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Boundary element method applied to certain structural-acoustic coupling problems
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 71 (1988) 225-234 NORTH-HOLLAND
BOUNDARY ELEMENT METHOD APPLIED TO CERTAIN STRUCTURAL-ACOUSTIC COUPLING PROBLEMS Masataka TANAKA Department of Mechanical Engineering, Shinshu University, 500 Wakasato, Nagano 380, Japan
Yoshifumi MASUDA Mitsubishi Heavy Industries Co., Ltd. Received 16 April 1988 Revised manuscript received 30 May 1988
This paper is concerned with an integral equation approach to certain structural-acoustic coupling problems. It is assumed that the structure is composed of plate components and is excited by the external or the internal noise source. The vibration of the cavity or the ambient atmosphere is governed by the Helmholtz equation, and that of the structure by the reduced out-of-plane bending vibration equation. These differential equations are transformed into the set of boundary integral equations, and they can be solved by means of the usual boundary element method and the boundary-domain element method. Taking ac~:ount of the coupling conditions on the surface of the ~tructure, we can finally obtain the coupled se~ of linear simultaneous equations with respect to nodal values. The solution of these equations enables us to investigate the properties of the structuralacoustic coupling system under consideration. The potential usefulness of the proposed method is demonstrated through some sam~'.,~ computations.
1. Introduction
In recent years there has been an increasing need to study the control or reduction techniques of noise emanating from vibrating structures. The acoustic properties of a homogeneous cavity with an arbitrary shape including also an infinite domain can be investigated by solving the Helmholtz equation. Many attempts have already been made to study such linear problems by meens of the boundary integral equation method, and much knowledge has been accumulated [1-4]. However, the noise in most cases is caused by a coupled vibrating effect between the structure and its ambient medium (air). Therefore, such structural-acoustic coupling problems must be analyzed to investigate more exactly the noise properties. Several attempts were made for these coupling problems from both the analytical and numerical standpoints. The analytical approach [5] is restricted to simpler systems, while the numerical one is in ~enera| very powerful and applicable to any cases with arbitrary shapes and boundary c~fl,d'itl0ns.' At present there is available the numerical approach in which the structure is discretized by the finite element method and the ambient medium is treated by the boundary element method [6-9]. In this study, the integral equation method is applied to the whole system under considera0045-7825188153.50 © 1988, Elsevier Y,cience Publisher B.V. (North-Holland)
226
M. Tanaka, Y. Masuda, BEM for swuctural-acoustic coupling
tion and resulting sets of integral equations are discretized by means of the boundary and also domain elements. The proposed method of solution consists of the following techniques: (l) The boundary element analysis system for three-dimensional acoustic problems; (2) The boundary-domain element method for the out-of-plane bending vibration problem of elastic plates using the corresponding static fundamental solution; (3) The coupling technique in which the equilibrium and compatibility conditions on the plate surface are taken into account. A new computer program is developed in this study and applied to some typical structuralacoustic coupling problems. Finally, the usefulness of the proposed method is demonstrated through such sample computations. 2. Structural-acoustic coupling system
Let us consider a cavity surrounded by six plate components as shown in Fig. 1. For the sake of simplicity, we assume that one plate is elastic and the other is rigid. The domain of the acoustic cavity is denoted by V and its surface by S. Using the Cartesian coordinate system O-xlxex3, we can express the governing equation for the vibration of the acoustic cavity as follows: 1
V2P(x, t) - ~o2 P(x, t) in V,
(1)
~.vhere V~ is the Laplacian operator and the superimposed dot denotes time derivative, while C0~.~the phase velocity (in this case the sound velocity in air) and P the (sound) pressure. It is assun,~, that the plate material is isotropic and homogeneous. Using the local Cartesian coordina~,_~,system O-x~x~x~, in which the axes x~ and x~ lie in the middle plane of the plate as shown in ~'~. 2, we can express the governing differential equation of the out-of-plane bending vibrat'Lon as follows:
DV4W(y, t) + gW(y, t) + pphW(y, t) = P(y, t) - Pe(Y, t) in Sp,
(2)
where V4 is the biharmonic differential operator, W the lateral deflection of the plate, D the flexural rigidity, pp the mass density of the plate material, h the plate thickness, g the damping coefficient, and Pe the external pressure. n+ --
Etastic Prate ~
!
[ Cevity'V
I x~
.x: .x' "'':"
Boundary S Fig. 1. Structural-acoustic system.
Corner Point At d / xl'
w
_ ,
At-I
Pe y "y' I1 f l
x.' A,.,I x," Boundary C
I
'
P
Fig. 2. Notation for out-of-plane bending of plate.
