含自由表面或雙材料介面之壓電物體受集中力或差排作用之暫態分析

行政院國家科學委員會專題研究計畫 成果報告

含自由表面或雙材料介面之壓電物體受集中力或差排作用

之暫態分析

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計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 97-2218-E-151-008- 執 行 期 間 ： 97 年 11 月 01 日至 98 年 10 月 31 日 執 行 單 位 ： 國立高雄應用科技大學土木工程系 計 畫 主 持 人 ： 陳世豪 計畫參與人員： 碩士班研究生-兼任助理人員：林建豪 碩士班研究生-兼任助理人員：黃威綸 處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 99 年 01 月 28 日

行政院國家科學委員會補助專題研究計畫

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□期中進度報告

含自由表面或雙材料介面之壓電物體受集中力或差排作用之

暫態分析

Transient analysis of a piezoelectric half-space with a free

surface or a piezoelectric bimaterial with the interface due to a

line force or dislocation

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執行單位：國立高雄應用科技大學

中 華 民 國 99 年 1 月 28 日

附件一

Two Dimensional Green’s Functions for the

Piezoelectric Half-Space

Shyh-Haur Chen, and Kuang-Chong Wu

Abstract—A novel formulation for two-dimensional self-similar anisotropic elastodyamic problems is generalized to piezoelectric materials. By making use of the formulation, the general solution of the displacements is expressed in terms of the eigenvalues and eigenvectors of a related eight-dimensional eigenvalue problem. Without the need of performing integral transforms as required in the well-known Cagniard-de Hoop method, the present formulation can be utilized to obtain expressions of analytical solutions directly. In the study, the method is applied to derive the explicit dynamic Green’s functions in the piezoelectric half-space. Numerical examples for the quartz of the semi-infinite region are illustrated.

Keywords：dynamic Green’s functions, piezoelectric half-space.

摘要：本研究將原本解二維自相似異向性彈性動力學問題的新方法推廣至解壓電 材料問題。應用此求解架構，壓電材料半空間波傳問題轉化成一個相關的八維特 徵植問題。藉由解此特徵植問題，位移的通解可由特徵值與特徵向量表示。使用 此求解架構，可以直接獲得問題的解析解而並不需要使用積分轉換的技巧—如著 名的 Cagniard-de Hoop 方法。本研究應用此求解架構推導獲得半無窮域壓電材料 的動態格林函數（Green’s function）。數值計算以壓電材料—石英為例計算壓電 材料半空間波傳問題。 關鍵字：動態格林（Green）函數，壓電材料半空間。 1. Introduction

Because of the intrinsic anisotropic elastic features of piezoelectric materials, many analytic methods for piezoelectric solids are derived from those for the general anisotropic elasticity. The Green’s functions, which relate the mechanical displacements and electric potential at a point to the concentrated forces or charges applied at another point, play an important role in understanding analytically mechanical or electrical behavior of loaded piezoelectric materials.

Lothe and Barnett [1] developed an integral formalism for surface waves in piezoelectric half-infinite solid. They [2] also considered the existence of surface waves in piezoelectric half-space subjected to various boundary conditions. Taylor and Crampin [3] considered the circumstances for the propagation of surface waves in a homogeneous anisotropic piezoelectric half-space. Their study revealed that the particular form of anisotropic symmetry with respect to the direction of propagation critically affects the properties of the surface waves. Peach [4] extended the results of Lothe and Barnett [1] for the anisotropic materials to those for the piezoelectric materials. He presented general existence theorems for surface waves on piezoelectric substrates. Gao and Noda [5] developed an exact solution for the static Green’s functions of a half-infinite piezoelectric solid. Their work showed that the normal component of the electric displacement on the solid surface is not zero and is dependent on the applied loads and the electro-elastic constants of the piezoelectric material and air.

