1. 1 Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model N. Fortin 1,2, J. Picaut 2, A. Billon 3, V. Valeau 4, A. Sakout 1 1 LEPTIAB (University of La Rochelle) 2 ESAR (Laboratoire Central des Ponts et Chaussées) 3 INTELSIG group (University of Liège) 4 LEA (University of Poitiers) The authors wish to thank the Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME) for providing financial support of this work.

2. 2 Introduction Sound field modeling in room acoustics Predicting sound level, reverberation time, acoustical parameters for Concert Hall, dwelling, building… Many propagation phenomena: reflection, absorption, diffusion, transmission, scattering, diffraction… Solutions: Solving the wave (or Helmholtz) equation: Analytical : no solution for “real rooms” Numerical: finite element method limited for low frequency only Others methods (energetic approaches, high frequency) Statistical theory of reverberation: “simple” geometries Ray-tracing (and similar): high computational time for “complex” rooms Alternative solution: diffusion model Good compromise acoustical results/computational time

3. 3 Diffusion model (1)  Diffusion model (MDF)  initially proposed by the authors for empty rooms with diffusely reflecting boundaries  following a diffusion process (diffusion equation)  validated in many room configurations: o rectangular rooms, long rooms, coupled rooms… o by comparison with others analytical models, numerical models (ray-tracing) experimental data

4. 4 Diffusion model (2)  Diffusion equation  Diffusion coefficient wacoustic energy density room mean free path (4V/S) csound speed

5. 5 Diffusion model (3)  Boundary condition wall (  ) hexchange coefficient nwall normal  wall absorption coefficient  transmission coefficient (Eyring’s absorption)(Sabine’s absorption) w out w in

6. 6 Diffusion model (4)  Atmospheric attenuation mcoefficient of atmospheric attenuation  Mixed specular-diffuse reflection Empirical correction swall scattering coefficient

7. 7 Diffusion model (5)  Diffusion by fitting objects D DtDt DfDf room fitted zone (n f, Q f,  f )

8. 8 Diffusion model (6)  Numerical solving  using FEMLAB with MATLAB®  now using COMSOL Multiphysics  Y. Jing and N. Xiang, Boundary condition for the diffusion equation model in room-acoustic prediction, Proceedings of the COMSOL Conference (2007)  Y. Jing and N. Xiang, On the use of a diffusion equation model for the energy flow prediction in acoustically coupled spaces, Proceedings of the COMSOL Conference (2008)

9. 9 Main objective  Developping an operational (acoustic) tool:  with acoustic knowledge (i.e. acoustical terms for materials, sound source, acoustic parameters…)  without COMSOL Multiphysics knowledge  Solution:  to develop a specific interface between the user (an acoustician) and COMSOL Multiphysics  manipulating all input data (geometry, acoustics…)  running calculation (multi-codes) like MDF (batch mode)  post-processing all output data: acoustical parameters

10. 10 I-Simpa  MDF interface

11. 11 I-Simpa  MDF interface

12. 12 I-Simpa  MDF interface  Python ™ script generation (.m COMSOL or MATLAB® script) 1. [GENERAL]Header 2. [GENERAL]Constants 3. [GEOMETRY]Vertices 4. [GEOMETRY]Faces 5. [GEOMETRY]Definition of domains 6. [GEOMETRY]Material (boundaries) 7. [GEOMETRY]Domain equations (PDE coefficients) 8. [RESULTS]2D Surface plots 9. [RESULTS]Definition of punctual receivers 10. [SETTINGS]Geometry analysis (FEM structure) 11. [SETTINGS]Mesh definition (FEM structure) 12. [SETTINGS]Application mode 13. [SETTINGS]ODE settings and description 14. [CALCULATION]Loading equations 15. [CALCULATION]Loading application 16. [CALCULATION]Meshing geometry 17. [CALCULATION]Solving problem 18. [CALCULATION]Saving results

13. 13 Results examples  Soundmap by frequency band / broadband:  Stationnary: steady state SPL  Temporal: time varying SPL  Acoustical parameters mapping SPL soundmap RT soundmap

14. 14 Results examples  Sound decay at receivers:  SPL by frequency band  Reverberation time  Rooms acoustical parameters  Energy flow Receiver spectrum Receiver sound decay

15. 15 Conclusion A fully operational tool for acoustic prediction in room, concert hall, building… has been developed Specific interface I-Simpa (user interface) Diffusion model (transparent).m script generation using Python™ COMSOL Multiphysics in batch mode

16. 16 Thank you for your attention Judicael.Picaut@lcpc.fr