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
227
Equations (1) and (2) are combined by taking into account the conditions on the plate region Sp, to obtain the governing equation of the structural-acoustic coupling system under consideration. The details of this procedure will later be discussed. 3. Boundary element analysis of acoustic field We shall show here the outline of boundary element analysis of a three-dimensional acoustic field with an arbitrary shape subject to arbitrary boundary conditions. It is assumed that the acoustic pressure P(x, t) is expressed as P ( x , t) -
p(x) exp(ieot),
(3)
where i = q'~"T. Then, (1) can be reduced to the Helmholtz equation expressed as
V2p(x) "[" k2p(x)
= 0,
(4)
where k = V to~Co and is referred to as the wave number. On the boundary of the acoustic field there are the following relationships [1-4]:
a__p.p=-itopv, an
ap = -itop/7_ On z '
(5)
where v is the particle velocity, z the acoustic impedance, p the mass density of the acoustic medium, and a( )/an the directional differentiation along the outward normal to the acoustic field boundary. Making use of (5) we can transform in the usual ~vay [1-4] the Helmholtz equation (4) into the following boundary integral equation:
c(y)p(y)+ fs q*(Y' y')p(y')dS(y')+itop fs --iwp
p(y') P*(Y' y')
fsv P*(Y' y ' ) v ( y ' ) d S ( y ' ) ,
z( y' ) dS(y') (6)
where Sv denotes the boundary portion of the acoustic field in which the particle velocity v is prescribed, while Sz stands for the portion with the given acoustic impedance. As is weli known [11, 12], the coefficient c in (6) is such that c(y) = 0/(4~r), in which 0 is the solid angle of the boundary point y with respect to the acoustic medium V. The asterisked functions in (6) are the fundamental solutions of the Helmholtz equation, and can be expres~,d for ~+he three-dimensional problems as follows: 1 exp(-ikr), p*(x, y ) = 4~rr
y) q*(x, y ) = ap*(x, On '
+(7)
where r denotes the distance between points x and y. If the boundary integral equation (6) is discretized by means of the usual boundary element
M. Tanaka, Y. Masuda, BEM for structural-acousticcoupling
228
method, the following set of equations can be derived:
[HI(p} =[Gl(v},
(8)
where {p} and {v} denote the column vectors including all the nodal values of the acoustic pressure p and the particle velocity v, respectively. The components h ij and gi~ of the coefficient matrices [H] and [G] in (8) can be computed from the fundamental solutions p* and q* in the following way:
hj~ = cSiJ+ I¢ q*(y'' ¢)t~(~)lJ(~)l d~, (9)
/,,
)1d~, where St denotes the jth boundary element and [J(~:)[ the Jacobian of the coordinate transformation from the refere::ce coordinate to the nondimensional intrinsic coordinates ~:of the element. Taking account of the prescribed boundary conditions, we can solve (8) for the unknown nodal values on the whole boundary.
4. Bending vibration analysis of elastic plates The so-called boundary-domain element method [13] will be briefly discussed for analyzing the out-of-plane bending vibration problem of elastic plates. It is assumed that the plate vibrdtes harmonically with time as in the acoustic field and that the deflection W(y, t), the external pressure Pe(~', t), and the acoustic pressure P(y, t) can be expressed as
W( y, t) = w( y) exp(io~t),
Pe( y, t)= p~( y) exp(ioJt),
P( y, t) = p( y) exp(io~t).
(10)
Then, (2) can be reduced to
DV4w(y) + itogw(y) - tO2pphw(y) = p(y) - p~(y) .
(11)
The integral equation formulation can be started with the following weighted residual statement:
fsp {DV4w(y)-itogw(y)-oJ2pphw(y)-p(y)+pe(y)}w*(y,
y')dSp=O.
(12)
Using the exact fundamental solution of the reduced bending vibration equation (11) as the weight function w* in (12) we can derive the integral equation which can be solved by means
M. Tanaka, Y. Masuda, BEM for structural-acousticcoupling
229
of the usual boundary element method. In this study, however, we use an approximate fundamental solution, that is, the fundamental solution of static bending problems given by 1 w*(y, y ' ) = 8=D
t .2
in r
(13)
It should be mentioned here that use of the exact fundamental solution leads to some complexity in mathematical treatment of the special functions included in the resulting integral equation, while such a difficulty is circumvented if the approximate fundamental solution is employed. Applying the above weight function to the weighted residual expression (12) and tracing the sa~e procedure as in the static bending problem of elastic plates [14-16], we can finally obtain the boundary integral equations which can be expressed in the following form:
Ck'(Y)u'(Y) + fc P~"(Y' y')u,(y') d C t y ' ) - fc U~,(y, Y')Pl(Y') dC(y') L
+ ~'. [M*,k(y , Am)Ul(Am)- U~2(y , Am)Mm(Am)] m=l
fsp (i¢og -- ¢02pph)U~2(y, x) dSp(x) +
fsp U~2(Y'x){p(x) - p~(x)} dSp(x),
(14)
where [G,] = '
L'
~,
=
,
=
M*,J
r.
'
(15)
The detailed expressions of the above notation can be found in the literature [14-16]. Since the approximate fundamental solution is employed for the present formulation, the first term of the fight-hand side of the boundary integral equation (14) includes the unknown variable u t (=deflection w) in the inner domain of the plate. To solve these equations by the usual boundary element method we must introduce an iterative solution procedure. To circumvent such difficulties we may use an additional integral equation. For this purpose we shall here take into account the integral equation with respect to the deflection w(y) at an internal point y expressed by
u,(x) + fc P~,(x, y')u,(y') dC(y') fc U~t(x' Y')Pt(Y') dC(y') -
L
+ mffil ~, [M*,(x, Am)Ul(Am)- U~2(X,Am)M,,(Am) ]
230
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling t"
= - Jsp (i¢0g - co2pph)U'~z(X, x')ui(x') dSp(x')
+ fsp U~lz(X' x'){p(x') - p¢(x')} dSp(x').
