Toronto, Canada International Symposium on Room Acoustics

2013 June 9-11

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Sound strength driven parametric design of an acoustic shell in a free field environment

Marco Palma (marcopalma.posta@gmail.com) Maddalena Sarotto (maddalena.sarotto@hotmail) Tomás Méndez Echenagucia (tomas.mendez@polito.it) Mario Sassone (mario.sassone@polito.it) Department of Architecture and Design (DAD) Politecnico di Torino Castello del Valentino, viale Mattioli 39, 10125, Torino, Italy Arianna Astolfi (arianna.astolfi@polito.it) Department of Energy (DENERG) Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

ABSTRACT

Focusing on the issues of sound propagation in a free field condition and on the concept of uniform sound energy in an outdoor performance environment, our research aimed to develop a computer aided process for the generation of reflective acoustic surfaces to be used as concert-shells, a computational design tool for acoustic form-finding. The project is ultimately aimed to investigate the acoustic potential of complex and doubly-curved surfaces through the analysis of the Total Relative Sound Level / Strength parameter (G), with reference to the proposed values set by M.Barron, based upon the source-receiver distance and the subsequent subjective judgments on loudness. A simplified and fast ray tracing acoustic simulation algorithm was developed in combination with parametrically controlled shape variations of the reflective surfaces. Sound energy uniformity evaluation function considering the direct and reflected sound components were written in order to define and evaluate the rate of distribution uniformity of sonic energy over the audience. This evaluation function was used in a genetic algorithm, that enabled us to explore a wide set of surface morphologies and to finally isolate the fittest one to our specific uniformity requirements. At the end of the genetic search, an acoustic simulation plug-in called Pachyderm was employed with both NURBS and mesh-based acoustic simulations, in order to validate the genetically selected surfaces with specific reference to G values. A further step of result data visualization and human selection was necessary to compare the output data and to evaluate the final surfaces from an architectural perspective.

1 INTRODUCTION

Outlining a reliable script-based process for the finding of the optimal shape for a sound-reflective shell to be located outdoor for musical performances was the main goal of this research, together with the understanding of the acoustic behavior of doubly curved and complex geometries. As a direct effect of the form-finding procedure, an optimal sound rays uniformity condition over the audience was considered to be the fundamental basis for the

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achievement of an acceptable Relative Sound Level or Strength (G) at mid-frequencies (500 Hz). A Genetic Algorithm (GA) was used, since the presence of the shell in an outdoor (reflection-free) environment and the consequent absence of lateral walls and ceiling structure addressed the research towards a massive optimization of sound energy by means of its intrinsic geometrical configuration. Since the absence of scientific data related to minimum values of G in outdoor spaces, M. Barron’s G minimum values for concert halls, considered as a function of the source-receiver distance, were used as a main reference 1. As shown in Figure 1, considering the energy optimization ambitions here pursued, Barron’s curve was set as both the minimum and maximum constraint for G values, thus expressing G optimal values. From this perspective, every G value above Barron’s curve was considered an energy waste, while every G below the minimum curve, in accordance with Barron’s analysis, was to be considered insufficient.

Figure 1: Barron’s curve was set as a both minimum and maximum constraint for G values, thus expressing G optimal values.

The research was related to a specific outdoor environment, a public square, in Barolo, Italy where music venues are usually held. Figure 2 shows Piazza Colbert during Bob Dylan's performance at Collisioni Festival 2012. The project shell position was assumed to be the same as the typical concerts stage one. The irregular topography of the square excluded a symmetry condition for the audience displacement, thus influencing the form-finding process.

Figure 2: Piazza Colbert in Barolo during Bob Dylan's performance at Collisioni Festival 2012.

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2 RELATED RESEARCHES

The utility of scripting and user customizable software interfaces for the generation of complex geometries and for specific room acoustic simulations was clearly introduced by A. Van der Harten 2. The use of coding and genetic algorithms for the design and optimization processes of acoustic diffusing elements has been widely discussed by T. Cox and P. D’Antonio 3, while S.Sato et al. have been investigating for several years the potential of genetic algorithms and computational tools for the shaping of theaters and concert halls 4. Analogous research topics have been deeply explored by T. I. Mendez et al., who specifically focused on the acoustic and structural qualities of folded plates and shells 5.