method in two-dimensional general anisotropic elastostatics [6]-[9]. A distinctive feature of the Stroh formalism is that the general solution is provided in terms of the eigenvalues and eigenvectors of a constant six-dimensional matrix. The general solution contains three arbitrary complex functions. These functions can often be determined by virtue of the orthogonality relations among the eigenvectors in conjunction with theories of analytic functions. The Stroh’s formalism has been applied to yield the static Green’s functions for various configurations (Ting, [8]). Generalization of the Stroh’s formalism to piezoelectric materials has been given by Ting [8], leading to an eigenvalue problem of a constant eight-dimensional matrix. Wu [9] extended the Stroh’s formalism to treat the self-similar elastodynamic problems for general anisotropic elastic material. The formulation is also based on a six-dimensional matrix, which, however, is a function of position and time. A major advantage of the novel formulation of Wu [9] is that solutions can be derived directly without the need of performing integral transforms. The formulation of Wu has been further extended to piezoelectric materials in the context of the quasi-static approximation to derive the dynamic Green’s functions for an infinite piezoelectric medium (Wu and Chen, [10]). In this study the dynamic surface Green’s functions for a general piezoelectric half-space is considered. The surface is assumed traction-free mechanically and insulating electrically.

2. Formulation

The formulation of Wu and Chen [10] for self-similar elastodynamic problems for general piezoelectric materials is outlined in this section. For a linear piezoelectric solid, the mechanical stress σij , the mechanical displacement u , the electric i

displacement D and the electric potential _i φ are related by

, , ij C uijks k s esij s σ = + φ , (1) , , i iks k s is s D =e u −ε φ , (2) where a subscript comma denotes partial differentiation with respect to spatial coordinates, repeated indices imply summation from 1 to 3, C_ijks are the elastic stiffness, and e , and _iks ε_is are, respectively, the piezoelectric stress constants and permittivity constants. In the absence of body forces and free charges the balance laws under quasi-static approximation require

σ_{ij j,} =  , (3) ρu_i

, 0

i i

D = , (4) where ρ is the density and an overhead dot designates derivative with respect to time t .

By virtue of letting φ = and u₄ D_i =σ_4i, (1) and (2) can be expressed in terms of the generalized stress and generalized displacement as

,

Ij EIjKsuK s

σ = , (5) where the upper case subscripts range from 1 to 4, lower case subscripts from 1 to 3 and generalized electric-mechanical constants E_IjKs are defined as

, , 1, 2, 3, , 1, 2, 3, 4, , 4, 1, 2, 3, , 4, 4. ijks sij IjKs iks is C I K e I K E e I K I K ε = ⎧ ⎪ _{= =} ⎪ = ⎨ _{= =} ⎪ ⎪ − = = ⎩ .

Equations (3) and (4) can also be combined as

* ,

Ij j IKuK

σ =ρδ  , (6) where δ_IK* =δ_IK, ,I K=1,2,3 , δ_IK being the Kronecker’s delta and

*

0, , 4

IK I K

δ = = . Substitution of (5) into (6) yields the governing equations in terms of the generalized displacement u as

* ,

IjKs K sj IK K

E u =ρδ u . (7) For two-dimensional problems in which the generalized displacement

[

1, 2, 3,

]

T

u u u φ =

u are independent of x , (7) can be expressed as ₃

,11 ( ) ,12 ,22 ˆ

T ρ

+ + + =

Qu R R u Tu Iu , (8) where ˆI , Q , R , and T are 4 4× matrices given by

ˆ 0 ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ I 0 I 0 , 11 11 11 E T ε ⎡ ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ Q e Q e , 21 12 12 E T ε ⎡ ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ R e R e , 22 22 22 E T ε ⎡ ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ T e T e , (9)

in which I is the 3 3× identity matrix and the elements of 3 3× matrices QE,

E

R , E

T , and 3 1× matrices e are _ij

1 1

E ik i k

Q =C , R_ikE =C_{i k1 2}, T_ikE =C_{i k2 2}, (e_{ij s}) =e_ijs. Consider the generalized displacement u in the following form

1 2 4

( ,x x t, )= ( )w +ψ( )t

u u e , (10) where w x x t( ,_{1 2}, ) is defined implicitly by

1 2 1 2

( , ,w x x t, ) wt x p w x( ) 0

Δ = − − = , (11) with ( )p w as an analytic function of w , ( )ψ t a function of t , and

[

]

