Estimation of Radiated Sound Power: A Case Study on Common Approximation Methods

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1 ACTA ACUSTICA UNITED WITH ACUSTICA DOI /AAA Estimation of Radiated Sound Power: A Case Study on Common Approximation Methods DennyFritze, Steffen Marburg, HansJürgen Hardtke Technische Universität Dresden, Institut für Festkörpermechanik, Dresden, Germany Summary The radiated sound power value is often used to evaluate the sound radiation of a machine or a product. Since its estimation requires the sound pressure on a surrounding surface of the radiating object, the sound power value is mostly computed under high numerical costs due to the acoustic field that has to be modeled. Therefore, approximations of the sound pressure are widely popular. In this article three common methods namely the equivalent radiated power, the lumped parameter model and an approximation based on the volume velocity are investigated. It is the goal of this paper to test these methods on realistic examples. The radiated sound power functions of the floor panel of acar and the radiation of adiesel engine under realistic load cases are estimated. PACS no Rj 1. Introduction The radiated sound power is often used to express the general radiation behavior of a component or a machine [1]. It represents the integral of sound intensity over aclosed surface surrounding the radiating object. This surface can also be the surface of the radiator. Thus, the product of sound pressure and particle velocity must be integrated overthe surface. In the case of aradiator with hard reflecting surface, the particle velocity is identical to the structural normal velocity. However, evaluation of the sound power requires an estimation of the sound pressure on the structure s surface. This means that a global quantity, i.e. sound pressure on the surface of the underlying structure, is computed. For alargescale surface mesh and alarge frequency range, this computation becomes very expensive especially if the structure is analyzed repeatedly as in structural acoustic optimization problems [2]. Usually,the radiated sound power is computed for exterior problems only.when considering interior problems, a local quantity such as the sound pressure at a single point or several internal points is used to estimate the acoustic properties. For the computation, we can use the adjoint operator approach which has different names in the literature [3, 4, 5], see also [2] and references therein. In [1], global quantities such as potential and kinetic energies are proposed to estimate the acoustic behavior. Cunefare and Engelstad et al. [6, 7, 8] use the sound pressure at many internal points. There are only afew academic cases for which the radiated sound power can be calculated exactly. Therefore, numerical approximations are very popular [9]. The Received 15January 2009, accepted 2April physics behind the sound radiation requires the modeling of the fluid structure interaction. Forheavy structures and light fluids, an unidirectional coupling approach is preferred. This still means that the structural dynamics and the acoustic field must be solved. The vibration of the structure is often numerical computed via the finite element method. The solution of the structural problem, i.e. the structural particle velocity accounts for the boundary condition of the acoustic field problem. The solution of the acoustic boundary value problem can be very complicated for radiation problems. The boundary element method appears as very popular approach [9, 10, 11, 12, 13] but becomes computationally expensive for largescale problems, cf. [14, 15]. For largescale problems, fast methods of the BEM are used instead, e.g. fast multipole techniques [16, 17, 18, 19, 20, 21, 22, 23, 24]). If the structure is modified during an optimization process or if alarge frequency range is investigated these methods remain problematic. Thus, simplified estimations are introduced. They are usually aimed to circumvent the solution of the full acoustic boundary value problem. In this paper,three different approximation methods are compared. Firstly, weconsider an approach which is called the equivalent radiated power (ERP). It is similar to the squared particle velocity and very popular for industrial problems due to its simplicity. It will mostly overestimate the sound power since it does not contain compensation effects by acoustical short circuits. Further, the radiated sound power can be approximated by the volume velocity. This includes compensation effects in a primitive way. The third method is referred to in the literature as the lumped parameter model (LPM), cf. [1, 25, 26], and as the Direct FEM [27, 28, 29]. It was shown in [30] that both approaches lead to the same result. The lumped parameter model uses the Rayleigh integral to evaluate the sound S.Hirzel Verlag EAA 833
