Transcript

Analysis of piezoelectric plates with a hole using nature boundaryintegral equation and domain decomposition

Xing-Yuan Miao, Guo-Qing Lin

School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan 430074, China

a r t i c l e i n f o

Article history:Received 5 May 2013Accepted 18 November 2013Available online 22 December 2013

Keywords:Piezoelectric mediumBoundary integration equationHarmonic analysisDefectsDomain decomposition

a b s t r a c t

In this paper, the plane problems of piezoelectricity are studied by using nature boundary integralequation and domain decomposition. A general displacement solution in terms of three potentialfunctions is adopted to solve exterior boundary value problems of piezoelectricity, and three mappingrelations corresponding to three potential functions are proposed for domain decomposition. Bysymbolic matrix inversion and derivation calculus, each potential function is governed by harmonicsecond-order partial differential equation in transformed domain with prescribed boundary condition.Therefore, three classic harmonic problems equivalent to the original plane piezoelectricity areestablished. Two cases of boundary conditions are considered, in which the displacement and electricpotential are prescribed or the traction and electric displacement are given on the boundary. Allproblems considered are equivalent to three independent harmonic problems, which are solved by usingnature boundary integration method proposed by Feng and Yu. A piezoelectric plate with a circular holeis analyzed as numerical examples. The results show that the proposed method is valid for thepiezoelectric plates with holes. The proposed method has potential applications to analyze multi-fieldcoupling problems.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The study on coupled fields involved with piezoelectricity,electrostriction and magnetostriction has aroused many researchers'interests, and the plane problems of piezoelectricity have beenwidely and deeply investigated in the past years. Sosa and Castro[16] first studied the concentrated loads acted on the boundary of apiezoelectric half plane and extended the complex potential functionmethod proposed by Lekhnitskii to analyze electroelastic problems,who also investigated the plane problems in piezoelectric mediumwith elliptic hole [15,17]. Ding et al. [3] derived the general solutionof plane problems of piezoelectric medium expressed by harmonicfunctions. Rajapakse [14] investigated the upper half plane problemof piezoelectric medium by means of the Fourier transform. Further-more, Benveniste [1], Chung and Ting [2], Dunn andWienecke [7], Luet al. [11], Pan and Yuan [13], Pan and Tonon [12] and Hu et al. [8]studied the Green's functions for a series of numerical computationsof plane problems. Li et al. [10] analyzed the piezoelectric planeproblem with fixed electrodes. As for computing methods, Yu andZhao [22] and Wu and Yu [19] applied the natural boundary integralequation in many numerical experiments based on natural boundaryreduction. Natural BIE has unique superiorities in multiple boundaryvalue problems. In recent years, the natural BIE has been applied to

solve a series of exterior problems for continuous or discrete domainin two-dimensional space. It can be referred to Wu and Yu [19], Duan Yu [5], Yu [21], Du and Yu [6], Yu and Zhao [22], Huang et al. [9]for more details.

In this paper, a coupling problem is decomposed into threesuccinct harmonic problems. First, the piezoelectric plane problemwith specified displacement and electric potential boundary con-ditions will be investigated. The general solutions of displacementand electric potential are expressed by three harmonic functions.Then, we separate the boundary conditions into three mappingregions, and eventually the original problem would be trans-formed into harmonic problem in three different mapping regions.Second, we discuss piezoelectric plane problem of the second kindwith specified distributed force and electric displacement bound-ary conditions. The general solutions of the stress and electricdisplacement are denoted by three harmonic functions. Hence, theequivalent three harmonic problems have been obtained by meansof separating the boundary conditions as well.

2. General solution to plane piezoelectricity

In this section, a general solution to plane piezoelectricity usingdisplacement potential functions will be briefly introduced, whichis first proposed by Ding et al. [4]. Considering the piezoelectricmedium occupying a plane without body force and electric charge,

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Engineering Analysis with Boundary Elements

0955-7997/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enganabound.2013.11.012

n Corresponding author: Tel.: +86 13971016840; fax: +86 027 87543538.E-mail address: [email protected] (G.-Q. Li).

