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Predictive Acoustics Software and Computer Aided Optimization in Room Acoustics

Olivier Warusfel

ICA 95, Trondheim (Norvège) 1995
Copyright © ICA 1995

Summary

The design of a new hall and the control of a variable acoustics hall adress similar problems. In both cases the goal is to optimize, under various constraints, the architectural parameters according to a desired acoustical result. On the one hand, the complex relationships between architectural parameters and the acoustical behaviour of the room limit the efficiency of an intuitive modification or tuning of the room. On the other hand, in spite of their improving prediction performances, room simulation softwares will partly fail as long as they do not include "inverse modules" that provide guides for the optimization step. This paper presents some examples of computer aided optimization based on inverse problem resolution techniques. The results and possible applications are discussed on practical examples : the automatic selection of the material characteristics for the design of a room, or the automatic control of movable elements in a variable acoustics hall.

Introduction

Considerable efforts have been made in various laboratories in order to develop computer softwares that describe the way the sound propagates within a room. Geometrical methods have been refined in order to describe different boundary conditions such as diffused or specular reflection on walls and diffraction on edges. Some of these software environments now offer auralization procedures and thus tend to appear as a promising alternative to scale model approach when aiming at the aided design of concert halls.

One of the possible advantages of predictive softwares is that the acoustical behaviour of the room is described via a physical formulation. We may then envision to extend the diagnostic step with methods that would help the optimization process of the project. The optimization is not discussed here in terms of subjective preference but in terms of estimating the optimal set of architectural parameters in order to fulfil the acoustical specifications of the hall project. Hence, the question may be seen as the inverse problem of predictive room acoustics.

The method described in this paper is an attempt to optimize the distribution of materials in a room with a given shape and according to a specific time and spatial distribution of early energy within the room.

Prediction and Optimization Synopsis

Figure 1 shows the general synopsis of the prediction process (solid arrows). It consists of various propagation models that use the following input data :

the transducer characteristics (source and receiver radiation pattern and frequency response),
the geometrical shape, usually discretised in the simulation model with plane surfaces,
the material characteristics that should relate both absorption and diffusion characteristics.

The time and spatial distribution of the sound energy is collected on receivers, under the form of impulse responses or energy time curves. A synthetic description of the acoustical quality of the room may be then expressed with the help of acoustical criteria. Finally the auralization step may be obtained through a convolution process of the estimated impulse response [Martin89][Blauert90], or from a synthetic room effect that reproduces the perceptualy significant cues of the room response [Jot95][Rindel94].

The optimization process (dashed arrows) consists of estimating the best set of all or part of the input data in order to obtain a given acoustical result. For the late time distribution of energy, the well known Sabine's formula, or it's derivatives, may be used: given the reverberation time, global architectural quantities like the volume or the mean absorption may be estimated. When considering non ideal diffused sound fields, one can use the geometrical methods described in [Schroeder80] and [Malcurt86]. Under Lambert's law reflection hypotheses, the spatial distribution during the steady state or the exponential decay may be obtained by the analysis of a matrix which represents the coupling between surfaces. The coupling coefficients integrate geometrical and absorption characteristics of the surfaces. Thus, given a reverberation time and a spatial distribution of sound energy one can get information on what the coupling between the surfaces should be.

Unfortunately, for the early distribution of energy, the classical prediction methods do not link directly the geometrical or material characteristics to acoustical criteria, for they are based on an iterative following of sound paths. Hence an optimization procedure requires to replace these models with an approximate analytical formulation that link architectural and acoustical quantities and that can be inverted.

In a previous study, conducted with Electricité de France [Raynal90], a collection of analytical formulae was introduced a priori and combined in a linear form in order to verify the relations observed on a set of configurations of a variable room (geometry and materials). The computation of the pseudo-inverse of these combinations provided relations from which both, the geometrical and material parameters of the room, could be optimized according to a desired acoustical quality.

Here we restrict our optimization procedure on the material characteristics, keeping the room shape constant. On a practical point of view, this question could arise at an early step of a project where one need to respect architectural shape constraints, or later on when the main structure of the hall is already built: the remaining degree of freedom is the choice of materials. The analytical formulation is derived from the observation of the results given by a first run of the prediction model. The material characteristics under optimization are the absorption coefficient and the diffusion coefficient that are used in the reflection model.