2 ACTA ACUSTICA UNITED WITH ACUSTICA power. Inthe papers of Herrin et al. [31, 32], the authors showed that the Rayleigh integral approximation might be very reliable for estimation of radiated sound power even for real applications such as radiation from an engine. In this article, we start with introducing these methods briefly.afterwards the methods are applied to realistic examples and, for better understanding, to some academic test cases. The first example is afloor panel of asedan s bodywork vibrating at low frequencies for arealistic load case. A higher frequency problem is investigated at a six cylinder diesel engine. There, we take into account a realistic load case and finally artificial boundary conditions including rigid body motion and pulsating behavior, both to satisfy academic interests rather than being relevant for practical applications. To appreciate the efficiency and the accuracy of these approximative methods we carry out a Multilevel Fast Multipole Analysis (MLFMA) for each example, cf. [16]. We use this result as our reference solution since there is no better estimation available for realistic examples. In practical applications efficient methods for the sound power computation are highly recommended, but there is alack of experience especially for complicated structures, i.e. very detailed surface meshes. The goal of this paper is to exercise a comparative study on such approximations. The results of the methods are often theoretically comprehensible butare consolidated by practical examples in this paper. Throughout this paper, aharmonic time dependence of e iωt is used. Furthermore, the term of coupling between structural and acoustic vibration is used in a one way sense, i.e. structural vibrations affect the acoustic radiation, but the sound field does not affect the structural vibrations. 2. Sound power evaluation and estimates The radiated sound power is often used to estimate the sound radiation of avibrating obstacle into the exterior. It is well known that the radiated sound power P is evaluated as the integral of sound intensity I in normal direction n over acircumscribing surface Γ. The intensity is calculated based on the sound pressure p and the particle velocity v.the circumscribing surface can be any surface including the surface of the radiating obstacle. The sound power is given by P = I n d Γ, where I = 1 2 pv (1) leads to P = 1 2 p(x)vn (x)dγ(x). (2) The asterisk denotes the conjugate complexvalue and {} is the real part of acomplex value. In what follows, we substitute the particle velocity v n = v n for v. Since v can be easily imported from structural dynamic analysis, the major focus for the efficient evaluation of the radiated sound power is situated in the estimation of the sound pressure. The formal way leads through the acoustic field analysis via numerical methods. (Herein, we concentrate on BEM) Evaluation based on BEM In numerical solutions, e.g. boundary element method and finite element method, the sound pressure and the particle velocity on the surface are usually interpolated by aset of basis functions assembled in an interpolation matrix Φ f as p(x) = Φ f (x)p and v(x) = Φ f (x)v. (3) Then, the sound power P can be formulated by the vector of sound pressure p and the particle velocity vector v as with P = 1 2 p T C ff v (4) C ff = Φ T f (x)φ f(x)dγ(x). In general, the matrix C ff is known as the boundary mass matrix; for the special case of constant elements, it is a diagonal matrix containing the element areas S µ as C ff = diag(s µ ) with S µ = dγ µ (x). (5) Consequently, the sound power can be computed by summing up over all N e elements as P = 1 2 N e µ=1 S µ p µ v µ, (6) wherein p µ and v µ represent the constant sound pressure and particle velocity values of the element µ.this summation approach can be used to estimate panel contribution on the overall radiating surface. Piecewise constant elements show excellent performance for BEM [11, 33, 34] and multilevel fast multipole analysis as discussed in the papers [16, 17, 18]. It is well known that simple boundary element solutions in the external domain suffer from the so called non uniqueness problem, also known as the problem of irregular or fictitious frequencies. There are many ways of solving this problem. Herein, we use the method of Burton and Miller [35] which is investigated in detail including investigation of the performance of constant elements in [36]. In general, the acoustic field solution provides an equation for the sound pressure. In the case of the boundary element method, the sound pressure vector p of the boundary of the acoustic domain can be formulated as p = Zv. (7) The matrix Z denotes the impedance condition of the boundary. It represents a complex nonsymmetric matrix. 834
