Proceedings of Meetings on Acoustics

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1 Proeedings of Meetings on Aoustis Volume 19, ICA 2013 Montreal Montreal, Canada 2-7 June 2013 Arhitetural Aoustis Session 4aAAa: Room Aoustis Computer Simulation I 4aAAa8. Time-domain formulation of an edge soure integral equation Peter Svensson* and Andreas Asheim *Corresponding author's address: Eletronis and Teleommuniations, Norwegian University oiene and Tehnology, O.S. Bragstads pl. 2B, Trondheim, NO-7491, -, Norway, svensson@iet.ntnu.no In omputer simulations of sound in enlosures, diffration omponents an be added to geometrial aoustis ones for inreased auray. A omputational problem with diffration is the large number of higher-order terms that is generated. A reent frequeny-domain edge soure integral equation ESIE effiiently handles the sum of all higher-order diffration for rigid, external sattering objets, while omputing firstorder diffration separately. Here, a time-domain formulation of the same ESIE is presented. An initial version handles higher-order diffration for separate sattering objets, suh as stage eiling refletors, and the extension to general geometries is outlined. With this approah, in a first step an inident transient sound field is omputed at disretized edge points, inluding the outgoing diretivity. In a seond step, the effets of diffration of arbitrarily high order is handled by solving the IE iteratively, yielding the omplete edge soure time signals. In a third step, the edge soure signals are propagated to reeiver points. Numerial issues will be disussed, inluding disretization strategies and how to handle shadow zone boundary singularities. Published by the Aoustial Soiety of Ameria through the Amerian Institute of Physis 2013 Aoustial Soiety of Ameria [DOI: / ] Reeived 23 Jan 2013; published 2 Jun 2013 Proeedings of Meetings on Aoustis, Vol. 19, Page 1

2 INTRODUCTION The omputation of sattering, and wave propagation in omplex geometries and for high frequenies, is often done with geometrial aoustis GA based methods. These methods have superior effiieny ompared to referene methods suh as the finite element or finite differenes method, or boundary element method. The auray of GA based methods is inreased by the addition of edge diffration omponents, and high auray has been demonstrated for rigid boundary onditions, both via omparisons with referene solutions and measurements. However, the addition of higher order diffration omponents is hallenging. Previous attempts to handle higher orders of diffration have employed asaded first-order diffration expression with a disretization of eah involved edge [1], [2], [3]. This implies a omputational load whih inreases exponentially with diffration order and it is ostly both beause of an inreasing number of diffration path ombinations, and beause of an inreasing ost per omponent. Reently, an edge soure integral equation ESIE was suggested as an alternative whih permits the omputation of arbitrarily high orders of diffration at little additional omputational ost. A frequeny-domain FD formulation was presented in [4], and in this paper, a orresponding time-domain formulation is presented. The FD formulation was presented for rigid, onvex sattering bodies, for whih exellent auray was found for all frequenies. It is still unertain how aurate the same approah is for non-onvex bodies, sine it has been pointed out that so-alled slope diffration needs to be handled for non-onvex geometries [5]. However, the inauray seems to be largest for low frequenies, and the worst-ase geometry is a hole in a thin sreen, where the ESIE inorretly predits zero higher-order diffration. Still, in this worst ase senario, the maximum inauray seems to be on the order of 2 db. Another open question is how to handle non-rigid boundary onditions for whih further work is needed. This paper fouses on sattering from bodies with rigid boundary onditions. THE EDGE SOURCE INTEGRAL EQUATION A frequeny-domain formulation of the edge soure integral equation ESIE has been presented earlier, in [4]. The relation between a time-domain TD and a frequeny-domain FD formulation has been demonstrated before for first-order diffration [6]. This relationship is quite straightforward, sine the involved frequeny dependene is of the form e jkr, whih translates diretly to a Dira pulse in the time-domain. Time-Domain Formulation of the Edge Soure Integral Equation The sound field will be modeled here as a time-domain formulation of a sum of geometrial aoustis GA omponents and edge diffration omponents, p total t = p diret t + p speular t + p D1 t + p Di t, 1 where subsript Di indiates diffration omponents of order i. The GA omponents inlude the diret sound and speular refletions, and all the omponents in Eq. 1 involve visibility fators, i.e., only when there is a free line of sight between a soure/reeiver and the refletion/diffration point on a surfae, or between suessive refletion/diffration points, will a omponent be inluded. A monopole is assumed as an ensonifying soure throughout. The GA omponents an be omputed with, e.g., the image soure method or beam traing, whereas the first-order diffration term is a sum of ontributions from edge portions that are visible to both the soure and the reeiver. In addition, first-order diffration might our in sequenes with preeding speular refletion, for whih the visibility for the image soure rather than the original soure i=2 Proeedings of Meetings on Aoustis, Vol. 19, Page 2