(16)
In the boundary-domain element method [13] the boundary integral equation (14) and the integral equation (16) are simultaneously discretized by means of the usual boundary element as well as the domain element which plays a similar role to that of the finite element. Application of the boundary-domain element method to (14) and (16) can yield the set of linear simultaneous equations with respect to nodal values on the plate boundary as well as in the inner domain of the plate. If the boundary conditions expressed as w ffi 0,
T, = 0
(clamped),
w = 0,
M~ - 0
(simply supported),
M. ffi 0,
V. = 0
(free),
(17)
are taken into consideration, we can finally obtain the following set of equations, that is, from
(14)
[ABI{Xe} ffi -(icog - co2pph)[Ds]{wo} + [Dsl{p - p.} ,
(18)
and from (16)
{ w o } + [A o]{ Xs} = - ( i c o g - oJ2pph)[Do]{ w o } + [Do]{ p -- p.} .
(19)
In the above equations {wo} denotes the column vector of nodal deflections in the inner domain of the plate. The other quantities [Aa], [Ds], [At,], and [Do] are the coefficient matrices which are related to the variables shown in (15) and can be computed from the fundamental solution w* given by (13). It is interesting to note that elimination of the unknown vector {X e} on the boundary can lead to the following set of equations:
[Gl{wo} =
+ [GI{p°},
(20)
where [1] being the unit matrix, [H~] ffi [1] - (ioJg -
~O2pph)[Gpl,
[Gp] =
[Aol[As]-~[Da]- [Do].
(21)
Suppose that in (20) and (21) the damping coefficient g is negligibly small and no pressure loadings are applied, we can obtain the algebraic set of eigenvalue equations expressed by
([I] + oj2pph[Gp]){wo} - 0 ,
(22)
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
231
from which we can calculate the eigenfrequency and also the eigenmode using a common-use library of subroutine subprograms available at almost all computing facilities.
S. Analysis of structural-acoustic coupling systems .The discretized sets of equations (8) and (20) are now combined to yield the set of equations for the structural-acoustic coupling system under consideration. By using the same nodal points on the interface of the two fields the acoustic field is discretized by the usual boundary element method and that of the elastic plate structure by the boundary-domain element method. The equilibrium and compatibility conditions must be satisfied on the interface of the acoustic field and the elastic plate. These conditions can be expressed such that pA = p ,
V =i(ow a ,
(23)
where p ^ denotes the acoustic pressure. To take into account these conditions, (8) is rewritten as follows:
,,,]{,:} where the superscriptsA and I denote the variables on the acoustic fieldand on the interface, respectively. Equation (24) is now combined with (20) to yield
"]{';I[°"i° ]I°'l I°] Go
=
0
-Hp
wa
+
Gp {P,}'
(25)
Equation (25) is solved numerically under the appropriate boundary conditions which can be expresseG as Pe - P~ on the interface, v- ~ on the other boundary,
(26)
where the superimposed bar denotes a prescribed value. These boundary conditions imply that the elastic plate is excited by the known external pressure. For our later use we now rewrite once again (25) together with the boundary conditions in the following form:
o6,
-iG'
H,,
o
0
6,
(27)
The solution of (27) can provide the response of the structural-acousticcoupling system under consideration.
232
M. Tanaka, Y. Masuda, BEM for structural.acoustic coupling
6. Computational results and discussion
A new computer code has been developed in this study. A couple of sample problems are calculated by using the computer code and the results obtained are discussed in the following. Now, we consider the acoustic cavity surrounded by six equal plates as shown in Fig. 3. It is assumed that one plate is elastic and excited by the external pressure, while the other plates are rigid, that is, subject to the boundary condition v = 0. The cavity is a cubic with the edge length 200 mm. The elastic plate is made of brass, its thickness is 0.941 mm, and mounted on the rigid plates with the simply supported boundary conditions, which correspond to the experiment by Guy and Bhattacharaya [5]. The element subdivision is shown in Fig. 4, where the eight-node isoparametric element is used for the whole boundary of the acoustic cavity (25 elements for the rigid boundary and 16 for the elastic plate), while the boundary of the elastic plate is divided into constant boundary elements of an equal size (totally 96 elements). In this numerical example the damping coefficient is assumed as g = 0. The sound pressure P2 on the rigid boundary opposite to the elastic plate as shown in Fig. 3 is calculated under these computational conditions. In Fig. 5 comparison is made with the analytical solution by Guy et al. [5]. The present BEM solutions are in good agreement with the analytical solutions. It is interesting to note that the experimental results of Guy et al. [5] also agree well with both the solutions mentioned above. The frequency % shown along the abscissa in Fig. 5 denotes the eigenfrequency of the elastic plate alone. The discontinuities appearing near the frequencies 90 Hz, 400 Hz, 700 Hz, and 850 Hz are the resonance points corresponding to the eigenmodes (°,t, c°~3,%3 of the elastic plate and the eigenmode 01 of the cavity under the coupling system, respectively. ELastic PLate
Fig. 3. Cavity-backed elastic plate. IIr
,
,
i , , , ,
,
w
•
tic PLate Fig. 4. Boundary element discretization.
