Transactions on Modelling and Simulation vol 3, 1993 WIT Press, ISSN X

Transcription

1 Boundary element method in the development of vehicle body structures for better interior acoustics S. Kopuz, Y.S. Unliisoy, M. Qali kan Mechanical Engineering Department, Middle East Technical University, Ankara, Turkey ABSTRACT In this paper, Boundary Element Method (BEM) is exploited for the numerical solution of interior acoustics problems governed by the Helmholtz's equation. A detailed literature survey on the application of the BEM to the radiation problems concerning the interior regions is presented. A formulation directed to the particular application of noise generation inside motor vehicle passenger compartments is developed. The solution of the problem is illustrated, in detail, on a classical acoustic problem, i.e. the acoustic response of a spherical cavity. The sound pressure variation along the radius is examined as a function of the dimensionless frequency number. The effect of an opening on the surface of the body is also examined. INTRODUCTION The prediction of the acoustic field due to arbitrarily shaped structures is an important research area in many disciplines. There are two main broad areas of radiation problems; namely exterior and interior acoustic radiation. This topic has a variety of applications, in particular in the aerospace engineering and automotive engineering fields in determining the interior noise levels of aircraft and ground vehicles. For these applications several different methods have been used, the finite element method, statistical energy analysis, and boundary element method (BEM) with integral formulations being the most prominent. Each method has its advantages and drawbacks depending on the particular frequency range of interest. In this paper, a formulation directed to the prediction of noise generation inside motor vehicle passenger compartments is introduced and a study of the boundary element method for the solution of the interior acoustics problems governed by the Helmholtz's equation is presented. The ultimate aim of the study is to produce a computational tool to be used in the prediction of the

2 58 Boundary Element Technology acoustical field inside a vehicle structure for which the vibration characteristics are available. The solution of the problem is illustrated on a classical acoustic problem, i.e. the acoustic response of a spherical cavity. The acoustic pressure variation along the radius is predicted as a function of the Helmholtz number. The effect of an opening on the surface of the vibrating body is also investigated. SURVEY OF PREVIOUS WORK The solution of the interior noise fields of cavities having arbitrary shapes by using the BEM has recently received much interest. Bell, Meyer, and Zinn [1] presented an integral solution of the Helmholtz equation for predicting acoustic properties of arbitrarily shaped bodies. They examined twodimensional problems of a circle and rectangle together with a duct having a right-angle bend. They also investigated the acoustic properties of a sphere using an axisymmetric formulation. Sestieri et al [2] discussed the structuralacoustic coupling problem by using the BEM for interior volumes having complex shapes. They investigated the importance of coupling and reached the conclusion that a fully coupled analysis did not seem to be justified. Seybert and Cheng [3] dealt with the application of the BEM to interior acoustics problems governed by the reduced wave (Helmholtz) differential equation. They applied the BEM formulation to an axisymmetric problem at which surface integrals could be reduced to line integrals along the generator of the cavity and to integrals over the angle of revolution. Therefore, the surface was discretized by using line elements, not surface elements and only the generator of the cavity needed to be discretized. They examined the acoustic response of a spherical cavity. In predicting the acoustical behavior of cavities, a different method was employed by Kipp and Bernhard [4]. They developed an Indirect Boundary Element Method (IBEM) which was basically derived by Chen and Schweikert [5]. In this method, a source density distribution is considered to exist at the boundary of the cavity and this boundary source distribution is determined by satisfying the boundary conditions. However, in the direct boundary element method, variables, namely, acoustic pressures and velocities are solved directly. The capability to include acoustic point sources within the cavity was also implemented in the study. The method was applied to the prediction of sound fields in spherical and rectangular cavities. All three types of boundary conditions, namely, pressure, velocity and impedance boundary conditions were verified. Fyfe [6] discussed the application of the BEM to predict the interior acoustic mode frequencies of an enclosed medium. He used a non-rectangular box and an automobile model to show the accuracy of the method. Cheng and

