Barrera Figueroa, Salvador

Cutanda Henriquez, Vicente

Juhl, Peter Møller

Many transducers (e.g. microphones) consist of a moving part facing the exterior on one side and facing a closed cavity on the other. In order to obtain and solve a model of such transducers, a fully coupled model can be used and the performance of this model will rely of the performance of the sub-models being coupled. Previous work has revealed that the performance of an interior lossless BEM formulation concerning a cavity with rigid walls can pose a limit on the accuracy of a fully coupled model of the transducer – in particular when including viscous and thermal losses in the cavity. This paper investigates the accuracy of a lossless acoustic BEM formulation for cavities with rigid walls at low frequencies and addresses possible solutions for restoring accuracy.<br/

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Downloaded from orbit.dtu.dk on: Sep 17, 2019On The Modelling Of Cavities At Low Frequencies With The BEMJuhl, Peter Møller; Cutanda Henriquez, Vicente; Barrera Figueroa, SalvadorPublished in:Proceedings of internoise 2019Publication date:2019Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Juhl, P. M., Cutanda Henriquez, V., & Barrera Figueroa, S. (2019). On The Modelling Of Cavities At LowFrequencies With The BEM. In Proceedings of internoise 2019 [2070] International Institute of Noise ControlEngineering.On The Modelling Of Cavities At Low Frequencies With TheBEMJuhl, Peter Møller1University of Southern DenmarkCampusvej 55, DK-5230 Odense M, DenmarkCutanda Heníquez, Vicente2Technical University of DenmarkBuilding 352, DK-2800 Kgs. Lyngby, DenmarkBarrea Figueroa, Salvador3Danish Fundamental Metrology A/SKogle Allé 5, DK-2970 Hørsholm, DenmarkABSTRACTMany transducers (e.g. microphones) consist of a moving part facing the exterioron one side and facing a closed cavity on the other. In order to obtain and solve amodel of such transducers, a fully coupled model can be used and the performanceof this model will rely of the performance of the sub-models being coupled. Previouswork has revealed that the performance of an interior lossless BEM formulationconcerning a cavity with rigid walls can pose a limit on the accuracy of a fullycoupled model of the transducer – in particular when including viscous and thermallosses in the cavity. This paper investigates the accuracy of a lossless acoustic BEMformulation for cavities with rigid walls at low frequencies and addresses possiblesolutions for restoring accuracy.Keywords: Numerical Acoustics, Low Frequencies, AccuracyI-INCE Classification of Subject Number: 71, 761. INTRODUCTIONThe Boundary Element Method (BEM) has been a popular and valuable tool forcalculation models of acoustic transducers like microphones. Initially, a major reason forits use was the easy handling of the infinite domains encountered in e.g. the calculation1pmjuhl@mci.sdu.dk2vcuhe@elektro.dtu.dk3sbf@dfm.dkof the free-field response [1]. However, more accurate models are required for veryaccurate calculations and for the design of transducers, and microphone models are nowbeing developed based on a full coupling of the exterior sound field, the diaphragmand the interior cavity for which viscous and thermal losses are taken into account [2].The end goal of this research is a unified full model including all relevant effects in thefull frequency range. As discussed in Ref. [2] the Finite Element Method (FEM) hasalso proven capable of handling viscous and thermal losses in acoustic models, but thehandling of large and very complex geometries is more of an issue with the FEM thanthe BEM.The problem addressed in this paper is the accuracy at very low frequencies. It iswell known that dealing with low frequencies in hard-walled cavities is a challenging tasksince a ’breathing’-type component in the excitation leads to very high sound pressuresin hard-walled cavities [3, 4]. Even though this behavior is entirely physically based,it poses challenges for the numerical implementation of problems involving cavities withrigid walls, since the solution becomes very sensitive to any imprecisions in the excitation.Therefore, very high precision of all parts of a coupled formulation of e.g. the microphoneproblem is required in contrast with the intuitive reasoning, that since the pressure insidethe cavity is almost constant with respect to the spatial coordinates, very few degrees offreedom are required to model the sound field in the cavity.An axisymmetric BEM implementation using the direct method and collocation [5, 6]is used to investigate the ’cavity problem’ in setups in which viscous and thermal lossesand coupling to the diaphragm and the external sound field are excluded in order to focusthe analysis.2. THE DIRECT BOUNDARY ELEMENT METHODThe BEM approach to acoustic radiation and scattering problems is based on theHelmholtz Integral Equation that relates the pressure p(Q) and normal velocity vn(Q)on the surface of a body of any