Proceedings of Meetings on Acoustics

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1 Proceedings of Meetings on Acoustics Volume 9, 20 ICA 20 Montreal Montreal, Canada 2-7 June 20 Noise Session pnsc: Joint Poster Session on Noise and Architectural Acoustics (Poster Session) pnsc6. Large-scale sound field rendering with graphics processing unit cluster for three-dimensional audio with loudspeaker array Takao Tsuchiya*, Yukio Iwaya and Makoto Otani *Corresponding author's address: Dept. of Information Systems Design, Doshisha University, - Miyakodani, Kyotanabe-shi, 60-02, Kyoto, Japan, ttsuchiy@mail.doshisha.ac.jp The sound field rendering is a technique to simulate the sound field from the three-dimensional numerical models constructed in the computer, and it is the same concept as the graphics rendering technique. In this paper, a GPU (Graphics Processing Unit) cluster system is applied to the sound field rendering. The compact explicit-finite difference time domain (CE-FDTD) method is implemented on the GPU cluster system. The CE-FDTD method is a kind of FDTD method in which the wave equation is directly discretized based on the central differences. The developed GPU cluster system consists of eight PC nodes in which four GPUs are mounted respectively. The rendering results are reproduced by a loudspeaker array system in which 57 loudspeakers surround on a anechoic room. The sound field renderings are performed for a large room with a volume capacity of about 5000 cubic meters, in which the impulse responses of two-second length with a sampling rate of 0 khz are calculated at 57 points corresponding to the loudspeaker positions. The impulse responses are then convoluted with dry music sources. The sound field rendering with the 57ch loudspeaker array system provides the realistic sound field reproduction with natural reverberation. Published by the Acoustical Society of America through the American Institute of Physics 20 Acoustical Society of America [DOI: 0.2/ ] Received 20 Jan 20; published 2 Jun 20 Proceedings of Meetings on Acoustics, Vol. 9, (20) Page

2 INTRODUCTION The graphics rendering technique is widely used in the field of computer graphics (CG). In CG, a two-dimensional image can be displayed from three-dimensional (D) models numerically constructed in the computer. The same approach can be applied to the sound field simulation which is called sound field rendering [-]. In the sound field rendering, auralization of the three-dimensional sound field can be realized with multi-speaker system. For more realistic rendering, it is necessary to include wave properties such as diffraction. In this study, the compact explicit-finite difference time domain (CE-FDTD) method [-6] is applied to the large-scale D sound field rendering to realize a natural sound field. The CE-FDTD method is one of the wave equation based FDTD methods [7] and is a high accuracy version of the standard FDTD methods [8]. In the CE-FDTD method, the wave equation is directly discretized on the basis of the central differences, not only along the axial directions but also the diagonal and the side-diagonal directions. There are some derivative schemes in the CE-FDTD method. The most accurate scheme is the interpolated wideband (IWB) scheme in which the cut-off frequency is in agreement with the Nyquist frequency. While the CE-FDTD method is highly suitable for the large-scale D sound field rendering, it requires enormous computer resources for the accurate analysis. To cope with this problem, a graphics processing unit (GPU) cluster system is developed and the CE-FDTD method is implemented on the GPU cluster. The GPU is a kind of many-core processors, which is capable of high-performance computation. Using GPUs, computing performance approximately 20-0 times faster than that of CPU can be easily achieved. We have developed a multi-gpu system [9] or a GPU cluster [0] system for the numerical analysis of a large-scale sound field. The developed GPU cluster system consists of eight PC nodes in which four GPUs are mounted respectively. In the cluster system, the sound field to be simulated is divided into 2 domains, which are respectively calculated by each GPU in parallel. Each GPU exchanges the boundary data for each time step via the PCI Express x6 bus in each node and via the InfiniBand link between nodes. The program code for GPU is described by the CUDA language and the code for the parallel programming is described by the OpenMP and the OpenMPI on the Linux platform. The CE-FDTD method is implemented on the GPU cluster system. The rendering results are then reproduced by a loudspeaker array system in which 57 loudspeakers are arranged in a rectangular room []. The realistic D sound field with natural reverberation is realized by the sound field rendering with the loudspeaker array system. THEORY CE-FDTD Method The-three dimensional wave equation on the sound pressure p is given as c p t 2 = 2 p x p y p z 2, () where c 0 is the sound speed. In the CE-FDTD method, the wave equation is discretized on the collocated grid on the basis of the central finite difference method. There are 27 grid points in a discretized cell of the CE-FDTD method as shown in Fig.. The grid intervals of x-, y- and z-directions are assumed to be all the same as Δ. Considering not Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 2