3 S r z,s ϕ S z θ W θ S θ R ϕ R rr,z R FIGURE 1: A single wedge. is to be used. Likewise, for sequenes of speular refletions after an edge diffration, the visibility for the image reeiver should be used. By letting S represent the original soure or an image soure, and R represent the reeiver or an image reeiver, this an be summarized as p D1 t = δ Γ t r R,z + r z,s VR,z V z,s ΩR, z, Sds z, 2 r R,z r z,s where the totality of edges in the polygonal model is represented by Γ, whih ould also be urved edges of thin diss. r z1,z 2 denotes the Eulidean distane between the points z 1 and z 2.It an be noted that a Dira pulse is assumed as a monopole soure signal whih implies that the quantity pt here is atually an impulse response. Furthermore, V a,b is a visibility fator whih is 1 if point b an be seen from point a, 0 if it an not be seen, 1/2 if the line from a to b exatly hits an edge, and a fration of 1 when this line hits a orner. Ω is a diretivity funtion for the path from soure S, via edge point z, to the reeiver R, given as ΩR, z, S = ν 4π where ν is the wedge index, ν = π/θ W, and the angles φ i are 4 sinνφ i oshνη osνφ i, 3 i=1 φ 1 = π + θ S + θ R, φ 2 = π θ S + θ R, φ 3 = π θ S θ R, φ 4 = π + θ S θ R. 4 The quantity η is given as η = osh 1 osϕ S osϕ R + 1, 5 sinϕ S sinϕ R The angles ϕ S and ϕ R refer to the angle between the edge or edge tangent and S and R, respetively, and the angles θ S, and θ R refer to the angle between the plane and S and R, respetively. This is illustrated in Figure 1. The expression for first-order diffration above have been given in many papers earlier, see e.g. [7], [1], albeit with a β diretivity funtion rather than an Ω as introdued here. Expressions for higher-order diffration omponents have also been presented earlier, [1], but here a reently Proeedings of Meetings on Aoustis, Vol. 19, Page 3

4 developed frequeny-domain integral equation formulation will be used as a basis for a time-domain expression. Consequently, the term i=2 p Dit in Eq. 1 is formulated as being aused by an effetive edge soure signal, qz 1, z 2, t whih aumulates all higher orders of diffration, p HOD t = p Di t = q z 1, z 2, t r R,z 1 + r z1,z 2 V R,z1 V z1,z 2 b z1,z Γ 1 Γ 2 2 ΩR, z 1, z 2 ds z2 ds z1, 6 r R,z1 r z1,z 2 i=2 where b z1,z 2 is 1/2 when the two edge points z 1 and z 2 are on the edges of the same polygonal fae, and 1 otherwise. The effetive edge soure signal, q, is found via a time-domain edge soure integral equation TD-ESIE, qz 1, z 2, t = q 0 z 1, z 2, t + Γ q z 2, z, t r z 2,z Vz2,z r z2,z Ωz 1, z 2, zds z, 7 where q 0 represents the omponent aused by first-order diffration from the original soure S. A more ompat version will be used for a matrix equation formulation, introduing a transfer funtion, hz 1, z 2, z, from point z, via edge point z 2, to point z 1, suh that qz 1, z 2, t = q 0 z 1, z 2, t + and this is the TD-ESIE to solve. hz 1, z 2, z = V z 2,z Ωz 1, z 2, z, 8 r z2,z Γ q z 2, z, t r z 2,z hz 1, z 2, zds z, 9 Solution Approah The Nyström method an be used for solving the integral equation. This simply means a spatial disretization of Eq. 9 following some quadrature sheme, qz 1, z 2, t q 0 z 1, z 2, t + w i q z 2, z i, t r z 2,z i hz 1, z 2, z i, 10 i where w i are weight funtions orresponding to the disrete sample points z i. It is assumed here that the geometry is desribed as a set of polyhedra and thin diss, whih inludes a set of edge segments, straight and urved, and eah segment will be sampled. The numbering will, however, be suh that i runs from 1 to the total number of sample points and subsets of i will belong to eah edge segment j. As the simplest example of disretization sheme, uniform sampling would imply that eah edge segment, j, of length l j would have n j sample points equally distributed, and then w i = l j /n j for the subset of sample points z i that belong to edge segment j. Superior onvergene will typially result from using Gauss-Legendre quadrature or even more sophistiated tehniques that employ the known singularity properties of the integrand. The ontinuous-time CT expressions above need to be onverted to disrete-time versions in order to formulate a matrix equation for the TD-ESIE. A standard approah is to let the disrete-time DT version, xn, of a quantity xt, be omputed as the integral of the CT version around eah sample instant, n+0.5 xn = xtdt, 11 n 0.5 where is the sampling frequeny. This integration is equivalent to onvolving the CT signal with a retangular window, of one sample s width, before sampling the signal, whih orresponds to filtering with an anti-alias filter of slope 6 db/otave, as given by the Proeedings of Meetings on Aoustis, Vol. 19, Page 4

