modeling and simulation of piezocomposites

Comput. Methods Appl. Mech. Engrg. 192 (2003) 3211–3232

www.elsevier.com/locate/cma

Modeling and simulation of piezocomposites

Jacob Fish *, Wen Chen

Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 4042 Johnnson Engineering Center,

Troy, NY 12180-3590, USA

Received 12 May 2003; accepted 12 May 2003

Abstract

Solution procedures for large-scale transient analysis of piezocomposites are developed. Computational efficiency of

unconditionally stable and conditionally stable implicit partitioned schemes as well as explicit partitioned schemes is

investigated in the context of multilevel solution methods applied to separate field equations.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

Multiphysics modeling and simulation has recently gained significant attention in computational science

and engineering community. Example problems falling into this category are coupled mechanical, thermal,

electrical, diffusion-reaction, fluid-structure interaction problems. In multiphysics problems, typically awide range of interacting spatial and temporal scales are involved resulting in large coupled time-dependent

system of equations.

Generally, there are three approaches for simulation of coupled systems [1]: field elimination, monolithic

and partitioned schemes. Field elimination is restricted to linear (or linearized) problems that permit effi-

cient decoupling. The monolithic scheme is in general computationally challenging, economically subop-

timal and software-wise unmanageable [2]. On the other hand, the partitioned treatment, which is the focus

of the present paper, is computationally efficient, flexible and software reusable.

In partitioned schemes, computational efficiency is counteracted by the fact that satisfactory numericalstability is hard to achieve. In fact the practical feasibility of the partitioned approach hinges entirely on the

stability analysis [1]. Stability analysis by standard Fourier techniques using a scalar test equation is not

generally possible because modes of individual subsystems are not modes of the coupled problem. More-

over, partitioned procedures are in general less accurate than their underlying subsystem time integrators.

In principle, higher order accuracy can be recovered by iterating the solution between the fields at each time

step [3]. However, the computational cost of interfield iterations may overshadow the gains from reduced

time steping [1].

* Corresponding author. Tel.: +1-518-276-6191; fax: +1-518-276-4833.

E-mail address: [email protected] (J. Fish).

0045-7825/03/$ - see front matter � 2003 Elsevier B.V. All rights reserved.

doi:10.1016/S0045-7825(03)00343-8

mail to: [email protected]

3212 J. Fish, W. Chen / Comput. Methods Appl. Mech. Engrg. 192 (2003) 3211–3232

In this paper we focus on the problem of piezoelectricity in periodic heterogeneous media. We start byformulating the monolithic equations and study stability of implicit time integrators (the Newmark

method) applied to the monolithic equations. It is demonstrated that taking the integration parameters

which render the Newmark method unconditionally stable for conventional structural dynamics problems

do not guarantee unconditional stability of the monolithic system. This is due to the fact that in structural

dynamics problems, the global mass matrix is positive definite and the stiffness matrix is at least positive

semi-definite, whereas for piezocomposites the global stiffness matrix of the monolithic scheme is indefinite,

and thus conventional implicit integrators directly applied to the monolithic system may not be stable.

From the monolithic equations, we construct the implicit partitioned schemes based on the block Gauss–Seidel method. It is shown that the partitioned scheme constructed directly from the monolithic equations

does not guarantee unconditional stability. Unconditionally stable partitioned scheme can be obtained by

augmenting the original equations of motion and subsequently splitting the field equations along the lines

of block Gauss–Seidel method.

Computational efficiency of solving the problem of piezoelectricity is the main focus of the present

paper. While partitioned approach reduces the solution of coupled physics problem into a sequence of

single field problems, the computational cost could be still insurmountable due to the existence of multiple

spatial scales within each physical process. For this type of problems the direct method may not compu-tationally optimal, while various multigrid methods though very efficient for differential equations with

constant coefficients, exhibit poor rates of convergence when applied to differential equations with rapidly

oscillating coefficients [4,5]. The representative volume element (RVE)-based multilevel method developed

by the authors has been shown to possess high rates of convergence for single-field structural dynamics

problems in periodic heterogeneous media [6]. In the present paper, we extend the framework of the RVE-

based multilevel method to transient analysis of piezocomposites.

Convergence properties of the RVE-based multilevel method are studied in the context of uncondi-

tionally stable and conditionally stable implicit and explicit partitioned schemes. We show that the rate ofconvergence of the RVE-based multilevel method for conditionally stable implicit and explicit partitioned

schemes is higher than for unconditionally stable scheme. This is in part due to the fact that the eigenspace

of the stabilized matrix cannot be reproduced by a linear combination of the RVEs. For high frequency

applications, the conditionally stable implicit and explicit partitioned schemes are advantageous, since the

choice of time step is governed by accuracy considerations, whereas the unconditionally stable schemes

might be better suited for low frequency applications.

2. Finite element formulation of piezoelectricity

2.1. Governing equations of piezoelectricity

Let X denote the domain of interest, and C the boundary of X. Consider the governing equations of

linear piezoelectricity, which include:

ii(i) Mechanical equilibrium

rij;j þ fi ¼ q€uui; ð1Þi(ii) Maxwell�s equation of equilibrium (assumed to be quasi-static and decoupled from mechanical fields)

Di;i ¼ 0; ð2Þ(iii) Constitutive equations

rij ¼ CijklSkl � ekijEk; Di ¼ eiklSkl þ eikEk; ð3Þ

J. Fish, W. Chen / Comput. Methods Appl. Mech. Engrg. 192 (2003) 3211–3232 3213

(iv) Strain–displacement relation and the curl-free electric field

Sij ¼ 12ðui;j þ uj;iÞ; Ei ¼ �/;i; ð4Þ

where r, S, E and D are the stress, strain tensors, electric field and electric displacement vectors, respec-tively; C , e, and e are the elastic tensor at constant electric field, dielectric tensor at constant strain field and

the piezoelectric tensor, respectively; u and / denote the displacement vector and electric potential (volt-

age), respectively; q is the mass density and f the body force. The superposed dot and comma denote

temporal and spatial differentiation, respectively. Since the piezoelectric tensor e is a third-rank tensor,

piezoelectric coupling exists in intrinsically anisotropic materials only [7].

To complete the description of the problem, the boundary conditions are prescribed as

ui ¼ �uui on Cu and / ¼ �// on C/; ð5Þ

rijnj ¼ T i on Cr and Dini ¼ �qq on Cq; ð6Þ

Cu [ Cr ¼ C; Cu \ Cr ¼ 0; C/ [ Cq ¼ C; C/ \ Cq ¼ 0; ð7Þwhere �uu and T are the prescribed mechanical displacement and surface traction, respectively; �// and �qq are

the prescribed electric potential and surface charge, respectively; n denotes the outward unit normal vector.