3 Boundary Element Technology 59 Seybert [7] examined both the interior and exterior acoustic radiation problems considering the application of the BEM. The acoustic response of a spherical cavity was determined and good agreement between the BEM results and the analytical solution was obtained. Wu et al [8] described a general boundary element code called BEMAP for acoustic analysis, along with the process of vectorizing and parallelizing the code on a vector parallel computer. Seybert, Cheng and Wu [9] presented an approach to the solution of coupled interior/exterior problems using the BEM. Recently, Seybert, Wu and Wan [10] rewieved the applications of boundary element modeling in acoustics. A general investigation of the application of the BEM for the numerical solution of noise problems inside a complex shaped cavity was performed by Suzuki et al [11]. A new formulation for complicated boundary conditions was proposed to solve practical noise problems inside a vehicle cabin. The effect of absorbent materials on the vibrating surfaces and the effect of leakage through an opening were considered in the analysis. They applied the method in determining the sound pressure inside a linear duct, and the transmission of sound through a cavity-backed plate, and in predicting the sound pressure field inside a rather simple sedan compartment model. These studies resulted in the development of a new computer code called ACOUST/BOOM to analyze the sound pressure radiated by a vibrating structure and to calculate the acoustical characteristics of the enclosed field [12]. Utsuno et al [13] studied a method for the analysis of the sound field in a full-size model of an automobile cabin. The acoustic resonant frequencies and the acoustic modes of the cabin were calculated by using the BEM. The results were then compared with an experiment conducted on a full-size cabin model made of plaster. The calculated modes and frequency response curves were found to be in good agreement with the experiments. FORMULATION The geometry of the problem is presented in Figure 1. An ideal, homogeneous, frictionless fluid fills an interior volume V of the cavity surrounded by a surface S. A generalfieldpoint is denoted by A, and a surface point by B. The positive unit normal n is directed from the point on S as shown in the figure. The problems to be considered are those dealing with acoustical behavior within cavities due to a vibrating source. For most practical problems a sinusoidal time dependence can be assumed so that the problem is simplified greatly. Then, the wave equation reduces to the well known Helmholtz equation : V^p+k^p-O, k = co/c (1)

4 60 Boundary Element Technology where p is the acoustic pressure, k is the wave number, 0) is the angular frequency, and c is the speed of sound in the medium. Figure 1. Geometry of the Physical Problem Solution of the Helmholtz equation can be obtained by seperation of variables [14]. This method involves the series expansions of the solutions in terms of the normal modes of the system and can only be used with special coordinate systems and boundary conditions. For arbitrarily shaped bodies, analytical solutions can only be obtained by numerical methods. Applying the Green's theorem together with the definition of Green's Function to the Helmholtz equation with the suitable boundary condition, one can obtain the following integral equation c(a)p(a)-lp(b) 1 r(a,b) jk coscc(a,b),-jkr(a,b) r(a,b) ds(b) v(b).-jkr(a.b) r(a,b) ds(b) (2) where z is the specific acoustic impedance, v is the surface velocity, r is the distance between the points A and B, <xis the angle between the unit normal and position vector r, and 47C for field points inside the body c(a) = { 271 forfieldpoints on the surface of the body 0 forfieldpoints outside the body

5 NUMERICAL SOLUTION TECHNIQUE Boundary Element Technology 61 In the previous section, the integral equation which describes the interior acoustic field of an arbitrarily shaped body is given.the numerical procedure for the BEM solution has been explained in detail in [3,7,11] and consists of the four main steps given below : 1.Discretization of the boundary surface into boundary elements. 2.Numerical integration to get an algebraic system of equations. 3.Solution of the system of equations to obtain the unknown boundary surface variables. 4. Solution of the field (interior) values. The two-dimensional surface of the body is discretized in the BEM rather than the three-dimensional field inside the body as in the Finite Element Method (FEM). Thus, the dimension of the problem is reduced by one. Another advantage of the BEM over the other domain methods is that the input data required needs less preparation effort when compared to the FEM. In order to represent both the geometry of the body and variation of the acoustic variables within each element, the quadratic isoparametric elements having six or eight nodes are used in this study. Variation of these variables can be represented as : 6-8 Xi(,T )= ZNj(^,n)Xi,, i=l,...,number of elements (3).1=1 where Nj are the second-order shape functions of the local coordinates (,T ). The discretized equations result in a matrix form as [A](X)={B) (4) where [A] is a square coefficient matrix,{b} is a right hand side vector and {X} is the unknown vector which contains the nodal pressures. APPLICATION OF BEM TO ACOUSTIC RESPONSE OF A SPHERICAL CAVITY The BEM is used to determine the acoustic response of a spherical cavity.the analytical solution for the acoustic pressure inside a spherical cavity is given as (5)