shape with the pressure at any point p(P) and the pressureof an incoming wave pinc(P). The harmonic time dependence ejωt is omitted, givingC(P)p(P) =∫S(∂G∂np(Q) + jkρcvn(Q)G)dS + 4pipinc(P), (1)where S is the surface of the body, Q a point on that surface and P any exterior or interiorpoint. The normal vector n is directed into the computational domain. The factor C(P)is the geometrical constant and represents the solid angle at P. The Green’s function for3-D free space isG(R) =e−jkRR, R = |P − Q|. (2)For fully axisymmetric problems a change to polar coordinates (r, θ) allows for therewriting of the surface integral to an integral along the generator L of the domain and arotational integral [6],C(P)p(P) =∫L(p(Q)∫ 2pi0∂G∂ndθ + jkρcvn(Q)∫ 2pi0Gdθ)r(Q) dL + 4pipinc(P), (3)in which the Jacobian r(Q) is introduced and use of the axisymmetry of p(Q) and vn(Q)has been made.Standard collocation results in a matrix equation [5],0 = Ap + jkρcBvn + 4pipinc, (4)where p is a vector of pressures at the surface nodes, vn is a vector of the normal velocityat surface nodes and pinc is a vector of incoming pressure at surface nodes.Setting up a calculation in BEM by placing a monopole of volume velocity Qm in thegeometrical center of an enclosure with rigid walls leads to solving the following equationAp = −4pipinc, (5)where pinc contains the nodal values ofpinc = jωρQmejkR4piR, (6)where R is the distance between the point source and the node. It is evident that themagnitude of the incoming sound field is directly proportional to ω.In the limit of low frequencies and assuming no losses the sound pressure in theenclosure becomes uniform with a magnitude given by the volume velocity, Qm, the cyclicfrequency ω and the volume of the enclosure V [7],p =ρc2QmjωV. (7)Hence, the sound pressure increases at low frequencies as 1/ω.In order for the solution of Equation (5) to obtain the 1/ω behavior in the limit of lowfrequencies with a right-hand side increasing as ω, the condition number of A must showa 1/ω2 behavior at least.Figure 1 shows the magnitude of the analytical and the BEM calculated sound pressureas well as the condition number as functions of the dimensionless frequency ka for a rigidsphere of radius a (left part of the figure) and for a circular cylinder of radius a and height2a (right part of the figure). The analytical solution is the low frequency approximationof Equation (7) and the excitation is a point source with unit volume velocity placed inthe centre of the cavity. The analytical and the BEM solution coincides in the plot andboth the 1/ω behavior of the solution and the 1/ω2 behavior of the condition number isconfirmed.3. THE SINGULAR VALUE DECOMPOSITIONThe singular value decomposition (SVD) is used to inspect the properties of the Amatrix. The SVD decomposes A into a product of three matrices:A = USV (8)in which U and V are unitary matrices: UUH = I and VVH = I where the superscript Hdenote conjugate transpose. Matrix S is a diagonal matrix, with its diagonal values σiiarranged in descending order. The condition number of A is the ratio of the largest (first)to the smallest (last) singular value. Figure 2 shows the singular values as a function ofdimensionless wavenumber and their index for the sphere (left) and the cylinder (right).It is evident that the increase of the condition number observed in Figure 1 is coursed byFigure 1: Sound pressure amplitude and condition number as function of dimensionlessfrequency ka for a cavity excited by a point source of unit volume velocity. Left: sphericalcavity of radius a; right: cylindrical cavity of radius a and height 2a.a decrease of the smallest singular value only - all other singular values are practicallyconstant in the range of frequencies observed. Therefore, the increase of the pressure atlow frequencies is directly linked to a decrease of the smallest singular value.In the limit of low frequencies, the oscillating part of the kernel in Equation (3)vanishes and the matrices in Equation (4) become real. Theoretically, the matrix Abecomes singular as the smallest singular value becomes zero. It follows that thecorresponding last column in V becomes the eigenvector corresponding to the eigenvaluezero. For a hard-walled cavity, it is well-known that this eigenvector correspond to aconstant pressure, and that this eigenvector is orthogonal to all other eigenvectors of thesystem. Inspection of the last column of matrix V shows that this eigenvector does notchange rapidly with frequency at low frequencies, which mean that the contribution ofthe cavity mode to the total solution is largely determined by the last column in V.Since, matrices U and V are unitary their inverse equal their complex transpose, andonce the SVD of A has been obtained, the solution of the system of equations Ax = b isreadily found:x = VHS−1UHb (9)where S−1 is diagonal matrix containing the reciprocal of the singular values, 1/σii. Thevector UHb is termed the solution coefficients, and due to the properties of