2.2. The weak formulation and semi-discrete equations of motion

From the governing equations (1)–(4) and boundary conditions (5)–(7), we establish the weak formu-

lation of the problem:

For t 2 ð0; T �, find uiðx; tÞ 2 V �uum and /ðxÞ 2 V �//

e , such thatZX½q€uui � ðrij;j þ fiÞ�wi dX ¼ 0 8wi 2 V 0

m ; ð8Þ

ZXDi;iWdX ¼ 0 8W 2 V 0

e ; ð9Þ

where

V �uum ¼ wðxÞ 2 H 1ðXÞ;w

n¼ �uu on Cu

o; ð10Þ

V�//

e ¼ /ðxÞ 2 H 1ðXÞ;/n

¼ �// on C/

o; ð11Þ

V 0m ¼ V �uu

m

��uu¼0

; V 0e ¼ V

�//e

��//¼0

; ð12Þ

H 1ðXÞ ¼ w�

¼ wðxÞ; x 2 Xjw;rwi 2 L2ðXÞ�

ð13Þ

with L2ðXÞ denoting the set of square-integrable functions over X.Integrating Eqs. (8) and (9) by parts and substituting the constitutive relations (3), the kinematics re-

lation (4) and the natural boundary conditions (6) into the resulting equations, and exploiting the symmetry

of the elastic tensor, we haveZXqwi€uui dXþ

ZXSijðwÞ CijklSklðuÞ

�þ ekij/;k

�dX ¼

ZXfiwi dXþ

ZCr

T iwi ds; ð14Þ


ZXW;i eiklSklðuÞ

�� eik/;k

�dX ¼

ZCq

�qqWds; ð15Þ

where SðwÞ ¼ rsw is the symmetric gradient of w.Subsequently the matrix notation is adopted. We employ finite element discretization with element shape

functions denoted as Nu and N/

u ¼ NuUe; / ¼ N/U

e; ð16Þwhere Ue and Ue are element nodal values. Differentiating Eq. (16) yields the element strain and electricfield vectors

S ¼ BuUe; E ¼ �r/ ¼ �B/U

e: ð17ÞInserting Eqs. (16) and (17) into (14) and (15) we obtain the semi-discrete equations of motion of piezo-

electricity

Muu€UU þ KuuU þ Ku/U ¼ F; K/uU � K//U ¼ Q; ð18Þ

where the assembly operators are defined in a usual manner:

Muu ¼Ne

e¼1

ZXe

qNTuNu dXe consistent mass matrix;

Kuu ¼Ne

e¼1

ZXe

BTuCBu dXe mechanical stiffness matrix;

Ku/ ¼Ne

e¼1

ZXe

BTu e

TB/ dXe; K/u ¼Ne

e¼1

ZXe

BT/eBu dXe piezoelectric stiffness matrices;

K// ¼Ne

e¼1

ZXe

BT/eB/ dXe dielectric stiffness matrix;

F ¼Ne

e¼1

ZXe

NTu f dXe þ

Sre

e¼1

ZCer

NTuT ds; Q ¼

Sqe

e¼1

ZCeq

NT/�qqds forcing vectors;

with Ne and Sre , S

qe denoting the total number of elements and the number of elements with prescribed

boundary conditions, respectively.

3. Field elimination and monolithic solution schemes

3.1. Static condensation

Suitably grounding the structure by specifying one or more nodal values of the potential renders K//

non-singular. From the second equation in (18), we have

U ¼ K�1// KT

u/U�

�Q�: ð19Þ

Substituting the above equation into the first equation in (18) yields

Muu€UU þ eKKU ¼ eFF ; ð20Þ


whereeKK ¼ Kuu þ Ku/K�1//K

Tu/;

eFF ¼ F þ Ku/K�1//Q: ð21Þ

Eq. (20) is obtained by field elimination or static condensation of the electric field. It can be observed

that eKK and Muu are positive definite, and Eq. (20) can be easily solved by using either implicit or explicit

time integrator. This approach has been adopted in [8,9]. While relatively easy to implement for small

problems, static condensation suffers a number of drawbacks. First, the condensed stiffness matrix eKK is

denser than Kuu, and second, it destroys spectral properties of Kuu, which could be taken advantage of in

designing efficient multilevel iterative solvers.

3.2. Monolithic scheme

Exploiting K/u ¼ KTu/, we can rearrange Eq. (18) in the monolithic form:

M€dd þ Kd ¼ F; ð22Þwhere

M ¼ Muu 0

0 0

� ; K ¼ Kuu Ku/

KTu/ �K//

� ; d ¼ U

U

� ; F ¼ F

Q

� : ð23Þ

It can be seen that M is positive semi-definite and K indefinite. The monolithic scheme for piezoelec-

tricity has been adopted by a number of authors [10–12]. However, prior to applying time integration

scheme with proven stability behavior to the monolithic equation (22) it is necessary to examine its suit-ability for problems of piezoelectricity. We focus on Newmark integrator

dnþ1 ¼ dn þ Dtvn þ Dt2 12

�� b

�an þ banþ1

�; ð24Þ

vnþ1 ¼ vn þ Dt ð1½ � cÞan þ canþ1�; ð25Þwhere dn, vn and an are the displacement, velocity and acceleration vectors at time tn, respectively; Dt is thetime step size; b and c are parameters which determine the stability and accuracy of the Newmark method.

For structural dynamics the Newmark scheme is unconditionally stable for 2bP cP 1=2.The stability of the Newmark method can be investigated by using the energy method (cf. [13,14]).

Eqs. (22) (with F ¼ 0), (24) and (25) can be combined to form the following identity:

aTnþ1Banþ1 þ vTnþ1Kvnþ1 ¼ aTnBan þ vTnKvn � ð2c� 1Þ½an�TB½an�; ð26Þ

where

B ¼ M þ Dt2 b

� 12c�K ; ½an� ¼ anþ1 � an: ð27Þ

From the above two equations, it can be observed that if cP 1=2 and B is positive definite, we have

aTnþ1Banþ1 þ vTnþ1Kvnþ1 6 aTnBan þ vTnKvn ð28Þ

which implies

aTnBan þ vTnKvn 6 aT0Ba0 þ vT0Kv0; n ¼ 1; 2; 3; . . . ð29ÞTherefore, an and vn are bounded. If K is non-singular, from Eq. (22) (with F ¼ 0), we have kdnk6

kK�1kkMkkank, and then dn is bounded. For M and K to be positive definite and at least positive semi-

definite respectively, as is the case of elasto-dynamics, the matrix B is positive definite for 2bP c, which is

the well-known condition for the Newmark scheme to be unconditionally stable. In the case of piezo-

electricity M and K are positive semi-definite and indefinite, respectively, and thus the condition of 2bP c


does not guarantee positive-definiteness of B. Nevertheless, in case of the average acceleration method (ortrapezoidal rule) with b ¼ 1=4 and c ¼ 1=2, the monolithic scheme applied to linear piezoelectricity is

unconditionally stable, in which case Eqs. (26) and (29) reduce to

aTnþ1Manþ1 þ vTnþ1Kvnþ1 ¼ aTnBan þ vTnKvn; ð30Þ

aTnMan þ vTnKvn ¼ aT0Ma0 þ vT0Kv0; n ¼ 1; 2; 3; . . . ð31Þ

4. Partitioned schemes

Applying the trapezoidal rule to the monolithic equation (22) yields the implicit unconditionally stable

integration scheme, as described in the previous section:

bKKdnþ1 ¼ bFF nþ1; vnþ1 ¼2

Dtðdnþ1 � dnÞ � vn; ð32Þ

where

bKK ¼ K þ 4

Dt2M ; bFF nþ1 ¼

4

Dt2M

�� K

dn þ

4

DtMvn þ Fn þ Fnþ1 ð33Þ

are the effective stiffness matrix and load vectors, respectively. First, we consider implicit partitioned

schemes directly constructed from the above monolithic scheme and show that the resulting partitionedscheme is not unconditionally stable. Unconditionally stable implicit partitioned scheme is then obtained

via augmentation. An explicit partitioned scheme is also constructed from the explicit integration of

mechanical equation.