5 sin funtion. This is a rude low-pass filter whih explains why quite high oversampling ratios might be needed for high auray. More sophistiated ways are possible, based on onvolving the CT signal with a more effiient anti-alias filter impulse response. Here, however, the simple filtering in Eq. 11 will be used throughout. Using an intermediate step where DT and CT signals are mixed, the disretized TD-ESIE an be written as n+0.5 qz 1, z 2, n q 0 z 1, z 2, n + n 0.5 i w i q z 2, z i, t r z 2,z i hz 1, z 2, z i dt = q 0 z 1, z 2, n + n+0.5 w i hz 1, z 2, z i q z n 0.5 2, z i, t r z 2,z i dt. 12 i The last integral integrates the CT signal q over a short time interval, but it is the DT signal q, and not the CT signal q, whih will be available. However, a CT signal an be reonstruted from its DT representation, as qt w r t n qn, n where w r is a reonstrution funtion. The ideal one whereby the approximation sign is replaed by an equals sign is the sin funtion but here, the simple boxar retangular window reonstrution funtion will be used instead, { 1, x <=0.5 w r x = 0, x >0.5, whih leads to that qt q t Therefore, the integral over a short time interval in Eq. 12 is where = a q n+0.5 n 0.5 q z 2, z i, t r z 2,z i z 2, z i, n 0.5 = a q dt n+0.5 n 0.5 q r z 2,zi a q z 2, z i, n r z2,z i a = a n r z2,z i + 1 a q n r z2,z i { 1 x x < 1 ax = 0 x >1. z 2, z i, t r z 2,z i z 2, z i, n z 2, z i, n r z2,z i +1 = a r z2,z i dt f Sr z2,z i, r z 2,zi + 0.5, 13 Aordingly the DT ESIE, on the Nyström form in Eq. 12, an be written as a onvolution sum qz 1, z 2, n q 0 z 1, z 2, n + i minn,n h,max w i hz 1, z 2, z i a n m f S r z2,z i q z 2, z i, n m. 14 m=0 The upper limit of the summation is the lowest of n and N h,max. The former is beause the system is assumed to be at rest before the time sample n = 0. The latter ondition is given by the longest possible delay within the system, between the two edge points that are the furthest apart. After introduing the disretized versions of all quantities, eah pair of edge sample points, z i 1, z j 2, that an see eah other will generate one unknown signal, qzi 1, z j 2, n, and all of these Proeedings of Meetings on Aoustis, Vol. 19, Page 5

