Acoustic-Structure Simulation of Exhaust Systems by Coupled FEM and Fast BEM

Transcription

1 Acoustic-Structure Simuation of Exhaust Systems by Couped FEM and Fast BEM Lothar Gau, Michae Junge Institute of Appied and Experimenta Mechanics, University of Stuttgart Pfaffenwadring 9, 755 Stuttgart, Germany ABSTRACT The sound properties of modern cars pay an increasingy important roe on the overa impression of a vehice. One of the major sound sources is hereby the exhaust system, which is exposed to arge pressure pusations resuting from the periodicay bown out exhaust gas of the car engine. These arge pressure pusations ead to vibrations of the thin was of the exhaust system and thus create another sound source which is additionay contributing to the sound radiation of the exhaust system besides the sound at the orifice. In this work, the sound radiation of an associated simpified exhaust system is anayzed. The sound transmission of the interior exhaust system is simuated by using conforming couping of FE-FE-modes. For the subsequent soution of the infinite exterior, probem the Fast-Mutipoe BEM is empoyed enabing a very fast and efficient computation. This work focuses on the contribution of the vibrating structure to the overa sound radiation. It is found that cose to some resonance frequencies of the in-vacuo structure there is a significant contribution of the vibrating structure to the overa radiated sound power. Nomencature F s, F f oad vectors u dispacement degree of freedom I a acoustic intensity vector W radiated soundpower I unity matrix v n norma veocity M s, K s mass and stiffness matrix of the structure Z specific acoustic impedance M f, K f mass and stiffness matrix of the fuid Γ cosed surface of radiating object M fs, K fs couping matrices κ wave number n unit norma vector on boundary surface ρ, c density and wave speed of fuid P fundamenta soution (V q)(x) singe ayer potentia p pressure degree of freedom (Kp)(x) doube ayer potentia q acoustic fux (K q)(x) adjoint doube ayer potentia R compex refection coefficient (Dp)(x) hyper-singuar operator 1 Introduction On a car there are severa sound sources each dominating at a different speed. At ow speeds up to 5 km/h the major sound source is the car engine. In most cases, the engine noise is very efficienty reduced by an exhaust system. However, the arge pressure pusations on the outet of the piston engine may ead to vibrations of the thin was of the exhaust system. These vibrations in the exhaust structure are additionay contributing to the sound radiation of the exhaust system. Besides, the propagation of the structure-borne sound induced in the was can acousticay short-circuit the siencer instaed in the duct as reported for inet and outet ducts of arge gas turbines by [1]. The excitation of structura vibration modes by the acoustic path is investigated and the sound radiation of an ideaized car exhaust system is anayzed. A specia focus is set on the contribution of the vibrating structure to the overa sound radiation of the exhaust system.

2 2 Simuation mode The simuation mode of the ideaized exhaust system consists of two sub-modes. In the first step, a fuy couped FE-FEmode is used to compute the interior acoustic probem of the exhaust system. The fuid-structure interface accounts for the infuence of the acoustics on the fexibe was. By means of this mode, the excitation of structura vibration modes by the acoustic path is computed. In the next step, the cacuated norma veocities on the boundary of the structura parts of the exhaust system and the pressure on the outet are used as Neumann and Dirichet boundary data respectivey for the BE-mode in order to sove the exterior acoustic probem. 2.1 Ideaized exhaust system In this work, an ideaized exhaust system is used. It consists of one expansion chamber, an ebow, an exhaust pipe and a tai pipe as shown in Fig. 1. The system is fixed at its ends by circuar soft rings. It is excited on the inet with a uniform, harmonic pusating pressure p = ˆpe jωt. This mode does not consider mean fow and vortices. It assumes that the inear acoustics theory is appicabe. The tota pipe ength is 2.4 m, the pipe radius is chosen as r =.34 m and the wa thickness is t w =.1 m. 3 Interior acoustic probem FSI-couped undamped probems are formuated by FE-discretization as presented in [2] [ ][ü ] [ ] [ ] [ Ms Ks K + fs u F s] =, (1) M f p K f p M fs with the noda dispacement degrees of freedom u, and the noda acoustic pressure degrees of freedom p. The mass couping matrix M fs and the stiffness couping matrix K fs are reated by M fs = ρk T fs. At the outet, a radiation impedance condition is appied using the resuts derived in [3]: The specific acoustic impedance Z is expressed in terms of the compex refection coefficient R by F f Z = p v n = ρc 1 + R 1 R. (2) The compex refection coefficient R is characterized by its magnitude R, the end correction factor δ and the wave number k R = R e j(π 2kδ). (3) Figure 1: Structure of the ideaized exhaust system