S−1, wherethe last entry is several magnitudes above all other entries, the last entry of UHb becomesimportant for the behavior of the system at low frequencies.Figure 3 shows the singular value, the solution coefficients and the resulting erroras functions of the dimensionless wavenumber for two test cases where analyticalsolutions are available: a pulsating and an oscillating sphere. Two calculations areshown using the axisymmetric formulation with nine nodes using linear and quadraticelements respectively. It is worth to note, that the A matrix is the same independent ofthe excitation, and that the behavior of the linear and the quadratic formulation is thesame, with the exception that the resulting error is more than one magnitude smaller forFigure 2: Singular values as function of dimensionless frequency ka for the BEMcoefficient matrix A. Left: spherical cavity of radius a; right: cylindrical cavity of radiusa and height 2a.the quadratic formulation, since the geometry of the sphere is much better representedwith quadratic elements. Note, that the error for the pulsating sphere is at a consistentlow level for both formulations whereas the error of the oscillating sphere increase tomore than 100 percent at low frequencies.In the limit of low frequencies, the solution coefficient of the pulsating sphere tendsto a finite value several orders of magnitude above the last singular value. This resultsin the solution being by far dominated by the corresponding cavity mode. On the otherhand, the oscillating sphere does not in theory excite the cavity mode (since the volumevelocity of this vibration should integrate to zero). Indeed the solution coefficient is quitesmall and decreases with decreasing frequency but not as fast as the last singular value.Hence, for very low frequencies, the solution of the oscillating sphere is ’polluted’ withthe cavity mode leading to the errors shown. Note, that due to the very high conditionnumber at low frequencies, corresponding to very small values of the last singular value,this effect is important even at very small values of the solution coefficient. Therefore, thecalculation is sensitive to even very small errors in the calculation of the matrices - anynumerical imprecision can potentially lead to very large errors.4. HANDLING THE CAVITY MODE WITH THE SVDThe observations that the cavity mode is connected to the last singular value and thatit is orthogonal to all other modes in the cavity may lead to a possible remedy of the lowfrequency difficulty. For the oscillating sphere, the cavity mode should not be present atall. Having the SVD of matrix A allows for removing the cavity mode simply by puttingthe reciprocal of the corresponding singular value to zero: 1/σMM = 0 before calculatingthe solution with Equation (9). Figure 4 shows the analytical and numerical (with andFigure 3: Singular values and solution coefficients of a spherical cavity with pulsatingor oscillating walls as function of dimensionless frequency ka. -*-, Singular values; -o-,solution coefficients for the pulsating sphere; -x-, solution coefficients for the oscillatingsphere; -+-, relative error for the pulsating sphere; --, relative error for the oscillatingsphere. Left: Linear elements, 9 nodes and 8 elements; right: Quadratic elements, 9nodes and 4 elementswithout the remedy) solution on the interior surface of the sphere at the frequency ka =10−6 as well as the relative error as a function of the frequency. It is evident that theoriginal solution is ’polluted’ with an amount of the cavity mode and that subtracting thismode lead to a good solution.The final test case is a step towards a more realistic senario for calculating thesensitivity of a condenser microphone. A cylindrical cavity with rigid walls of radiusa = 10 mm and a height of 20 µm is considered. Such a geometry is representative fora one inch microphone and besides the small volume it also gives rise to the problemof close surfaces, which has to be carefully dealt with [8]. The cavity is excited bythe movement of one of the plane end-faces of the representing the diaphragm. Themovement is given by,un(r) = u0(J0(z0ar)− z1), (10)in which u0 is an arbitrary amplitude, z0 ≈ 3.83 is the first zero of the derivative of thecylindrical Bessel function J0 and z1 ≈ −0.40 is the value of J0 at z0. In other words: theprescribed velocity follows the movement of a diaphragm so that the displacement andits derivative with respect to the radial direction is zero at the rim. The test case has beenselected so that an analytical solution in terms of two modes can easily be calculated, andthe BEM calculations are made using linear elements with mesh sizes from 7 to 259 nodesat a frequency of 54 Hz corresponding to k = 1 m−1.Having a discretized geometry and a prescribed normal velocity, it is a simple matterto calculate the volume and the volume velocity injected in the cavity. Thereby, the soundpressure due to the cavity mode is easily calculated by Equation (7). Figure 5 showsthe