Engineering Analysis with Boundary Elements 40 (2014) 71–77

the governing equations for this piezoelectric problem are given asfollows:

Dfu;w;ϕgT ¼ 0; ð1Þwhere

c11 ∂2∂x21

þc44 ∂2∂x23

ðc13þc44Þ ∂2∂x1x3

ðe15þe31Þ ∂2∂x1x3

ðc13þc44Þ ∂2∂x1x3

c44 ∂2∂x21

þc33 ∂2∂x23

e15 ∂2∂x21

þe33 ∂2∂x23

�ðe15þe31Þ ∂2∂x1x3

� e15 ∂2∂x21

þe33 ∂2∂x23

� �ɛ11 ∂2

∂x21þɛ33 ∂2

∂x23

2666664

3777775: ð2Þ

In the above equations, u and w are the mechanical displacementcomponents along x1 and x3 directions, ϕ is the electric potential,D represents a differential matrix operator. Cij; eij; ɛij are the elastic,piezoelectric and dielectric constants, respectively.

Based on the derivation in Ding's work [4], the fundamentalsolution to Eq. (1) is expressed in terms of three quasi-harmonicfunctions:

∂2

∂y2k1þ ∂2

∂y2k3

!ψ k ¼ 0; ð3Þ

where subscript k¼ 1;2;3, which will be always implicit impliedin this paper, and ψk are potential functions, yk1 ¼ x1þαkx3,yk3 ¼ βkx3 and sk ¼ αkþ iβk are the three roots of the followingequation:

as6�bs4þcs2�d¼ 0; ð4Þwhere

a¼ c44ðe233þc33ɛ33Þ;b¼ c33½c44ɛ11þðe15þe31Þ2�þɛ33½c11c33þc244�ðc13þc44Þ2�

þe33½2c44e15þc11e33�2ðc13þc44Þðe15þe31Þ�;c¼ c44½c11ɛ33þðe15þe31Þ2�þɛ11½c11c33þc244�ðc13þc44Þ2�

þe15½2c11e33þc44e15�2ðc13þc44Þðe15þe31Þ�;d¼ c11ðe215þc44ɛ11Þ: ð5Þ

It has been studied on sk by Ding et al. [4] and by [18] that, theeigenvalue sk has a positive imaginary part, i.e. βk40, and sk canalso be a pure imaginary number. For simplicity, we only considerthe sk as imaginary in this paper. In the rest of this paper, yk1 ¼ x1,yk3 ¼ yk ¼ skx3 will be always established.

Therefore, the displacement components and electric potentialcan be obtained as

u¼ ∑3

k ¼ 1a1k

∂ψ k

∂x1; w¼ ∑

3

k ¼ 1a2k

∂ψ k

∂yk; φ¼ ∑

3

k ¼ 1a3k

∂ψ k

∂yk; ð6Þ

where

a1k ¼ 1;

a2k ¼ðc11ɛ11�m3s2kþc44ɛ33s4k Þ

ðm1�m2s2k Þsk;

a3k ¼ðc11e15�m4s2kþc44e33s4k Þ

ðm1�m2s2k Þsk; ð7Þ

in which

m1 ¼ ðc13þc44Þɛ11þðe15þe31Þe15;m2 ¼ ðc13þc44Þɛ33þðe15þe31Þe33;m3 ¼ c11ɛ33þc44ɛ11þðe15þe31Þ2;m4 ¼ c11e33�c13ðe15þe31Þ�c44e31: ð8Þ

Applying the constitutive equations of the plane piezoelectricity tothe displacement solution, the stress components and electric

displacement can be obtained by

s11 ¼ ∑3

k ¼ 1b11

∂2ψ k

∂x23; s33 ¼ ∑

3

k ¼ 1b12

∂2ψ k

∂x21; s13 ¼ ∑

3

k ¼ 1b13

∂2ψ k

∂x1∂x3;

ð9Þ

D1 ¼ ∑3

k ¼ 1b14

∂2ψ k

∂x21; D3 ¼ ∑

3

k ¼ 1b15

∂2ψ k

∂y2k; ð10Þ

where

b11 ¼ C11a1kþC13a2kþe31a3k;

b12 ¼ C13a1kþC33a2kþe33a3k;

b13 ¼ C44ða1kþa2kÞþe15a3k;

b14 ¼ e15ða1kþa2kÞ�ɛ11a3k;

b15 ¼ e31a1kþe33a2k�ɛ33a3k:

9>>>>>>=>>>>>>;

ð11Þ

In summary, there are three harmonic functions ψ k in transformeddomain Ωkðx1; ykÞ satisfying the governing equation of planepiezoelectricity, and all physical fields can be expressed in termsof the functions ψ k. In order to obtain the solution of ψ k fulfillingthe prescribed boundary equations, the equations on ∂Ωðx1; x3Þwill be separated into three independent boundary conditions onthe transformed boundary ∂Ωkðx1; ykÞ, which will be presented inthe following sections.