3 2Δ 2Δ 2Δ FIGURE : A cell of CE-FDTD method consists of 27 grid points. (a) SLF (b) CCP (c) OCTA FIGURE 2: Various stencils for the CE-FDTD method. only the axis directions but also the diagonal and the side-diagonal directions, eq. () is discretized as {( ) ( ) } δt 2 p n i,j,k = χ 2 δx 2 + δy 2 + δz 2 + a δxδ 2 y 2 + δyδ 2 z 2 + δzδ 2 x 2 +bδxδ 2 yδ 2 z 2 p n i,j,k, (2) where a and b denote two independent numerical parameters, p n i,j,k represents the sound pressure on the grid point (x, y, z) =(iδ,jδ,kδ) at time t = nδt, Δt is the time step and χ = c 0 Δt/Δ is the Courant number. δ 2 is an operator on the central finite difference. For examples, δ 2 t p n i,j,k p n+ i,j,k 2pn i,j,k + p n i,j,k, () δ 2 xp n i,j,k p n i+,j,k 2p n i,j,k + p n i,j,k. () δy 2 and δz 2 are given in the same manner. Equation (2) is then rewriten as ) p n+ i,j,k = d (p n i+,j,k + p n i,j,k + p n i,j+,k + p n i,j,k + p n i,j,k+ + p n i,j,k +d 2 (p n i+,j+,k + p n i+,j,k + p n i+,j,k+ + p n i+,j,k + p n i,j+,k+ + p n i,j+,k ) +p n i,j,k+ + p n i,j,k + p n i,j+,k + p n i,j,k + p n i,j,k+ + p n i,j,k +d (p n i+,j+,k+ + p n i+,j,k+ + p n i+,j+,k + p n i+,j,k ) +p n i,j+,k+ + p n i,j,k+ + p n i,j+,k + p n i,j,k where d d are the coefficients given as + d p n i,j,k p n i,j,k, (5) d = χ 2 ( a +b), d 2 = χ 2 (a 2b), d = χ 2 b, d = 2( χ 2 +6aχ 2 bχ 2 ). (6) In eq.(5), d corresponds to the stencil along the axis directions as shown in Fig.2 (a), which is called standard leapfrog (SLF, i. e. WE-FDTD), d 2 to the stencil along the side-diagonal directions called cubic close packed (CCP) as shown in Fig.2 (b), and d to Proceedings of Meetings on Acoustics, Vol. 9, (20) Page

4 TABLE : Numerical parameters for the derivative schemes in the CE-FDTD method. method a b d d 2 d d χ m fc SLF 0 0 CCP OCTA IISO IISO2 IWB the stencil along the diagonal directions called octahedral (OCTA) as shown in Fig.2 (c). Other stencils can be configured by the combination of these stencils. The numerical accuracy is controlled by adjusting these parameters as shown in Table I. The upper limit of Courant number χ m and the normalized cut-off frequency are also tabulated in Table I. As shown in Table I, the IWB scheme has the widest bandwidth. In this paper, the IWB scheme is therefore used for the sound field rendering. Figure shows the normalized cut-off frequency against the Courant number for the axial, the side-diagonal and the diagonal directions, respectively. The normalized cut-off frequency becomes large when the Courant number increases for all schemes. It is found that the Courant number should be set as the upper limit for the accurate analysis. For the (, 0, 0) direction, the characteristic accords by all schemes, but the upper limit of the cut-off frequency is restricted as shown in Table I. The IWB scheme also has the comprehensively widest bandwidth. As shown in Fig., the cut-off frequency depends on the Courant number, so when desired cut-off frequency f c is given, the maximum grid interval is derived as Δ m = c 0 f c χf c. (7) Table II shows the minimum computer resources to calculate an impulse response with time length of s and volume capacity of m when the desired cut-off frequency f c is 20kHz in air (c 0 = 0 m/s). In the table, N =(/Δ m ) denotes the total number of grid points which is proportional to the memory usage, and N f denotes the total number of the TABLE 2: Minimum computer resources to calculate an impulse response with time length of s and volume capacity of m when the desired cut-off frequency f c is 20kHz in air. method fc f s Δ m N memory N f calculation (khz) (mm) 0 6 usage 0 2 time SLF IWB Proceedings of Meetings on Acoustics, Vol. 9, (20) Page