3 R δ f f Figure 2: Radiation impedance condition: Magnitude and end correction factor of a circuar pipe (radius r =.34 m) according to [3]. The vaues of R and δ for the system at hand are shown in Fig. 2. For ow frequencies the magnitude of the refection coefficient is cose to unity whie its phase ange is amost π, hence an incoming wave is amost fuy refected at the tai pipe s end. In the FE-program ANSYS, the compex impedance condition (2) is modeed in the harmonic case by adding massess surface eements onto the outet, assigned with viscosity and added mass per unit area. In the simuation the expansion chamber shows a typica transmission oss behavior with a maximum vaue of 15 db as shown in Fig. 3. These resuts are in good agreement with those presented in [4]. Transmission oss [db] f [Hz] Figure 3: Transmission oss of ideaized exhaust system with a maximum vaue of 15 db. The moda anaysis of the in-vacuo system (exhaust system without fuid eements) shows resonance frequencies as indicated by the vertica ines in the ower pot of Fig. 6. The eigenmode of the exhaust system at f = Hz is depicted in Fig. 4. It is evident that the exhaust pipe and the expansion chamber are moving in opposite phase. The ampitude of the taipipe is much ower than the ampitude of the exhaust pipe and the expansion chamber.

4 4 Exterior acoustic probem Figure 4: Eigenmode of the in-vacuo exhaust system at f = Hz Having soved the interior acoustic probem, the computed dispacements on the pipe and the pressure vaues on the outet are used as boundary data for the exterior acoustic probem, yieding vaues for the pressure and acoustic fux respectivey on the surface of the exhaust system. This probem is computed by means of the Fast Mutipoe Mutieve Boundary Eement Method (FMM-BEM) [5] which wi be descriped briefy in the next two subsections. 4.1 The Boundary Eement Method (BEM) The Boundary Eement Method is we suited for acoustic computations, offering high accuracy and a simpe mesh generation, especiay for modeing exterior acoustic probems [6]. Having once computed the acoustic pressure and the acoustic fux on the boundary of the acoustic domain, the acoustic fied pressure and the fied fux are computed at any arbitrary fied point with ow numerica costs. For time-harmonic behavior, the pressure p in an acoustic fied is governed by the Hemhotz equation 2 p(x) + κ 2 p(x) =, (4) where κ = ω/c is the acoustic wavenumber and c is the sound veocity. The acoustic fux q on the boundary is prescribed as q(x) := p(x), x Γ, (5) n which is proportiona to the surface veocity and thus to the dispacement ampitudes of a vibrating structure. For the exterior probem, the Hemhotz equation is soved in an exterior domain Ω e, with the boundary Γ = Ω e. The fundamenta soution P (x,y) for the Hemhotz equation with spatia Dirac source is given by P (x,y) = 1 e iκr 4π r, (6) where r = x y is the Eucidean distance between the fied point x and the oad point y. Appication of Green s second theorem to the weak form of the Hemhotz equation Ω e P (x,y) ( 2 p(x) + κ 2 p(x) ) dω e = (7) yieds the representation formua p(x) = Γ P P (x,y) (x,y)q(y)ds y + p(y)ds y, x Ω e. (8) Γ n y By shifting the fied point onto the smooth boundary, one obtains the boundary integra equation p(x) = 1 2 p(x) P P (x,y) (x,y)q(y)dγ y + p(y)dγ y, x Γ. (9) Γ Γ n }{{} y }{{} (V q)(x) (Kp)(x)

