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Sound Field Modeling in Architectural Acoustics using a Diffusion Equation Based Model

N. Fortin1,2, J. Picaut2, A. Billon3, V. Valeau4, A. Sakout1

1LEPTIAB (University of La Rochelle)2ESAR (Laboratoire Central des Ponts et Chaussées)

3INTELSIG group (University of Liège)4LEA (University of Poitiers)

The authors wish to thankthe Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME)

for providing financial support of this work.

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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

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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

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Diffusion model (2)

Diffusion equation

Diffusion coefficient

tFt

twtwD ,

,, r

rr

3

cD

w acoustic energy density

room mean free path (4V/S)c sound speed

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Diffusion model (3)

Boundary conditionwall()

outinin

4w

cwh

wD

n

4

ch

h exchange coefficientn wall normal wall absorption coefficient transmission coefficient

)1ln(4

c

h

(Eyring’s absorption)(Sabine’s absorption)

wout win

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Diffusion model (4)

Atmospheric attenuation

tFtwmct

twtwD ,,

,, rr

rr

m coefficient of atmospheric attenuation

Mixed specular-diffuse reflection

DsDc )( Empirical correction

s wall scattering coefficient

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Diffusion model (5)

Diffusion by fitting objects

tFtwmcc

t

twtwD

f

ft ,,

,, rr

rr

DD

DDD

f

ft

D

DtDf

roomfitted zone

(nf, Qf, f)

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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)

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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

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I-Simpa MDF interface

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I-Simpa MDF interface

Geometry(.3ds file)

Material(acoustic properties)

Sound source(location, spectrum, orientation...)

Punctual receivers(location, orientation...)

Surface receiver(soundmap)

Fitting zone(acoustic properties)

COMSOL Script or

Matlab Script

NumericalResults

Scriptgeneration

Output formatconversion

Resolution ofdiffusion equations

I-SIMPA (user interface) COMSOL Multiphysics(batch mode)

Diffusion Model(script Python™)

XML

Acousticpost-treatment

Binary file

.m file

ASCII file

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I-Simpa MDF interface

Python™ script generation (.m COMSOL or MATLAB® script)1. [GENERAL] Header2. [GENERAL] Constants3. [GEOMETRY] Vertices4. [GEOMETRY] Faces5. [GEOMETRY] Definition of domains 6. [GEOMETRY] Material (boundaries)7. [GEOMETRY] Domain equations (PDE coefficients)8. [RESULTS] 2D Surface plots9. [RESULTS] Definition of punctual receivers10. [SETTINGS] Geometry analysis (FEM structure)11. [SETTINGS] Mesh definition (FEM structure)12. [SETTINGS] Application mode13. [SETTINGS] ODE settings and description14. [CALCULATION] Loading equations15. [CALCULATION] Loading application16. [CALCULATION] Meshing geometry17. [CALCULATION] Solving problem18. [CALCULATION] Saving results

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Results examples

Soundmap by frequency band / broadband: Stationnary: steady state SPL Temporal: time varying SPL Acoustical parameters mapping

SPL soundmap

RT soundmap

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Results examples

Sound decay at receivers: SPL by frequency band Reverberation time Rooms acoustical parameters Energy flow

Receiver spectrum

Receiver sound decay

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Conclusion

A fully operational tool for acoustic prediction in room, concert hall, building… has been developedSpecific interface I-Simpa (user interface)

Diffusion model (transparent)

.m script generation using Python™ COMSOL Multiphysics in batch mode

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Thank you for your attention

Judicael.Picaut@lcpc.fr