradient

smooth-ing technique in order to weaken the consistence requirement for

the field functions, allowing the use of certain types of

discontinu-ous 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

CS-FEM) [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

edge-based 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

effi-ciency, and spatial and temporal stability These attractive

proper-ties 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

fash-ion[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

insta-bility is defined to have spurious non-zero eigen modes Such

mod-els 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

em-ploy the Lagrangian kernels as Rabczuk et al [34,35] proposed

Also, Beissel and Belytschko[36]have developed a scheme to

sta-bilize these nodal integrated methods by the addition to the

poten-tial 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

sta-bilization term over the problem domain regulated by a

stabiliza-tion 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 FEM models Further, the stabilized NS-FEM 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 piezo-electric 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 X2 R2 bounded by CðC¼CuþCt;

Cu\Ct¼ 0Þ as

divD þ qs¼ 0



whereris the Cauchy stress tensor, b represents the vector of body force applied in the problem domain, D denotes the electric dis-placement and qsis the free point charge density

For dynamics problems of linear electroelastic solids, the strong form of the governing equation is

where q is the density of the mass, and g is a set of viscosity parameters

The straineand the electric field E are, respectively, derived from the displacement u and the electric potentialu, and could

be written by the vector form

e¼rsu;

wherersis the symmetric gradient operator,

rs¼

@

x 0 @ y

y

@ x

Writing the stress tensorras the vector form, the constitutive equations have the following form

r

D

 

T

E

 

where cE denotes the elastic matrix measured at constant electric field,jSis the dielectric matrix at constant mechanical strain, and

e is the piezoelectric matrix These tensors are known experimen-tally for various kinds of piezoelectric materials They are usually not isotropic To be specific, Eq.(5)can also be written in a compo-nent form for the 2D plane piezoelectric problem

rxx

ryy

rxy

Dx

Dy

2 6 6 6 6

3 7 7 7 7

¼

2 6 6 6 6

3 7 7 7 7

exx

eyy

cxy

Ex

Ey

2 6 6 6 6

3 7 7 7 7

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

E

 

T

r

D

 

ð7Þ

in which use is made of the relationships d ¼ ec1

E ; sE¼ ec1

E , and

eS¼jSþ ec1

E eT, where,eSis the dielectric matrix measured at

con-stant stress, d stands for the piezoelectric strain matrix and sEis the

elastic compliance matrix

At partCu, the essential boundary condition is given by

where uCis the vector of the prescribed displacement, anduC

de-notes the prescribed electrical potential, whereas at the partCt,

the natural boundary condition is given by

r n ¼ tC

where tCis the vector of prescribed tractions, qCdenotes the

sur-face change on Ct, and nj is the surface outward normal of the

boundaryCt

2.2 Galerkin weak form and finite element formulation

In this section, a finite element formulation for piezoelectricity

is established via a variational formulation, in which the following

general energy functionalpis used to express a summation of the

following two parts: (1) mechanical contribution including kinetic

energy, strain energy, and mechanical external work, and (2)

elec-trical contribution involving dielectric energy and electric external

work[38]

Z

X

1

2q_uTu þ_ 1

2eTðuÞrðuÞ  uTb

Z

C t

uTtCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

mechanical part



Z

X

1

T

ðuÞEðuÞ uqs

Z

C t

uqCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

electric part

ð10Þ

Consider the domain discretized into Neof non-overlapping and

non-gap elements and Nn nodes, such that X¼ [N e

m¼1Xe

m and

Xei\Xej ¼ 0; 8i – j, the approximation of displacement field uh

and electric potential for a 2D electroelastic problem is given by:

i2n e

n

NiðxÞui;