6 qz i 1, z j 2, n an be staked in one large matrix q. Now, the TD-ESIE in Eq. 14 will need to be solved using a time-marhing approah. Thus, one olumn of q will have to be solved for at a time, starting with the first olumn orresponding to n = 0 et. One suh olumn will be denoted q n and then q n = q 0,n + h a 0 w q n + h a 1 w q n where denotes the Hadamard produt, i.e., an element-wise produt in Matlab.. Note that the matrix a 0 has non-zero values only for those edge-to-edge ombinations that reah eah other within the first sample slot. This is a very small fration of all the edge-to-edge ombinations and therefore the h a 0 matrix is a "sifted" version of h. In total, the h matrix is spread out aross a number of suh sifted versions, and eah non-zero value of h appears only in two of the sifted matries, beause Eq. 13 shows that eah edge-to-edge ombination is spread out aross only two time samples. Eq. 15 will ontain N h,max terms. One important detail relates to the sampling frequeny vs. the shortest distane between two edge points. The sampling frequeny an be hosen high enough that a 0 is zero, and then the time-marhing in Eq. 15 is straightforward. However, if a lower sampling frequeny is hosen, then the RH side in Eq. 15 will ontribute also to q n, suh that a rewriting into a matrix inversion form will be neessary. In the numerial examples below, high enough sampling frequenies are hosen in order to avoid the involved matrix inversion. In [4] it was pointed out that the FD ESIE ould be solved either diretly, via a matrix inversion, or using iteration whereby eah iteration step orresponded exatly to one diffration order. For the TD ESIE formulation presented here, however, the time-marhing approah will not separate diffration orders. Rather, the omplete solution will result, sample by sample. Sine the response will be infinitely long, a trunation at a suitably low amplitude will have to be employed. It an be noted that this time-marhing approah appears not to possess any instability tendenies for the tested sampling frequenies. Previous approahes to seond-order diffration, and beyond, have onstruted eah omplete path from soure to edge point possibly via speular refletions, via n th order diffration to another edge point, and finally to the reeiver possibly via yet another sequene of speular refletions. The ruial differene here is that the all paths are onstruted from the soure to an edge point possibly via speular refletions, but then stored at that point. For eah edge point, however, the amplitude in the diretion of eah visible other edge point is omputed. After that, all possible edge-to-edge transfers are omputed using iteration, ending with yet another set of amplitudes for eah edge point, in the diretion of eah visible other edge point. As a final step, these amplitudes, alled edge soure signals, are propagated to the reeiver points possibly via a sequene of speular refletions. This approah indiates that all the visible edge-to-edge paths must be known when the first soure term q 0 is omputed. Furthermore, the inlusion of speular refletions in-between edge diffration events requires additional pre-proessing before the soure term: all the relevant speular refletions must be established. Apparantly, this is a huge ompliation but in this study, only onvex geometries are studied. This fous on onvex geometries follows the method in [4] where it was aknowledged that non-onvex geometries would require the introdution of so-alled slope diffration terms. NUMERICAL EXAMPLES Numerial results will be presented here for one speifi ase: the sattering from a irular rigid dis for an on-axis inident wave. A six-sided polygonal approximation of a dis is used, a monopole soure is plaed at a height of 10 radii, and a reeiver is plaed entrally on the dis, right at the surfae of the soure-side of the dis, see Fig. 2. Proeedings of Meetings on Aoustis, Vol. 19, Page 6

7 FIGURE 2: A six-sided thin dis with a reeiver at the enter of the dis, and an axisymmetri monopole soure at a distane of 10 radii. Comparison between the frequeny- and the time-domain ESIE results The TD ESIE, as given by Eq. 15, and the following propagation integral in Eq. 6, were implemented in Matlab. Eah edge was disretized with 8 or 16 points, following a uniform or a Gauss-Legendre weighting sheme. A sampling frequeny of 55 khz was hosen, whih gave a minimum of one sample s delay between all edge point pairs. Suffiiently long time histories were omputed suh that the edge soure signals had deayed to insignifiant levels. As referene solution, a FD omputation was arried out with a 32-point Gauss-Legendre disretization of eah edge, following the method in [4]. The TD results were onverted to the FD using FFT. Gauss-Legendre quadrature was used for all the results in Fig. 3. Two types of errors will affet the results. One is aused by the spatial disretization, whih is idential for the TD and the FD method. This type of error initially grows relatively slowly with frequeny, up to a point where there are too many osillations over the integration range, and the error inreases rapidly. Around ka = 25, where the error for the "FD 8 GL"ase starts to inrease, and orrespondingly around ka = 50, for the "FD 16 GL" ase, the disretization then gives 1.8 points per λ on average; sine the GL points are distributed non-uniformily. A seond soure is the time-disretization whih obviously ours only for the TD results. The simple time-disretization sheme whih is employed here is expeted to give an error whih inreases as +6 db per otave for this ase. These two error mehanisms an be observed in the results. At very low frequenies, the TD error is negligible, and the TD and the FD formulations give idential auray. At some frequeny, the inreasing TD error starts to dominate. Then, at high enough frequenies, either of the two mehanisms might be dominating. A omplementary view is offered by the results in Fig. 3b, where the results with 16 GL points in Fig. 3a are repeated, plus one additional urve. The added results are omputed with the TD method, using the same spatial disretization, but with a 4 times oversampling. This oversampling a sampling frequeny of 220 khz then leads to an 80-times oversampling at the frequeny ka = 25, whih is the highest that is presented here. For many appliations, an auray of 10 3 is not required and a muh lower oversampling ration an be aepted. One of the attrative features of the TD formulation is that it provides results for many frequenies at one. It is therefore highly desirable that the sought frequeny range is a large fration of the sampling frequeny. Clearly, however, these results indiate that higher-order time disretization shemes are needed. The impulse response funtion One example of an impulse response omputed with the TD-ESIE is presented in Fig. 4a. Also plotted are impulse responses omputed with the order-by-order omputation desribed in [1]. All impulse responses were omputed with the same spatial disretization 16 Gauss-Legendre points, a sampling frequeny of 220 khz, and were subsequently lowpass filtered by onvolving with a 99-sample wide retangular window. This lowpass filtering redues the amplitude by 6 db around 1340 Hz, ka = What appears as noise between 4 and 8 ms Proeedings of Meetings on Aoustis, Vol. 19, Page 7