4 0, 0, 0, 1 T = e . With (10), (8) becomes 2 2 1 _ˆ 1 [ ( )( T) ( ) ] ( ) w p w p w w w ρ ∂ ⎧ _{− + + + ′} ⎫₌ ⎨ ⎬ ′ ′ Δ ∂ ⎩Q I R R T Δ u ⎭ 0, (12) where u′( )w denotes the derivative of u( )w with respect to w, Δ is given by ′

1 2 2 ( , , , ) ( ) w x x t t p w x w ∂Δ ′ ′ Δ = = − ∂ , (13) and ( )p w′ is the derivative of p w with respect to ( ) w. Equation (12) shows that for the generalized displacement u given by (10) to be a solution to (8), u( )w must satisfy (12) and ( )ψ t is arbitrary.

By letting u′( )w have the following form ( )w f w( ) ( )w

′ =

u a , (14) where ( )f w is an arbitrary scalar function of w. It follows that u is a solution of (8) if ( , ) ( )p w w = , D a 0 (15) where ( , )D p w is given by 2 2_ˆ ( , ) ( T) p w = + p + + p −ρw D Q R R T I . (16) For non-trivial solutions of ( )a w we must have

( , )p w =0,

D (17) where D is the determinant of D . Equation (17) provides eight eigenvalues of p

as a function of w, denoted by p w_α( ),α =1, 2,...,8. The corresponding function

1 2

( , )

w_α =w_α y y , where y₁=x t₁/ , and y₂ =x₂/t, can be determined from (11) with ( )

p w replaced by p w_α( ).

A graphical way for finding real p s′ can be achieved by making use of the slowness surface, ( ,s s space, where _{1 2}) s₁=1/w and s₂ = p w/ . No real p_α exists for t→ ∞ or w→0. In this case p_α appear in four complex conjugated pairs. On the other hand as t→0 or w→ ∞, there are six real p_α. From (16) the other two complex roots and the corresponding a may be shown to be *

* 12 * * 4 22 , , i p ε ε p ε − + = a =e , (18) where ε = ε ε_{11 22}−ε₁₂2 and i= −1.

The general solution of the generalized displacement satisfying (8) may be represented as 8 1 2 ,1 1 ( ) ( ,x x t, ) fα wα _α(w_α) α= α = ′ Δ

∑

u a , (19) 8 1 2 ,2 1 ( ) ( ,x x t, ) p wα α f_α(w_α) _α(w_α) α= α = ′ Δ

∑

u a , (20) 8 1 2 4 1 ( ,x x t, ) wα f_α(w_α) _α(w_α) ( )t α α ψ = = − + ′ Δ

∑

u a  e . (21) By substituting (19) and (20) into the constitutive laws, the general solutions of the generalized stress vectors t and ₁ t , where ₂ t and ₁ t are given by ₂

1 ( 11, 21, 31, 1) T D σ σ σ = t and 2 ( 12, 22, 32, 2) T D σ σ σ =

t , can be expressed, respectively, as

8 2 1 1 2 1 1 _ˆ ( ,x x t, ) wα α(wα) p wα( α) α(wα) fα(wα) α α ρ = ⎡ ⎤ = _⎣ − _⎦ ′ Δ

∑

t Ia b , (22) 8 2 1 2 1 ( ) ( ,x x t, ) fα wα _α(w_α) α= α = ′ Δ

∑

t b , (23) where

(

2

)

1 _ˆ ( )w ( T p w( ) ) ( )w w p w( ) ( )w p α = + α α = − −ρ + α α b R T a Q I R a , (24)

The second identity in (24) follows from (15).

An alternative method for determining p_α( ,y y_{1 2}) is given by substituting (11) into (16) and rewrite D as

2 1 2 ˆ ˆ ˆ ˆ ( ,p y y, )= +p( + T)+p D Q R R T, (25) where 2 1 11 11 11 ˆ E T y ρ ε ⎡ − ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ Q I e Q e , 1 2 21 12 12 ˆ E T y y ρ ε ⎡ − ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ R I e R e , 2 2 22 22 22 ˆ E T y ρ ε ⎡ − ⎤ = ⎢ ₋ ⎥ ⎣ ⎦ T I e T e .