4.1. Partitioning directly from the monolithic scheme

Partitioned schemes can be constructed directly from the monolithic scheme by splitting the effective

stiffness matrix. The first equation of the monolithic scheme (32) can be written as

bKK uu Ku/

KTu/ �K//

" #Unþ1

Unþ1

� ¼

bFF unþ1

Qnþ1

" #; ð34Þ

where

bKK uu ¼ Kuu þ4

Dt2Muu; bFF u

nþ1 ¼ Muu4

Dt2Un

�þ 4

Dt_UUn þ €UUn

þ Fnþ1: ð35Þ

We start by considering a partitioned scheme where the electric field is predicted first. Splitting the

coefficient matrix of (34) based on the block Gauss–Seidel method yields

bKK uu 0

KTu/ �K//

" #Unþ1

Unþ1

� ¼ � 0 Ku/

0 0

� Unþ1

Unþ1

� þ

bFF unþ1

Qnþ1

" #: ð36Þ

Utilizing the last-step-value predictor on the right hand side of (36), we havebKK uu 0

KTu/ �K//

" #Unþ1

Unþ1

� ¼ � 0 Ku/

0 0

� Un

Un

� þ

bFF unþ1

Qnþ1

" #: ð37Þ


Expanding the above equation yields the electric field predicted partitioned scheme:

bKK uuUnþ1 ¼ bFF unþ1 � Ku/Un; K//Unþ1 ¼ �Qnþ1 þ KT

u/Unþ1: ð38Þ

Alternatively, we can predict the mechanical field. The monolithic equation (34) can be written as

�K// KTu/

Ku/bKK uu

" #Unþ1

Unþ1

� ¼ Qnþ1bFF u

nþ1

" #: ð39Þ

Splitting the coefficient matrix of the above equation based on the block Gauss–Seidel method yields

�K// 0

Ku/bKK uu

� Unþ1

Unþ1

� ¼ � 0 KT

u/

0 0

� Unþ1

Unþ1

� þ Qnþ1bFF u

nþ1

" #: ð40Þ

Utilizing the last-step-value predictor and expanding the resulting equation lead to the mechanical fieldpredicted partitioned scheme

K//Unþ1 ¼ �Qnþ1 þ KTu/Un; bKK uuUnþ1 ¼ bFF u

nþ1 � Ku/Unþ1: ð41Þ

Both partitioned schemes solve the two field equations independently at each time step.

4.1.1. Stability analysis of the implicit partitioned schemes

Consider the partitioned procedures (38) and (41) which predict the electric field and mechanical field,

respectively. Both partitioned procedures compute the velocity and acceleration fields based on the fol-

lowing equations:

_UUnþ1 ¼ _UUn þDt2

€UUn

hþM�1

uu ðFnþ1 � KuuUnþ1 � Ku/Unþ1Þi; ð42Þ

€UUnþ1 ¼ M�1uu ðFnþ1 � KuuUnþ1 � Ku/Unþ1Þ: ð43Þ

For the purpose of stability analysis, it suffices to consider the force-free case, i.e., Fnþ1 ¼ Qnþ1 ¼ 0.

Stability analysis is carried out using standard procedures [15,16]. Non-trivial solutions of the partitioned

procedures are sought in the form

Unþ1 ¼ kUn; _UUnþ1 ¼ k _UUn; €UUnþ1 ¼ k €UUn; Unþ1 ¼ kUn: ð44ÞStability requires jkj6 1. When the modulus is unity, the root must be a simple one [17]. Replacing k

by ð1þ zÞ=ð1� zÞ, stability condition reduces to

ReðzÞ6 0: ð45ÞFirst, we consider the mechanical field predicted partitioned procedure. Substituting Eq. (44) into (41)–

(43) yields

� 1þ z1� z

K// KTu/

Ku/ Kuu þ4z2

Dt2Muu

264375 Un

Un

� ¼ 0: ð46Þ

The characteristic polynomial associated with (46) is

z34

Dt2Muu

�� þ z24

Dt2Muu þ z Kuu

�� Ku/K

�1//K

Tu/

�þ Kuu þ Ku/K

�1//K

Tu/

�� ¼ 0; ð47Þ


where j � j denotes the matrix determinant. The necessary condition for stability is that all coefficient ma-

trices of the polynomial in z should be positive definite (the first part of Routh–Hurwitz conditions [17,18]).

Since Muu and Kuu þ Ku/K�1//K

Tu/ are positive definite, but Kuu � Ku/K

�1//K

Tu/ is indefinite, the necessary

condition for stability cannot be satisfied.

Next, we investigate the electric field predicted partitioned scheme. Substituting Eq. (44) into (38), (42)

and (43) yields

K// �KTu/

ð1� z2ÞKu/ Kuu þ4z2

Dt2Muu

24 35 Un

Un

� ¼ 0: ð48Þ

The characteristic polynomial of (48) is

Muu

�� Dt2

4Ku/K

�1//K

Tu/

z2 þ Dt2

4Kuu

�þ Ku/K

�1//K

Tu/

�� ¼ 0: ð49Þ

To investigate the stability properties of the above equation it is convenient to utilize the theorem of

Bellman [18], which states: If A, B and C are non-negative definite, and either A or C positive definite, then

z2A�� þ 2zB þ C

�� ¼ 0 ð50Þ

has no roots with positive real parts. The proof is given in [18].In view of the above theorem, it is obvious that the necessary and sufficient condition for stability of the

electric field predicted partitioned scheme is that the matrix Muu � Dt2

4Ku/K

�1//K

Tu/ is non-negative definite,

which can be achieved with sufficiently small Dt. Therefore, the scheme is conditionally stable.

We proceed to investigate the critical time step of the above scheme. Consider the eigenvalue problem

Muu

�� Dt2

4Ku/K

�1//K

Tu/

x ¼ cx: ð51Þ

The necessary and sufficient condition for stability becomes

cP 0: ð52Þ

From the generalized eigenvalue problem

Ku/K�1//K

Tu/

h� ðximÞ2Muu

ix ¼ 0; ð53Þ

we have

M�1uu Ku/K

�1//K

Tu/x ¼ ðximÞ2x ð54Þ

Premultiplying (51) by M�1uu yields

x� Dt2

4M�1

uu Ku/K�1//K

Tu/x ¼ cM�1

uu x: ð55Þ

Substituting (54) into the above equation yields

cM�1uu x ¼ 1

h� ðximDtÞ2=4

ix: ð56Þ

If c ¼ 0, we have from the above equation

Dt ¼ 2=xim: ð57Þ


If c 6¼ 0, we have from (56)

M�1uu x ¼

1� ðximDtÞ2=4h i

cx: ð58Þ

The positive definiteness of M�1uu and Eq. (52) lead to

1� ðximDtÞ2=4 > 0: ð59Þ

From Eqs. (57) and (59), it follows that

Dt6 2=xim: ð60Þ

The above stability condition must be satisfied for each system mode. Consequently, the highest fre-

quency ximn is critical and must satisfy (60). Therefore, the stability condition becomes