6 62 Boundary Element Technology where K is a constant to be determined by the boundary conditions at the surface (r=a) of the sphere. If the surface of the spherical cavity of radius a pulsates radially with a uniform velocity (i.e. v=l at r=a), the solution takes the following form p(r)=j (6) sin(ka)-kacos(ka)l kr J The acoustic pressure distribution along the radius of the spherical cavity is only function of ka (dimensionless frequency). Then the magnitude of the acoustic pressure at the inner wall of the spherical cavity p(r=.) = sin(ka)- kacos(ka) and the magnitude of the acoustic pressure at the half radius (r=a/2) of the spherical cavity p(r =./2) = 2kazsin(ka/2) (8) sin(ka)- kacos(ka) and the magnitude of the acoustic pressure at the center of the spherical cavity U_2 p(r = 0)1 = - - (9) sin(ka)-kacos(ka) The magnitude of the acoustic pressure at the inner wall, at the half radius, and at the center of the spherical cavity pulsating with a uniform surface velocity of v = 1 m/s are obtained from both the analytical solution and the BEM. For the BEM solution, the sphere is discretized into 16 quadrilateral quadratic isoparametric elements each having 8 nodes. The total number of nodes is 50. The computer code is developed on both the IBM 3090 mainframe using the FORTRAN 77 compiler and on a (33 MHz) personal computer using the LAHEY compiler. The ESSL and IMSL subroutines are used in the solution of the complex set of simultaneous equations. The results obtained are displayed in Figures 2, 3, and 4. The BEM results agree quite well with the analytical solutions as can be seen from the given figures. Figure 4 represents the acoustic pressure distribution at the center of the spherical cavity as a function of ka. The magnitude of the pressure is in good agreement with the analytical result. The point where the pressure magnitude goes to infinity corresponds to the acoustical resonance of the spherical cavity under the Neumann boundary

7 Boundary Element Technology 63 condition and is found from the solution of tan(ka) = ka. Note that this point is predicted accurately. The BEM code is also verified [15,16] on the solution of a classical exterior acoustic radiation problem, i.e. the sound pressure on the surface of a uniformly pulsating sphere. The results obtained were in good agreement with the theoretical results. 10 cd 8 BEM 2 * "o, 4 2 \ \ Analytical *^* ^ ^ ^ m *_^^ *-. 0 ^ () I ka Figure 2. Magnitude of p / zka at the inner wall of a pulsating spherical cavity 10 1 \ 8 CO V BEM j 5 * % 4 \ Analytical 2 \ /-< "*- # A A A * m- # 0 () ^5 ka Figure 3. Magnitude of p / zka at the half radius of a pulsating spherical cavity To see the effect of the opening on the acoustic pressure, a part of the surface of the spherical cavity is considered to have an impedance (mixed) boundary condition. The opening on the surface of the spherical cavity is 1/16 of the total surface. The pressure variation at the center of the spherical cavity is

8 64 Boundary Element Technology displayed in Figure 5. As one notes, the pressure magnitudes drop down in case of an opening for ka less than 2. The assumed impedance relation holds for small frequencies where acoustic pressure and particle velocity are uniform over the opening. Additional resonance of the cavity caused by the opening is also observed from the figure at ka= BEM CO \ Analytical 3 * \ Z 4 /' K \\ 2 /*»- _*. * * -# _, # -#'* _*_ *-tr-*' 0 C) ka Figure 4. Magnitude of p / zka at the center of a pulsating spherical cavity *. * \. ' '. \\\# 30 CQ ~~~ "N 20 # CL ~" 10 _ 0 C) 0.5 #». ' i No opening Opening ka 2 Figure 5. BEM results at the center of a spherical cavity. CONCLUSION In this paper thefirststage of a research programme directed towards the prediction of sound pressure levels inside vehicle structures, under realistic operating conditions, is introduced. A literature survey on the application of the BEM to the radiation problems concerning the interior regions is presented. The