3. Plane piezoelectricity with displacement and electricpotential conditions

In this section, the displacement and electric potential bound-ary conditions will be spliced into three independent conditions intransformed domain.

It is assumed that the original conditions for static elasto-electric are given as follows:

uj∂Ω ¼ u0ðx1; x3Þ; wj∂Ω ¼w0ðx1; x3Þ; φj∂Ω ¼φ0ðx1; x3Þ: ð12ÞSubstituting Eq. (6) into Eq. (12) yields the boundary conditions forthe displacement and electric potential, that is

∑3

k ¼ 1a1k

∂ψ k

∂x1¼ u0ðx1; x3Þ;

∑3

k ¼ 1a2k

∂ψ k

∂yk¼w0ðx1; x3Þ;

∑3

k ¼ 1a3k

∂ψ k

∂yk¼φ0ðx1; x3Þ: ð13Þ

Considering a simply connected region surrounded by one smoothcurve, which can be defined by the following t-parametric equa-tions:

x1 ¼ f 1ðtÞ; x3 ¼ f 3ðtÞ; ð14ÞFor purpose of separating the boundary conditions into thetransformed domains, the coordinate transformation relation isused. Then, Eq. (13) can be rewritten with respect to t-parameteras follows:

∑3

k ¼ 1

a1kf ′1

∂ψ k

∂t¼ u0ðx1; x3Þ;

∑3

k ¼ 1

a2kskf

′3

∂ψ k

∂t¼w0ðx1; x3Þ;

∑3

k ¼ 1

a3kskf

′3

∂ψ k

∂t¼φ0ðx1; x3Þ: ð15Þ

Rewriting Eq. (15) into matrix form leads to

AðtÞ∂ψ k

∂t¼UðtÞ; ð16Þ

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 71–7772

where A1k ¼ a1k=f′1, A2k ¼ a2k=skf

′3, A3k ¼ a3k=skf

′3 and U¼

½u0;w0;φ0�T . Therefore,∂ψ k

∂t¼ A�1ðtÞUðtÞ ¼ μk; ð17Þ

where A�1 is the inverse matrix of A and will be obtainedby symbolic matrix inversion operation. The partial differential∂ψ k=∂t has connection with the normal differential ∂ψ k=∂nk,which is

∂ψ k

∂nk¼ ∂ψ k

∂t∂t∂nk

¼ f ′1f ′3skμk

f ′3sk cos θkþ f ′1 sin θk; ð18Þ

where θk is the direction angle shown in Fig. 1.

cos θ¼ dx1ds

¼ n2; sin θ¼ dx3ds

¼ �n1;

cos θk ¼dyk1dsk

¼ nk2; sin θk ¼dyk3dsk

¼ �nk1: ð19Þ

Finally, three equivalent problems used to describe the originalproblem are established by

Δψ k ¼ 0 on Ωk;

∂ψ k

∂nk¼ f ′1f ′3skμk

f ′3sk cos θkþ f ′1 sin θk

on ∂Ωk:

9>=>; ðk¼ 1;2;3Þ ð20Þ

It should be noticed that each of the equivalent problems isdefined in a certain ðx1; ykÞ coordinate system. During the symbolicmatrix operations on the right side of the second equation of Eq.(17), the prescribed boundary conditions shown in Eq. (13) shouldbe simultaneously replaced by u0ðx1; ykÞ, w0ðx1; ykÞ and φ0ðx1; ykÞfor k¼ 1;2;3 respectively. Obviously, the equivalent problemsshown in Eq. (20) are three classic harmonic problems withNeumann conditions.

4. Plane piezoelectricity with traction and electricdisplacement conditions

In this section, the traction and electric displacement boundaryconditions will be studied using the same approach as the lastsection.