Dt6Dtimcr ¼ 2=ximn : ð61Þ

4.1.2. Consistency and accuracy of the electric field predicted implicit scheme

By employing the equilibrium equation at tn to eliminate €UUn, we can express the electric field predicted

partitioned scheme (38), (42) and (43) in the following form

dnþ1 ¼ Bdn þ Pn; ð62Þ

where B ¼ B�11 B2,

B1 ¼

bKK uu 0 0

KTu/ 0 �K//

Dt2M�1

uu Kuu IDt2M�1

uu Ku/

266664377775; B2 ¼

4

Dt2Muu � Kuu

4

DtMuu �2Ku/

0 0 0

�Dt2M�1

uu Kuu I �Dt2M�1

uu Ku/

2666664

3777775;

dn ¼

Un

_UUn

Un

26643775; Pn ¼ B�1

1

Fn þ Fnþ1

Qnþ1

Dt2M�1

uu ðFn þ Fnþ1Þ

266664377775:

ð63Þ

Inserting the exact solution dðtnÞ into (62), we have

dðtnþ1Þ ¼ BdðtnÞ þ Pn þRðtnÞ; ð64Þ

where RðtnÞ ¼ ½RTu ðtnÞ _RRT

u ðtuÞ RT/ðtnÞ �

Tis the local truncation error. Inserting the following Taylor

expansions:

dðtnþ1Þ ¼ dðtnÞ þ Dt _ddðtnÞ þDt2

2€ddðtnÞ þ

Dt3

6

vdðtnÞ þOðDt4Þ; ð65Þ

Fnþ1 ¼ F þ Dt _FFn þDt2

2€FFn þ

Dt3

6

vFn þOðDt4Þ; ð66Þ


Qnþ1 ¼ Qn þ Dt _QQn þDt2

2€QQn þ

Dt3

6

vQn þOðDt4Þ ð67Þ

into Eq. (64) and expanding the resulting equation, it can be shown that

RðtnÞ � OðDt3Þ and RðtnÞ ! 0 as Dt ! 0: ð68ÞTherefore, the electric field predicted partitioned scheme is consistent and second-order accurate.

4.2. Unconditionally stable implicit partitioned scheme

4.2.1. Stabilization by semi-algebraic augmentation

The stabilization is carried out by a semi-algebraic augmentation technique proposed by Farhat et al.

[15] for thermoelastic problems.

First, the structural equation in (18) is integrated using the trapezoidal rule:

_UUnþ1 ¼ _UUn þDt2ð €UUn þ €UUnþ1Þ ¼ _UUn þ

Dt2

€UUn

hþM�1


Unþ1 ¼ Un þ Dt _UUn þDt2

4ð €UUn þ €UUnþ1Þ ¼ Un þ Dt _UUn

Dt2

4€UUn

hþM�1

uu ðFnþ1 � KuuUnþ1 � Ku/Unþ1Þi:

ð70ÞConsider the electric equation

KTu/Unþ1 � K//Unþ1 ¼ Qnþ1: ð71Þ

Substituting Eq. (70) into the above equation yields

bKK /uUnþ1 þ bKK //Unþ1 ¼ bFF/nþ1; ð72Þ

where

bKK /u ¼Dt2

4KT

u/M�1uu Kuu; bKK // ¼ K// þ Dt2

4KT

u/M�1uu Ku/;

bFF/nþ1 ¼

Dt2

4KT

u/M�1uu Fnþ1 þ KT

u/ Un

�þ Dt _UUn þ

Dt2

4€UUn

�Qnþ1:

ð73Þ

Rearranging (70) yields

bKK uuUnþ1 þ Ku/Unþ1 ¼ bFF unþ1; ð74Þ

where bKK uu and bFF unþ1 are given in (35). Eqs. (72) and (74) can be written as

bKK //bKK /u

Ku/bKK uu

" #Unþ1

Unþ1

� ¼

bFF/nþ1bFF unþ1

" #: ð75Þ

Splitting the coefficient matrix in (75) using the block Gauss–Seidel method and last-step-value pre-

dictor, we have

bKK // 0

Ku/bKK uu

" #Unþ1

Unþ1

� ¼ � 0 bKK /u

0 0

� Un

Un

� þ

bFF/nþ1bFF unþ1

" #: ð76Þ


Expanding the above equation and using (69) yield the following partitioned algorithm:

1. Predict the mechanical field

Upnþ1 ¼ Un: ð77Þ

2. Solve for the electric field

bKK //Unþ1 ¼ bFF/nþ1 � bKK /uU

pnþ1: ð78Þ

3. Correct the mechanical field

bKK uuUnþ1 ¼ bFF unþ1 � Ku/Unþ1: ð79Þ

4. Compute velocity and acceleration fields

_UUnþ1 ¼ _UUn þDt2

€UUn

hþM�1


€UUnþ1 ¼ M�1uu ðFnþ1 � KuuUnþ1 � Ku/Unþ1Þ: ð81Þ

4.2.2. Stability and accuracy of the stabilized implicit scheme

4.2.2.1. Stability. We consider the force-free case, i.e. Fnþ1 ¼ Qnþ1 ¼ 0. The stability of the proposed

partitioned scheme can be examined by seeking a non-trivial solution in the form of (44). Substituting (77)

into (78), and (44) into (78)–(81), we have after some algebraic manipulations

Kuu þ4z2

Dt2Muu Ku/

Dt2

2ð1� z2ÞVKuu � zð1� zÞK�1

//KTu/ zðzþ 1ÞI þ Dt2

2VK u/

264375 Un

Un

� ¼ 0; ð82Þ

where

V ¼ K�1//K

Tu/M

�1uu : ð83Þ

The characteristic polynomial associated with (82) is

det Muuz3�

þMuuz2 þDt2z4

Kuu

�þ Ku/K

�1//K

Tu/ I

�þ Dt2

2M�1

uu Kuu

þ Dt2

4Kuu

�þ Ku/K

�1//K

Tu/

��¼ 0:

ð84ÞRecall that Muu, Kuu and K// are positive definite; Ku/K

�1//K

Tu/ is positive definite provided that Ku/ has

full column rank and positive semi-definite if Ku/ is column rank deficient. In any case, all coefficient

matrices of the characteristic polynomial (84) are positive definite. Therefore, the first part of the Routh–

Hurwitz criterion for unconditional stability is satisfied. In order to check the second part of this criterion,

we consider a 2-dof model problem. The corresponding scalar form of (84) is

a3z3 þ a2z2 þ a1zþ a0 ¼ 0; ð85Þwhere

a3 ¼ a2 ¼ 1; a1 ¼Dt2

4x2

�þ e2

mu1

�þ Dt2

2x2

; a0 ¼

Dt2

4x2

�þ e2

mu

: ð86Þ


Since Dt, u, x2, e2 and m > 0, all coefficients of the polynomial (85) are positive. Moreover

a1a2 � a0a3 ¼Dt4

8

e2x2

mu> 0; ð87Þ

which verifies that the second part of the Routh–Hurwitz criterion is satisfied and therefore the partitionedprocedure is unconditionally stable.