The function p_α( ,y y_{1 2}) can be directly obtained by D( ,p y y₁, ₂) =0 . The corresponding w_α( ,y y_{1 2}) is simply given by (11) and the associated eigenvector

1 2

( ,y y )

α

1 2 1 _ˆ ˆ _{( , ) (}ˆT ˆ_{) (} ˆ₎ y y p p p α α α α α α = + = − + b R T a Q R a , (26) The second line of (26) follows from (15). The vector bˆ ( , )_α y y_{1 2} is related to

( )w α b by 1 2 2 ˆ _{( ,y y ) ( )w wy} ˆ _{( )w} α = α −ρ α b b Ia . (27) Equation (26) can be cast into the following eight-dimensional eigenvalue problem

ˆ ˆ ˆ _{= p} Nξ ξ , (28) where 1 2 3 1 ˆ ˆ ˆ ˆ ˆT ⎛ ⎞ = ⎜_⎜ ⎟_⎟ ⎝ ⎠ N N N N N , ˆ ˆ ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ a ξ b , 1 1 ˆ = −ˆ− ˆT N T R , ˆ2 ˆ 1 − = N T , ˆ3 ˆ ˆ 1ˆ ˆ T − = − N RT R Q.

The p and ˆξ are the eigenvalue and right eigenvector, respectively, of Nˆ . Since

2 ˆ

N and Nˆ₃ are symmetric, the left eigenvector, ηˆ , of Nˆ defined by ˆ ˆT ˆ p = N η η , (29) is given by ˆ ˆ = ⎜ ⎟⎛ ⎞ ⎝ ⎠ b a η .

If the eigenvalues p_α and p_β are distinct, the corresponding left and right eigenvectors satisfy orthogonality relations

ˆ ˆ ˆ

ˆT T T 0,

α β =a bα β +b aα β = α β≠

η ξ . (30) 3. Dynamic Surface Green’s Function

Consider a piezoelectric half-spacex₂ ≥ . The surface at 0 x₂ = is assumed 0 traction-free mechanically and insulating electrically. A line impulse force hδ( )t and a line impulse charge qδ( )t , δ( )t being the Dirac delta function, are applied at the origin. The surface is mechanically traction-free (t₂E =0 ) and electrically insulating ( D₂ = ), where 0 t₂E =(σ₁₂ σ₂₂ σ₃₂)T . Then the corresponding conditions on the boundary x₂ = are given by 0

2( , )x t1 = −δ( ) ( )x1 δ t

t F , (31) where F=(h h₁, ₂,h₃,−q)T.

As in the case of unbounded media (Wu and Chen, [10]), the generalized stresses are homogeneous of degree -2 and the generalized displacement u homogeneous of degree -1. Thus the fictitious generalized displacement u given by *

* 1 2 1 2 ( ,x x t, ) t ( ,x x , )τ τd −∞ =

∫

u u , (32) is homogeneous of degree 0. The conditions for the corresponding fictitious generalized stress t is *2 * * 2( , )1 2( , )1 ( )1 ( ) t x t x τ τd δ x H t −∞ =

_∫

= − t t F. (33) The general expression for the generalized stress vector t as given by (23) can be *₂ rewritten in the following matrix form:

* 2 1 2 1 ( ,x x t, )=2 Re⎛_⎜ ( ) ( )⎞_⎟ ′ Δ ⎝ ⎠ t B w f w , (34) where B w( )=

[

b₁(w₁) b₂(w₂) b₃(w₃) b₄(w₄)

]

, 1 2 3 4 1 1 1 1 1 diag⎡ ⎤ = _{⎢ ⎥} ′ ′ ′ ′ ′ Δ _⎣Δ Δ Δ Δ _⎦ ,

[

1 1 2 2 3 3 4 4

]

( )= f w( ) f w( ) f w( ) f w( ) T

f w . For t>0, (33) and (34) yield

(

1

)

1 2 Re q( )y = −δ( )y F, (35) where 1 1 1 ( )y = ( ) ( )y y q B f . (36) The analytic function q( )η with η= y₁+iy₂ satisfying (35) is given by