At first, applying the parameter equation of the boundaryshown in Eq. (14) to Eqs. (9) and (10) yields the stress componentsand electric displacement, these are

s11 ¼ ∑3

k ¼ 1bk11

∂2ψ k

∂t2; s33 ¼ ∑

3

k ¼ 1bk12

∂2ψ k

∂t2; s13 ¼ ∑

3

k ¼ 1bk13

∂2ψ k

∂t2;

ð21Þ

D1 ¼ ∑3

k ¼ 1bk14

∂2ψ k

∂t2; D3 ¼ ∑

3

k ¼ 1bk15

∂2ψ k

∂t2; ð22Þ

where

bk11 ¼ C11a1kf ′21

þC13a2kskf ′23

þe31a3kskf ′23

;

bk12 ¼ C13a1kf ′21

þC33a2kskf ′23

þe33a3kskf ′23

;

bk13 ¼ C44a1kþa2kskf ′1f ′3

þe15a3k

skf ′1f ′3;

bk14 ¼ e15a1kþa2kskf ′1f ′3

�ɛ11a3k

skf ′1f ′3;

bk15 ¼ e31a1kf ′21

þe33a2kskf ′23

�ɛ33a3kskf ′23

:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

ðk¼ 1;2;3Þ ð23Þ

Next, the stress and electric displacement boundary conditionswill be separated into three independent conditions. It is assumedthat the original conditions for static elastoelectric are given asfollows:

pj∂Ω ¼ p0ðx1; x3Þ; qj∂Ω ¼ q0ðx1; x3Þ; dj∂Ω ¼ d0ðx1; x3Þ; ð24Þ

where p, q, d represent the normal traction, the tangentialtraction and the electric displacement, respectively, and they arewritten by

s11n1þs13n3 ¼ p;

s13n1þs33n3 ¼ q;

D1n1þD3n3 ¼ d; ð25Þ

where n1;n2 represent the direction cosines on the boundary.Substituting Eqs. (21) and (22) together with Eq. (25) into

Eq. (24) yields

∑3

k ¼ 1tk1∂2ψ k

∂t2¼ p0ðx1; x3Þ;

∑3

k ¼ 1tk2∂2ψ k

∂t2¼ q0ðx1; x3Þ;

∑3

k ¼ 1tk3∂2ψ k

∂t2¼ d0ðx1; x3Þ; ð26Þ

where

tk1 ¼ n21b

k11þn2

2bk12þ2n1n2b

k13;

tk2 ¼ n1n2ðbk12�bk11Þþðn21�n2

2Þbk13;tk3 ¼ n1b

k14þn2b

k15:

9>>>=>>>;

ðk¼ 1;2;3Þ ð27Þ

Rewriting Eq. (26) into matrix form leads to

TðtÞ∂2ψ k

∂t2¼ PðtÞ; ð28Þ

Fig. 1. Direction angle in two domains.

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 71–77 73

where P¼ ½p0; q0; d0�T . Taking the inverse matrix operation on Eq.(28) leads to

∂2ψ k

∂t2¼ T�1ðtÞPðtÞ; ð29Þ

where T�1ðtÞ is the inverse matrix of TðtÞ, which is obtained usingsymbolic inverse matrix operation similar to Eq. (17). In order toobtain the functions ψ k using natural boundary integral equation,the second-order partial differential equation must be degener-ated into a first-order partial differential equation on the boundaryΓk. Thus, the integration of Eq. (29) gives

∂ψ k

∂t¼ ψ̂ k0þCk ¼ λk; ðk¼ 1;2;3Þ ð30Þ

where ψ̂ k0 ¼RΓkT �1ik ðtÞPiðtÞ dt, and the unknown constants Ck can

be determined from the following displacement and electricpotential integral conditions (see Appendix for more detail), i.e.ZΓu ds¼ 0;

ZΓw ds¼ 0;

ZΓϕ ds¼ 0: ð31Þ

Once the constants Ck are determined, Eq. (30) plays the same roleas that of Eq. (18). The separated boundary conditions similar toEq. (18) can be obtained by

∂ψ k

∂nk¼ f ′1skf ′3λkskf ′3 cos θkþ f ′1 sin θk

: ð32Þ

In the end, a displacement expression which is analogous to Eq.(20) can be obtained as

Δψ k ¼ 0 on Ωk;

∂ψ k

∂nk¼ f ′1f

′3skλk

f ′3sk cos θkþ f ′1 sin θk

on ∂Ωk:

9>>=>>; ðk¼ 1;2;3Þ ð33Þ

Comparing Eq. (33) with Eq. (20) shown that the only differ-ence between them is the boundary function μk in Eq. (20)replacing by λk in Eq. (33), in which constants Ck are involved.

