Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 9, 23 ICA 23 Montreal Montreal, Canada 2-7 June 23 Architectural Acoustics Session 4aAAa: Room Acoustics Computer Simulation I 4aAAa5 Simulation of non-locally reacting boundaries with a single domain boundary element method Robertus Opdam*, Diemer De Vries and Michael Vorländer *Corresponding author's address: Institute of Technical Acoustics, RWTH Aachen University, Neustrasse 5, Aachen, 5266, NRW, Germany, robopdam@akustikrwth-aachende The significance of taking into account non-locally reacting behavior of boundaries compared to the often-used locally reacting assumption in room acoustics has not been extensively investigated To make this possible a boundary element method is developed, inspired on a seismic simulation method known as the WRW method The novelty of this method compared to other boundary element methods (BEM) is that the calculation is performed in only one domain There is no need for a fluid-structure coupling, which in general allows faster simulation times The theory of the method is presented and some example structures are simulated with both locally and non-locally reacting behavior The results are shown and discussed Published by the Acoustical Society of America through the American Institute of Physics 23 Acoustical Society of America [DOI: 2/ ] Received 2 Jan 23; published 2 Jun 23 Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page

2 INTRODUCTION The computational modelling of acoustical fields in enclosed spaces (rooms) is in practice done with algorithms based on the mirror image source model (MISM) [], the ray tracing model (RTM) [2] or a mixture of both (hybrid method) [3, 4] In the most common hybrid solutions the first reflections (specular components) are calculated by the mirror image source method and the later reflections (scattered components) by the ray tracing method Those two methods are in the mean time able to deal with diffraction (non-specular components) [5, 6, 7] But these methods are still not taking into account all physical wave phenomena Including these is important from a listeners point of view and in the calculation of room acoustical parameters, which strongly depend on the early part of the room impulse response (RIR) [8, 9] The average value of room acoustical parameters over time and space is sufficient for most room acoustical parameters, but not at the individual listeners positions [9, ] Round robin tests of room acoustical computer simulations have pointed out that there is room for improvement of the predictions [, 2] Therefore, in this paper a wave field extrapolation method is presented, which is derived from an algorithm that is widely applied in seismic exploration [3] This seismic algorithm is called the WRW model, where the W stands for wave propagation and the R for reflection [4] The model is wave based and most similar to BEM methods, but the advantage is that it can incorporate non-locally reacting boundary properties by using a single domain, where other wave based methods as boundary element (BEM) and finite element methods (FEM) need to solve two domains (acoustical and structural) The model was suggested for 2-D simulations [5] and in this paper a further developed model is presented, including the implementation of non-locally reacting boundaries and 3-D capability WAVE FIELD EXTRAPOLATION (a) Kirchhoff-Helmholtz integral equation (b) Rayleigh integral equation FIGURE : Integral equation domains with primary sources ( ), volume V and boundary S The underlying theory of the WRW method starts with the Huygens principle Huygens stated that a propagating wave in a homogeneous medium can be described by adding all contributions of secondary sources positioned along a wavefront This interpretation was later quantitatively derived and described by the Kirchhoff-Helmholtz integral: P( r A,ω) = 4π S jk [jωρ V n ( r S,ω) + P( r S,ω) + jk cos(φ) jk ] ds () where = = r A r S is the distance from a secondary source point on the boundary S and the reconstruction point A within the enclosed volume V, φ is the angle between the normal vector n pointing inward to the volume and the vector between a secondary source and the reconstruction point, V n is the particle velocity in the normal inward direction, P the pressure Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 2