5 f c f c χ SLF CCP OCTA IISO IISO2 IWB (a) (, 0, 0) (c) (,, ) χ f c (b) (,, 0) χ FIGURE : Cut-off frequency against Courant number. floating point operation which is proportional to the calculation time. The memory usage and the calculation time are expressed in the ratios to those calculated by the SLF scheme. It is found that the memory usage of the IWB scheme is less than one-third of that of the SLF scheme to achieve the same bandwidth. When the scheme is implemented on the GPU, the calculation speed in the small memory scheme is faster than the large memory scheme because the calculation time is almost consumed by the data transfer in GPU. It is also found that the calculation time of the IWB scheme is about 0% compared with that of the SLF scheme to achieve the same bandwidth. SOUND FIELD RENDERING GPU Cluster System The GPU cluster system developed in this study consists of eight PC nodes. Figure shows the block diagram of each PC node and the cluster system. Four GPUs are mounted in each node, so the total number of GPUs is 2 in the system. The GPUs used in the calculation are NVIDIA Tesla M2075 (RAM: 6 GB, processor clock:.5ghz, processor cores: 8). The total memory of the system is 92 GB. Motherboard is Node Node 2 Node Node InfiniBand HUB PCI Express x6 GPU CPU GPU CPU GPU CPU GPU CPU Main Memory Node 5 Node 6 Node 7 Node 8 FIGURE : Block diagram of PC node and cluster system. Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 5

6 unit in m FIGURE 5: Rendering model. TABLE : Source positions for each orchestra part. part (x, y, z) m part (x, y, z) m Vn (5, 6.5, ) Cb (6.5, 7.5, ) Vn2 (5, 9, ) Fl, Cl, Tp (6.5, 22,.) Va (, 9, ) Ob, Fg, Hr (.5, 22,.) Vc (, 6.5, ) Tim (9, 2,.8) Supermicro SYS-2026GT-TRF on which four PCI Express x6 interfaces are available. CPU is Intel Xeon E5620 (processor cores: 6, processor clock: 2.0GHz). The version of CUDA is., that of OpenMP is 2.5 and that of OpenMPI is.6 on the CentOS 5.5 platform. Each GPU exchanges the boundary data for each time step via the PCI express bus with OpenMP and each PC node exchanges the data via the InfiniBand link with OpenMPI. Rendering model Figure 5 shows the rendering model, whose volume capacity is,22 m. The medium is assumed to be air whose sound speed c 0 is 0 m/s. The reflection coefficient R of all boundaries are assumed to be In the case of f s =0kHz, the grid interval Δ is 8.5 mm, so the numerical domain is divided into cells whose memory requirement is 55 GB. Table shows the positions of point sources corresponding to musical instruments of the orchestra. The unidirectional directivity is assumed for the point sources. To avoid the long-time ringing caused by the numerical dispersion error, the eight points driving is devised in the one-eighth cell. This driving method acts as a spacial low-pass filter whose cut-off frequency is the Nyquist frequency. Rendering results and auralization For the auralization, the rendering results are reproduced by the 57-channel loudspeaker array system. Figure 6 shows the loudspeaker array system, in which 57 loudspeakers are surrounded on the walls and ceiling of a rectangular room []. The array is.58 m deep, 2.78 m wide and.72 m high. The spacing of the loudspeaker is 0.5m. The receiving points in the numerical models are assumed at the positions corresponding to the loudspeakers in the center of the numerical model. The impulse responses are calculated at 57 receiving positions for 2 seconds. The unidirectional directivity of cardioid type is assumed for the virtual receivers, which directs to the outward normal direction to receive incident sound waves. To give Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 6