, . , ,
50
_~ 50
30 2O
~30
+o
0
m 0 -10 .--Guy eL. ot.~( -20 ,° BEM , , ,~2~,~., 10 J . l O O ~ r Tl~k~ (0141" +T+k Frequency (Hz) ~u w=4 Fig. 5, Transmission of sound through simply supported elastic plate.
J
m 0 . . ~ , , . ~ ~ -I0 , , ,BEM ~ , ~ ° -20
1~,t
i
Frequency (Hz) ~m w~wz~, M Ik Fig. 6. Transmission of sound through clamped elastic plate.
M. Tanaka, Y. Masuda, BEM for swuctural.acoustic coupling
233
Figure 6 shows the results obtained for the same structural-acoustic coupling system wh the elastic plate is clamped to the rigid boundary. 7. Concluding remarks This paper has presented an integral equation approach to the structural-acoustic coupling problems. Some typical examples were computed by using the computer code developed in this study. The potential usefulness and availability of the proposed method were demonstrated through comparison of the results obtained with other solutions. Although it was assumed in this paper that the structure was composed of plate components, the present method can also be applied to other types of the structure with arbitrary boundary conditions. It could be recommended as future research work to explore a similar integral equation approach to a wide variety of the structural-acoustic coupling problems. Numerical computations of this study were carried out by using mainly the HITAC computer installed at Shinshu University and partly the HITAC machine of the Computing Center at University of Tokyo. Acknowledgment The authors wish to thank an unknown reviewer of the journal for his useful comments and suggestions to improve the manuscript. References [1] H.A. Schenck, Improved integral equation formulation for acoustic radiation problems, J. Acoust. Soc. Amer. 44 (1968) 41-58. [2] A.F. Seybert, B. Soenaarko, F.J. Rizzo and D.J. Shippy, Application of the BIE method to sound radiation problems using an isoparametric element, ASME J. Vibration, Acoustics, Stress, and Reliability in Design 106 (1984) 414-420. [3] M.A. Latcha and A. Akay, Application of the Helmholtz integral in acoustics, ASME J. Vibration, Acoustics, Stress and Reliability in Design, 108, (1986) 447-453. [4] M. Tanaka and Y. Masuda, A general purpose BEM computer code for acoustic problems, Trans. Japan Soc Mech. Engrs. Ser. C53 (1987) 387-391 (in Japanese). [5] R.W. Guy and M.C. Bhattacharaya, The transmission of sound through a cavity-backed finite plate, J. Sound and Vibration 27 (1973) 207-223. [6] A. Sestieri, D.D. Vescovo and P. Lucibello, Structural-acoustic coupling in complex shapec~ cavities, J. Sound and Vibration 96 (1984) 219-233. [7] S.H. Sung and D.J. Nefske, A coupled structural-acoustic finite element model for vehicle interior noise analysis, ASME J. Vibration, Acoustics, Stress, and Reliability in Design 106 (1984) 314-318. [8] M. Bessho, H. Kawabe and Y. lwasaki, Underwater sound radiated from a vibrating ship hull, in: M. Tanaka and C.A. Brebbia, eds., Boundary Elements VIII (Springer, Berlin, 1986) 83-93. [9] S. Suzuki, M. lmai and S. lsyiyama, Coupling of the boundary element method and modal analysis for structural acoustic problems, Trans. Japan Soc. Mech. Engrs. Ser. C 52 (1986) 310-317 (in Japanese). [10] M. Tanaka, K. Miyazaki and K. Yamagiwa, Integral equation approach to free vibration problems of assembled plate structures, in: M. Tanaka and Q. Du, eds., Theory and Applications of Boundary Element
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M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
Methods, Proceedings 1st Japan-China Symposium on Boundary Element Methods, 1987, Karuizawa, Japan, (Pergamon Press, Oxford, 1987). [11] C.A. Brebbia, The Boundary Element Method for Engineers (Pentech Press, London, 1978). [12] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques---Theory and Applications in Engineering (Springer, Berlin, 1983). [13] M. Tanaka and K. Tanaka, On a new boundary element solution scheme for elastoplasticity, Ing. Arch. 50 (1982) 289-295. [14] M. Tanaka and K. Miyazaki, A direct BEM for elastic plate-structures subjected to arbitrary loadings, in: C.A. Brabbia and G. Maier, eds., Boundary Elements VII (Springer, Berlin, 1985) 4/3-4/16. [15] M. Tanaka and K. Miyazaki, A direct boundary element method for elastic bending analysis of plates, Trans. Japan Soc. Mech. Enos. Set. A 51 (1985) 1636-1641 (in Japanese). [16] M. Tanaka, Bending problems of elastic bodies, in: Boundary Element Methods--Theory and Applications, Corona, Tokyo (1986) Ch. 4, 69-92 (in Japanese).
