Adv. Theor. Appl. Mech., Vol. 3, 2010, no. 4, 189 - 194
The Study of Three-Dimensional Structural
Acoustic Radiation Using FEM and BEM
Zhisheng Xu1 and Qibai Huang
The State Key Laboratory of Digital Manufacturing Equipment and Technology
Huazhong University of Science and Technology
Wuhan, 430074, China
1
xuzhisheng@gmail.com
Abstract
In this paper, the acoustic pressure distribution on the surface of a threedimensional structure and the total acoustic power radiated by the structure
resulting from a known surface velocity distribution has been obtained by FEM
and BEM. The pressure distribution is computed by Helmholtz Boundary Integral
Equation. The sound radiation power of structure can be expressed as a positive
definite quadratic form of the Hermitian by impedance matrix. Finally, taking two
problems as simulation examples, the analytical results and experimental results
show the validity and accuracy of the presented method for calculating the
acoustic radiation.
Keywords: Sound radiation, FEM-BEM, Acoustic radiation, Helmholtz Boundary
Integral Equation
1. Introduction
Noise control has been becoming a greater importance [1] everyday. The
control of the noise radiated by a structure represents the control of the acoustic
power radiated by the structure over some frequency range of interest. Many
physical noise sources radiate a substantial portion of their total acoustic power
output at a limited number of discrete frequencies. Reducing the radiated power at
these discrete frequencies would greatly reduce the overall noise level. The
determination of acoustic power radiated from structures is very important in
many vibrating engineering applications.
190
Zhisheng Xu and Qibai Huang
The Finite Element Method (FEM) [2] and Boundary Element Method (BEM)
[3] are used very often to solve acoustical problems. Comparing to FEM, BEM is
a semi-analytical method which can deduce one dimension and it has good
precision and little computation, so BEM is a more effective method solving the
acoustical radiation problems than FEM.
Because analytical solutions do not exist for an arbitrary shape structure,
numerical methods are used to model the radiation characteristics of structures.
The Helmholtz Integral Equation constitutes an integral relationship between the
acoustic pressure and velocity distribution on the surface of a structure and the
acoustic pressure at any point in space, including on the surface of the structure
itself. The BEM formulation of the Helmholtz Integral Equation enabled the
determination of surface acoustic pressure distributions on the surface given a
known harmonic motion of the surface.
The research presented in this paper targets the acoustic radiation of
structures. Firstly, the velocity distribution of structure is solved by FEM.
Secondly, it requires determine the acoustic pressure and the acoustic intensity
distribution on the surface of a structure based on a known surface velocity
distribution. Finally, a formulation for the total acoustic power radiated by a threedimensional structure expressed solely in terms of the structure's surface velocity
distribution should be proposed.
2. Finite element analysis of structural vibration
For an arbitrary shape structure, the motion equation on time harmonic
mechanical loading can be expressed as
(1)
MX&& + CX& + KX = Fa eiωt
Where the symbols M , C and K denote the N dimensional structural mass,
damping and stiffness matrices, respectively.
It is assumed that displacement response on time harmonic mechanical loading
can be expressed as
X = X a e − iωt
(2)
The amplitude of velocity response can be stated as
Va = ω X a = ω ( −ω 2 M + jωC + K ) Fa
−1
(3)
3. Boundary element analysis of structural vibration
Consider the acoustic pressure field in the exterior unbound domain, for an
arbitrary shape structure, the governing differential equation in steady-state linear
acoustics is the classical Helmholtz equation as follows
∇2 p + k 2 p = 0
(4)
Acoustic radiation using FEM and BEM
191
where p is the amplitude of sound pressure which may be a complex number,
ω
denotes the wavenumber. ω and c are the angular frequency and
c
speed of sound, respectively.
For arbitrary enclosed structures, including those with corner and edges,
applying the Green’s second identity, Helmholtz integral equation is given by
∂G ( P, Q)
(5)
+ i ρω vn (Q)G ( P, Q))dS (Q)
C ( P) p ( P) = ∫ ( p (Q)
∂n
S
Where n is the unit normal vector pointing into the acoustic domain, C ( P) is
and k =
e − ikr
the corner coefficient, and G =
is the fundamental solution of the
4π r
inhomogeneous Helmholtz equation, namely the Green’s function of free space.
If the structure is divided into N n nodes and N e elements and four-node
isoparametric elements are used in the mesh. The finite element interpolation
function of the sound pressure and velocity over entire element can be expressed as
4
4
l =1
l =1
p = ∑ N l pl , v = ∑ N l vl
(6)
Where pl , vl represent the sound pressure and the normal velocity of nodes of
1
element, respectively. N = (1 + ξlξ )(1 + ηlη ) is an interpolation shape function.
4
The numerical evaluation of Eq. (5) for the nodes of each element ultimately
yields a system of algebraic equation
H H P = GH V
(7)
Where P = ( p1 , p2 ,L , pN n )T , V = (v1 , v2 ,L , vNn )T .
It is well known that the classical boundary element method fails to provide a
unique solution at certain frequency. Several modified integral formulations have
been developed to overcome the problem. By far, the CHIEF method is the most
popular method, and it is adopted in this study. One may get
H C P = GCV
(8)
Eq. (7) and Eq. (8) form the following simultaneous equations
⎡HH ⎤
⎡GH ⎤
+
(9)
⎢ H ⎥ P = ⎢G ⎥ V ⇒ HP = GV ⇒ P = H GV = AV
⎣ C⎦
⎣ C⎦
4. Numerical calculation of acoustic power from the surface of a
three-dimensional vibrating structure
For an arbitrary shape structure, when the excitation load is a time harmonic
mechanical loading with prescribed excitation frequency, the acoustic intensity
distribution on the surface of structure
192
Zhisheng Xu and Qibai Huang
1
I = Re ⎡⎣ p (Q)vn∗ (Q) ⎤⎦
2
(10)
where P is the node on the surface of structure, p ( P) and vn ( P ) are the sound
pressure and normal velocity of node P , respectively. *(asterisk) indicates complex
conjugation of the quantity.