4.2.2.2. Consistency and accuracy. Using the equilibrium equation at tn to eliminate €UUn in expressions (35),

(73) and (80) for bFF unþ1,

bFF/nþ1 and

_UUnþ1, respectively, and substituting the resulting effective load vectors into

(78) and (79), then writing the resulting equations in a matrix form yield

dnþ1 ¼ Rdn þ Ln; ð88Þ

where dn is given in (63) and

R ¼ R�11 R2; Ln ¼ R�1

1 L0; ð89Þ

R1 ¼I þ Dt2M�1

uu Kuu=4 0 Dt2M�1uu Ku/=4

DtM�1uu Kuu=2 I DtM�1

uu Ku/=2

0 0 K// þ Dt2KTu/M

�1uu Ku/=4

264375;

R2 ¼I � Dt2M�1

uu Kuu=4 DtI �Dt2M�1uu Ku/=4

�DtM�1uu Kuu=2 I �DtM�1

uu Ku/=2

KTu/ � Dt2KT

u/M�1uu Kuu=2 DtKT

u/ �Dt2KTu/M

�1uu Ku/=4

264375;

L0 ¼Dt2M�1

uu ðFn þ Fnþ1Þ=4DtM�1

uu ðFn þ Fnþ1Þ=2Dt2KT

u/M�1uu ðFn þ Fnþ1Þ=4�Qnþ1

264375:

ð90Þ

Inserting the solution dðtnÞ into (88), we have

dðtnþ1Þ ¼ RdðtnÞ þ Ln þ sðtnÞ; ð91Þ

where sðtnÞ ¼ sTu ðtnÞ _ssTu ðtnÞ s/ðtnÞ� �T

is the local truncation error. Inserting the Taylor expansions (65)–

(67) into (91) and expanding the resulting equation yield

suðtnÞ � OðDt3Þ; _ssuðtnÞ � OðDt3Þ; s/ðtnÞ � OðDt3Þ; ð92Þ

sðtnÞ � OðDt3Þ and sðtnÞ ! 0 as Dt ! 0: ð93Þ

Therefore, the partitioned scheme is consistent and second-order accurate.

4.3. Explicit partitioned scheme

The explicit partitioned scheme can be also constructed by integrating the mechanical equation directly

using an explicit integrator, e.g. the central difference scheme for the velocity and acceleration fields

_UUn ¼1

2DtðUnþ1 �Un�1Þ; €UUn ¼

1

Dt2ðUnþ1 � 2Un þUn�1Þ: ð94Þ


The mechanical equation of motion at tn is

Muu€UUn þ KuuUn þ Ku/Un ¼ Fn: ð95Þ

Substituting the second equation in (94) into (95) yields

1

Dt2MuuUnþ1 ¼ bPPn; ð96Þ

where

bPPn ¼ Fn þ2

Dt2Muu

�� Kuu

Un �

1

Dt2MuuUn�1 � Ku/Un: ð97Þ

The resulting algorithm for the explicit partitioned scheme is:

1. Evaluate the mechanical field by Eq. (96). Note that if lumped mass matrix is employed, there is no

solution of equations for the mechanical field.

2. Evaluate the electric field by

K//Unþ1 ¼ KTu/Unþ1 �Qnþ1: ð98Þ

3. If required, compute the velocity and acceleration fields at tn using Eq. (94).

Based on Eq. (94), the starting value of displacements is taken as

U�1 ¼ U0 � Dt _UU0 þDt2

2€UU0: ð99Þ

4.3.1. Stability of the explicit partitioned scheme

As before, we seek for a non-trivial solution in the form of (44). Substituting Eq. (97) into (96), and (44)

into (96) and (98) yields

�K// KTu/

Ku/ Kuu þ4z2

ð1� z2ÞDt2 Muu

24 35 Un

Un

� ¼ 0: ð100Þ

The characteristic polynomial of (100) is

det z2 Muu

�� Dt2

4Kuu

�þ Ku/K

�1//K

Tu/

�þ Dt2

4Kuu

�þ Ku/K

�1//K

Tu/

��¼ 0: ð101Þ

Since Kuu þ Ku/K�1//K

Tu/ is positive definite, by the theorem of Bellman, the stability condition of the

explicit partitioned scheme becomes

Muu �Dt2

4Kuu

�þ Ku/K

�1//K

Tu/

�P 0: ð102Þ

By similar arguments described in Section 4.1.1, it follows that the explicit partitioned scheme is con-

ditionally stable and the condition for stability is governed by

Dt6Dtexcr ¼ 2=xexn ; ð103Þ

where xexn is the largest value of the following generalized eigenvalue problem

Kuu

�hþ Ku/K

�1//K

Tu/

�� ðxexÞ2Muu

ix ¼ 0: ð104Þ


Since Kuu is positive definite, by comparing Eqs. (53) and (104), we have

xexn > xim

n ; and thus Dtexcr < Dtimcr : ð105Þ

4.3.2. Consistency and accuracy

Substituting Eq. (97) into (96) and (98), we have

ynþ1 ¼ A1yn þ A2yn�1 þ Ln; ð106Þwhere

yn ¼Un

Un

� ; A1 ¼

2I � Dt2M�1uu Kuu �Dt2M�1

uu Ku/

K�1//K

Tu/ 2I � Dt2M�1

uu Kuu

��Dt2K�1

//KTu/M

�1uu Ku/

� ; ð107Þ

A2 ¼�I 0

�K�1//K

Tu/ 0

� ; Ln ¼

Dt2M�1uu Fn

Dt2K�1//K

Tu/M

�1uu Fn � K�1

//Qnþ1

� : ð108Þ

Inserting the exact solution yðtnÞ into (106) yields

yðtnþ1Þ ¼ A1yðtnÞ þ A2yðtn�1Þ þ Ln þ rðtnÞ; ð109Þwhere rðtnÞ ¼ rTu ðtnÞ rT/ðtnÞ

� �Tis the local truncation error. Inserting the Taylor expansions

yðtnþ1Þ ¼ yðtnÞ þ Dt _yyðtnÞ þDt2

2€yyðtnÞ þ

Dt3

6vyðtnÞ þOðDt4Þ; ð110Þ

yðtn�1Þ ¼ yðtnÞ � Dt _yyðtnÞ þDt2

2€yyðtnÞ �

Dt3

6vyðtnÞ þOðDt4Þ; ð111Þ

and (67) into (109), yield

ruðtnÞ � OðDt4Þ; r/ðtnÞ � OðDt4Þ: ð112ÞSubstituting the exact solutions into (94), we have

_UUðtnÞ ¼1

2DtUðtnþ1Þ½ �Uðtn�1Þ� þ _rruðtnÞ; ð113Þ

€UUðtnÞ ¼1

Dt2Uðtnþ1Þ½ � 2UðtnÞ þUðtn�1Þ� þ €rruðtnÞ; ð114Þ

where _rruðtnÞ and €rruðtnÞ are local truncation errors in velocity and acceleration, respectively. Substituting the

expansions (110) and (111) into (113) and (114), yield

_rruðtnÞ � OðDt2Þ; €rruðtnÞ � OðDt2Þ: ð115ÞThe truncation errors vanish as Dt ! 0. Therefore, the explicit partitioned scheme is consistent.

5. The RVE-based multilevel method for partitioned scheme

The explicit and implicit conditionally stable as well as implicit unconditionally stable partitionedschemes proposed in the previous section entail solving linear systems of equations at each time station.