i2n e

n

where ne

n is the set of nodes of the element containing

x; ui¼ u½ xi uyiTis the vector of nodal displacements, respectively,

in x axis and y axis,uiis the nodal electric potential, and Niis a

ma-trix of shape functions

ð12Þ

in which NiðxÞ is the shape function for node i Substituting the

approximations of Eq.(11)into Eq.(3), we obtain

e¼rsu ¼X

i2n e

n

Bu

i2n e

n

where

Bu

i ¼

Ni;y Ni;x

2

6

3

7

Ni;y

ð15Þ

Taking Hamilton’s variational principle yields to

Z

X

duTqu þ d€ eTðuÞrðuÞ  duTb

Z

C t

duTtCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

mechanical part

 Z

X

dDT

ðuÞEðuÞ  duqs

Z

C t

duqCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

electric part

ð16Þ

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

m 0

€

u



kTuu kuu

u

u



q



where

m ¼ Z

X

Z

X

ku u¼ Z

X

Z

X

f ¼ Z

X

NT

Z

C t

NT

q ¼  Z

X

NT

ðxÞqsdX

Z

C t

NT

3 NS-FEM for the piezoelectricity 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 the-ory[19,20] Consider the 2D domainX discretized into Ns non-overlapping smoothing domains as shown inFig 1, the smoothing operation on the gradient of a field / for a point xkin a smoothing domainXs

kis given as follows

 /ðxkÞ ¼ Z

Fig 1 Division of problem domainXinto non-overlapping smoothing domainsXs

where Wðx; x  xkÞ is a smoothing function that generally satisfies

the following properties[19]:

Z

The Heaviside-type piecewise constant function is employed in this

research:

s

k x 2Xsk;

(

ð26Þ

where As

kis the area of the smoothing domainXsk

Assuming /;j exists (the assumed field / is continuous), and

introducing divergence theorem to Eq.(24), we shall have



/;jðxkÞ ¼

Z

X s/;jðxÞWðx; x  xkÞdX¼ 1

Ask

Z

whereCs

kis the segments of boundary of the smoothing domainXsk

Substituting the smoothed gradient /i;jin Eq.(24)to the

follow-ing smoothfollow-ing operation of strain vector e in Eq.(3) yields the

smoothed strain as follows



ekðxkÞ ¼

Z

X sekðxÞWðx; x  xkÞdX¼

Z

X srsuWðx; x  xkÞdX

Ask

Z

where Lnis the matrix of unit outward normal which can be

ex-pressed as

2

6

3

7

Similarly, the smoothed electric field can be expressed by

EðxkÞ ¼

Z

X sEðxÞWðx; x  xkÞdX

¼

Z

X su;jðxÞWðx; x  xkÞdX¼ 1

Ask

Z

C suðxÞnjdC: ð30Þ

3.2 Construction of smoothing domains

Smoothing domain of the NS-FEM is constructed based on the

nodes of elements, as illustrated inFig 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-ically for problems with complicated geometry, which is also employed as the mesh platform in this work

Consider the domainXdiscretized into Nenon-overlapping and non-gap triangular elements and Nnnodes, the local smoothing do-mains in the NS-FEM are constructed with respect to the nodes of triangular elements, such thatX¼ [N n

k¼1XskandXsi\Xsj¼ 0; 8i – j,

in which Nnis 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:Ns¼ Nn For the triangular elements, the smooth-ing domainXskfor node k is created by connecting sequentially the mid-edge-points and the centroids of the surrounding triangles of the node as shown inFig 2

3.3 Smoothed Galerkin weak form and discrete equations Because a smoothed Galerkin weak form with smoothed gradi-ent over smoothing domains is variationally consistgradi-ent as proven

in[22], using this smoothed or weakened weak form with dis-placement field and electrical potential satisfying the essential boundary conditions, we have

dpsðuÞ ¼ 0 ¼

Z

X

duTqu þ d€ eTðuÞrðuÞ  duTb

Z

C t

duTtCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

mechanical part

 Z

X

dDTðuÞEðuÞ  duqs

Z

C t

duqCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

electric part

Employing the strain smoothing operation, the smoothed strain

ekoverXs

kfrom the displacement approximation in Eq.(3)can be written in the following matrix form

i2n s

Bu

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

i2n s

where ns

kis the set of nodes associated with the smoothing domain

Xsk Bu

iðxkÞ is the smoothed strain matrix for the mechanical part, and

BuiðxkÞ corresponds to the electric part, i.e., the smoothed electric field matrix Those two matrix operations can be written as follows