8 FIGURE 3: Sound pressure amplitude, re. free-field, omputed with the ESIE for the ase in Fig. 2. The errors are absolute errors differene from the referene solution. FD = frequeny-domain results and TD = time-domain results. The frequeny axis is given as ka where a is the equivalent radius for a irle of the same area as the six-sided dis. a Comparison between the TD and FD formulations, and between 8 and 16 quadrature points. b Same results as in a, with the addition of a urve with four times higher sampling frequeny for the TD formulation. are inauraies at high frequenies, aused by the limited number of quadrature points. Computational load The dominating omputational part of the formulation above is the setting up of the h a i matries, and the iterative matrix multipliation and summation proedure, desribed by Eq. 15. In the same way as for the Boundary Element Method, the subsequent propagation of the sound field to a number of reeiver positions, by Eq. 6, is typially a small part of the total omputational load. The FD formulation of the ESIE has a omputational omplexity whih is dominated by the number of elements in a transfer funtion matrix, whih is On 3 edge points for eah frequeny. The TD formulation has the same number of non-zero elements in the large transfer matrix, but these elements are distributed aross a number of very sparse h a i matries. The number of suh matries is determined by the number of unique integer propagation delays for the totality of edge-to-edge point set. When oversampling is used, this number of unique delays tends to grow more than linearly with the number of edge points. If oversampling is not used, then the number of unique sample delays would not hange muh with number of edge points. In the prototype Matlab implementation, where oversampling was employed, a omplexity of On 5 has been observed, but this number will be affeted by edge points how effiiently the set of sparse matries an be set up and handled. Ultimately, a omplexity loser to On 3 ould be targeted for the TD ESIE formulation. Sine the number of edge edge points points would be proportional to the highest frequeny of interest, the omplexity ould potentially ome lose to Ofmax 3. CONCLUSIONS A time-domain formulation of the edge soure integral equation ESIE has been presented. The auray is omparable to frequeny-domain results, with the additional error soure aused by the time-disretization. It has been shown that a simple time-disretization sheme requires high oversampling ratios for high auray. The omputational ost of the time-domain formulation has been analyzed for a prototype Matlab implementation. The learly dominating Proeedings of Meetings on Aoustis, Vol. 19, Page 8

9 FIGURE 4: a Impulse response funtions, re. free-field, for the ase in Fig. 2. The diret sound, inluding the speular refletion, is a pulse at 0 ms. The first-order diffration omponent appears with negative amplitude around 3.5 ms. b Relative error, with the TD ESIE results as referene solution. ost is the setting up of the transfer matries, and the time-marhing use of these matries. A ost of On 5 was observed, and the high exponent was aused by the high oversampling edge points used, whih lead to too sparsely distributed transfer matries. REFERENCES [1] U. P. Svensson, R. I. Fred, and J. Vanderkooy, An analyti seondary soure model of edge diffration impulse responses, J. Aoust. So. Am. 106, [2] M. Okada, T. Onoye, and W. Kobayashi, A ray traing simulation of sound diffration based on analyti seondary soure model, in Proeedings of the 19th European Signal Proessing Conferene EUSIPCO 2011, Barelona, Spain, 29 Aug. - 2 Sep [3] L. Antani, A. Chandak, M. Taylor, and D. Manoha, Effiient finite-edge diffration using onservative from-region visibility, Appl. Aoust. 73, [4] A. Asheim and U. P. Svensson, An integral equation formulation for the diffration from onvex plates and polyhedra, Tehnial Report TW610, Katholieke Universiteit Leuven, Dept. of Computer Siene [5] J. Summers, Reverberant aousti energy in auditoria that omprise systems of oupled rooms, Ph.D. thesis, Rensselaer Polytehni Institute, Troy, NY, USA [6] U. P. Svensson, P. T. Calamia, and S. Nakanishi, Frequeny-domain edge diffration for finite and infinite edges, Ata Austia/Austia 95, [7] H. Medwin, Shadowing by finite noise barriers, J. Aoust. So. Am. 69, Proeedings of Meetings on Aoustis, Vol. 19, Page 9