5. Solution procedure using nature boundary integrationequations

From the above sections, the solution to plane piezoelectricitycan be reduced to solving three independent Laplace problemswhich is shown in Eq. (20) or Eq. (33). It is well known that thereare many methods available to solving Eq. (33) or Eq. (20) in theliterature. In this paper, the method called as nature boundaryintegral equation (NBIE) established by Feng and [20] is adopted.Therefore, the solution procedure will be briefly introduced asfollows. Without lost of generality, the solution procedure isdesigned for piezoelectric plate with a unit circular hole, whoseresults will be discussed in detail in the next section.

Prior to the solution procedure, the natural boundary integralmethod for harmonic problem in exterior elliptic region will bebriefly introduced [22]. Taking problem described by Eq. (20) as anexample, the original domain ∂Ω is a unit cycle and the trans-formed domain ∂Ωk is a elliptic defined by x21þy2ks

�2k ¼ 1, and the

final transformed domain is a unit cycle again defined by μk ¼ 1using elliptic ðμk;ϕkÞA∂Ω′

k coordinates. The transform relationsfrom Cartesian coordinates to elliptic coordinates are

x1 ¼ f 0 cosh μ cos ϕyk ¼ f 0 sinh μ sin ϕ: ð34ÞIn elliptic coordinates, the Laplace operator remains unchanged forthe new domain.

Another important issue is concerned with computationmeshes. It is worth noting that the uniform or equal spacingmeshes cannot be inherited after coordinate transformations. In

another word, if meshes in ∂Ω′k are uniform, meshes in ∂Ωk and in

∂Ω are all non-uniform, and vice versa. Fig. 2 shows the uniformmeshes in ∂Ω but non-uniform meshes in ∂Ωk and ∂Ω′

k. In fact, theuniform meshes in ∂Ω′

k are always needed for NBIE solution asstated by Yu and Zhao [22]. In this case, the interpolationcalculation will be needed, in which piecewise linear basis func-tions LjðθÞ on boundary will be used, that is

LjðθÞ ¼

N2π

ðθm�θj�1Þ; θj�1rθmrθj;

N2π

ðθjþ1�θmÞ; θjrθmrθjþ1;

0; else;

8>>>><>>>>:

ð35Þ

which can be found in Yu and Zhao [22] or Yu [20].The seven main steps in solution procedure will be addressed

in the following.

� Step 1: Input the problem definition including material con-stants, geometry and load conditions.

� Step 2: Solve the eigen value sk and associated matrix AðtÞ orTðtÞ using Eqs. (4), (15) and (27) respectively. If the boundarytraction are given, the solution of Ck shown in Appendix willbe done before TðtÞ calculation.

� Step 3: Make symbolic matrix inverse operation on AðtÞ or TðtÞ,and obtain the three harmonic problems shown by Eq. (33) orEq. (20).

� Step 4: Solve Eq. (33) or Eq. (20) using NBIE and uniform gridsϕj ¼ j=2πN; j¼ 1;2;…;N, where N is the number of nodes, andthe solving procedure follows the same procedure as describedby Yu and Zhao [22]. For the considered problem of a unit cycle,the nodal solution of U∂Ω′

kj is obtained.

� Step 5: Define the grids on physical boundary ∂Ω, i.e. uniformdistributed grids ð1;θnÞ on unit cycle, and then compute thegrids ðxn1; ynk Þ in the transformed domain ∂Ωk , and finally thegrids ð1;ϕn

k) in elliptic coordinate system ðμk;ϕkÞA∂Ω′k are

obtained by interpolation using

ψ ∂Ω′kk ¼ψ ∂Ω′k

k ðϕikÞ ¼ ∑

N

j ¼ 1Ljðϕk;ϕ

ikÞU∂Ω′k

j : ð36Þ

where Lj is shown in Eq. (35). Finally, three potential compo-nents values ψn

k on each boundary node are obtained.� Step 6: Composite the node solutions on boundary using Eqs.

(15) and (26).� Step 7: Compute the physical fields at the given internal nodes.

Similar to Step 5, the coordinates of internal nodes are

transformed from ðrj;θjÞAΩ into ðxj1; yjkÞAΩk and finally into

ðμik;ϕ

ikÞAΩ′

k coordinates, while the internal value of

ψΩ′kk ðμj

k;φjÞ is obtained by using Poisson integration equation,

and finally Eqs. (6), (9) and (10) are used to calculate thequantities at the internal nodes.