Figure 1. Synopsis of prediction and optimization processes

Reflection Model

The reflection model used by the predictive environment is based on a combination of two ideal models : a specular model and a Lambert's law diffusion model. Figure 2 illustrates the behaviour of the boundary. When a ray strikes the surface its energy is split into two parts which will be controlled by different algorithms. The specular part goes on the classical ray tracing method, while the diffused one will be picked up by a Ma_rkov process which relates the diffused coupling between the room boundaries. The first order diffused contribution is imputed to the different receivers by taking into account the exact geometrical path (i.e. from the impact of the ray on the boundary to the receiver location) while further diffused contributions of a given surface are all radiated from its barycentre. The balance between these two models is controlled on each surface by a diffusion coefficient that is proper to the material or geometrical specificities of the boundary. The diffusion coefficient, together with the absorption coefficient, may be measured using the method described in [warusfel92].

Figure 2. Reflection model used in the predictive environment

Inversion Method [Dauchez92][Bastia93]

The acoustical specifications are expressed by the values of sound energy cumulated in the following time intervals, the transitions beeing smoothed by weighted functions : [0;20], [20;40], [40;80] (msec). This description may be seen as a simplified time distribution of the energy on different receiver locations, and could be derived from given values of classical acoustic criteria (Dir/Ref, C40, C80, Edt...). The analytical formula that links the material characteristics and the energy quantities is derived from the observation of specular reflections up to order three and from the first order diffused reflections. Equation 1 describes the energy value for a given receiver r and a given time interval k in terms of cumulated contributions of first, second and third order rays, the term R_rk integrates the remaining contributions of upper order rays or from non first order diffusion contributions.

Eest_rk = D^I_rk + D^II_rk + D^III_rk + R_rk (1)

For example the term D^III_rk may be written as the sum of elementary incident contributions IS_rkl_mn that were specularly reflected by the ordered triplet of material l, m, n plus the first order diffusion contribution ID_rkl_mn of the last material n of this triplet (Equ. 2).

D^III_rk = l_mn ( IS_rkl_mn (1-l)(1-dl)(1-m)(1-dm)(1-i>n)(1-dn) +

ID_rkl_mn (1-l)(1-dl)(1-m)(1-dm)(1-i>n)(dn) ) (2)

From this analytical formula we may derive an error function between the desired Eobj_rk and the estimated Eest_rk simplified time and spatial distributions of the early energy (Equ. 3). The material coefficients are optimized by a classical gradient algorithm aiming at minimizing the error function J. In order to limit the convergence domain, the function J may integrate a term that constraints the global absorption of the room to be constant.

J = _rk ( w_rk * [ Eest_rk - Eobj_rk ]²) (3)

The optimization process is organized according to the following steps. The whole process (a-b-c) may be repeated in order to verify the consistency of the analytical formula.

a) Execution of the prediction program.
b) Extraction of the analytical formula and of the error function.
c) gradient iterative algorithm until error converges to a local minimum.

Validation on an Existing Room

The validation consists of estimating the material characteristics of an existing room (Auditorium du Louvre in Paris) from the knowledge of its shape and of the time distribution of energy measured at different receiver locations. The geometrical model is discretised in 200 plane surfaces and is supposed to be made of 11 different materials. The acoustical specifications are cumulated energy in the above mentionned time intervals measured on 6 omnidirectionnal microphones located in the audience area. A directive source (DI = 7dB) is located at the middle of the stage and is pointing towards the audience.