3 ACTA ACUSTICA UNITED WITH ACUSTICA If (7) is applied into (4), itisworth to mention that the radiated sound power P = 1 2 v T R ZT R C ffv R + v T I ZT R C ffv I (8) requires the real part of the impedance Z R only. The subscripts () R and () I denote the real and imaginary part, respectively Equivalent radiated power (ERP) A very common approach for the sound pressure can be found in the local relation p f c f v, (9) where f and c f represent the fluid smass density and the speed of sound, respectively. The relation between sound pressure and particle velocity is reduced to the fluid s characteristic impedance Z 0 = f c f.this is atypical approximation for high frequencies and for the far field. The acoustic field evaluation is avoided, so that the sound power is finally approximated by P ERP = 1 2 fc f v T C ff v. (10) This is completely equivalent to the squared velocity value integrated overthe radiating surface P ERP = 1 2 fc f v(x) 2 dγ(x). (11) If the surface is discretized by piecewise constant elements, we gain P ERP = 1 2 N e fc f S µ v µ vµ. (12) µ=1 The ERP does not contain any local acoustic effect since all sources (herein: all elements) have the same radiation efficiency of σ=1. Thus, the ERP will usually overestimate the radiation, but will give a qualitatively good approximation for structure induced acoustical fields especially as an upper bound. However: Note that the radiation efficiency can exceed the value of one. It should be mentioned that the sound power estimation by ERP is very similar to the high frequency solution for scattering, i.e. the Plane Wave Approximation [37]. Both approaches assume the radiation efficiencytobeσ=1and perform best for convex rigid bodies and high frequencies Lumped parameter model (LPM) The lumped parameter model presented by Fahnline and Koopmann [1, 25, 26] is based on an approximation of the Rayleigh integral with p(x) = ik f c f G(x, y)v(y)dγ(y) (13) G(x, y) = 1 2π x y e ik x y. The Green s function G(x, y) represents the fundamental solution of the Helmholtz equation for the three dimensional half space. The wavenumber k is the quotient of the circular frequencyand the speed of sound k = ω/c. The Rayleigh integral is gained as asimplification of the sound pressure representation formula known from integral methods. It is exact for plain radiating surfaces which are embedded in arigid baffle. The idea behind the lumped parameter model is to develop the Green s function as ataylor series for the source at x µ and the receiveraty ν G(x, y) = G(x µ,y ν )+(x x µ ) G(x, y ν) x x=xµ + (y y ν ) G(x µ,y) +... (14) y y=yν This can be understood as a multipole expansion. If we use piecewise constant interpolation again and include only the monopole (the first)term of equation (14) as G(x, y) G(x µ,y ν ) = G µν,wecan formulate the impedance matrix as Z = (z µν ) with z µν = ik f c f G µν S ν. (15) Finally,the sound power approximation via the lumped parameter model yields P LP M = 1 2 k N e N e fc f S µ S ν G µν vµ vν with µ=1 ν=1 (16) sin(k x y ) G µν =. (17) 2π x y Herein, the interaction of the radiating sources is weighted with the imaginary part of the Green s function. Note that the imaginary part of G µν remains finite (and negative) for x y 0 rather than the Green s function itself. Further, it can be seen that the interaction of two sources has no effect on the sound power if their distance is amultiple of half an acoustical wavelength. The double summation turns out to be computationally much more expensive than the single sum for the equivalent radiated power but much more efficient than the BEM solution of the full boundary value problem even when using the MLFMA. The accuracy of this approach significantly depends on the compliance with the Rayleigh integral assumptions and on the mesh refinement. 835
4 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 2. Floor panel: sound power in terms of frequency. 3. Radiating floor panel Figure 1. Floor panel: original finite element model (upper) and boundary element half space model (lower) Volume velocity The volume velocity u is defined as the integral of the particle velocity over the radiating surface [1] N e u = v dγ = v µ S µ. (18) µ=1 Avolume velocity based radiated sound power P VV is formulated as P VV = k2 f c f 4π uu. (19) The authors are not aware of areference about sound power estimation based on volume velocity. However, since uu can be rewritten as N e N e uu = v µ vν Sµ S ν, (20) µ=1ν=1 this sound power estimate can be understood as areduction of P LP M,cf. Equation (16) with the weighting by the Green s function G µν and afactor of k/2π. Note that this factor is equal 1for µ = ν,i.e. x y,cf. [1]. The sound power estimation based on volume velocity contains local acoustic effects based on antiphase vibration of the sources (dipole effects) but requires the evaluation of a single sum only, ifequation (19) is solved on the basis of (18). The floor panel of asedan s bodywork accounts for the first example. The original FE model of this structural component is shown in the upper part of Figure 1. In order to compute a coupled FE BE solution, we need to enhance this model. The MLFMA realization of the boundary element solution requires a closed fluid surface, so that we introduce auxiliary elements creating aclosed obstacle with the half space plane, i.e. the