The whole sound power radiated from the structure surface can be obtained by
integrating the intensity over the plate surface area as
W=
1
2
Ne
Re ∑ ( Pi T Vi ∗ )dSi
∫
(11)
i =1 S
i
Where P = ( p1 , p2 ,L , pNn )T , V = (v1 , v2 ,L , vNn )T , denote the sound pressure
and the normal velocity of the four nodes of element Si , respectively.
Substituting Eq. (10) into Eq. (11)
W=
1
2
Ne
Re ∑
∫ (V
T
i
i =1 S
i
AiTVi ∗ )dSi
(12)
Where AiT the matrix of the four nodes of element Si , eg, if element Si is
composed of node 1,2,3and 4, matrix AiT = [ A1,Ti , A2,T i , A3,T i , A4,T i ] . A4,T j denotes the jth
column of matrix A corresponding the node number of sound pressure pj in global
coordinate.
Substituting Eq. (6) into Eq. (12)
W=
1
2
Ne
4
Re ∑ vi T ∑
i =1
∫(N
T
l
l =1 S
i
AiT N l )dSi vi ∗
(13)
Introduce a notation
ci =
∑ ∫(N
4
l =1 Si
T
l
AiT N l )dSi
(14)
Eq. (14) can be rewritten as
W=
1
2
Ne
Re ∑ vi T ci vi ∗
Eq. (16) may be rewritten in matrix form as:
W=
1
2
(15)
i =1
Re (V T CV ∗ ) =
(V CV
4
1
T
∗
+ (V T CV ∗ )
*
)
(16)
According to V T CV ∗ = (V T CV ∗ ) , the sound power can be rewritten as
T
W =V H
CT + C
V = V H MV
4
(17)
where M is an impedance matrix, superscript H denotes conjugation
transpose. The matrix M is a Hermitian matrix. For arbitrary velocity vector, the
sound power is positive, and matrix M is positive definite and Hermitian matrix M
is nice because it has n positive eigenvalues and the eigenvectors of it form a basis
of n dimensional space.
Acoustic radiation using FEM and BEM
193
5. Numerical examples
In order to verify the accuracy of the proposed method, an example of acoustic
radiation problem is tested. For radiation problems, we get numerical solutions and
experimental solutions to compare the reliability and accuracy of proposed method.
The example is a simplified cab with the thickness of h = 1.5mm (see figure 1).
A time-harmonic concentrated loading with prescribed frequency
f = 200,500, 750, 1000Hz and amplitude Fa = 100 N is applied to the point
(250mm, 125mm, 0mm). The cab has the Young’s modulus E = 2.1× 1011 N / m 2 ,
Poisson’s ratio υ = 0.3 , and the specific mass ρ = 7800 Kg / m3 . The air density
and the sound speed are ρ 0 = 1.225Kg / m3 and c = 343m / s , respectively.
The acoustic pressures of selected points and total acoustic power at different
frequencies using numerical method and experimental method are listed in TABLE
I. Figure 2 and Figure 3 show experimental system and experimental equipment.
The results show that the difference between numerical results and experimental
results is in the acceptable difference rang.
Figure 1 Mesh model of simplified cab
The simplified cab
Microphone (B&K4193)
and Sound intensity probe
Vibration exciter (B&K4825)
Force tranceducer
(PCB288D01)
B&K 3560C
Figure 2 Experimental system
Power amplifier (B&K2716)
Signal analyzer (B&K Pulse 10.0)
194
Zhisheng Xu and Qibai Huang
Figure 3 Experimental equipment
TABLE I Pressure of surface nodes of radiation cube
Numerical results (dB)
Frequencies
Acoustic power
1. (0.25, 0, 0.15)
2. (0, 0.125, 0.15)
3. (0.25, 0.25, 0.15)
4. (0.5, 0.125, 0.1)
5. (0.15, 0.125, 0.3)
200Hz
63.1
68.8
77.0
68.6
72.0
74.1
500Hz
88.2
89.8
86.0
89.8
81.3
89.1
750Hz
92.4
96.5
92.0
96.8
80.5
94.6
1000Hz
89.7
94.6
88.2
94.5
81.1
88.6
Experimental results (dB)
200Hz
64.6
69.0
77.6
69.8
73.5
75.6
500Hz
89.4
90.7
86.2
90.6
83.1
90.1
750Hz
93.8
97.1
92.6
97.5
82.2
96.2
1000Hz
91.2
95.5
89.3
95.5
81.6
89.8
6. Conclusion
The sound radiation of three-dimensional structures is studied using FEM and
BEM in this paper. Taking a cube and a simplified cab as simulation examples, we
calculate the acoustic pressure and acoustic power using numerical method,
analytical method and experimental method. The simulation results show that the
numerical method proposed in this paper can overcome the nonuniqueness and
compares very well with the analytical results and experimental results.
References
[1] S. J. SHARP, P. A. NELSON, and G. H. KOOPMANN, a theoretical
investigation of optimal power absorption as a noise control technique.
Journal of Sound and vibration, vol. 251, p. 927-935, 2002.
[2] Wu SW, Lian SH, Hsu LH. A finite element model for acoustic radiation. J
Sound Vib 1998;20(3):489–98.
[3] L. H. Chen and D. G. Schweikert, Sound radiation from an arbitrary body. J.
Acoust. Soc Am., 35(10), 1626-1632 (1963).
Received: October, 2009