6. Plane piezoelectric plate with a unit circular hole

To verify the validity of this algorithm, an infinite mediumoccupied with PZT6B with a unit circular hole is discussed. Theboundary between the hole and the piezoelectric medium isdefined as

x1 ¼ cos t; x3 ¼ sin t: ð37ÞThe displacement boundary condition is assumed as Eq. (17),which is equal to the Dirichlet boundary condition.

u0 ¼ 0; w0 ¼ 0; φ0 ¼ 1: ð38Þ

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 71–7774

The stress boundary condition is assumed as Eq. (29), which isequal to the Neumann boundary condition.

p0 ¼ 1; q0 ¼ 0; d0 ¼ 0: ð39Þ

In this paper, the piezoelectric material has been poled in x3direction, and the material properties are listed as follows:

½C� ¼

16:8 6:0 6:0 0 0 06:0 16:8 6:0 0 0 06:0 6:0 16:3 0 0 00 0 0 2:7 0 00 0 0 0 2:7 00 0 0 0 0 5:4

2666666664

3777777775ð1010 N=m2Þ;

½e� ¼0 0 0 0 4:6 00 0 0 4:6 0 0

�0:9 �0:9 7:1 0 0 0

264

375ðC=m2Þ;

½ɛ� ¼3:6 0 00 3:6 00 0 3:4

264

375ð10�9 C=VmÞ:

Considering an infinite piezoelectric plate containing a circularhole under generalized combined mechanical-electric fields, asshown in Fig. 3.

In order to demonstrate the validity of the functions derivedabove, we turn to the finite element analysis and the commercialsoftware ANSYS is used. In each case, 20 finite elements are usedfor the quarter circular loading area. Figs. 4–7 compare thedisplacement components on the edge of the hole betweennumerical solution and the FEM solution in two types of boundaryconditions. It is observed that the FEM results are relatively closeto the numerical results. In each figure, the maximum differencebetween the two solutions is smaller than 10%. In fact, moreaccurate or exact solution should be chosen to comparing with theresults in this paper, but they are rarely reported in the literatureand the error estimation of the proposed method will be studiedin the coming future.

7. Conclusions

In this paper, the plane problems of piezoelectricity with acircular hole are investigated by using nature boundary integralequation and domain decomposition. A general displacementsolution in terms of three potential functions is adopted and threemapping relations associated with three potential functions areproposed for domain decomposition. By symbolic matrix inversionand calculus derivation, each potential function is governed byharmonic second-order partial differential equation in trans-formed domain with one prescribed boundary condition. Threeclassic harmonic problems equivalent to the original plane piezo-electricity are established, which are solved using nature boundaryintegration method proposed by Feng and Yu. The proposed

Fig. 2. Meshing process in physical and transformed domains.

U or V

x1

x3

Fig. 3. Displacement and voltage on the boundary of the hole.

x 10−12

App

roxi

mat

e so

lutio

n of

dis

plac

emen

t u

on hole

NBIE solutionFEM solution

Fig. 4. Horizontal displacement u along the hole with the first type boundarycondition.

on hole

Fig. 5. Vertical displacement v along the hole with the first type boundarycondition.

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 71–77 75

approach has been successfully implemented for piezoelectricplate with single circular hole. Meanwhile, the solution proceduresfor the two different boundary conditions on the edge of the holehave been established. Comparing with the FEM results it showsthat the proposed method is valid and accurate.

One potential application of the proposed method is that, themultiple coupling fields including electromechanical, magnetostrictiveand even magnetoelectroelastic coupling will be solved by using thenature boundary integral equations and domain decomposition.

Acknowledgments

This project is supported by the NSFC under the Grant Nos.10972083 and 10932004.

Appendix A

As the unknown constants Ck can be determined from thedisplacements and electric potential integral conditions, then

∮Γ ∑3

k ¼ 1a1kψ̂ k0

∂t∂x1

dsþ∮Γ ∑3

k ¼ 1a1kCk

∂t∂x1

ds¼ 0;

∮Γ ∑3

k ¼ 1a2kψ̂ k0

∂t∂yk

dsþ∮Γ ∑3

k ¼ 1a2kCk

∂t∂yk

ds¼ 0;

∮Γ ∑3

k ¼ 1a3kψ̂ k0

∂t∂yk

dsþ∮Γ ∑3

k ¼ 1a3kCk

∂t∂yk

ds¼ 0: ðA:1Þ

Based on curve integral of the first kind,

∮L ∑3

k ¼ 1a1kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qf ′1ðtÞ

dtþ∮L ∑3

k ¼ 1a1kCk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qf ′1ðtÞ

dt ¼ 0;