BOUNDARY ELEMENT METHOD APPLIED TO CERTAIN STRUCTURAL-ACOUSTIC COUPLING PROBLEMS Masataka TANAKA Department of Mechanical Engineering, Shinshu University, 500 Wakasato, Nagano 380, Japan
Yoshifumi MASUDA Mitsubishi Heavy Industries Co., Ltd. Received 16 April 1988 Revised manuscript received 30 May 1988
This paper is concerned with an integral equation approach to certain structural-acoustic coupling problems. It is assumed that the structure is composed of plate components and is excited by the external or the internal noise source. The vibration of the cavity or the ambient atmosphere is governed by the Helmholtz equation, and that of the structure by the reduced out-of-plane bending vibration equation. These differential equations are transformed into the set of boundary integral equations, and they can be solved by means of the usual boundary element method and the boundary-domain element method. Taking ac~:ount of the coupling conditions on the surface of the ~tructure, we can finally obtain the coupled se~ of linear simultaneous equations with respect to nodal values. The solution of these equations enables us to investigate the properties of the structuralacoustic coupling system under consideration. The potential usefulness of the proposed method is demonstrated through some sam~'.,~ computations.
1. Introduction
In recent years there has been an increasing need to study the control or reduction techniques of noise emanating from vibrating structures. The acoustic properties of a homogeneous cavity with an arbitrary shape including also an infinite domain can be investigated by solving the Helmholtz equation. Many attempts have already been made to study such linear problems by meens of the boundary integral equation method, and much knowledge has been accumulated [1-4]. However, the noise in most cases is caused by a coupled vibrating effect between the structure and its ambient medium (air). Therefore, such structural-acoustic coupling problems must be analyzed to investigate more exactly the noise properties. Several attempts were made for these coupling problems from both the analytical and numerical standpoints. The analytical approach [5] is restricted to simpler systems, while the numerical one is in ~enera| very powerful and applicable to any cases with arbitrary shapes and boundary c~fl,d'itl0ns.' At present there is available the numerical approach in which the structure is discretized by the finite element method and the ambient medium is treated by the boundary element method [6-9]. In this study, the integral equation method is applied to the whole system under considera0045-7825188153.50 © 1988, Elsevier Y,cience Publisher B.V. (North-Holland)
226
M. Tanaka, Y. Masuda, BEM for swuctural-acoustic coupling
tion and resulting sets of integral equations are discretized by means of the boundary and also domain elements. The proposed method of solution consists of the following techniques: (l) The boundary element analysis system for three-dimensional acoustic problems; (2) The boundary-domain element method for the out-of-plane bending vibration problem of elastic plates using the corresponding static fundamental solution; (3) The coupling technique in which the equilibrium and compatibility conditions on the plate surface are taken into account. A new computer program is developed in this study and applied to some typical structuralacoustic coupling problems. Finally, the usefulness of the proposed method is demonstrated through such sample computations. 2. Structural-acoustic coupling system
Let us consider a cavity surrounded by six plate components as shown in Fig. 1. For the sake of simplicity, we assume that one plate is elastic and the other is rigid. The domain of the acoustic cavity is denoted by V and its surface by S. Using the Cartesian coordinate system O-xlxex3, we can express the governing equation for the vibration of the acoustic cavity as follows: 1
V2P(x, t) - ~o2 P(x, t) in V,
(1)
~.vhere V~ is the Laplacian operator and the superimposed dot denotes time derivative, while C0~.~the phase velocity (in this case the sound velocity in air) and P the (sound) pressure. It is assun,~, that the plate material is isotropic and homogeneous. Using the local Cartesian coordina~,_~,system O-x~x~x~, in which the axes x~ and x~ lie in the middle plane of the plate as shown in ~'~. 2, we can express the governing differential equation of the out-of-plane bending vibrat'Lon as follows:
DV4W(y, t) + gW(y, t) + pphW(y, t) = P(y, t) - Pe(Y, t) in Sp,
(2)
where V4 is the biharmonic differential operator, W the lateral deflection of the plate, D the flexural rigidity, pp the mass density of the plate material, h the plate thickness, g the damping coefficient, and Pe the external pressure. n+ --
Etastic Prate ~
!
[ Cevity'V
I x~
.x: .x' "'':"
Boundary S Fig. 1. Structural-acoustic system.
Corner Point At d / xl'
w
_ ,
At-I
Pe y "y' I1 f l
x.' A,.,I x," Boundary C
I
'
P
Fig. 2. Notation for out-of-plane bending of plate.
