Structural Acoustics Applications of the BEM and the FEM

Transcription

1 Structural Acoustics Applications of the BEM and the FEM A. F. Seybert, T. W. Wu and W. L. Li Department of Mechanical Engineering, University of Kentucky Lexington, KY U.S.A. SUMMARY In this paper we discuss how the boundary element method (BEM) may be used by itself or in conjunction with the finite element method (FEM) for the solution of problems in structural acoustics. Problems in this category include: the scattering of sound from an elastic solid or shell submerged in a heavy fluid such as sea water, and the sound field generated within a vehicle passenger compartment or aircraft cabin due to vibration of the surrounding structure. In the latter case, the modes of the structure and the acoustic fluid may be coupled even if the fluid is air. Two approaches to structural acoustics modeling are considered. In one approach, the BEM is used to model both the elastic and the acoustic parts of the problem. In the second approach, the FEM is used to model the structure, while the BEM is used to model the fluid. INTRODUCTION The interaction of sound and vibration is an important subject in acoustics. If a light fluid is contained within a vibrating structure, such as the air within a vehicle passenger compartment, there may be significant coupling between the structural vibration and the sound field. In such cases, the coupled structural/acoustics problem must be solved. When a structure is submerged in a heavy fluid, such as sea water, the vibration of the structure will be affected by the surrounding sound pressure, and the whole system becomes a coupled structural/acoustics problem. There have been many papers published using a variety of analytical methods to examine structural/acoustic interaction [1-5]. However, when the structure has an arbitrary shape, methods amenable to numerical implementation are usually more desirable than approximate analytical methods. In this paper we use two numerical techniques to solve problems in structural acoustics. In the first approach, the coupled elastic/acoustic boundary value problem is reformulated into a pair of integral equations, one for the interior (elastic) domain and one for the exterior (acoustic) domain. In its numerical form the integral equation method is referred to B. S. Annigeri et al. (eds.), Boundary Element Methods in Engineering Springer-Verlag Berlin Heidelberg 1990

2 537 as the boundary element method (BEM). The coupled elastic/acoustic technique used herein is referred to as the BEM/BEM approach. The second numerical technique uses a finite element method (FEM) for the structure, but retains the BEM for the fluid. We refer to this method as the FEM/BEM approach. The FEM/BEM approach is preferred for structural acoustics problems in which the structural geometry is complex or when the structure is a thin shell. In the latter case, a BEM model of the structure would not be practical because the elements would be very small to avoid the near-singular behavior caused by the thin structural domain. FORMUlATION For acoustic radiation or scattering from an elastic structure, we must solve a boundary value problem involving coupling between the elastic structure B2, bounded by the surface S, and the surrounding acoustic fluid Bl, see Fig. 1. (Similarly, we may pose a problem consisting of an acoustic fluid, bounded by a surface S, contained within an elastic structure B2 bounded on the exterior by a surface s). For scattering problems, we denote the prescribed incident wave as Pi. The elastic structure is assumed to be homogeneous, isotropic and linearly elastic, and the fluid is assumed to be inviscid. Body forces are assumed zero. Boundary conditions may be specified along the surfaces sand S. Along s, where Bl and B2 are not in contact, we may impose surface tractions, ---APi 5 Fig. 1. Nomenclature for radiation or scattering of acoustic waves from an elastic structure.

3 538 surface displacements, or a linear combination of the two, corresponding to a well-posed boundary value problem. The appropriate boundary conditions at the interface S are continuity of normal displacement of the structure and the particle displacement of the fluid and equilibrium of the normal stress on the structure and the acoustic pressure. of the problem may be found in Reference 6). (A complete statement BEM/BEM FORMULATION To reformulate the problem in Fig. 1 in terms of integral equations valid at the boundary S + s, we use the boundary integral equation for the exterior acoustics problem and a companion interior integral equation the elastic structure. The resulting boundary integral equation formulation for our coupled problem is [6]: for TIT C (P2)y(P 2 ) - [y (P2,Q)~(Q) - T (P2,Q)y(Q)] ds(q), s+s (1) which is valid for P2 in B2 or on S + sand C(Pl)P(Pl ) = I [ p(q) :~(Pl,Q) - ~(Q) ~(Pl,Q)J ds(q) + 4~Pi(Pl)' (2) S which is valid for PI in HI or on S. In the above equations, ~ and! are the Stokes' displacement and traction tensors, respectively, ~(Pl,Q) is the free-space Green function, ~ and t are the displacement and the stress, respectively, in the elastic structure and p and ap/an are the sound pressure and the normal derivative of the sound pressure, respectively, in the acoustic fluid. Special care has to be used in integrating Eqs. (1) and (2) numerically due to the singularities in the integrands. However, it is noted that both integrals exist. A numerical solution to Eqs. (1) and (2) can be achieved by discretizing the boundary into a number of surface elements. In this paper, we use isoparametric quadratic elements [6]. In numerical form Eq. (1) becomes [H]lu) - [a]lt) (3) where H and a are the assembled coefficient matrices resulting from the numerical integration. The momentum equation is used to replace the normal derivative of the pressure in Eq. (2) with displacement, and the resulting numerical form of Eq. (2) is

4 539 where T and D are the assembled coefficient matrices. The coupled elastic/acoustic system of equations can be obtained by combining Eqs. (3) and (4) and replacing the tractions on the surface with pressure. The resulting linear system of equations can be solved for the nodal values of displacements and acoustic pressure on the elastic/acoustic interface as well as the displacements (or tractions) on the inner solid surface. FEM/BEM FORMULATION Although, theoretically, there is no limitation on the closeness between the outer surface S and the inner surface s, the BEM/BEM approach will require a large number of integration points and elements if the elastic structure is an extremely thin shell. An alternative way to model an elastic thin shell is the finite element method [7]. If we use the FEM to model the elastic thin shell and the BEM to model the acoustic field, the resulting matrix representation is L -D ] [ :] [ f ] (5) where K is the stiffness matrix, M is the mass matrix, f is the forcing vector due to the applied loading, and L takes care of the loading from the sound pressure at the interface. X, Fig. 2. One octant of the outer surface of the sphere showing nodes and element configuration.