half space plane is used as a symmetry plane. The structure consisting of multi layered sheets of steel is reduced to its outer surface. The final fluid surface model can be found in the lower part of Figure 1. The radiating elements of the FE structural model match with the elements of the fluid model so that acoupling is easily generated. The auxiliary elements are assumed to have zero particle velocity. The FE structural model s surface consists of nodes with adof of 6each and linear continuous shell elements. In the simulation the entire structural model is solved but only the elements on its surface are taken into account for the subsequent coupled acoustical simulation. The fluid boundary element model contains elements and nodes. Since constant elements are preferred for the BE solution, we end up with unknowns for the half space solution. The vibration of the model is excited by two single harmonic point forces at the rear part of the floor panel. The sound power solutions which are gained by the previously explained methods are computed over a frequency range of 10 to 200 Hz. They are shown in Figure 2. As expected, the ERP overestimates the actual radiated sound power. Moreover, it contains more peaks than the other curves because acoustic short circuit effects are not considered in this function. As one example, the peak at 55 Hz in the LPM or PVV functions is not as much developed as in the ERP function. The structural mode shape belonging to this peak defines atorsional vibration, i.e. quadrupole so 836
5 ACTA ACUSTICA UNITED WITH ACUSTICA lution. Such amode shape is well known to have lowradiation efficiency. The LPM and PVV solutions give higher values than the BEM reference solution in the frequency range up to approximately 80 Hz. Above this frequency they seem to normally underestimate the sound power. This is supposedly due to the Rayleigh integral assumption, which is quite contrary to the tub like shape of the floor panel. If we compare the LPM and the PVV functions, they show high discrepancy in the frequency range from 130 Hz to 180 Hz. The interaction between the discrete sources on the surface (i.e. the elements), which is only included in the LPM value, affects the sound power significantly in this frequency range. The error for LPM, PVV and ERP compared to the BEM solution is plotted in Figure 3. If the BE solution is understood as the reference solution, the lumped parameter model becomes highly interesting due to its efficient computation and good agreement. One has to admit and consequently be aware of the fact, that there is no error estimate for the applied plain radiator assumption yet. However, the ERP and PVV can be computed even faster, but the implemented approximative assumptions can lead to questionable results. It is suggested to use both values (P VV and P ERP )incombination to qualitatively estimate the radiated sound power,in particular to identify resonance peaks and to distinguish between resonances with lowand with high radiation efficiency Δ P in [db] PERP PBEM PVV PBEM PLP M PBEM f in Hz Figure 3. Floor panel: absolute error of sound power estimates in terms of frequency. 4. Radiating diesel engine The second example investigates the sound radiation of a diesel engine. Herein, the sound power solutions of the BEM, the lumped parameter model and the ERP approximation are compared. Again, the finemesh of the structure is directly used for the fluid. The fluid surface model contains nodes and constant elements. The problem is solvefor the frequency range up to 3000 Hz. The model is presented in Figure 4. In what follows, we consider one case of realistic excitation and further cases of artificial excitations using the entire engine as an elementary radiator, i.e. monopole or dipole source. The excitation of the acoustic field is applied by defining the particle velocity over the surface at each investigated frequency Realistic excitation The particle velocity distribution over the engine s surface for a certain operations condition was computed and provided by the AVL/ACC Graz (Austria). In Figure 4, the geometry mesh of the surface is shown. Originally, the particle velocity was given on the mesh of linear continuous elements. The piecewise constant particle velocity data which is used for our simulations can be understood as an average of the normal velocity on each element. To provide the reader with avivid impression and a comparison of the level distributions of the particle velocity, the sound pressure and the sound intensity, these data Figure 4. Diesel engine: boundary element model. are visualized for two specified frequencies, i.e. 503 Hz, cf. Figure 5, and 2196 Hz, cf. Figure 6. It can be realized at the lower frequency of503 Hz that the intensity contribution does not match with the velocity contribution. At 2196 Hz, the contributions show less differences. This means that the intensity is dominated by the velocity contribution. The same effect is revealed by the sound power spectrum in Figure 7. There, the ERP values agree with the BEM solution better and better the higher the frequencies are. The differences between the ERP and LPM solutions with respect to the BEM reference are shown in Figure 8. Actually, it is quite surprising that the simple approximations of ERP and LPM catch the behaviour of the reference solution with this accuracy, not only in the high frequency range. Even in the lower frequencyrange, the difference of approximate solutions and reference are a couple of decibel. However, peaks and valleys in the curves are found at the same frequencies. 837