Bu

iðxkÞ ¼



0 biyðxkÞ



biyðxkÞ bixðxkÞ

2 6

3 7 5; BuiðxkÞ ¼



bixðxkÞ



biyðxkÞ

where bihðxkÞ; h ¼ x; y, is computed by



bihðxkÞ ¼ 1

Ask

Z

C snhðxÞNiðxÞdC: ð35Þ

Using the Gaussian integration along the segments of boundary

Cs

k, we have:



bih¼ 1

Ask

XN seg

m¼1

XN gau

n¼1

wm;nNiðxm;nÞnhðxm;nÞ

where Nsegis the number of segments of the boundaryCs

k; Ngauis 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 boundaryCs

k Substituting the approximated displacements and electric potential in Eq.(3), and the smoothed strains and electric field,

Fig 2 Construction of node-based strain smoothing domains based on 3-node

triangular elements.

respectively, from Eqs.(32) and (33)into the smoothed Galerkin

weak form leads to the following equation

duT

Z

X

qNTN€udXþ duT

Z

X

ðBuÞTcEBuudXþ duT

Z

X

ðBuÞTeTBuudX

Z

X

ðBuÞTeBuudX du

Z

X

ðBuÞTjsBuudX

 duT

Z

X

NTbdX duT

Z

C t

NTtCdCþ du

Z

X

NTqsdX

Z

C t

Eliminating du and duyields the following two discrete

equilib-rium equations

m 0

€

u



þ



kuu ku u



kT

u u kuu

u

u



q



where f and q are computed similarly by Eqs.(22) and (23),

respec-tively, 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



kuu¼XN s

k¼1

Z

X sðBuÞTcEBudX; ð39Þ



ku u¼XN s

k¼1

Z

X sðBuÞTeTBudX; ð40Þ



kuu¼ XN s

k¼1

Z

Xs

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



kuu¼XN s

k¼1



ku u¼XN s

k¼1



kuu¼ XN s

k¼1

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

½mf€ug þ ½kuufug þ ½ku ufug ¼ 0;

½kT

Eliminating theuyields the following equation

½mf€ug þ ½kuu ku uGkT

where G denotes the Moore–Penrose pseudoinverse of kuu

There-fore the natural frequency x and mode k can be computed by

solving the following eigenvalue problem

x2½m þ kuu ku uGkT

u u

4 Stabilization of NS-FEM

4.1 Governing equations and variational principle

With regard to the nodal integrated methods, direct nodal

inte-gration 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-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 spa-tially 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

psðuÞ ¼ Z

X

1

2qu_Tu þ_ 1

2eTðuÞrðuÞ  uTb

Z

C t

uTtCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed potential functionalmechanical part

 Z

X

1

TðuÞEðuÞ uqs

Z

C t

uqCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed potential functionalelectric part

þal2c E

Z

X

ðdiv rþ bÞTðdiv rþ bÞdX

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed squared residualmechanical part

al2c



j

Z

X

ðdivD þ qsÞ2dX

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed squared residualelectric part

;

ð48Þ

whereais the dimensionless, real, finite and non-negative stabiliza-tion parameter; lcis the characteristic length of the elements in the mesh that is determined by

lc¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2areaðXÞ

Ne

s

where areaðXÞ is the area of the problem domain, and Ne is the number of elements E is the effective Young’s modulus of the mate-rial, and the average diagonal value in the elastic matrix is em-ployed for anisotropic material as E ¼ ðc11þ c22þ c33Þ=3 Likewise,



j is the effective dielectric coefficient of the material as



j¼ ðj11þj22Þ=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

(i) whilea! 0, the functional in Eq.(48)converges to the ori-ginal smoothed potential energy functional;