3 TOOLS AND PROCEDURES

Two simulation models were produced. Model 1 was used for a geometrical evaluation of rays distribution uniformity, Model 2 for proper, physically reliable, acoustic simulations.

Figure 2: Schematics of the followed workflow and basic operations executed in the research.

Figure 2 schematically introduces the followed workflow and the basic operations executed in the research, remarking the fact that the entire sequence of operations was executed within the same software platform, Rhinoceros3D ™, a NURBS three-dimensional modeler 6.

3.1 Model 1

As shown in Figure 3, Model 1 is a basic geometrical ray-tracer, a three elements system composed of an omnidirectional source, a receiving surface (the audience) and a reflecting surface (the shell). Since the model is embedded in a parametrically controlled environment, the shell morphology can be altered according to the limits imposed to its geometrical domain. In this case the acoustic potential of a single free-shaped surface was investigated. The shell surface is generated by an array of points in the three-dimensional space, while the points spatial coordinates, which ultimately constitute the parametric variables, define its domain. Model 1 output is a numerical count of the rays distribution over the receiving surface, considering both the first order of reflections and the direct sound rays. The attenuation of the reflected rays, due to the absorbing coefficient of the reflecting surface, was considered. Since each variation of the surface points coordinates produced a morphological variation of the reflecting surface and, consequently, a different distribution of the sound rays over the receiving surface, an objective criterion was needed in order to evaluate each generated morphology. Indeed the GA, taking advantage of the parametric capabilities of the model, was used to explore the given geometric domain and to rate each morphological variation by means of a fitness function.

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Figure 3: Screen capture of Model 1, a basic geometrical ray tracer composed of a source, a receiving surface (the audience) and a reflecting surface (the shell).

3.1.1 Surface parameterization and search space definition

Given the coordinate system shown in Figure 4, the reflecting surface was parametrically generated by an ordered array of three-dimensional points, grouped into curvilinear clusters along the X direction. A previous form-finding session highlighted that a concave surface would have been more performative. The results of that session provided the initial points spatial coordinates configuration. To set the points local search space, each surface point was thought to be the geometric center of a virtual square belonging to XZ plane, with edge length equal to a user-defined d value, meaning that each point was free to move of ± d/2 along the X and Z axes. Considering that in this case d was set equal to 2 m, each possible combination of point coordinates defined a precise surface morphology. The control points were located within a volumetric constraint measuring 12 m x 14 m x 12 m (approximately the occupancy of a medium sized outdoor stage), defining the global search space.

Figure 4: Shell surface parametric generation and visualization of the local search space.

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3.1.2 Genetic algorithms (GA) general behavior

Without focusing on the technical details of GA, a synthetic definition of the functioning procedure is introduced. Genetic Algorithms (GA) are computational tools for the research of solutions to optimization problems, firstly introduced by John Holland at the University of Michigan in the 1960s 7. As stated by Koza “The GA is a highly parallel mathematical algorithm that transform a set (population) of individual mathematical objects [...], each with an associated fitness value, into a new population (i.e., the next generation) using operations patterned after the Darwinian principle of reproduction and survival of the fittest and after naturally occurring genetic operations (notably sexual recombination)” 8.

Figure 5: Schematics of Genetic Algorithms (GA)

Given a specific problem, the GA is able to produce a high number of possible solutions to the problem (individuals) and rate them, according to the result that the fitness function provides for each individual analysis. In each generation the fittest individuals are chosen for reproduction, and their genes mixed in order to breed the next generation population. Generation by generation, the population becomes fitter and fitter to the required solution, until the solution is found (or approximate) and the process stops (Figure 5).

3.1.3 Fitness function and casted rays uniformity

In order to evaluate and rate the shell surfaces, their acoustic behavior was related to a fitness value considering both the direct and the reflected sound components. With reference to equation (1), after the receiving surface was subdivided in a grid of n sub-surfaces, a numerical count of direct (D) and reflected rays (R) was made for each sub-surface; a value equal to 1 to was assigned the first and a value equal to 1*k (k = 0.2) to the latter, simulating an absorption coefficient. In order to conceptually simulate the best acoustic uniformity condition for the audience, the reference value J was set as ideal ray count for all the receivers. E ultimately represents the total deviation from the ideal hypothesized uniformity condition.