∮L ∑3

k ¼ 1a2kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dtþ∮L ∑3

k ¼ 1a2kCk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ 0;

∮L ∑3

k ¼ 1a3kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dtþ∮L ∑3

k ¼ 1a3kCk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ 0:

ðA:2ÞLet be

∮Lka1kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qf ′1ðtÞ

dt ¼ ψ̂ k1;

∮Lka2kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ ψ̂ k2;

∮Lka3kψ̂ k0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ ψ̂ k3; ðA:3Þ

and

∮Lka1k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qf ′1ðtÞ

dt ¼ Gk1;

∮Lka2k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ Gk2;

∮Lka3k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif ′21 ðtÞþs2k f

′23 ðtÞ

qskf ′3ðtÞ

dt ¼ Gk3: ðA:4Þ

Then Eq. (A.1) can be written as

∑3

k ¼ 1ψ̂ k1þC1G11þC2G21þC3G31 ¼ 0;

∑3

k ¼ 1ψ̂ k2þC1G12þC2G22þC3G32 ¼ 0;

∑3

k ¼ 1ψ̂ k3þC1G13þC2G23þC3G33 ¼ 0: ðA:5Þ

Finally, the unknown constants Ck can be obtained by solving theabove three equations.

0

1

2

3

4

5

6

7

8x 10−11

App

roxi

mat

e so

lutio

n of

dis

plac

emen

t u

NBIE solutionFEM solution

on hole

Fig. 6. Horizontal displacement u along the hole with the second type boundarycondition.

−7

−6

−5

−4

−3

−2

−1

0

1

2x 10−11

App

roxi

mat

e so

lutio

n of

dis

plac

emen

t w

NBIE solutionFEM solution

on hole

Fig. 7. Vertical displacement v along the holewith the second type boundary condition.

C1 ¼∑3

k ¼ 1ψ̂ k1G23� ∑

3

k ¼ 1ψ̂ k3G21

!ðG32G23�G33G22Þ� ∑

3

k ¼ 1ψ̂ k2G23� ∑

3

k ¼ 1ψ̂ k3G22

!ðG31G23�G33G21Þ

ðG11G23�G13G21ÞðG32G23�G33G22Þ�ðG12G23�G13G22ÞðG31G23�G33G21Þ;

C2 ¼∑3

k ¼ 1ψ̂ k1G13� ∑

3

k ¼ 1ψ̂ k3G11

!ðG32G13�G33G12Þ� ∑

3

k ¼ 1ψ̂ k2G13� ∑

3

k ¼ 1ψ̂ k3G12

!ðG31G13�G33G11Þ

ðG21G13�G23G11ÞðG32G13�G33G12Þ�ðG22G13�G23G12ÞðG31G13�G33G11Þ;

C3 ¼∑3

k ¼ 1ψ̂ k1G13� ∑

3

k ¼ 1ψ̂ k3G11

!ðG22G13�G23G12Þ� ∑

3

k ¼ 1ψ̂ k2G13� ∑

3

k ¼ 1ψ̂ k3G12

!ðG21G13�G23G11Þ

ðG31G13�G33G11ÞðG22G13�G23G12Þ�ðG32G13�G33G12ÞðG21G13�G23G11Þ: ðA:6Þ

X.-Y. Miao, G.-Q. Li / Engineering Analysis with Boundary Elements 40 (2014) 71–7776

The two types of displacement expressions can be easilycalculated on the basis of the natural integral equation. TheDirichlet–Neumann boundary relation is

KU¼ ∂ψ∂n

; ðA:7Þ

and in accordance with the Poisson integral equation, the domainand Dirichlet boundary relation is

ψ ¼ PU; ðA:8Þwhere K is the natural integral operator and P is the Poissonintegral operator. Then, the problems can be explained as solvingthe natural integral equation.

For this paper's problem, the function of any node of theoutside region can be approximated using the known boundarynodal values. Suppose the Poisson integral formula is given asfollows:

u0j ¼

ZΩk

Pu0 ds¼ ∑N

i ¼ 1Ui

ZΩk

PLiðθÞ ds; ðA:9Þ

Let LiðθÞ; i¼ 1;…;N, be the boundary element basis functions onΩk. Then, the function of exterior nodal values can be obtained.

ukðx1; x3Þ �∑ju0j ðx1; x3Þ: ðA:10Þ

References

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