(ii) while lc! 0, the functional in Eq (48) also converges to the original smoothed potential energy functional, for any finitea;

(iii) for a finite model (lcis finite positive constant), the strong-form equilibrium system equation is better enforced by using a largera, and the weakened weak form[22]is better enforced using a smallera Therefore, adjusting the stabil-ization 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 possiblea to obtain desired number of smallest eigen-modes for a given 2D solids, so that we

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

dpsðuÞ ¼ 0 ¼

Z

X

d _uTqu þ d_ eTðuÞrðuÞ  duTb

Z

C t

duTtCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed potential functionalmechanical part



Z

X

dDT

ðuÞEðuÞ  duqs

Z

C t

duqCdC

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed potential functionalelectric part

þ2al2c

E

Z

X

ðdivdrÞTðdiv rÞdXþ2al2c

E

Z

X

ðdivdrÞTbdX

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed square residualmechanical part

2al2c



j

Z

X

ðdivdDÞðdivDÞdX2al2c



j

Z

X

ðdivdDÞqsdX

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

smoothed square residualelectric part

4.2 Discretization

In the proposed variational principle in Eq.(50), the

discretiza-tion of smoothed total potential funcdiscretiza-tional manifested by the first

two terms can directly use the procedure in Section3.3 Therefore,

we now construct specifically the discretization on the last two

stabilization terms of Eq.(50), which consist of the

square-residu-als 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-resid-ual of mechanical stress equilibrium includes the

second-deriva-tives 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;mnðxÞ ¼ 0 ðm; n ¼ x; yÞ, hence, the stabilization term will have

no contribution for the stabilization if N;mnðxÞ 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

displace-ment fields (the assumed displacedisplace-ment 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 domainXs

k



u;mnðxkÞ ¼

Z

X su;mnðxÞWðx; x  xkÞdX¼ 1

Ask

Z

C su;mðxÞnndC

Ask

Z

C s

X

i2n s

Ni;mðxÞui

0

@

1

AnndC

i2n s

1

Ask

Z

C sNi;mðxÞnndCui: ð51Þ

In the same way, the smoothed second-order derivatives of

electric potential, u;ijover the smoothing domainXs

k can be ex-pressed by:



u;mnðxkÞ ¼

Z

Xs

Ask

Z

Cs

u;mðxÞnndC

i2n s

1

Ask

Z

C sNi;mðxÞnndC ui: ð52Þ

From Eqs.(51) and (52), it is clear that smoothing operation relaxes the requirement of field function Consequently, the smoothed sec-ond-order derivative only requires C0continuity, so that the square-residuals can be evaluated using linear elements

Next, the smoothing operation is applied to the divergence of the Cauchy stress tensorr In this regard, one can arrive at the smoothed divergence of the Cauchy stress tensor that can be ex-pressed in the following vector form:

div r¼ rxx;xþ rxy;y



ryy;yþ ryx;x

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

m

as

Cs

m¼ c11 c12 c13 c31 c32 c33 e11 e21 e13 e23

we have

div r¼ rxx;xþ ryx;x



ryy;yþ rxy;y

where Ksis defined as

Ks¼ exx;x eyy;x cxy;x exx;y eyy;y cxy;y Ex;x Ey;x Ex;y Ey;y

: ð56Þ

Employing the strain–displacement and electric field-potential relationships in Eq.(5), then

Ks¼ ½ux;xx uy;yx ux;yxþ uy;xx ux;xy uy;yy ux;yyþ uy;xy u;xx u;yx u;xy u;yyT:

ð57Þ Substituting Eqs.(51) and (52)into Ksleads to the smoothed the smoothed divergence of the Cauchy stress tensor over the smooth-ing domainXsk

div r¼ rxx;xþ ryx;x



ryy;yþ rxy;y

¼ CsmKs¼ CsmX

i2n s

Bs

idi; ð58Þ

where ns

kis the set of nodes associated with the smoothing domain

Xsk; di¼ u½ i uiT, and Bs

i is expressed by

Bs

i¼ Ni;xx 0 Ni;yx Ni;xy 0 Ni;yy

where

Ask

Z

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

Dy;y

¼ CseKs¼ CseX

i2n s

Bs

where Cs

eis defined by

Cs

e¼ e½ 11 e12 e13 e21 e22 e23 j11 j12 j21 j22:

ð62Þ

Also, it is worth noting that Bsis the same as that given in Eq.(59)

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

equation

duTZ

X

qNTN€udXþ duTZ

X

ðBuÞTcEBuudXþ duT



Z

X

ðBuÞTeTBuudXþ du

Z

X

ðBuÞTeBuudX

Z

X

ðBuÞTjsBuudX duT

Z

X

NTbdX

 duT

Z

C t

NTtCdCþ du

Z

X

NTqsdXþ du

Z

C t

NTqCdCþ2al2c

T



Z

X

ðBsÞTCsm T

CsmBsd þ ðBsÞTCsm T

bdX2al2c



j dd

T



Z

X

ðBsÞT Cse T

CseBsd þ ðBsÞT Cse T

Eliminating du and duyields

m 0

€

u



þ



kuu ku u



kT

u u kuu

þ ks m

þ ks e

u



q



þ ffmg þ ffeg

ð64Þ

in which ks

m; ks

e;fmand feare the newly introduced matrices in the

discretized algebraic equations of system that are then assembled by



ks

m¼2al2c

E

XN s

k¼1

Z

Xs

ðBsÞTðCsmÞTCsmBsdX

¼2al2c

E

XN s

k¼1

ðBsÞTCsm T



ks

e¼ 2al2c



j

XN s

k¼1

Z

X sðBsÞT Cse T

CseBsdX

¼ 2al2c



j

XN s

k¼1

ðBsÞT Cse T

fm¼ 2al2c

E

XN s

k¼1

Z

X sðBsÞTCsm T

fe¼2al2c



j

XN s

k¼1

Z

X sðBsÞT Cse T

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

Section3.2;

(3) loop over smoothing domains

a determine the node connecting information of the

smoothing domainXskassociated with node k;

b calculate the outward unit normal for each boundary

segment of the smoothing domainsXs

k;

c compute the smoothed strain matrix BuðxkÞ and the

smoothed electric field matrix BuðxkÞ by using Eq.(34);

d evaluate the smoothed stiffness matrix kuu for the

mechanical field, kuufor the electric field, and ku u for

the mechanical-electric coupling field over the current

smoothing domain by using Eqs.(39)–(41);

e compute the smoothed matrix for the divergence of stress BsðxkÞ by Eq.(59), and obtain the stabilized stiff-ness matrices ks

m and ks

e from the square-residuals, respectively, of mechanical stress equilibrium and elec-tric 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;

(4) calculate the consistent mass matrix m;

(5) implement essential boundary conditions;

(6) solve the linear system of equations to obtain the nodal dis-placements and electric potentials (static analysis); and eigenmodes and frequencies (eigenvalue problems); (7) post-processing of desired results

6 Numerical examples Benchmark problems are examined to demonstrate the validity

of the proposed stabilization scheme within the framework of NS-FEM for the piezoelectricity The strain energy used in this research

is defined as

EðXÞ¼1 2

Z

X

Numerical errors are then calculated by the following equations

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PN n

i¼1 uexact

i  unumerical i

PN n

i¼1 uexact i

v u

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

PN n

i¼1 uexact

i unumerical i

PN n

i¼1 uexact i

v u

ð71Þ

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 numer-ical method

6.1 Patch test

A standard patch test is first considered, whose nodal distribu-tion and geometry are presented inFig 3 The piezoelectric mate-rial PZT-4 as listed inTable 1is employed in this patch test The boundary conditions for the mechanical displacements and the electric potential are assumed to be[5]

wherer0is an arbitrary stress parameter For the given boundary

conditions, the corresponding analytical solutions for the stresses

rand for the electric displacements D are obtained as

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

fol-low ‘‘exactly’’ (to machine precision) the same linear function of

the imposed boundary conditions of the patch It shows inTable 2

that the desired results gained by the present method with stabilization parametera¼ 0:05 match the exact solutions (other parameters are found to match as well, however, the correspond-ing results are not listed here due to the length limit), and hence the method successfully passes the patch test