3 The integral terms are basically a weighted mono- and dipole Wavenumber k is defined as the angular frequency ω divided by the propagation velocity c and ρ is the air density With the Kirchhoff-Helmholtz equation as given in Eq () it is possible to predict the pressure anywhere in the source free volume V if the normal particle velocity and pressure are known everywhere at the boundary S as indicated in Figure (a) In the derivation of the Kirchhoff- Helmholtz integral (see [4] for the full details) it is possible to expand the Green s function for a homogeneous medium linearly with an arbitrary solution of the source free Helmholtz equation, say Γ( r,ω) This solution Γ( r,ω) can be chosen in such a way that the dipole or monopole term in Eq () is cancelled out These two solutions are given as: Γ ( r,ω) = Γ 2 ( r,ω) = e jk (2) jk (3) respectively, where the subscripts and 2 point out that respectively the monopole or the dipole term in Eq () drops out These particular solutions are only valid for all cases if the boundary S of the volume V has specific conditions In this particular case the volume V will be enclosed by a boundary S, which is divided in two parts S and S S is characterised as an infinite plane, with on top a half-space S that is infinitely far away as shown in Figure (b) The primary sources are all located below the infinite plane in such a way that there is only a contribution of secondary sources at the boundary part S The infinite half-space S has no secondary source contribution and can therefore be left out of the integration surface In this case the pressure at any point within the volume V can be described by the knowledge of whether the pressure or the normal particle velocity along the boundary S This results in the following two equations to describe the pressure at a certain point A in the source-free half space: P( r A,ω) = jωρ 2π P( r A,ω) = jk 2π S S [ P( r S,ω) [ ] jk V n ( r S,ω) ds (4) + jk cos(φ) ] jk ds (5) where Eq (4) is called the Rayleigh-I and Eq (5) the Rayleigh-II integral equation The method described in this paper makes use of the Rayleigh-II integral equation In fact this equation allows to extrapolate a wave field to a further point in space if the pressure at the boundary is known WRW model When an extrapolation in space from a plane z to a plane z is performed as in Figure 2, the Rayleigh-II equation can be expressed as: + P(x, y, z,ω) = W(Δx,Δy,Δz,ω) P(x, y, z,ω)dxdy (6) where Δx = x x, Δy = y y and Δz = z z In fact this represents a spatial convolution FIGURE 2: Spatially discretised geometry for extrapolation of the wave field from a plane z = z to a plane z = z Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 3

4 in x and y with W(Δx,Δy,Δz,ω) as the integral kernel that represents the propagation from each point on the z plane to the z plane The propagation kernel is defined as: W(Δx,Δy,Δz,ω) = jk 2π ( + jk jk )cos(φ) (7) In numerical simulations it is necessary to perform a discretisation step and the spatial convolution, as defined in Eq (6), can then be expressed as: P(z ) = W(z, z ) P(z ) (8) where the pressure is calculated for one frequency at all the points at the z = z plane Keep in mind that if the propagation is reversed, so extrapolation from the plane z = z to z = z, the propagation matrix W(z, z ) has to be inverted The new propagation operator can be defined as W(z, z ) = [W(z, z )] T Another important process of wave propagation is the reflection at a boundary In case of a geometry with one boundary at z = z, the relation between the incoming wave field P + (z ) and the reflected wave field P (z ), as drawn in Figure 3, can be expressed as: P (z ) = R(z ) P + (z ) (9) where the reflection matrix R(z ) is defined as a diagonal matrix with the reflection coefficients of FIGURE 3: Reflection of a wave field at a boundary each grid point on the boundary for the locally reacting case The reflection coefficient is defined as R = α(ω), with α(ω) the absorption coefficient of the boundary The reflection coefficients can be complex valued in case of, for instance, porous absorber material To make the process of propagation from a source to a boundary, reflection, and propagation from the boundary to a detector complete, the two previous processes are merged together The source(s) can be described by the source vector S(z s ) at a certain plane z = z s, the propagation from the source(s) to the boundary by the propagation matrix W(z b, z s ), the reflection process by the reflection matrix R(z b ) and as last step the propagation from the boundary to the detector(s) by W(z d, z b ) This process is schematically shown in Figure 4 The resulting sound pressure at the detector is given by: P(z d ) = P + (z d ) + P (z d ) = W(z d, z s ) S(z s ) + [W(z d, z b )R(z b )W(z b, z s )] S(z s ) () which is known in the field of seismic exploration as the WRW model FIGURE 4: Propagation from sources ( ) to boundary to detectors ( ) Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 4

5 WRW for multiple boundaries In the previous section the WRW method is derived for one reflection at one boundary In this section the basic model is extended to multiple boundaries and multiple reflections to be able to simulate an enclosed space To get a better overview over the matrix operations the notation is slightly changed The lower indices of the matrices indicate the propagation direction, for example W(z d, z s ) is indicated as W ds for the propagation from the source(s) to the detector(s), or the boundary they are working on, for example R(z ) is indicated as R b This also takes away the restriction that the propagation can only take place between parallel planes The orientation of the boundaries is unrestricted and does not even have to be closed The example case that is used is a rectangular room with four sidewalls (boundaries b to b 4 ), a ceiling (boundary b 5 ) and a floor (boundary b 6 ), which also encloses sources (s and s 2 ) and detectors (d to d 4 ) 2-D cross section of this geometry is given in Figure 5 FIGURE 5: Rectangular room with walls (boundaries), sources ( ) and detectors ( ) The contribution to the wave field by the reflective part can be defined as an iterative process, which starts with propagation from the source(s) to the walls followed by a reflection and propagation to the other walls and finally contributes to the detector point(s) The calculation of the reflected wave field is schematically shown in the top part of Figure 6 The matrix operators that FIGURE 6: Diagram of the model with the total P, reflected Prefl and direct wave field Pdir include all the reflection and propagation properties for the complete geometry will be defined as follows The source vector S includes the number of sources and their strength for the specific frequency ω and is defined as (with the total number of sources i): S = ( S (ω) S 2 (ω) S i (ω) ) T () The matrix operator for the propagation from source(s) to detector(s) W ds, source(s) to the boundaries W bs and boundaries to the detector(s) W db are defined as (with the total number of sources, detectors and/or boundaries indicated as i and j): W x y W x y 2 W x y i W x2 y W xy = W x2 y 2 W x2 y i (2) W x j y W x j y 2 W x j y i The boundary properties are collected in one matrix R, where on the diagonal the matrices of Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 5