5 y a z a D z b x b Figure 5: The action of sources at y a are bunded in z a, then transated to another center z b, and evauated at the fied points x b. The expressions (V q)(x) and (Kp)(x) are caed the singe and doube ayer potentias, respectivey. The hypersinguar boundary integra equation [5] is obtained by taking the norma derivative and the imit x Γ of Eq. (8) q(x) = 1 2 q(x) P (x,y) q(y)dγ y Γ n } x {{} (K q)(x) + 2 P (x,y) p(y)dγ y, Γ n x n y }{{} x Γ, () (Dp)(x) where (K q)(x) and (Dp)(x) are the adjoint doube ayer potentia and the hyper-singuar operator, respectivey. A more detaied description of the Boundary Eement Method for acoustics can be found in [7, 8, 9]. The standard BEM has one significant drawback. For frequencies that correspond to the eigenfrequencies of the associated interior probem, the soution is not unique. Burton and Mier [] proposed a combination of the standard and the hyper-singuar boundary integra equation (Eqs. (9) and ()) to avoid this probem at the so caed critica frequencies. Using an isoparametric Gaerkin formuation with piecewise inear shape functions for the pressure and constant shape functions for the acoustic fux, the Neumann probem considered in this paper is written as Burton-Mier BEM formuation by superposition and i κ -weighting ( 1 2 I + K + i ) κ D p = ( V i 2κ I i ) κ K q. (11) 4.2 Fast Mutipoe-BEM (FBEM) A major imitation of cassica BEM formuations is the numerica compexity of order O(N 2 ) for setting up and storing the fuy popuated system matrices. Thus the appication of cassica BEM methods is imited to the ow frequency regime with moderate numbers of unknowns. Since modern iterative sovers ike GMRES [11] ony require the evauation of matrix-vector products, the fuy popuated system matrices do not need to be set up. The key idea of the fast mutipoe method is the efficient evauation of the matrixvector products by using a series expansion of the fundamenta soution, which combines the effect of sources far away from a fiedpoint in a farfied representation. Foowing Rokhin [12], the mutipoe expansion reads with d = x b z b + z a y a and D as depicted in Figure 5 e iκ x y x y = eiκ D+d D + d = iκ (2 + 1)( 1) j (κ d )h (1) (κ D )P ( ˆD ˆd), D > d, (12) = where j ( ) denotes spherica Besse functions, h ( ) Hanke function, P ( ) Legendre poynomias and ˆ ( ) indicates normaized vectors. A diagona form of the mutipoe expansion is obtained by using the orthonormaity of the Legendre poynomias on the unit sphere and the expansion of spherica waves e iκ D+d D + d = iκ 4π = (2 + 1)i h (1) (κ D ) e iκd s P (s ˆD)ds. (13) S 2 The summation must be truncated to the expansion ength L, since the Hanke functions diverge for arge. Therefor,

6 transation operators are defined M L (s,d) = L = (2 + 1)i h (1) (κ D )P (s ˆD), (14) which ony depend on the distance vector D. Thus, a hierarchica custering strategy is appied to achieve an efficient mutieve scheme. A custers which are within some distance of a custer form the nearfied. Custers, whose father custers fufi the nearfied condition, but are not in each others nearfied themseves, form the interaction ist. In the foowing, the evauation of the matrix-vector product for the singe ayer potentia is discussed. For the singe ayer potentia with constant shape functions, the v-th component of the matrix-vector product is now approximated by G ν v ν (V nearfied u) ν + w ν,j ν j=1 µ farfied w µ,i µ P (x ν,j,y µ,i ), (15) u µ G µ i=1 } {{ } evauation with mutipoe method where µ, w µ,i and y µ,i are the Jacobi determinant, the Gauss weights, and the integration points for eement µ, respectivey. The terms are defined anaogous for the eement ν. The expression on the right side are evauated efficienty using the mutipoe agorithm, which can be sketched as foows: First, a farfied signature F γ max (s) is computed for a custers γ on the owest eve. The farfied signature is transated to the interaction ist using the transation operators (14) N γ (s) = interactionist M L (s,d)f γ (s). (16) At the next step, F γ is shifted to the center of the father custer. The ast two steps must be repeated unti the interaction ist is empty. In the downward pass, the nearfied signatures N γ (s) are shifted to the chid custers. Finay, the soution in the integration points is recovered on the owest eve. For the compete mutipoe agorithm, the computing cost is of order O(N og 2 N) [5]. Thus the FMM agorithm enabes a fast and efficient computation of arge scae BEM systems up to more than degrees of freedom using standard desktop PC. 4.3 Sound radiation The sound radiation of the exhaust system is evauated in a post-processing step by separatey computing the radiated sound power through the cross-section of the outet W ΓD and through the was of the exhaust system W ΓN W Γ = W ΓD + W ΓN = I a n dγ + Γ D I a n dγ Γ N, (17) where I = pv is the oca intensity vector on the surface and n the unit norma vector pointing outwards from the cosed surface Γ = Γ D + Γ N. The soid ine in the upper pot of Fig. 6 shows the tota radiated sound power W Γ scaed to its maximum vaue. The oca maxima and minima every 4-6 Hz resut from the constructive and destructive interference of the refected wave on the outet. The dash-dotted ine indicates the magnitude of the sound power radiated by the vibrating structure W ΓN, whie in the ower pot of Fig. 6 the ratio of the magnitude of W ΓN and W Γ is depicted. Both pots ceary show that W ΓN does not contribute significanty to the tota radiated sound power for the vast majority of frequencies. Yet, for some frequencies marked with an asterisk, energy is absorbed by the was, though the vaues are very sma. In contrast to this, it is found that for some frequencies cose to resonance frequencies of the in-vacuo exhaust system the ratio of the magnitude of W ΓN and W Γ is higher than %. Thus W ΓN significanty contributes to the tota radiated sound power. Some of the acoustic energy inserted into the system by the appied uniform, harmonic pusating acoustic pressure is transferred via the fuid-structure interface onto the was of the exhaust system, generating structura vibrations. So far, the radiated sound power of the exhaust system has been considered. In this paragraph the sound fied of the exhaust system at a certain frequency wi be investigated. Figure 7 shows the computed sound fied at a frequency f = 288 Hz. The vertica pane cose to the outet of the exhaust system shows the circuar radiation of the sound at the outet. In this area the tai pipe sound is the dominating source. The vertica pane in the rear is chosen at a distance of 2 cm from the ongitudina axis of the exhaust system, giving a good impression how the sound pressure fied might ook ike at the underfoor of a car. Two dark spots (one bue spot and one red spot) are observed representing areas of amost the same magnitude but opposite sign resuting from the motion of the exhaust system simiar to the eigenmode shown earier in Fig. 4.