6 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 5. Diesel engine: surface distribution of particle velocity, sound pressure and sound intensity at 503 Hz Engine as an elementary radiator According to the Rayleigh integral assumptions the engine can not be understood as a plain radiator. Although we can simply model the engine as a cuboid by six half space plains it is questionable to explain the good agreement between the LPM and the BE solutions. Therefore, two simple cases of the engine s surface vibrations are investigated. Firstly, we exercise a harmonically pulsating vibration of the engine, i.e. a constant unit particle velocity is applied to all elements. This is similar to a monopole radiation. The resulting sound power levels are shown in Figure 9. The second case investigates the harmonic rigid body translation of the engine which can be interpreted as 838 a dipole source. Furthermore, we divide this case into two subcases which comprise two directions of motion. These are the two horizontal directions x and y, cf. Figure 4. We assume a unit vibration amplitude of the rigid body motion. Figures 10 and 11 show the resulting sound power levels. In all cases, the ERP estimation yields a constant sound power level. It provides a good approximation for higher frequencies. At low frequencies the ERP approximate is not able to represent the acoustic elimination eﬀects. However, the LPM fails to give a good estimation of the radiated sound power for the entire frequency range. This is caused by the incorrect calculation of the distances Rµν between the source and receiver elements. Again, this is
7 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 6. Diesel engine: surface distribution of particle velocity, sound pressure and sound intensity at 2196 Hz. due to the violation of the Rayleigh integral presumption. For concave ﬂuid boundaries such as for the engine model, this problem can be avoided by the visibility test. This test checks the visibility between source and receiver point. Unfortunately, the test becomes quite expensive for large scale models [31]. (It is only required once, though.) When carrying out the visibility check, only the elements which have an unobstructed connection to each other are considered for the summations. This means, that each of the six sides of the cuboid like engine is separately investigated and then summed up to the overall radiated sound power value. Alternatively, an eﬀective distance could be calculated via the periphery of the engine. This means, that the distance of the sound wave around the obstacle is taken into account. However, it is assumed that such an additional algorithm will be computationally ineﬃcient and, thus, destroying the eﬃciency impact of the LPM compared to the MLFMA. 5. Interpretation of the results According to piecewise constant elements of the radiating surface, the previously presented sound power approximations can be reformulated in the following way, cf. [29] P = Ne $ Ne $ µ=1 ν=1 Pµν = Ne $ µ=1 Pµµ + 2 N e 1 $ Ne $ Pµν. (21) µ=1 ν=µ+1 839
8 ACTA ACUSTICA UNITED WITH ACUSTICA Figure 7. Diesel engine: sound power in terms of frequency for realistic loadcase. Figure 8. Diesel engine: sound power difference between LPM / ERP and reference solution in terms of frequency. Figure 9. Diesel engine as a monopole radiator: sound power in terms of frequency. Figure 10. Diesel engine vibrating as a rigid body in x direction: sound power in terms of frequency. This means that each of the N e constant elements acts as an acoustical monopole source (i.e. apiston) providing the partial sound power, i.e. the sound power por Figure 11. Diesel engine vibrating as arigid body in y direction: sound power in terms of frequency. tion P µν.the portions can be distinguished into independently radiating sources P µµ and the interaction between the sources P µν (µ = ν). Due to reciprocity, the interaction matrix with entries P µν is symmetric. In general, each portion P µν can be determined by P µν = 1 2 fc f S µ σ µν v µ v ν, (22) where σ µν represents the radiation efficiency ofthe portion. Following the derivation of the presented methods, the radiation efficiency can be written as σ µν = z µν Sν general, f c f S µ σ µν = δ µν for ERP, σ µν = k2 S ν sin(kr µν ) for LPM and 2π kr µν σ µν = k2 S ν 2π for volume velocity. (23) Note that σ µν is indeed dimensionless but generally non