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 inFig 4is considered The pie-zoelectric 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 stressr0¼ 1:0 Pa in the y direction on its top and bottom boundaries and an applied voltage V0¼ 1000 V

to the left and right boundaries as shown inFig 4 The piezoelectric material PZT-5 is taken and its related parameters are provided as follows

sE¼

2 6

3 7

 108N=V2:

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 ex-panded 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

ð75Þ

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

ux¼ s13r0x; uy¼d15V0x

L

The numerical simulations of the proposed stabilized NS-FEM are carried out using a regular mesh with nodal distribution of

7  7 as shown inFig 5(a) The mechanical displacements and the electric potential at the central line ðy ¼ 0Þ with stabilization parametersa¼ 0:05 as depicted inFigs 6–8are compared directly

Table 1 Piezoelectric material properties of PZT-4 and PVDF.

cE¼

2 6

3 7

2 6

3 7

5 GPa

0:046 0:046 0

Coulomb=m2

 109F=m

Table 2

The results of patch test.

Variable Results

Fig 4 Piezo-strip under a uniform stress and an applied voltage.

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 parametersa

are employed for the numerical simulations The numerical errors

in displacement and electric potential solutions calculated by Eqs

(70) and (71)respectively are presented inTable 3, which demon-strates that the stabilized NS-FEM using different stabilization parametersain 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 gen-erated in the following fashion

x0¼ x þDx  rcair;

whereDx andDy are initial regular element sizes in x and y direc-tions, respectively rc is a computer-generated random number between 1.0 and 1.0, and air is a prescribed irregularity factor whose value is chosen between 0.5 in this research (seeFig 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

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

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

-5

x (mm)

ux

stabilized NS-FEM α=0.05 Exact

Fig 6 Variation of horizontal displacement u at the central line ðy ¼ 0Þ of the

single-layer piezoelectric strip.

0

0.2

0.4

0.6

0.8

1

1.2x 10

-3

x (mm)

uy

stabilized NS-FEM α=0.05 Exact

Fig 7 Variation of vertical displacementvat the central line ðy ¼ 0Þ of the

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

1x 10

-6

x (mm)

stabilized NS-FEM α=0.05 Exact

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

6.3 Cook’s membrane

A benchmark problem, shown inFig 9, a clamped tapered panel

subjected to a distributed tip load F ¼ 1 N, resulting in deformation

dominated by a bending response, is then analyzed The piezoelec-tric material PZT-4 whose parameters listed inTable 1is 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 toFig 9 Four discretizations (3-node triangular elements) with uniform nodal distribution: (5  5; 9  9; 17  17, and 33  33Þ, are used for the present stabilized NS-FEM (sNS-FEM-T3) For comparison, such

Table 3

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

e u

e u

Fig 9 Geometry and boundary conditions of Cook’s membrane.

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3x 10

-4

DOF

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.

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

(a2 ½0:0; 0:1Þ The lower bound solution is obtained using the FEM-T3 and the

FEM-Q4.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

x 10-8

DOF

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.

Fig 11 Comparison of electric potential at point A of Cook’s membrane.

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5x 10

-5

DOF

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.

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

With regard to the nodal integrated methods, direct nodal

inte-gration leads to a numerically spatial instability in meshfree settings... 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

Section3.2;... node as shown inFig

3.3 Smoothed Galerkin weak form and discrete equations Because a smoothed Galerkin weak form with smoothed gradi-ent over smoothing domains is variationally consistgradi-ent