7 (a) layout (b) loudspeaker array inside an anechoic room FIGURE 6: 57-channel loudspeaker array system. the unidirectional directivity for the virtual receiver in the standard FDTD method, sum of the sound pressure and the particle velocity multiplying by the characteristic impedance at the receiving point is calculated. However in the CE-FDTD method, the particle velocity is not stored on the memory, so the following equation is introduced to derive the particle velocity. u n+ i,j,k = u n i,j,k + χ 2{ (p n i+,j,k p n i,j,k) sin θ cos φ } +(p n i,j+,k p n i,j,k) sin θ sin φ +(p n i,j,k+ p n i,j,k ) cos θ /2 (8) where u = Z 0 u, Z 0 is the characteristic impedance, θ is elevation angle, and φ is horizontal angle. Figure 7 shows the impulse response of the sound pressure and corresponding reverberation curve calculated at the position of a certain loudspeaker when the sound impulse is radiated from the center position of the stage (Tim). The calculation time is about 2 hours for each point source, so the total calculation time is days. For auralization, the calculated impulse responses are convolved with the orchestra parts which were recorded separately at anechoic room in Helsinki university [2]. In the convolution, the dry sources are downsampled from 8 khz to 0 khz because the sampling rate of the calculated impulse responses (f s =0kHz) is different from that of the dry sources (f s =8kHz). The convolved signals are then low-pass filtered with the cut-off frequency of 8 khz to remove the numerical dispersion noise. The filtered signals are finally upsampled to 8 khz. The reverberation curve well agrees with the theory of the reverberation in rectangular rooms with specular reflections []. As a result of the sound field rendering, it is confirmed that the sound localization was achieved in the speaker array, and that two or more people can hear the music at the sound pressure (μpa) (a) impulse response time (s) energy density level (db) FDTD theory (b) reverberation curve time (s) FIGURE 7: Impulse response and corresponding reverberation curve for the large room model. Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 7

8 same time. The spatial impression was achieved by considering the position of orchestra part. CONCLUSIONS The D sound field rendering was performed on the basis of the compact explicit-finite difference time domain method by 2 GPU cluster system. The impulse responses were calculated for a large room with a volume capacity of 700 m in which sampling rate was 0 khz. The rendering results were then reproduced by a speaker array system in which 57-channel loudspeaker array are arranged in a rectangular room. The realistic D sound field with natural reverberation was realized by the sound field rendering with the loufspeaker array system. ACKNOWLEDGMENTS This project is partly supported by the National Institute of Information and Communication Technology. REFERENCES [] M. Boone, Acoustic rendering with wave field synthesis, Proc. ACM SIGGRAPH and EUROGRAPHICS CAMPFIRE: Acoustic Rendering for Virtual Environments, (200). [2] T. Tsuchiya, and M. Otsuka, Sound Field Rendering with Digital Boundary Condition, Proc. Inter Noise 200, 68 (200). [] T. Tsuchiya, Three-Dimensional Sound Field Rendering with Digital Boundary Condition using Graphics Processing Unit, Jpn. J. Appl. Phys. 9, 07HC0-2, (200). [] M. V. Walstijn, and K. Kowalczyk, On the numerical solution of the 2D wave equation with compact FDTD schemes, Proc. of the th Int. Conf. on Digital Audio Effects (DAFx-08) (2008). [5] K. Kowalczyk and M. V. Walstijn, Room Acoustics and Acoustic System Modeling-Wideband and Isotropic Room Acoustics Simulation Using 2-D Interpolated FDTD Schemes, IEEE Trans. on Audio Speech and Lang. Process. 8, (200) 78. [6] K. Kowalczyk, and M V. Walstijn, Room acoustics simulation using -D compact explicit FDTD schemes, IEEE Trans. Audio Speech and Lang. Process. 9, (20). [7] Y. Miyazaki, and T. Tsuchiya, Perfectly Matched Layer for the Wave Equation Finite Difference Time Domain Method, Jpn. J. Appl. Phys. 5, 07GB02-5 (202). [8] K. S. Yee, Numerical Solution of Initial Boundary Value Problems Involving Maxwell s Equations in Isotropic Media, IEEE Trans. Antennas Propag, AP-, (8), 02 07, (966). [9] T. Tsuchiya, and M. Otsuka, A high-performance numerical simulation of sound field in time domain using multi-gpu system, Proc. 6 th Int. Conf. Sound & Vib., (2009). Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 8

9 [0] T. Tsuchiya, A. Morikochi, and T. Ishii, A High Performance Numerical Simulation of Sound Field in Time Domain Using a GPU Cluster System, Proc. Forum Acousticum, , (20). [] J. Trevino, T. Okamoto, Y. Iwaya, and Y. Suzuki, High order Ambisonic decoding method for irregular loudspeaker arrays, Proc. 20th Int. Cong. Acoust., ICA200, 8 (202). [2] [] T. Sakuma, Approximate theory of reverberation in rectangular rooms with specular and diffuse reflections, J. Acoust. Soc. Am. 2, (202). Proceedings of Meetings on Acoustics, Vol. 9, (20) Page 9