1 ( ) 2 i η π η = q F . (37) Therefore, 1 1 ( ) ( ) 2 w w iw α α α π − = f B F . (38) Let e be the unit vector in _α α -direction and the matrix I_α =e e . The analytic _{α α}T function ( )f w is obtained as 4 1 1 1 1 ( ) ( ) 2πi w α α wα − = =

∑

f w I B F , (39) where the 4 4× matrix B(w_α)=

[

b₁(w_α) b₂(w_α) b₃(w_α) b₄(w_α)

]

, the vector

(

)

(w ) T p w( ) (w )

β α = + β α β α

b R T a , (p w_{β α}) and a_β(w_α) are the eigenvalues and eigenvectors, respectively, of N =ˆξˆ pξ with w wˆ = _α. Equation (39) can also be expressed as the following form:

4 1 1 1 1 1 ( ) ( ) ( ) 2 2 T T k k k k k k k f w w w iw α α α iw π = − π − ⎛ ⎞ = _{⎜ ⎟} = ⎝

∑

⎠ e I B F e B F . (40) The fictitious generalized velocity u is given by *

4 4 * 1 * 1 2 4 1 1 4 1 * 4 1 1 1 ( , , ) Im ( ) ( ) ( ) 1 1 Im ( ) ( ) ( ) T k k k k k T k k k k k k x x t w w t w w t α α α ψ π ψ π − = = − = ⎧ ⎛ ⎞⎫ = − _{⎨ ⎜ ⎟⎬} + ′ Δ _{⎝ ⎠} ⎩ ⎭ ⎧ ⎫ = − _{⎨ ⎬} + ′ Δ ⎩ ⎭

∑

∑

∑

u a e I B F e a e B F e    , (41)

The function ψ*( )t is determined by requiring u*→0 as t→0+ in (41). The result is * 4 * 1 2 4 ( ,x x t, ) F ( )t t ψ π ε ⎡ ⎤ = −_⎢ + _⎥ ⎣ ⎦ u  e , (42) or * 4 * ( )t F ( )t t φ ψ π ε = − +  _{ , (43)} where w*= +y₁ p y* ₂.

If φ*( )t is required to be bounded at t=0, the function ψ*( )t must be in the following form

*_{( )t} F4 _{c t( )} t ψ π ε = +  , (44) where ( )c t is a regular function of t . Since only the spatial variation of the electric potential φ*( )t is of interest, we can let ( )c t = (Wu and Chen, [10]). The actual 0 generalized displacement u, which is the same as the fictitious generalized velocity

* u , is obtained as * 1 2 1 2 1 2 ( ,x x t, )= ( ,x x t, )= _sf+( ,x x t, ) u u G F, (45) where G+_sf is the free surface Green’s tensor for t>0 and can be expressed as

4 1 1 2 4 4 1 1 1 4 1 1 1 1 ( , , ) Im ( ) ( ) 1 1 1 Im ( ) ( ) n T T sf k k k k k n T k k k k k k x x t w w t w w t α α α π π ε π π ε + + + − = = − = ⎧ ⎡ ⎛ ⎞⎤⎫ ⎪ ⎪ = − _{⎨ ⎢ ⎜ ⎟⎥⎬}+ ′ Δ _{⎝ ⎠} ⎪ ⎣ ⎦⎪ ⎩ ⎭ ⎧ ⎫ = − _{⎨ ⎬}+ ′ Δ ⎩ ⎭

∑

∑

∑

G a e I B e e a e B I . (46)

Since as t→0+, the fictitious generalized displacement is

{

}

* 4 * 1 2 1 2 4 ( ,x x t, ) F Re log(x p x ) πε = + u e , (47) while u*( ,x x t_{1 2}, )=0 as t→0−. The Green’s function G_sf( ,x x t_{1 2}, ) for t>0− is given by

{

*

}

1 2 1 2 1 2 4 ( ) ( , , ) ( , , ) Re log( ) sf sf t x x t x x t δ x p x πε + = + + G G I . (48) 4. Numerical Examples

The Green’s functions given by (48) were computed next for quartz, which is a crystal of trigonal 32 symmetry class. The Green’s functions may be expressed in the following dimensionless form:

0 0 1 2 0 0 1 2 0 0 1 2 ( / ) ( , , ), , 1, 2, 3, ( , ) ( / ) ( , , ), 4, 1, 2, 3 or 1, 2,3 , 4 ( / ) ( , , ), 4, 4 ij ij ij ij C r c G x x t i j G e r c G x x t i j i j r c G x x t i j π ψ τ π πε ⎧ = ⎪ =_⎨ = = = = ⎪ _{= =} ⎩ , (49)

where c₀ = C₀/ρ τ, =tc₀/ ,r r= x₁2+x₂2, ψ =tan (−1 x₂/x₁). Here C , ₀ e and ₀

2 0 e0/C0

ε = , respectively, are certain reference elastic constant, piezoelectric stress constant and permittivity. The elastic stiffness constants C, the piezoelectric stress constants e, and dielectric constants ε of quartz used for calculations were from [11].

Figure 1 displays the wave surface of quartz in the infinite region. The three bulk wavefronts are denoted by L, FT, and ST. Some head wavefronts are designated as Hi

(i=1 ~ 5). Figure 2 shows the components of Green’s functions, ( )G ₁₁, ( )G ₁₂, and

13

( )G for the observational angle ψ =0D. In figure 2, the surface wave is denoted by SAW and the pseudo-surface wave is denoted by PSAW. Figure 3 shows the components of Green’s functions (( ) ,G 1j j=1, 2, 3) for the observational angle

36

ψ = D

Figure 1. Wavefronts and the angle of observation for quartz.

Figure 2. The components of Green’s functions ( )G , 11 ( )G , and 12 ( )G for 13

0

ψ = D

.

Figure 3. The components of Green’s functions ( )G , ₁₁ ( )G , and ₁₂ ( )G for ₁₃ 36

ψ = D

5. Concluding Remarks

A novel formulation developed by Wu [9] for two-dimensional anisotropic elastodynamics is extended to treat general piezoelectric materials. The present formulation does not require integral transforms and can be used to acquire the general solutions of displacement or stress fields in the time domain directly. The formulation is applied to derive analytic expressions for dynamic Green’s functions of general half-space piezoelectric solids. The Green’s functions can be simply calculated using the eigenvalues and eigenvectors of a related eight by eight matrix. Numerical examples provided for the piezoelectric material-quartz show that the dynamic responses can be accurately computed by the proposed formulation.

6. Acknowledgements

The research was supported by the National Science Council of Taiwan under grant NSC 97-2218-E-151-008 and NSC 98-2221-E-151-057.

References：

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crystals. Existence considerations.” J. Appl. Phys. 47, pp. 1799-1807, 1976. [3] D. B. Taylor, and S. Crampin, “Surface waves in anisotropic media: propagation

in a homogeneous piezoelectric halfspace.” Proc. R. Soc. Lond. A 364, pp. 161-179, 1978.

[4] R. Peach, “On the existence of surface acoustic waves on piezoelectric substrates.” IEEE Trans. Ultrason. Ferroelect. Freq. Contr.. Vol 48, No. 5, pp. 1308-1320, 2001.

[5] C. F. Gao, and N. Noda, “Green’s functions of a half-infinite piezoelectric body: exact solutions.” Acta Mechanica 172, pp. 169-179, 2004.

[6] A. N. Stroh, “Dislocations and cracks in anisotropic elasticity,” Phil. Mag. 3, pp. 625-646, 1958.

[7] A. N. Stroh, “Steady state problems in anisotropic elasticity,” J. Math. Phys. 41, pp. 77-103, 1962.

[8] T. C. T. Ting, ”Anisotropic elasticity: theory and application.” Oxford University Press, 1996.

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New York, 1975. 計畫成果自評部份： 本計畫之執行已完成壓電材料半空間物體暫態問題研究分析，解得在任意半 無限域之異向性彈性壓電體內之任意方向集中力及差排作用下的暫態解以及可 計算此類問題解的數值程式。本計畫之成果將可對複雜之異向性彈性壓電波傳問 題 ， 提 供 一 個 簡 單 方 便 之 分 析 工 具 。 初 步 成 果 已 參 與 ” The 2010 IAENG International Conference on Scientific Computing (ICSC'10)” 國際會議，未來預計 論文整理後投稿國際期刊。