error as a function of the mesh fineness (in terms of nodes) for the original BEMFigure 4: Solution of an oscillating sphere at low frequencies. Left: Magnitude of ofthe sound pressure as a function of the angle at ka = 10−6; -*-, analytical solution; -o-, original numerical solution; -+-, numerical solution with the cavity mode removed.Right: relative error as a function of dimensionless frequency; --, original solution;-^-, solution with the cavity mode removed. Quadratic elements, 9 nodes and 4 elementsformulation and for the remedy suggested, in which the cavity mode is removed from theBEM solution as described above, and then added back by the expression in Equation (7).It is evident that the proposed strategy is much more numerically stable and that is restoresthe convergence of the linear BEM formulation.5. DISCUSSION AND CONCLUSIONThe accuracy of the Boundary Element Method at low frequencies in lossless cavitieshas been addressed with an axisymmetric BEM implementation using linear and quadraticelements. Using the Singular Value Decomposition on the BEM coefficient matrix, it hasbeen shown that the physical behavior of the sound pressure - i.e. the increase of soundpressure at low frequencies due to the ’cavity mode’ - is reflected in a rapid decrease ofthe last singular value. The corresponding right singular vector is also the eigenvector- i.e. the discretized representation of the cavity mode. Even for problems where theretheoretically is no ’breathing’ component in the excitation - and therefore no theoreticalcomponent of the cavity mode - the strong ill-conditioning of the BEM coefficient matrixcourses very small imprecisions to ’excite’ the cavity mode, and thereby corrupt thesolution.In the problems studied here, the interior BEM domain is not coupled to other domainsor a model of the diaphragm. Therefore, errors are only found at very low frequencies.In coupled problems the level of numerical approximation and imprecision is expected tobe larger, and it can be expected that the low frequency problem will be more severe andbe found at higher frequencies. On the other hand it is worth to note, that the problem isnot a numerical breakdown, in the sense that the behavior observed is physically based.Figure 5: Relative error as function of the number of nodes when solving for the interiorsound pressure of a cylindrical cavity of radius a = 10 mm and height h = 20 µm withone end-face vibrating. -*-, using the original formulation; -o-, substracting and addingback the cavity mode. The frequency is about 54 Hz corresponding to a wavenumber ofk = 1m−1Therefore, any numerical model or method that deals with calculation of lossless cavitiesat low frequencies, can potentially suffer of similar issues.A possible remedy is to subtract the cavity mode by removing it in the SVD beforecalculating the matrix inverse, and then to add the component back calculated analyticallyin the limit of low frequencies. In the problems considered in this paper, this has beensuccessful in a range of frequencies below the first non-trivial mode, but it should be notedthat this strategy is not expected to be successful in the full range of frequencies.6. REFERENCES[1] Peter M. Juhl. A numerical investigation of standard condenser microphones. Journalof Sound and Vibration, 177(4):433 – 446, 1994.[2] Vicente Cutanda-Henríquez and Peter Møller Juhl. An axisymmetric boundaryelement formulation of sound wave propagation in fluids including viscous andthermal losses. The Journal of the Acoustical Society of America, 134(5):3409–3418,2013.[3] K.A. Hussain and K.S. Peat. Boundary element analysis of low frequency cavityacoustical problems. Journal of Sound and Vibration, 169(2):197 – 209, 1994.[4] Sergey Soroki and Søren Tørholm Christensen. Low-frequency breakdown ofboundary element formulation for closed cavities in excitation conditions with a‘breathing’-type component. Communications in Numerical Methods in Engineering,16:325 – 334, 2000.[5] Vicente Cutanda Henriquez and Peter Møller Juhl. Openbem - an open sourceboundary element method software in acoustics. In INTER-NOISE and NOISE-CONCongress and Conference Proceedings, volume 2010, pages 5873–5882. Institute ofNoise Control Engineering, 2010.[6] Peter M. Juhl. An axisymmetric integral equation formulation for free space non-axisymmetric radiation and scattering of a known incident wave. Journal of Soundand Vibration, 163(3):397 – 406, 1993.[7] Finn Jacobsen and Peter Møller Juhl. Fundamentals of General Linear Acoustics.John Wiley & Sons, New York, 2013.[8] Vicente Cutanda Henriquez, Peter Møller Juhl, and Finn Jacobsen. On the modelingof narrow gaps using the standard boundary element method. The Journal of theAcoustical Society of America, 109(4):1296–1303, 2001.

On The Modelling Of Cavities At Low Frequencies With The BEM

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On The Modelling Of Cavities At Low Frequencies With The BEM

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