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
227
Equations (1) and (2) are combined by taking into account the conditions on the plate region Sp, to obtain the governing equation of the structural-acoustic coupling system under consideration. The details of this procedure will later be discussed. 3. Boundary element analysis of acoustic field We shall show here the outline of boundary element analysis of a three-dimensional acoustic field with an arbitrary shape subject to arbitrary boundary conditions. It is assumed that the acoustic pressure P(x, t) is expressed as P ( x , t) -
p(x) exp(ieot),
(3)
where i = q'~"T. Then, (1) can be reduced to the Helmholtz equation expressed as
V2p(x) "[" k2p(x)
= 0,
(4)
where k = V to~Co and is referred to as the wave number. On the boundary of the acoustic field there are the following relationships [1-4]:
a__p.p=-itopv, an
ap = -itop/7_ On z '
(5)
where v is the particle velocity, z the acoustic impedance, p the mass density of the acoustic medium, and a( )/an the directional differentiation along the outward normal to the acoustic field boundary. Making use of (5) we can transform in the usual ~vay [1-4] the Helmholtz equation (4) into the following boundary integral equation:
c(y)p(y)+ fs q*(Y' y')p(y')dS(y')+itop fs --iwp
p(y') P*(Y' y')
fsv P*(Y' y ' ) v ( y ' ) d S ( y ' ) ,
z( y' ) dS(y') (6)
where Sv denotes the boundary portion of the acoustic field in which the particle velocity v is prescribed, while Sz stands for the portion with the given acoustic impedance. As is weli known [11, 12], the coefficient c in (6) is such that c(y) = 0/(4~r), in which 0 is the solid angle of the boundary point y with respect to the acoustic medium V. The asterisked functions in (6) are the fundamental solutions of the Helmholtz equation, and can be expres~,d for ~+he three-dimensional problems as follows: 1 exp(-ikr), p*(x, y ) = 4~rr
y) q*(x, y ) = ap*(x, On '
+(7)
where r denotes the distance between points x and y. If the boundary integral equation (6) is discretized by means of the usual boundary element
M. Tanaka, Y. Masuda, BEM for structural-acousticcoupling
228
method, the following set of equations can be derived:
[HI(p} =[Gl(v},
(8)
where {p} and {v} denote the column vectors including all the nodal values of the acoustic pressure p and the particle velocity v, respectively. The components h ij and gi~ of the coefficient matrices [H] and [G] in (8) can be computed from the fundamental solutions p* and q* in the following way:
hj~ = cSiJ+ I¢ q*(y'' ¢)t~(~)lJ(~)l d~, (9)
/,,
)1d~, where St denotes the jth boundary element and [J(~:)[ the Jacobian of the coordinate transformation from the refere::ce coordinate to the nondimensional intrinsic coordinates ~:of the element. Taking account of the prescribed boundary conditions, we can solve (8) for the unknown nodal values on the whole boundary.
4. Bending vibration analysis of elastic plates The so-called boundary-domain element method [13] will be briefly discussed for analyzing the out-of-plane bending vibration problem of elastic plates. It is assumed that the plate vibrdtes harmonically with time as in the acoustic field and that the deflection W(y, t), the external pressure Pe(~', t), and the acoustic pressure P(y, t) can be expressed as
W( y, t) = w( y) exp(io~t),
Pe( y, t)= p~( y) exp(ioJt),
P( y, t) = p( y) exp(io~t).
(10)
Then, (2) can be reduced to
DV4w(y) + itogw(y) - tO2pphw(y) = p(y) - p~(y) .
(11)
The integral equation formulation can be started with the following weighted residual statement:
fsp {DV4w(y)-itogw(y)-oJ2pphw(y)-p(y)+pe(y)}w*(y,
y')dSp=O.
(12)
Using the exact fundamental solution of the reduced bending vibration equation (11) as the weight function w* in (12) we can derive the integral equation which can be solved by means
M. Tanaka, Y. Masuda, BEM for structural-acousticcoupling
229
of the usual boundary element method. In this study, however, we use an approximate fundamental solution, that is, the fundamental solution of static bending problems given by 1 w*(y, y ' ) = 8=D
t .2
in r
(13)
It should be mentioned here that use of the exact fundamental solution leads to some complexity in mathematical treatment of the special functions included in the resulting integral equation, while such a difficulty is circumvented if the approximate fundamental solution is employed. Applying the above weight function to the weighted residual expression (12) and tracing the sa~e procedure as in the static bending problem of elastic plates [14-16], we can finally obtain the boundary integral equations which can be expressed in the following form:
Ck'(Y)u'(Y) + fc P~"(Y' y')u,(y') d C t y ' ) - fc U~,(y, Y')Pl(Y') dC(y') L
+ ~'. [M*,k(y , Am)Ul(Am)- U~2(y , Am)Mm(Am)] m=l
fsp (i¢og -- ¢02pph)U~2(y, x) dSp(x) +
fsp U~2(Y'x){p(x) - p~(x)} dSp(x),
(14)
where [G,] = '
L'
~,
=
,
=
M*,J
r.
'
(15)
The detailed expressions of the above notation can be found in the literature [14-16]. Since the approximate fundamental solution is employed for the present formulation, the first term of the fight-hand side of the boundary integral equation (14) includes the unknown variable u t (=deflection w) in the inner domain of the plate. To solve these equations by the usual boundary element method we must introduce an iterative solution procedure. To circumvent such difficulties we may use an additional integral equation. For this purpose we shall here take into account the integral equation with respect to the deflection w(y) at an internal point y expressed by
u,(x) + fc P~,(x, y')u,(y') dC(y') fc U~t(x' Y')Pt(Y') dC(y') -
L
+ mffil ~, [M*,(x, Am)Ul(Am)- U~2(X,Am)M,,(Am) ]
230
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling t"
= - Jsp (i¢0g - co2pph)U'~z(X, x')ui(x') dSp(x')
+ fsp U~lz(X' x'){p(x') - p¢(x')} dSp(x').