Attention is restricted to piezocomposites where the size of the microstructure is comparable to that of the

wavelength of a traveling signal necessitating discretization on the scale of heterogeneity. The classical

multigrid approach with standard linear interpolation operators applied to the resulting discrete system has


been show to exhibit poor rate of convergence, mainly because the lower frequency subspace is not geo-metrically smooth for problems in heterogeneous media [6].

We focus on the RVE-based multilevel method developed by the authors for structural dynamics

problems [6], in which the special intergrid transfer operators are constructed from the solution of the RVE.

The so called RVE-based multilevel approach has been shown to possess an increasing rate of convergence

with increase in mismatch of properties between microconstituents and decrease in time step size. In the

case of homogeneous media and static analysis, the method has been shown to recover the rate of con-

vergence of the classic multigrid method, i.e., 1/3.

We note that in the mechanical field equations (38) and (79), the coefficient matrix is identical to thatobtained from the conventional structural dynamics problems. Convergence characteristics of the RVE-

based multilevel method for mechanical equations has been studied in [6]. In the remaining of this section

attention is restricted on convergence studies of the RVE-based multilevel method applied to the electric

field equations (38) or (98), and (78).

5.1. Two-level iteration matrices

We consider the two-level iteration process consisting of t SSOR pre- and post-smoothing iterations.The coarse-model dielectric and effective dielectric stiffness matrices are obtained by restricting the corre-

sponding source grid matrices:

K0// ¼ QTK//Q; bKK 0// ¼ eQQT bKK //eQQ; ð116Þ

where Q and eQQ are the prolongation operators for the corresponding electric equations. The resulting

two-level iteration matrices are given by

L// ¼ St//T//S

t//;

bLL// ¼ bSS t//

bTT//bSS t

//; ð117Þ

where

S// ¼ I � P�1//K//; bSS// ¼ I � bPP�1

//bKK //; ð118Þ

T// ¼ I �QK�10//Q

TK//; bTT// ¼ I � eQQ bKK�10//

eQQT bKK //; ð119Þ

P// and bPP// are smoothing pre-conditioners.

5.2. Prolongation operators

5.2.1. One-dimensional case

The RVE is composed of three elements as shown in Fig. 1. Between adjacent RVEs a ‘‘soft’’ (withsmaller stiffness) interface element is placed. For simplicity, we focus on the case of constant mass density,

q1 ¼ q2 ¼ q, and the volume fraction a ¼ 0:5.The effective dielectric and dielectric stiffness matrices of the RVE are evaluated as

bKK r// ¼ K r

// þDt2

4K rT

u/ðM ruuÞ

�1K r

u/; ð120Þ

K r// ¼

a1 �a1 0 0

�a1 a1 þ b1 �b1 0

0 �b1 a1 þ b1 �a10 0 �a1 a1

26643775; a1 ¼ e1A=le; b1 ¼ e2A=le; ð121Þ

RVE

le le

c1 e1 ε1, , c2 e2 ε2, , ρ1 ρ2=

Fig. 1. The 1D model with RVE.


K ru/ ¼

a2 �a2 0 0

�a2 a2 þ b2 �b2 0

0 �b2 a2 þ b2 �a20 0 �a2 a2

26643775; a2 ¼ e1A=le; b2 ¼ e2A=le; ð122Þ

M ruu ¼ qleAdiagð1=2; 1; 1; 1=2Þ; ð123Þ

where bKK r//, K

r// and K r

u/ are the effective dielectric, dielectric and piezoelectric stiffness matrices of the

RVE, respectively; M ruu is the lumped mass matrix of the RVE; q, A and le are the mass density, cross-

sectional area and element length, respectively; e1, e2, e1 and e2 are dielectric and piezoelectric constants of

the two material constituents, respectively. The prolongation operators for the RVE are constructed basedon the corresponding constrained minimization problems:

Minimize : P1r ¼ 1

2wT

r Kr//wr subjected to kwrk2 ¼ 1; ð124Þ

Minimize : P2r ¼ 1

2/T

rbKK r

///r subjected to k/rk2 ¼ 1 ð125Þ

which yields the following eigenvalue problems:

K r//wr ¼ lrwr; kwrk2 ¼ 1; ð126Þ

bKK r///r ¼ kr/r; k/rk2 ¼ 1: ð127Þ

The global prolongation operators Q and eQQ are block diagonal matrices:

Qn�m ¼ diagfQr;Qr; . . . ;Qrg; eQQn�m ¼ diagf eQQr;eQQr; . . . ;

eQQrg; ð128Þ

Qr ¼ ½ p1 p2 �; eQQr ¼ ½ q1 q2 �; ð129Þwhere p1 and p2 are eigenvectors corresponding to the first two smallest eigenvalues of the eigenvalue

problem (126), whereas q1 and q2 are eigenvectors corresponding to the first two smallest eigenvalues of the

eigenvalue problem (127).

5.2.2. Two-dimensional case

We consider a 2D structured mesh in x1 and x3 plane as shown in Fig. 2. Plane stress assumption is made.

The RVE consists 9 elements as illustrated in Fig. 2. The total electric field degrees of freedom for the RVE

problem is 16.

As in the 1D case, the global prolongation operators Q and eQQ are assembled from the prolongation

matrices, Qr andeQQr, of RVEs. The RVE prolongation matrix consists of eigenvectors of the eigenvalue

The 9-element RVE

(1) (2)

Two phases of the RVE

Fig. 2. Global and RVE (local) finite element meshes.


problems (126) and (127), respectively. In the numerical examples considered, we take the first 8 eigen-

vectors corresponding to the lowest eigenvalues.

The two constituent phases are assumed to be made of PZT ceramic with transversely isotropic pro-

perties. The constitutive equation (3) in matrix form can be written as

r ¼ CS � eTE; D ¼ eS þ eE; ð130Þwhere

r ¼ ½ r1 r2 r3 r4 r5 r6 �T; S ¼ ½ S1 S2 S3 S4 S5 S6 �T; ð131Þ

D ¼ ½D1 D2 D3 �T; E ¼ ½E1 E2 E3 �T; ð132Þ

Ckl ¼

c11 c12 c13 0 0 0

c12 c11 c13 0 0 0

c13 c13 c33 0 0 0

0 0 0 c44 0 0

0 0 0 0 c44 0

0 0 0 0 0 ðc11 � c12Þ=2

2666666664

3777777775; eij ¼

e11 0 0

0 e11 0

0 0 e33

264375;

eik ¼0 0 0 0 e15 0

0 0 0 e15 0 0

e31 e31 e33 0 0 0

264375; k; l ¼ 1; 2; . . . ; 6; i; j ¼ 1; 2; 3: ð133Þ

Tensor and matrix indices are related by

11 ! 1; 22 ! 2; 33 ! 3; 23 ! 4; 31 ! 5; 12 ! 6

or alternatively

k ¼ 12ðiþ jÞdij þ ½9� ðiþ jÞ�ð1� dijÞ;


where dij is the Kronecker delta. Components of stress and strain tensors and matrices are related as

Stress: rij ! rk;

Strain: Sij ! Sk for i ¼ j (k ¼ 1; 2; 3); 2Sij ! Sk for i 6¼ j (k ¼ 4; 5; 6).

For plane stress in x1 and x3 plane r2 ¼ r4 ¼ r6 ¼ 0 and E2 ¼ 0.