Table 1. illustrates the optimization process for the 1khz octave band. At the initial step all materials are declared with the same values of absorption and diffusion (0 = 0.25 and d0 = 0.1). After 2 iterations of the complete process, the estimation error converges to a local minimum which module is 1/8 of the module of the initial error. The different material characteristics of the model have converged to specific values which may be compared with the actual material of the existing room. The estimated absorption coefficients seem relevant except for the surfaces covered with carpet. The diffusion coefficients show good tendancies although the estimated values are caricatural. An interesting observation is for the surfaces corresponding to the coffered ceiling and the audience : there diffusion coefficients converge to high values which compensate for the fact that they were approximated in the model with flat surfaces.

model
material Mat1 Mat2 Mat3 Mat4 Mat5 Mat6 Mat7 Mat8 Mat9 Mat10 Mat11
alpha0 initial step
alpha2 final step 0.25
0.01 0.25
0.8 0.25
0.01 0.25
0.3 0.25
0.42 0.25
0.01 0.25
0.25 0.25
0.21 0.25
0.01 0.25
0.2 0.25
0.28
alpha0 initial step
alpha2 final step 0.1
0.05 0.1
0.5 0.1
0.43 0.1
0.2 0.1
0.85 0.1
0.2 0.1
0.2 0.1
0.05 0.1
0.85 0.1
0.2 0.1
0.1
room material stone carpet wooden
stage absorb.
doors seats concrete mixing
concrete
absorb.
panel wood
panel coffered
concrete projec-
tion
screen metallic
struct.

These results are encouraging, but some considerations on gradient algorithms must be outlined. The error function presents several minima and, therefore, it does not ensure a stable result of the optimization procedure. Different parameters influence the convergence process: the number of receivers, the choice of the time intervals and weightings, or the integration of practical or acoustical constraints in the error function. However, according to the definition of the error function, each local minimum appear to represent an alternative architectural solution that satisfies the acoustical specifications.

Conclusion

An example of optimization procedure of architectural parameters has been presented and discussed. The basic idea is to derive from a prediction process a set of analytical formulae that link the acoustical quantities with the architectural parameters under optimization. It could be shown that, given the geometry and simple acoustical specifications, the material characteristics could be automatically optimized. Although the stability conditions of the method have not been fully investigated, these preliminary results allow to envision further developments that would help for the design of concert halls, the automation of a variable acoustic hall, or the tuning of an electroacoustic system.

References

[Schroeder80] M.R. Schroeder, D. Hackman Iterative calculation of reverberation time Acustica 45 (1980)

[Malcurt86] PhD thesis. Université de Toulouse. 1986

[Martin89] J. Martin, J.P. Vian Binaural sound simulation of concert halls by mean of ray tracing method. ICA89. Vol 2

[Blauert90] J. Blauert, H. Lehnert, W. Pompetzki & N. Xiang Binaural room simulation Acustica 72 (1990)

[Raynal90] Détermination approchée de la relation entre les paramètres de variabilité acoustique de l'Espace de Projection de l'Ircam et les caractérisations acoustiques de la salle. Université du Maine, Le Mans 1990.

[Warusfel92] Validation of a computer model environnment for room acoustics prediction Proceedings ICA Beijing 1992

[Dauchez92] N. Dauchez Optimisation des coefficients d'absorption. Rapport de stage. Université Valenciennes 1992

[Bastia93] J. Bastianelli Problèmes d'inversion en acoustique des salles Rapport Ecole Polytechnique 1990.

[Rindel94] J.H. Rindel, C. Lynge, G. Naylor, K. Rishoj Combining a computer model and commercial sound studio hardware for on-line 'sketch' auralization. W.C. Sabine Cent. Symposium Cambridge 94. pp105-108

[Jot95] J.M. Jot, V. Larcher, O. Warusfel Digital signal processing issues in the context of binaural and transaural stereophony Proceedings of 98th AES Paris 1995..

model material	Mat1	Mat2	Mat3	Mat4	Mat5	Mat6	Mat7	Mat8	Mat9	Mat10	Mat11
alpha0 initial step alpha2 final step	0.25 0.01	0.25 0.8	0.25 0.01	0.25 0.3	0.25 0.42	0.25 0.01	0.25 0.25	0.25 0.21	0.25 0.01	0.25 0.2	0.25 0.28
alpha0 initial step alpha2 final step	0.1 0.05	0.1 0.5	0.1 0.43	0.1 0.2	0.1 0.85	0.1 0.2	0.1 0.2	0.1 0.05	0.1 0.85	0.1 0.2	0.1 0.1
room material	stone	carpet	wooden stage	absorb. doors	seats	concrete	mixing concrete absorb. panel	wood panel	coffered concrete	projec- tion screen	metallic struct.