5 540 EXAMPLE PROBLEMS We tested the BEM/BEM and the FEM/BEM methods by solving several problems in structural acoustics having spherical geometry. these problems is shown in Fig. 2. A typical mesh used in The first problem that we consider is the radiation from a hollow steel sphere in sea water with inner radius b = 0.5 m and outer radius a - 1 m. The inner surface is loaded with a uniform time-harmonic pressure of amplitude Po. shell, the BEM/BEM technique is used. As this is a thick, homogeneous Figure 3 shows the sound pressure at a distance of 3a from the center of the hollow sphere plotted versus ka where k is the wave number in the fluid. The data in Fig. 3 show the rapid convergence that may be expected with the BEM/BEM.. The numerical solution, however, is not as good at the resonance frequency (ka-8) as it is at other frequencies. Our numerical study showed that the error at ka=8 will be reduced to 0.7 percent if a mesh of 386 nodes is used. Next, we consider a BEM/BEM test case consisting of the scattering of an incident plane wave of amplitude Pi by the hollow sphere described above. Figure 4 shows the ratio of the scattered pressure to the incident pressure at a distance of 3a from the center of the sphere for a sphere of various composition when ka = 1. from the right in Fig. 4. The plane wave is incident on the hollow sphere The results for the rigid sphere are also shown , Analytical o Bem 114 Nodes Bem 58 Nodes Fig. 3. ka Normalized sound pressure at a distance of 3a from the center of a hollow steel sphere in sea water excited by an uniform internal pressure Po.

6 in Fig. 4 for comparison. the 114 node BEM/BEM model discussed above. All of the scattering data were obtained with The effect of material composition on the scattered pressure may be seen quite clearly in Fig , , , - RiOid -- Steel --- Aluminum ---- Rubber tH-f-1:---+-l--f-f--t '-- --L....L...L- ~ Fig. 4. Ratio of the scattered to the incident pressure at a distance of 3a from the center of a hollow sphere of various compositions in sea water when ka - 1. AN INTERIOR STRUCTURAL/ACOUSTICS PROBLEM As a test of the FEMjBEM technique, consider an air-filled spherical aluminum shell of radius I m and thickness 5 rom excited externally by a uniform pressure of 1 Pa (94 db sound pressure level). Figure 5 shows the sound pressure level at the center of the spherical cavity as a function of frequency. The frequency regions where the sound pressure level is high correspond to the resonances of the air cavity within the shell. DISCUSSION AND SUMMARY Two numerical methods have been used to solve structural acoustics problems. In the BEM/BEM method, we have reformulated the acoustic radiation and scattering problems for elastic structures in terms of a pair of integral equations, one for the elastic structure, the other for the host fluid and both valid at the interface. These integral equations were reduced to a numerical form using quadratic isoparametric elements. The BEM/BEM formulation was tested with good results for several radiation

7 542 iii" e '"" so 1'4 II: ~ fii gj 40 IE Q :z: ~ 230 *\* I. *V' * BEMAP/ANSYS ANALYTICAL 20 Fig FREQUENCY (HZ) Sound pressure level at the center of a air-filled aluminum shell of radius 1 m and thickness 5 mm due to a uniform external pressure of 94 db. and scattering problems of practical importance. For elastic thin shells, the BEM/BEM approach is still quite reliable if more integration points are used to integrate the integral equations for very close elements. However, a coupled FEM/BEM technique also provides excellent results for thin shell problems. In addition, the FEM/BEM method is more practical in cases where the structure has a complex geometry. approach showed good results. A test problem using the FEM/BEM 1. J. J. Faran, Jr., "Sound Scattering by Solid Cylinders and Spheres," J. Acoust. Soc. Am. 23, , (1951). 2. M. C. Junger, "Sound Scattering by Thin Elastic Shells," J. Acoust. Soc. Am. 24, (1952). 3. R. Hickling, "Analysis of Echoes from a solid Elastic Sphere in Water," J. Acoust. Soc. Am. 34, 1582 (1962). 4. K. Yosioka and Y. Kawasima, "Acoustic Radiation Pressure on a Compressible Sphere," Acustica 5, (1955). 5. T. Hasegawa and K. Yosioka, "Acoustic-Radiation Force on a Solid Elastic Sphere," J. Acoust. Soc. Am. 46, (1969). 6. A. F. Seybert, T. W. Wu and X. F. Wu, "Radiation and Scattering of Acoustic Waves from Elastic Solids and Shells using the Boundary Element Method," J Acoust. Soc. Am. 84, (1988). 7. D. T. Wilton, "Acoustic Radiation and Scattering from Elastic Structures," Int. J. Num. Meth. En&.. 13, (1978).