The constitutive equation (130) in the matrix form can be simplified as

r1

r3

r5

24 35 ¼~cc11 ~cc13 0~cc13 ~cc33 00 0 c44

24 35 S1S3S5

24 35�0 ~ee310 ~ee33e15 0

24 35 E1

E3

� ; ð134Þ

D1

D3

� ¼ 0 0 e15

~ee31 ~ee33 0

� S1S3S5

24 35þ e11 0

0 ~ee33

� E1

E3

� ; ð135Þ

where

~cc11 ¼ c11 � c212=c11; ~cc13 ¼ c13 � c12c13=c11; ~cc33 ¼ c33 � c213=c11;

~ee31 ¼ e31 � e31c12=c11; ~ee33 ¼ e33 � e31c13=c11; ~ee33 ¼ e33 þ e231=c11:ð136Þ

The spectral radii of the two-level iteration matrices qðL//Þ and qðbLL//Þ with t ¼ 3 are evaluated nu-

merically and the results are presented in Tables 1–4, in which n is the size of the source mesh andDtcr ¼ 2=xim

n is the critical time step for the implicit conditionally stable partitioned scheme; ~ee11 ¼ e11,~cc44 ¼ c44 and ~ee15 ¼ e15.

For the explicit and implicit conditionally stable partitioned schemes, the coefficient matrix of the electric

field equation is simply the dielectric stiffness matrix. It only depends on the dielectric constants of the

media. From Tables 1 and 2, it can be observed that the rate of convergence improves with increasing

heterogeneity in dielectric constants and is independent of the problem size.

The coefficient matrix of the unconditionally stable partitioned scheme is the effective dielectric

matrix, which depends on the time step, the dielectric and piezoelectric constants. From Tables 3 and 4,

Table 1

Spectral radius of the 1D two-level iteration matrix qðL//ÞqðS3

//T//S3//Þ n ¼ 100 n ¼ 200 n ¼ 300

e1=e2 ¼ 1 0.0243 0.0243 0.0243

e1=e2 ¼ 50 0.0206 0.0206 0.0206

e1=e2 ¼ 100 0.0116 0.0116 0.0116

e1=e2 ¼ 1000 0.0013 0.0013 0.0013

Table 2

Spectral radius of the 2D two-level iteration matrix qðL//ÞqðS3

//T//S3//Þ n ¼ 14� 14 n ¼ 22� 22 n ¼ 30� 30

~eeð1Þii =~eeð2Þii ¼ 1 0.0216 0.0356 0.0420

~eeð1Þii =~eeð2Þii ¼ 50 0.0204 0.0212 0.0214

~eeð1Þii =~eeð2Þii ¼ 100 0.0136 0.0138 0.0139

~eeð1Þii =~eeð2Þii ¼ 1000 0.0020 0.0021 0.0021

Table 4

Spectral radius of the 2D two-level iteration matrix qðbLL//Þ

q bSS 3//

bTT//bSS 3

//

� �n ¼ 22� 22 n ¼ 30� 30

Dt=Dtcr ¼ 1 Dt=Dtcr ¼ 10 Dt=Dtcr ¼ 1 Dt=Dtcr ¼ 10

r1 ¼ 1, r2 ¼ 1 0.0726 0.4414 0.0808 0.4901

r1 ¼ 1, r2 ¼ 100 0.0360 0.2121 0.0366 0.2254

r1 ¼ 1, r2 ¼ 1000 0.0055 0.1322 0.0056 0.1386

r1 ¼ 10, r2 ¼ 1 0.0659 0.5116 0.0749 0.5621

r1 ¼ 10, r2 ¼ 100 0.0963 0.7231 0.0983 0.7872

r1 ¼ 10, r2 ¼ 1000 0.0249 0.6088 0.0252 0.6598

r1 ¼ 100, r2 ¼ 1 0.1073 0.0386 0.1290 0.0416

r1 ¼ 100, r2 ¼ 100 0.0187 0.9997 0.0195 0.9999

r1 ¼ 100, r2 ¼ 1000 0.0386 0.9999 0.0402 1.0000

r1 ¼ ~eeð1Þik =~eeð2Þik , r2 ¼ ~eeð1Þii =~ee

ð2Þii .

Table 3

Spectral radius of the 1D two-level iteration matrix qðbLL//Þ

q bSS 3//

bTT//bSS 3

//

� �n ¼ 100 n ¼ 200


e1=e2 ¼ 1, e1=e2 ¼ 1 0.0767 0.8558 0.0767 0.8569

e1=e2 ¼ 1, e1=e2 ¼ 100 0.0301 0.2846 0.0302 0.2846

e1=e2 ¼ 1, e1=e2 ¼ 1000 0.0039 0.1170 0.0039 0.1170

e1=e2 ¼ 4, e1=e2 ¼ 1 0.0783 0.6484 0.0784 0.6485

e1=e2 ¼ 4, e1=e2 ¼ 100 0.0638 0.8378 0.0639 0.8380

e1=e2 ¼ 4, e1=e2 ¼ 1000 0.0102 0.4262 0.0102 0.4262

e1=e2 ¼ 10, e1=e2 ¼ 1 0.9976 0.9999 0.9994 1.0000

e1=e2 ¼ 10, e1=e2 ¼ 100 0.0871 0.9185 0.0871 0.9187


we can observe that the rate of convergence improves with increasing heterogeneity in dielectric con-stants, but increasing heterogeneity in piezoelectric constants tends to affect the rate of convergence

adversely. In all cases, the rate of convergence improves with decreasing time step and is independent of

the problem size.

To this end we compare the critical time steps of the implicit conditionally stable and the explicit par-

titioned schemes in 2D case. Numerical results reveal that the ratio of critical time steps in implicit and

explicit schemes is only slightly affected by the heterogeneity in dielectric and piezoelectric constants, but is

extremely sensitive to the heterogeneity in elastic constants. In Fig. 3, we plot the ratio of critical time steps

between the implicit and explicit schemes as well as the spectral radius of the two-level iteration matrixfor the mechanical equation versus the ratio of elastic constants ~ccð1Þkl =~cc

ð2Þkl .

It can be observed from Fig. 3 that with increasing ratio of elastic constants of the two constituent

phases, the spectral radius of the two-level iteration matrix for the mechanical equation decreases and the

ratio of critical time steps increases. Thus the implicit conditionally stable partitioned scheme can use a

larger time step than that used for the explicit scheme at the expense of solving a linear system of mecha-

nical equations at each time station. When the mismatch of mechanical properties is considerable, the

100 200 300 400 500 600 700 800 900 10006

8

10

12

14

16

18

20

22

24

Rat

io o

f crit

ical

tim

e st

eps

Ratio of critical time steps ∆tim/∆tex

Spectral radius for the

mechanical equation

100 200 300 400 500 600 700 800 900 10000.005

0.0094

0.0139

0.0183

0.0228

0.0272

0.0317

0.0361

0.0406

0.045

Ratio of elastic constants c(1)/c(2)

Spe

ctra

l rad

ius

of it

erat

ion

mat

rix fo

r m

echa

nica

l equ

atio

n

Fig. 3. Ratio of critical time steps and spectral radius of mechanical equation versus the ratio of elastic constants.


spectral radius of the two-level iteration matrix for the mechanical equation could be sufficiently small for

the two-level iteration process to converge in a couple of cycles.