symmetric in all formulations. It can be transformed into asymmetric form σ µν by multiplying with S µ as σ µν = S µ σ µν for all formulations. The ERP neglects all the interaction between the sources but each source portion P µ µ has aradiation efficiency of 1.In the method using the volume velocity (PVV), all interactions are considered but have aconstant frequency dependent radiation efficiency. The LPM radiation efficiencyadditionally contains information about the distance between the single sources. Thus, the PVV will give questionable results for higher frequencies, since the interaction efficiencywill vanish much faster with the distance between the sources than modeled by the method. Despite of this, the PVV will provide a good approximation of the average sound power value in the lower frequency range, what can not be interpreted from the ERP function. The partial radiation efficiency is illustrated in Figure 12 schematically. Another important fact can be found in the dependency of the approximative methods on the used mesh refinement. The equivalent radiated power and the sound power value provided by the volume velocity are independent on the element size, if the refinement provides no additional 840
9 ACTA ACUSTICA UNITED WITH ACUSTICA Table I. Overview onthe three sound power approximations. Method estimated time mesh refinement frequencyrange ERP O(N e ) no effect no prediction available PVV O(N e ) no effect low frequencies LPM O(Ne 2 ) convergence for finer mesh low/mid frequency range ERP PVV LPM σ σ µµ =1 σ σ µ ν σ µµ = k 2 S ν 2π R µν µ ν σ µµ = k 2 S ν 2π R µν µ ν Figure 12. Radiation efficiencyofthe three approximation methods (schematic visualization). information but subdivision of elemental data. This means if the genuine mesh is refined by subdividing the elements, i.e. no additional information for velocity contribution is created, the ERP and the PVV will each provide the results of the coarser mesh again. The LPM depends on the mesh refinement since it shows convergence behavior for finer meshes in this case. The radiation efficiency, cf. Figure 12, in terms of R µν can be approximated more accurately for finer meshes. If the sound power is estimated from avery coarse mesh, the ERP or PVV solution can provide sufficient results. R µν R R R on the structural vibration. Consequently, the coupling between the structure and the fluid is modeled unidirectionally. To determine the sound power, the velocity distribution on the structure s surface acts as input information of the acoustic field. The particle velocity is approximated by piecewise constant interpolation. Two ofthe methods, the ERP and the PVV, require evaluation of asingle summation of order O(N e )ofthe velocity distribution over the N e constant elements only. Since therefore these methods are computationally very fast, theycan be used in combination to estimate the sound radiation in the lower frequency mainly in aqualitative manner. The lumped parameter model contains a double summation of order O(N 2 e )overall N e elements since the interaction between the discretized piston sources is considered. This model is based on the Rayleigh integral but even may provide acceptable results if the Rayleigh integral assumption is violated. The LPM will mostly fail if the structural vibration contains rigid body motion or if no phase information is included in the velocity distribution. The general statements about the estimated computation time, the dependency on mesh refinement and the recommended frequencyrange of the approximation methods are compiled in Table I. It was shown in this article that these approximation methods can be successfully used for realistic problems. An expensive and highly detailed boundary element computation wascarried out to provide areference solution for the investigated examples. Acknowledgments These investigations were extracted from the research project P 579. The research project P 579 " Minimum sound emission of steel plates" was carried out by the Institut für Festkörpermechanik with technical and scientific support by the FOSTA Research Association for Steel Application, Düsseldorf (Germany), with funds of the Stiftung Stahlanwendungsforschung, Essen (Germany). Wefurther thank the Audi AG Ingolstadt (Germany) and AVL/ACC in Graz (Austria). The computation was run on the SGI Orig0 at the Zentrum für Hochleistungsrechnen of the Technische Universität Dresden. 6. Conclusions The presented sound power approximations have in common that the acoustic field is not solved to estimate the radiated sound power of astructural component. It was presumed in these methods that the acoustic field has no effect References [1] G. H. Koopmann, J. B. Fahnline (eds.): Designing quiet structures: A sound power minimization approach. Academic Press, San Diego, London, [2] S. Marburg: Developments in structural acoustic optimization for passive noise control. Archives of Computational 841
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