1. Computation of the complete acoustic field with Finite-Differences algorithms. Adan Garriga Carlos Spa Vicente López Forum Acusticum Budapest31/08/2005

2. Forum Acusticum Budapest La UPF a Ca l’Aranyó Parc Barcelona Media Summary: INTRODUCTION STATEMENT OF THE PROBLEM FINITE-DIFFERENCES ALGORITHMS: The MacCormack Method 2D AND 3D RESULTS: APPLICATIONS CONCLUSIONS AND FUTURE WORK Contents

3. 31/08/2005Forum Acusticum Budapest Parc Barcelona Media Goal: Simulate the propagation of sound waves in 3D virtual environments. Physical renderization of sound fields. Introduction Why?: New emerging multimedia technologies applications: digital cinema, video games, virtual reality, communications, music… Real Time: Many applications require renderization of the acoustic field in real time. Balance between accuracy and time of computation.

4. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Statement of the Problem Characterization of the acoustic field (4 quantities): –Pressure of the fluid (air): P –Three components of the air velocity: u Classical linear equations for the acoustic field:

5. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Parc Barcelona Media Finite-Differences Numerical Methods: Numerical Methods can be divided in two groups: Geometrical-based methods. Decomposition of the sound field in elementary waves: Image Source, Ray- Tracing, Beam-Tracing… Physical-Based methods. Exact numerical solution of the differential equations: Boundary Elements (BE), Finite Elements (FE) and Finite Differences (FD).

6. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Why FD? They give an accurate physical solution for the acoustic field For multimedia applications both the sound source and the receiver can move around. Therefore, we need to compute the sound field in the whole space at each time. Easy to implement in different geometries. Easy to parallelize. Finite-Differences

7. 31/08/2005Forum Acusticum Budapest Numerical Equations Finite-Differences1 2

8. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Numerical Parameters: Air density: 1,21 Kg/m The speed of sound: c = 330 m/s Space discretization: x = 0,01 m (valid for =100-1500 Hz) Time discretization: t = 0,00002 s Number of float operations per second for a square room of 2 X 2 meters: N=56 GFLOPS REAL TIME ! 2D Results 3

9. 31/08/2005Forum Acusticum Budapest 2D Results

10. 31/08/2005Forum Acusticum Budapest 2D Results

11. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó 3D Results: For the same quality results, the number of floating point operations per second (FLOPS) is: N = 16 TFLOPS. Only supercomputers work at this speed. NOT AT REAL TIME! 3D Results and ApplicationsApplications: 1D or 2D Real-Time rendering sound applications. 3D Non-Real-Time applications: digital cinema (RACINE)

12. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Conclusions: We have found upper bounds for high quality rendering of acoustic fields. For 1D and 2D applications, the algorithm works at Real-Time for frequencies = 100-1500 Hz. Conclusions

13. 31/08/2005Forum Acusticum Budapest La UPF a Ca l’Aranyó Future Work Future Work: For 3D applications we have to reduce the number of FLOPS: we have to introduce approximations. We are developing new hybrid algorithms: using geometric-based algorithms for high frequencies.