6 separate boundary properties R b, as mentioned in the previous section, are placed: R b R b2 R = (3) R bn Propagation between boundaries is collected in the matrix W, which includes the individual propagations between the boundary parts (with N the total number): W b b W b b 2 W b b N W b2 b W = W b2 b 2 (4) W bn b W bn b N The matrix components W pp represent the interaction of the boundary with itself This can be interpreted as propagating bending waves In general cases these matrices will be zero W pp = Furthermore, it can be easily seen that W pq = W qp Analogue to Eq (), the reflected wave field at the detector location(s) in case of one reflection with multiple boundaries is calculated by: P = [W db RW bs ] S (5) and if this is extended to the m th -order reflection, the expression in Eq (9) can be generalised to: P m = [ W db (RW) m ] RW bs S (6) The resulting reflected wave field at the detector locations Prefl is given by the summation of all reflection orders M, which gives the following expression in case M = : P refl = [ Wdb (RW) m ] RW bs S m= = [ W db (I RW) RW bs ] S (7) where use is made of the fact that the summation is a Neumann series, which can be written as a matrix inversion The matrix I here is the unity matrix Furthermore, it is necessary to add the direct wave field Pdir from the source(s) to the detector(s) to get the total wave field P at the detector(s), schematically shown in the bottom part of Figure 6 The total wave field is therefore given by: P = Pdir + Prefl = [ W ds + W db (I RW) ] RW bs S (8) Non-locally reacting reflection The structure of the matrix with the boundary properties R b allows to incorporate non-locally reacting (or angle dependent) reflection properties Each row in the R b matrix can be interpreted as a reflectivity convolution operator [6] In case of a locally reacting surface the convolution operator is represented by a finite impulse in the space-frequency domain and gives unity in the angle-frequency domain as shown in Figure 7(a) The transformation between the two domains is done by means of a spatial Fourier transform: R(k x,ω) = R(x,ω)e i(kxx) dx (9) Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 6

7 R(x,ω) R(k x,ω) R(x,ω) R(k x,ω) x k x x k x (a) Locally reacting (b) Non-locally reacting FIGURE 7: Reflection function in space-frequency R(x,ω) and angle-frequency R(k x,ω) domain where k x = ω c sinα, with α the angle of incidence In case of a non-locally reacting boundary the convolution operator is represented as a function, for example only reflection for small angles and no reflection at wider angles (block function), giving a sinc-function in the k x,ω-domain as given in Figure 7(b) The reflection matrix is defined in this case by a band matrix: R(x,ω) R(x 2,ω) R(x 2,ω) R(x 22,ω) R b = (2) R(x NN,ω) Numerical aspects One of the prerequisites of obtaining a valid outcome of the model is to have enough boundary sample points In general there is the minimum requirement of satisfying Nyquist s spatial sample theorem The expression for the maximum frequency in relation to the sample distance is given as f max = c 2Δξ with Δξ the distance between two boundary sample points SIMULATIONS The simulations shown here are performed in the 2-D domain to better illustrate the differences between locally and non-locally reacting boundary effects In all the simulations both an angle independent reflection coefficient and an angle dependent reflection coefficient with a block function in the k x -domain, as given in Fig 7(b) are used Of course a block function for the reflectivity (small angles of incidence are fully reflected and larger angles fully absorbed) is an extreme case, but it is suited to show the influence of non-locally reacting effects L-wall structure A simple L-shaped wall geometry is taken, where a single broadband point source is placed in front and a microphone array in between as in Fig 8(a) The results and differences are given in Fig 8(b) for the time domain, where the following effects are visible: The edge diffraction of the wall ends (arrow a), first order reflection of the wall parallel to the detector array (arrow b) and the second order reflection of the sidewall to the parallel wall (arrow c) The first reflection does not show much difference, because the incoming wave field is in an almost perpendicular angle So it reacts as a quasi locally reacting surface due to the block reflection coefficient function in the k x -domain On the other hand the second order reflection is a result of a wave field coming in under a larger angle This explains the difference between the second order reflection and the first order reflection, caused by non-locally reacting boundary properties What can also be Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 7