7 normaized sound power -5 WΓ WΓN f [Hz] WΓN /WΓ -1 power ratio f [Hz] Figure 6: Top: Normaized radiated sound power of the exhaust system W Γ and the contribution of the was of exhaust structure W ΓN. Bottom: Ratio of W ΓN to W Γ. The vertica ines represent the resonance frequencies of the in-vacuo exhaust system. W ΓN does not contribute significanty to the overa radiated sound power W Γ, except for the case of pressure induced resonance of the exhaust structure. Figure 7: Sound fied of vibrating exhaust system at f = 288 Hz.

8 5 Concusion In this work a method to efficienty compute the sound radiation of an exhaust system is presented. It consists of a consecutive computation of the acoustic interior and exterior probem. The simuation resuts of the acoustic interior probem soved with a FE-FE-formuation are used as boundary data for a BE-formuation of the acoustic exterior probem. For the computation of the acoustic exterior probem, the Fast Mutipoe Boundary Eement Method is appied enabing a fast and efficient computation of arge scae systems. For the presented ideaized exhaust system the simuation resuts suggest a minor contribution of the sound generated by the vibrating structure to the tota radiated sound power. However, cose to resonance frequencies of the in-vacuo system an energy exchange on the FE-FE interface is observed and structura vibration of the structure of the exhaust system is induced. In the case of a more reaistic exhaust system, especiay for the case of a more effective muffer with a much higher transmission oss, the infuence of the sound radiated by the vibrating structure wi pay a more important roe. In this work ony the excitation of structura vibration by the acoustic path is considered. Vibrations induced by the movement of the attached engine are not taken into account. The presented method proves viabe for the computation of the sound radiation of exhaust systems and can easiy be extended to account for the additiona effects mentioned. 6 Acknowedgement Funding of this project by the Friedrich-und-Eisabeth-Boysen-Stiftung is gratefuy acknowedged. References [1] A. Facian, A. Nisson, L. Feng, and E. Nisson. Propagation of structure-borne sound in siencers used in power pants. In Proceedings of the Eeventh Internationa Congress on Sound and Vibration, pages 69 76, 24. [2] O.C Zienkiewicz and R.L. Tayor. The Finite Eement Method, voume 1. Butterworth-Heinemann, Oxford, 5th edition, 2. [3] H. Levine and J. Schwinger. On the radiation of sound from an unfanged circuar pipe. Physica Review, 73(4):383 46, [4] M.L. Munja. Acoustics of Ducts and Muffers. John Wiey & Sons, New York, [5] M. Fischer. The Fast Mutipoe Boundary Eement Method and its Appications to Structure-Acoustic Fied Interaction. PhD thesis, University of Stuttgart, 24. [6] L. Gau and Wagner M. Kög M. Boundary Eement Methods for Engineers and Scientists. Springer, Berin, 23. [7] O. v. Estorff, editor. Boundary Eements in Acoustics: Advances and Appications. WIT Press, Southampton, UK, 2. [8] T.W. Wu. Boundary Eement Acoustics: Fundamentas and Computer Codes. WIT Press, Southampton, UK, 2. [9] L. Gau and M. Fischer. Large Scae Simuations of Acoustic Structure Interaction Using the Fast Mutipoe BEM. In R. Hemig, A. Mieke, and B. Womuth, editors, Mutified Probems in Soid and Fuid Mechanics, pages , 26. [] A.J. Burton and G.F. Mier. The appication of the integra equation method to the numerica soution of some exterior boundary-vaue probems. Proc. Roya Soc. Lond, A, 323:21 2, [11] Y. Saad. Iterative Methods for Sparse Linear Systems. SIAM, Society for Industria and Appied Mathematics, Phiadephia, 23. [12] V. Rokhin. Diagona forms of transation operators for the hemhotz equation in three dimensions. Appied and Computationa Harmonic Anaysis, 1:82 93, 1993.