(1)

Equation (1) was chosen as fitness function and E was considered as an error value to be minimized by means of the GA. A minimum reflected rays number constraint was coupled with (1) in order to improve the average rating standard of individuals in the initial population. Since the major energetic contribution of the direct sound to the receivers near the stage, the GA selection criterion was expected to privilege those shell surfaces able to concentrate reflected rays far from the stage.

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3.2 Model 2

To acoustically evaluate the shell surfaces, an accurate simulation model was needed. This could take advantage of Pachyderm, an open source acoustic engine embedded into the Rhinoceros 3D™ modeling environment 9. All commands and setup were directly launched and edited from the Iron Python script editor, allowing a high degree of customization and control over the simulation software. Model 2 is basically composed of a three-dimensional acoustic model of the environment (necessary to properly simulate the real acoustic conditions) and of a 100% sound absorbing bounding box containing the entire scene (used to simulate the free-field conditions in the virtual environment). Stage area was 12 meters deep by 14 meters long, covering a total area of 168 square meters. The receiving surface had an irregular trapezoidal-like shape, with the short edge close to the stage, with a total area of 1200 m2 and with a central axis measuring 36 m. Two separate codes were used in sequence to complete Model 2 and to edit and launch the acoustic simulations.

Figure 7: Screen capture of Model 2

3.2.1 Source position and receivers density

The first code was needed to automatically set the source and the receiver points in the desired number and position. The omnidirectional source was positioned 1.25 m above the stage floor and 3.5 m from the front edge of the stage, along the central axis. Receivers density was set at 1 every 3m2, for a total of 380 receivers, set at a height of 1.70 m from the floor plan (a standing audience was imagined). A higher number of receivers (1 receiver per m2) would have better simulated a real standing audience condition, but too much computational power would have been required.

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3.2.2 Pachyderm setup

A series of pre-simulations were run in order to find the optimal software setup related to the utilized machine 10. For all the simulations a “Ray Tracing + Image Source” simulation algorithm was used, an amount of 500'000 casted rays, a cutoff time of 2'000 ms and reflection order set to 1. With this setup the average simulation time with 380 receiver points was around 45 minutes. 4 RESULTS

More than 50 shapes were produced by Model 1 and analyzed with Model 2. Among those, 3 final shells were chosen according to their G values.

4.1 Model 1 results

The morphologic differences among the generated surfaces were mainly due to variations in the initial settings of Model 1. The bigger roles were played by the surface parameterization settings (local domain of point coordinates and number of control points defining the surface), together with a constraint imposed to the fitness function (minimum reflected rays number). The chosen surfaces morphologies are shown in Figure 8.

Figure 8: Perspective view of A, B and C (from left to right) surfaces morphology

Both surfaces A and B were generated starting from a 5 by 5 grid of points, but their degree of freedom was one the half of the other: A points could move inside a 1m by 1m domain space, while B points domain space was the double wide. C points had the same search domain of B points, but they were defined by a 3 by 5 initial grid, in order to exclude the presence of S-shaped sections in the XZ plane. Surfaces A and B had the top control points fixed in the positive direction, in order to strictly respect the global search space boundaries, while Surface C could extend its top section both in the X and Z directions over the upper limit of the research domain. Their resulting morphologies are directly related to the initial conditions. In general, bigger the search domain, more complex the resulting shape (B), and higher the initial restrictions, smaller the range of possibilities (A). The number of surface control points plays a major role: given the same search domain, less control points will result in a simpler surface morphology (C). The consequent distribution of first reflected rays over the audience is shown in Figure 9.

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Figure 9: Plan view of the three chosen surfaces first reflected rays distributions over the audience (A, B and C).