8 observed is the edge diffraction at the two ends of the wall in both cases, being similar for the locally and non-locally reacting case time (ms) Locally a b c Non locally Difference 3 2 distance from array centre (m) (a) Geometry with source and detector array (b) Reflected pressure fields (in Pa) of non-locally and locally reacting boundary and the difference FIGURE 8: L-shaped wall structure Closed rectangular structure A simple rectangular geometry enclosing a sound source, as in Fig 9(c), is simulated to show that the WRW algorithm can also handle closed geometries Again the aforementioned boundary conditions are used, but this time the results are presented in the frequency domain to show that the eigenfrequencies (modes) of the structure can also be simulated accurately, see Figs 9(a), 9(b) and 9(d) The mode [2 ] at 29 Hz is visible and a difference between a locally reacting (Fig 9(a)) and non-locally reacting boundary (Fig 9(b)) of at most 8 db is shown (Fig 9(d)) Width (m) Length (m) 2 Width (m) Length (m) 2 (a) Abs pressure (in db) for a locally reacting boundary (b) Abs pressure (in db) for a non-locally reacting boundary Width (m) Length (m) (c) Geometry with source position (d) Pressure difference (in db) FIGURE 9: Closed rectangular geometry simulated at 29 Hz (mode [2 ]) Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 8

9 CONCLUSIONS A wave based 3-D acoustical simulation algorithm (WRW method) is presented, which is able to handle non-locally reacting (angle dependent) boundary conditions Preliminary results are shown for a 2-D L-shaped wall structure and a small rectangular domain It is shown that non-locally reacting boundaries do affect the whole acoustical space and not only close to the boundaries, though it is necessary to further investigate the quantitative influence of non-locally reacting boundary properties on the total wave field compared to the assumption of locally reacting ones ACKNOWLEDGMENTS This research was funded by the Deutsche Forschungsgemeinschaft (German Research Foundation) DFG as part of the SEACEN research unit ( REFERENCES [] J Borish, "Extension of the image model to arbitrary polyhedra", J Acoust Soc Am 75(6), (984) [2] A Krokstad, S Strøm, S Sørsdal, "Calculating the acoustical room response by the use of a raytracing technique", J Sound Vib 8(), 8-25 (968) [3] M Vorländer, "Simulation of the transient and steady-state sound propagation in rooms using a new combined ray-tracing/image-source algorithm", J Acoust Soc Am 86(), (989) [4] T Lentz, D Schröder, M Vorländer, I Assenmacher, "Virtual Reality System with Integrated Sound Field Simulation and Reproduction", EURASIP J App Sig Proc 27(), (27) [5] D Schröder, UP Svensson, A Pohl, U Stephenson, M Vorländer, "On the accuracy of edge diffraction simulation methods in geometrical acoustics", Proc of Internoise 22, New York City, (22) [6] UP Svensson, "Modelling acoustic spaces for audio virtual reality", Proc st IEEE Benelux Workshop on Model based Processing and Coding of Audio, 9-6 (22) [7] UP Svensson, RI Fred, J Vanderkooy, "An analytic secondary source model of edge diffraction impuls responses", J Acoust Soc Am 6(5), (999) [8] B-I Dalenbäck, M Kleiner, UP Svensson, "A macroscopic view of diffuse reflection", J Audio Eng Soc 42(), (994) [9] RR Torres, UP Svensson, M Kleiner, "Computation of edge diffraction for more accurate room acoustics auralization", J Acoust Soc Am 6(5), (999) [] J Baan, D de Vries, "Reflection and diffraction in roomacoustic modelling", Proc Forum Acusticum, Berlin (999) [] M Vorländer, "International round robin on room acoustical computer simulations", Proc 5th Intl Congress on Acoustics (ICA), Trondheim, II, (995) [2] I Bork, "A Comparison of Room Simulation Software - The 2nd Round Robin on Room Acoustical Computer Simulation", ACUSTICA united with acta acustica 86(6), (999) [3] AJ Berkhout, DJ Verschuur, "Estimation of multiple scattering by iterative inversion, Part I: Theoretical considerations", Geophysics 62(5), (997) [4] AJ Berkhout, "Applied Seismic Wave Theory", Elsevier, Amsterdam, 987 [5] AJ Berkhout, D de Vries, J Baan, BW van den Oetelaar, "A wave field extrapolation approach to acoustical modeling in enclosed spaces", J Acoust Soc Am 5(3), (999) [6] CGM de Bruin, CPA Wapenaar, AJ Berkhout, "Angle-dependent reflectivity by means of prestack migration", Geophysics 55(9), (99) Proceedings of Meetings on Acoustics, Vol 9, 5 (23) Page 9