(16)
In the boundary-domain element method [13] the boundary integral equation (14) and the integral equation (16) are simultaneously discretized by means of the usual boundary element as well as the domain element which plays a similar role to that of the finite element. Application of the boundary-domain element method to (14) and (16) can yield the set of linear simultaneous equations with respect to nodal values on the plate boundary as well as in the inner domain of the plate. If the boundary conditions expressed as w ffi 0,
T, = 0
(clamped),
w = 0,
M~ - 0
(simply supported),
M. ffi 0,
V. = 0
(free),
(17)
are taken into consideration, we can finally obtain the following set of equations, that is, from
(14)
[ABI{Xe} ffi -(icog - co2pph)[Ds]{wo} + [Dsl{p - p.} ,
(18)
and from (16)
{ w o } + [A o]{ Xs} = - ( i c o g - oJ2pph)[Do]{ w o } + [Do]{ p -- p.} .
(19)
In the above equations {wo} denotes the column vector of nodal deflections in the inner domain of the plate. The other quantities [Aa], [Ds], [At,], and [Do] are the coefficient matrices which are related to the variables shown in (15) and can be computed from the fundamental solution w* given by (13). It is interesting to note that elimination of the unknown vector {X e} on the boundary can lead to the following set of equations:
[Gl{wo} =
+ [GI{p°},
(20)
where [1] being the unit matrix, [H~] ffi [1] - (ioJg -
~O2pph)[Gpl,
[Gp] =
[Aol[As]-~[Da]- [Do].
(21)
Suppose that in (20) and (21) the damping coefficient g is negligibly small and no pressure loadings are applied, we can obtain the algebraic set of eigenvalue equations expressed by
([I] + oj2pph[Gp]){wo} - 0 ,
(22)
M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
231
from which we can calculate the eigenfrequency and also the eigenmode using a common-use library of subroutine subprograms available at almost all computing facilities.
S. Analysis of structural-acoustic coupling systems .The discretized sets of equations (8) and (20) are now combined to yield the set of equations for the structural-acoustic coupling system under consideration. By using the same nodal points on the interface of the two fields the acoustic field is discretized by the usual boundary element method and that of the elastic plate structure by the boundary-domain element method. The equilibrium and compatibility conditions must be satisfied on the interface of the acoustic field and the elastic plate. These conditions can be expressed such that pA = p ,
V =i(ow a ,
(23)
where p ^ denotes the acoustic pressure. To take into account these conditions, (8) is rewritten as follows:
,,,]{,:} where the superscriptsA and I denote the variables on the acoustic fieldand on the interface, respectively. Equation (24) is now combined with (20) to yield
"]{';I[°"i° ]I°'l I°] Go
=
0
-Hp
wa
+
Gp {P,}'
(25)
Equation (25) is solved numerically under the appropriate boundary conditions which can be expresseG as Pe - P~ on the interface, v- ~ on the other boundary,
(26)
where the superimposed bar denotes a prescribed value. These boundary conditions imply that the elastic plate is excited by the known external pressure. For our later use we now rewrite once again (25) together with the boundary conditions in the following form:
o6,
-iG'
H,,
o
0
6,
(27)
The solution of (27) can provide the response of the structural-acousticcoupling system under consideration.
232
M. Tanaka, Y. Masuda, BEM for structural.acoustic coupling
6. Computational results and discussion
A new computer code has been developed in this study. A couple of sample problems are calculated by using the computer code and the results obtained are discussed in the following. Now, we consider the acoustic cavity surrounded by six equal plates as shown in Fig. 3. It is assumed that one plate is elastic and excited by the external pressure, while the other plates are rigid, that is, subject to the boundary condition v = 0. The cavity is a cubic with the edge length 200 mm. The elastic plate is made of brass, its thickness is 0.941 mm, and mounted on the rigid plates with the simply supported boundary conditions, which correspond to the experiment by Guy and Bhattacharaya [5]. The element subdivision is shown in Fig. 4, where the eight-node isoparametric element is used for the whole boundary of the acoustic cavity (25 elements for the rigid boundary and 16 for the elastic plate), while the boundary of the elastic plate is divided into constant boundary elements of an equal size (totally 96 elements). In this numerical example the damping coefficient is assumed as g = 0. The sound pressure P2 on the rigid boundary opposite to the elastic plate as shown in Fig. 3 is calculated under these computational conditions. In Fig. 5 comparison is made with the analytical solution by Guy et al. [5]. The present BEM solutions are in good agreement with the analytical solutions. It is interesting to note that the experimental results of Guy et al. [5] also agree well with both the solutions mentioned above. The frequency % shown along the abscissa in Fig. 5 denotes the eigenfrequency of the elastic plate alone. The discontinuities appearing near the frequencies 90 Hz, 400 Hz, 700 Hz, and 850 Hz are the resonance points corresponding to the eigenmodes (°,t, c°~3,%3 of the elastic plate and the eigenmode 01 of the cavity under the coupling system, respectively. ELastic PLate
Fig. 3. Cavity-backed elastic plate. IIr
,
,
i , , , ,
,
w
•
tic PLate Fig. 4. Boundary element discretization.