5.3. Convergence of the two-level method

Consider the problem illustrated in Fig. 1. The bar is fixed at (x ¼ 0) and free at (x ¼ 1) with no externalload and electric charge acting on the bar. The following initial disturbance in the displacement field is

considered:

f ðxÞ ¼ f0a0½x� ðx0 � dÞ�4½x� ðx0 þ dÞ�4f1� H ½x� ðx0 þ dÞ�g½1� Hðx0 � d� xÞ�;where a0 ¼ 1=d8 and HðxÞ is the Heaviside step function; f0, x0 and d are the magnitude, the location of the

maximum value and the half width of the initial pulse. This pulse is similar in shape to the Gaussian

distribution function. In the computations, we take x0 ¼ 1=2 and f0 ¼ 0:05.For both electric and mechanical equations, we employ three SSOR pre- and post-smoothing iterations.

The stopping criterion is taken as

krk2kf k2

6Tol ¼ 10�6; ð137Þ

where krk2 and kf k2 are the 2-norms of the residual and the right-hand-side vectors, respectively.

For parametric study of convergence characteristics of the RVE-based multilevel solver, we carry out

computations with different ratios of material constants. The results are summarized in Tables 5 and 6 for

the implicit conditionally and unconditionally stable partitioned schemes, respectively. In both tables, the

numbers in the parentheses correspond to the cycle count of the mechanical equation, while the numbers

without parentheses denote the cycle numbers for the electric field equation. For the conditionally stablescheme, we take Dt ¼ Dtimcr .

From Tables 5 and 6, it can be seen that the RVE-based multilevel method converges faster for the

mechanical equation than for the electric equation. Moreover, increasing the ratios of elastic and dielectric

constants, and cutting the time step size tend to reduce the number of cycles.

Table 6

Numbers of cycles for the unconditionally stable scheme (Tol ¼ 10�6)

Iteration numbers n ¼ 400 n ¼ 1000


c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 1 6(2) 64(4) 5(2) 87(4)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 10 5(2) 26(4) 5(2) 29(4)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 100 4(2) 9(4) 4(2) 9(4)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 1000 3(2) 6(4) 3(2) 6(4)

c1=c2 ¼ 1, e1=e2 ¼ 4, e1=e2 ¼ 100 5(2) 55(4) 5(2) 65(4)

c1=c2 ¼ 10, e1=e2 ¼ 2, e1=e2 ¼ 2 4(4) 37(4) 4(3) 37(4)

c1=c2 ¼ 100, e1=e2 ¼ 2, e1=e2 ¼ 2 4(4) 37(3) 4(3) 36(3)

c1=c2 ¼ 1000, e1=e2 ¼ 2, e1=e2 ¼ 2 4(3) 35(2) 4(2) 35(2)

Table 5

Numbers of cycles for the conditionally stable scheme (Tol ¼ 10�6)

Iteration numbers n ¼ 400 n ¼ 1000 n ¼ 2000

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 1 4(2) 4(2) 4(2)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 10 4(2) 4(2) 4(2)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 100 3(2) 3(2) 3(2)

c1=c2 ¼ 1, e1=e2 ¼ 1, e1=e2 ¼ 1000 2(2) 2(2) 2(2)

c1=c2 ¼ 1, e1=e2 ¼ 4, e1=e2 ¼ 100 3(2) 3(2) 3(2)

c1=c2 ¼ 10, e1=e2 ¼ 2, e1=e2 ¼ 2 4(4) 3(3) 3(3)

c1=c2 ¼ 100, e1=e2 ¼ 2, e1=e2 ¼ 2 4(3) 3(3) 3(3)

c1=c2 ¼ 1000, e1=e2 ¼ 2, e1=e2 ¼ 2 4(2) 3(2) 3(2)


Acknowledgements

The support of the Sandia National Laboratories under Contract DE-AL04-94AL8500 and the Office

of Naval Research through grant number N00014-97-1-0687 are gratefully acknowledged.

References

[1] C.A. Felippa, K.C. Park, C. Farhat, Partitioned analysis of coupled mechanical systems, Comput. Methods Appl. Mech. Engrg.

190 (2001) 3247–3270.

[2] S. Piperno, C. Farhat, Partitioned procedures for the transient solution of coupled aeroelastic problems, Part II: energy transfer

analysis and three-dimensional applications, Comput. Methods Appl. Mech. Engrg. 190 (2001) 3147–3170.

[3] E. Tursa, B.A. Schrefler, On consistency, stability and convergence of staggered solution procedures, Rend. Mat. Acc. Lincei,

Rome 5 (9) (1994) 265–271.

[4] J. Fish, V. Belsky, Multigrid method for a periodic heterogeneous medium. Part I: Convergence studies for one-dimensional case,

Comput. Methods Appl. Mech. Engrg. 126 (1995) 1–16.

[5] J. Fish, V. Belsky, Multigrid method for a periodic heterogeneous medium. Part 2: Multiscale modeling and quality control in

multidimensional case, Comput. Methods Appl. Mech. Engrg. 126 (1995) 17–38.

[6] J. Fish, W. Chen, RVE based multilevel method for periodic heterogeneous media with strong scale mixing, Journal of

Engineering Mathematics 46 (1) (2003) 87–106.

[7] H.F. Tiersten, Linear Piezoelectric Plate Vibrations, Elements of the Linear Theory of Piezoelectricity and the Vibrations of

Piezoelectric Plates, Plenum Press, New York, 1967.

[8] Y.H. Lim, V.V. Varadan, V.K. Varadan, Finite-element modeling of the transient response of MEMS sensors, Smart Mater.

Struct. 6 (1997) 53–61.

[9] C.Q. Chen, X.M. Wang, Y.P. Shen, Finite element approach of vibration control using self-sensing piezoelectric actuators,

Comput. Struct. 60 (3) (1996) 505–512.


[10] Y. Kagawa, T. Tsuchiya, T. Kataoka, Finite element simulation of dynamic responses of piezoelectric actuators, J. Sound Vib. 191

(4) (1996) 519–538.

[11] D.A. Saravanos, P.R. Heyliger, D.A. Hopkins, Layerwise mechanics and finite element for the dynamic analysis of piezoelectric

composite plates, Int. J. Solids Struct. 34 (3) (1997) 359–378.

[12] O.J. Aldraihem, R.C. Wetherhold, Mechanics and control of coupled bending and twisting vibration of laminated beams, Smart

Mater. Struct. 6 (1997) 123–133.

[13] T.J.R. Hughes, The Finite Element Method, Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs,

NJ, 1987.

[14] T.J.R. Hughes, Analysis of transient algorithms with particular reference to stability behavior, in: T. Belytschko, T.J.R. Hughes

(Eds.), Computational Methods For Transient Analysis, North-Holland, Amsterdam, New York, Oxford, 1983.

[15] C. Farhat, K.C. Park, Y.D. Pelerin, An unconditionally stable staggered algorithm for transient finite element analysis of coupled

thermoelastic problems, Comput. Methods Appl. Mech. Engrg. 85 (1991) 349–365.

[16] O.C. Zienkiewicz, D.K. Paul, A.H.C. Chan, Unconditionally stable staggered solution procedure for soil-pore fluid interaction

problems, Int. J. Numer. Meth. Engrg. 26 (1988) 1039–1055.

[17] J.D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, London, 1973.

[18] R. Bellman, Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.

modeling and simulation of piezocomposites

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