4.2 Model 2 results

G value at medium frequencies (500 Hz) was the chosen evaluation parameter. The output of Model 2 permitted a local comparison between G actual values, derived by the acoustic simulation of each single shell surface, and G optimal values, derived by Barron's function, at each receiver position, allowing a precise mapping of satisfied and unsatisfied receivers. Each receiver that showed a higher G value than Barron's optimal values was considered satisfied. On the other hand, not all receivers with lower values were considered unsatisfied: assuming 1 dB as the reference value for the G Just Noticeable Difference (J.N.D.) 11 and assuming Δ as the difference between G optimal value and G actual value at each receiver position, all those receivers showing 0 dB < Δ < +1 dB were considered satisfied. Referring to Figure 10, A1, B1 and C1 represent the mapping of G values over the receiving surface expressed as a color gradient. A2, B2 and C2 express an intuitive visual feedback of the receivers satisfaction rate: the more the fading to blacks, the higher the distance from Barron’s optimal G values. Only 100% white receivers have to be considered as fully satisfied. Surface A presents the highest satisfied receivers percentage (54%) with a radial and uniform decreasing of G from the stage to the furthest seats. Surface B counts the 51% of satisfied receivers, but shows a discontinuous G values distribution over the audience. Surface C covers the furthest distance from the stage with optimal sound strength values, but only satisfies the 44% of receivers.

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Figure 10: Colour map of G values over the receiving surface (A1, B1 and C1) and spotted visualization map of the receivers satisfaction rating (A2, B2 and C2) of A, B and C surfaces.

5 CONCLUSIONS

The usage of concave shaped elements for acoustic purposes is often discouraged. On the other hand, this research frames them to be the most performative solution for passive acoustics in outdoor or reflection-free environments. Since the presence of a limited amount of available energy and the simultaneous absence of a built enclosure, acoustic diffusion and convex shapes need to be avoided. Vice-versa energy concentration has to be pursued, remarking the importance of an accurate control over concave shapes, so that acoustic focus phenomena are excluded 12. A close interrelation between the GA research behavior and the definition process of the final shell surfaces was noticed. As a matter of fact, the initial approximate concave shape and the limits of the first reflections covered area were defined during the first few generations, while the biggest part of the genetic search was committed to the modelling of the shell small scale curvatures and to the definition of the related minor adjustments of the casted rays distribution over the single receiving sub-surfaces. Considering Surface A as the most performative in terms of both satisfied receivers percentage and wave front uniform distribution, its morphological configuration would cover with an optimal G a distance of 30 m distance from the stage covering an audience area of about 600 m2 .

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REFERENCES

1 M. Barron, When is a concert hall too quiet? (19th ICA, Madrid, 2007)

2 A. v.d. Harten, Customized Room Acoustics Simulations Using Scripting Interfaces (161th ASA, Seattle, 2011)

3 T.J. Cox, P. D’Antonio, Acoustic Absorbers and Diffusers.Theory Design and Applications- Second Edition (Taylor&Francis, Abingdon, 2009)

4 S.Sato, K. Ohtori, A. Takizawa, H. Sakai, Y.Ando, H. Kawamura, Applying genetic algorithms to the optimum design of a concert hall (Journal of Sound and Vibration, vol. 258, pp. 517-526, 2002)

5 T.I. Mendez, A. Astolfi, M.J. Jansen, M. Sassone, Architectural, Acoustic and Structural Form (Journal of the International Association for Shell and Spatial Structures, vol. 49, no. 3, pp. 181-186, 2008)

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University of Michigan, 1975) 8 J.R. Koza, Genetic Programming: On the programming of computers by means of natural

selection - First Edition (The MIT Press, Cambridge,1992, p. 18) 9 A. v.d. Harten, Pachyderm Acoustical Simulation: An Open Source Geometrical Acoustics

Laboratory. (http://www.perspectivesketch.com/pachyderm/) 10 The machine runs a 2.53 GHz Intel Core 2 Duo processor with 4 GB 1067 MHz DDR3 of

available memory 11 ISO 3382:1997, Acoustics — Measurement of the reverberation time of rooms with reference

to other acoustic parameters (1997) 12 M. Vercammen, Sound Concentration caused by Curved Surfaces (PhD thesis, Eindhoven

University of Technology, 2012)