, . , ,
50
_~ 50
30 2O
~30
+o
0
m 0 -10 .--Guy eL. ot.~( -20 ,° BEM , , ,~2~,~., 10 J . l O O ~ r Tl~k~ (0141" +T+k Frequency (Hz) ~u w=4 Fig. 5, Transmission of sound through simply supported elastic plate.
J
m 0 . . ~ , , . ~ ~ -I0 , , ,BEM ~ , ~ ° -20
1~,t
i
Frequency (Hz) ~m w~wz~, M Ik Fig. 6. Transmission of sound through clamped elastic plate.
M. Tanaka, Y. Masuda, BEM for swuctural.acoustic coupling
233
Figure 6 shows the results obtained for the same structural-acoustic coupling system wh the elastic plate is clamped to the rigid boundary. 7. Concluding remarks This paper has presented an integral equation approach to the structural-acoustic coupling problems. Some typical examples were computed by using the computer code developed in this study. The potential usefulness and availability of the proposed method were demonstrated through comparison of the results obtained with other solutions. Although it was assumed in this paper that the structure was composed of plate components, the present method can also be applied to other types of the structure with arbitrary boundary conditions. It could be recommended as future research work to explore a similar integral equation approach to a wide variety of the structural-acoustic coupling problems. Numerical computations of this study were carried out by using mainly the HITAC computer installed at Shinshu University and partly the HITAC machine of the Computing Center at University of Tokyo. Acknowledgment The authors wish to thank an unknown reviewer of the journal for his useful comments and suggestions to improve the manuscript. References [1] H.A. Schenck, Improved integral equation formulation for acoustic radiation problems, J. Acoust. Soc. Amer. 44 (1968) 41-58. [2] A.F. Seybert, B. Soenaarko, F.J. Rizzo and D.J. Shippy, Application of the BIE method to sound radiation problems using an isoparametric element, ASME J. Vibration, Acoustics, Stress, and Reliability in Design 106 (1984) 414-420. [3] M.A. Latcha and A. Akay, Application of the Helmholtz integral in acoustics, ASME J. Vibration, Acoustics, Stress and Reliability in Design, 108, (1986) 447-453. [4] M. Tanaka and Y. Masuda, A general purpose BEM computer code for acoustic problems, Trans. Japan Soc Mech. Engrs. Ser. C53 (1987) 387-391 (in Japanese). [5] R.W. Guy and M.C. Bhattacharaya, The transmission of sound through a cavity-backed finite plate, J. Sound and Vibration 27 (1973) 207-223. [6] A. Sestieri, D.D. Vescovo and P. Lucibello, Structural-acoustic coupling in complex shapec~ cavities, J. Sound and Vibration 96 (1984) 219-233. [7] S.H. Sung and D.J. Nefske, A coupled structural-acoustic finite element model for vehicle interior noise analysis, ASME J. Vibration, Acoustics, Stress, and Reliability in Design 106 (1984) 314-318. [8] M. Bessho, H. Kawabe and Y. lwasaki, Underwater sound radiated from a vibrating ship hull, in: M. Tanaka and C.A. Brebbia, eds., Boundary Elements VIII (Springer, Berlin, 1986) 83-93. [9] S. Suzuki, M. lmai and S. lsyiyama, Coupling of the boundary element method and modal analysis for structural acoustic problems, Trans. Japan Soc. Mech. Engrs. Ser. C 52 (1986) 310-317 (in Japanese). [10] M. Tanaka, K. Miyazaki and K. Yamagiwa, Integral equation approach to free vibration problems of assembled plate structures, in: M. Tanaka and Q. Du, eds., Theory and Applications of Boundary Element
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M. Tanaka, Y. Masuda, BEM for structural-acoustic coupling
Methods, Proceedings 1st Japan-China Symposium on Boundary Element Methods, 1987, Karuizawa, Japan, (Pergamon Press, Oxford, 1987). [11] C.A. Brebbia, The Boundary Element Method for Engineers (Pentech Press, London, 1978). [12] C.A. Brebbia, J.C.F. Telles and L.C. Wrobel, Boundary Element Techniques---Theory and Applications in Engineering (Springer, Berlin, 1983). [13] M. Tanaka and K. Tanaka, On a new boundary element solution scheme for elastoplasticity, Ing. Arch. 50 (1982) 289-295. [14] M. Tanaka and K. Miyazaki, A direct BEM for elastic plate-structures subjected to arbitrary loadings, in: C.A. Brabbia and G. Maier, eds., Boundary Elements VII (Springer, Berlin, 1985) 4/3-4/16. [15] M. Tanaka and K. Miyazaki, A direct boundary element method for elastic bending analysis of plates, Trans. Japan Soc. Mech. Enos. Set. A 51 (1985) 1636-1641 (in Japanese). [16] M. Tanaka, Bending problems of elastic bodies, in: Boundary Element Methods--Theory and Applications, Corona, Tokyo (1986) Ch. 4, 69-92 (in Japanese).