9 Boundary Element Technology 65 BEM formulation has been given for the interior acoustic radiation problems together with the numerical solution technique and illustrated on a classical acoustic problem. The application of the BEM to the numerical solution of the acoustic response of a spherical cavity has been performed. The acoustic pressure distribution along the radius of the spherical cavity is found from the applied numerical method and then compared with the analytical results. The results obtained from the BEM agree quite well with the analytical results. Finally, the effect of an opening on the acoustic pressure at the center of the sphere has been investigated. As a next step in this ongoing research, the experimental verifications will be performed. A non symmetric box structure will used in the experiments and the vibration data from this box will be obtained. The obtained vibration data will be used as boundary condition for the BEM code developed. Finally, the sound pressure levels obtained from the experiments and the BEM will be compared. If the results are satisfactory the method will be applied to a vehicle body structure whose vibration characteristics will be measured and/or calculated using a FEM model. ACKNOWLEDGEMENTS Middle East Technical University Research Fund Project No is gratefully acknowledged for the support and the computational facilities provided. REFERENCES 1. Bell, W.A., Meyer, W.L., and Zinn, B.T., "Predicting the Acoustics of Arbitrarily Shaped Bodies Using an Integral Approach", AIAA Journal, 1977, Vol. 15, No. 6, pp Sestieri, A., Vescovo, D.D., and Lucibello, P., "Structural-Acoustic Coupling in Complex Shaped Cavities", J.Sound and Vibration, Vol. 96, No. 2, 1984, pp Seybert, A.F., and Cheng, C.Y.R., " Application of the Boundary Element Method to Acoustic Cavity Response and Muffler Analysis",.J.Vib. Ac. St. Rel. Des, Vol. 109, 1987, pp Kipp, C.R., and Bernhard, R.J., "Prediction of Acoustical behavior in Cavities Using an Indirect Boundary Element Method",.J.Vib. Ac. St. Rel. Des., Vol. 109, 1987, pp Chen, L.H. and Schweikert, D.G., "Sound Radiation from an Arbitrary Body", J.Acoust. Soc. Am., Vol. 35, No. 10, 1963, pp Fyfe, K.R., "Determination of Acoustic Modal Properties from Boundary Element Modelling", Dynamic Engng., 1988, Heverlee, Belgium.

10 66 Boundary Element Technology 7. Cheng, C.Y.R, and Seybert, A.F., " Recent Applications of the Boundary Element Method to Problems in Acoustics", SAE Paper No , 1988, pp Wu, T.W., et. al, "Vectorization and Parallelization of the Acoustic Boundary Element Code BEMAP on the IBM ES / 3090 VF", International Congress on Recent Developments in Air-and Structure-Borne Sound and Vibration, 1990, Auburn University, USA. 9. Seybert, A.F., Cheng, C.Y.R., and Wu, T.W., "The Solution of Coupled Interior / Exterior Acoustic Problems Using The Boundary Element Method", J.Acoust. Soc. Am., Vol. 88, No. 3, 1990, pp Seybert, A.F., Wu, T W, and Wan, G.C., "Recent Applications of Boundary Element Modelling in Acoustics", Second International Congress on Recent Developments in Air-and Structure-Borne Sound and Vibration, 1992, Auburn University, USA. 11.Suzuki, S., Maruyama, S., and Ido, H, "Boundary Element Analysis of Cavity Noise Problems with Complicated Boundary Conditions", J.Sound and Vibration, Vol. 130, No. 1, 1989, pp shiyama, et. al., "The Applications of ACOUST / BOOM - A Noise Level Predicting and Reducing Computer Code", SAE Paper No , 1989, pp Utsuno, et. al., "Analysis of the Sound Field in an Automobile Cabin by Using the Boundary Element Method", SAE Paper No , 1990, pp Morse, P.M. and Feshbach, H, "Methods of Theoretical Physics",Mc-Graw Hill Book Company, Kopuz, S., Unlusoy, Y.S., and Caliskan, M, "Formulation of the interior acoustic fields for passenger vehicle compartments", Proceedings of the Second International Congress on Recent Developments in Air-and Structure-Borne Sound and Vibration, pp , Auburn University, Auburn, USA, Kopuz, S., Unlusoy, Y.S., and Caliskan, M, "The Use of BEM in the Prediction of Interior Acoustics of Vehicle Body Structures", to be presented at the 6th National Conference on Machine Theory, Trabzon, Turkey, September 15-17, 1993 (in Turkish).