Excerpt from the Proceedings of the COMSOL Conference 2010 Paris Optimal Design of Slit Resonators for Acoustic Normal Mode Control in Rectangular Room Sergio E. Floody*1, Rodolfo Venegas2 and Felipe C. Leighton3 1Universidad de Chile, Facultad de Artes, Departamento de Música, Licenciatura en Sonido, 2University of Salford, Acoustics Research Centre, Ingeniería Civil en Sonido y Acústica, Universidad Tecnológica de Chile Inacap *Compañía 1264, Santiago, Santiago, Chile, [email protected] Abstract: The present article presents a method the supporting plate, the width of the supported to redistribute the acoustic modes of a plate and the distance between the nearest rectangular enclosure in the low frequency range extremes of the supported plates are denoted as using slit resonators. The objective of the present x1, x2 and x3 respectively. This is shown in Fig1. work is to compare different strategies of optimal The resonant frequency and absorption design in order to determine the dimensions of characteristics of this type of devices have been the resonators. The method of the finite elements studied by Pedersen [1]. Mechel [2] has included will be used to model the acoustic physical viscous and thermal losses to this formulation. behavior of the room. In addition a neuronal Geometric modifications and the effects of network will estimate the loudness level grazing flow have been accounted for in [3-5]. perceived by the auditor. The different strategies of design are: First, a strategy of design will be implemented based on the minimization of the fluctuations of the sound level pressure. Second, the optimization will be based on the diminution of the variations of the loudness level. Finally, two methods of optimization, genetic algorithm x and differential evolution will be compared. The 2 three different strategies from optimization will be compared generally and of it will determine x3 the design variables that are critics in this x1 process. Keywords: Slit resonators, normal mode control, optimal design. 1. Introduction Figure 1. Dimensional characteristics of slit resonators. The sound field of an enclosure is characterized by the interaction between the The room dimensions’ optimization has been source and the acoustic properties of the room. studied by Cox et al [6] and Zu et al [7-8]. In The frequency response and the balance of the these works, the fluctuations of the sound timbre depend on the geometry and the materials pressure level have been minimized. Instead, of the enclosure. The objective of this article is Floody and Venegas [9-10] have proposed the to decrease the effects of the resonances at low optimization of the room dimensions based on frequencies and to suitably distribute the normal minimizing the loudness level fluctuations. In modes of vibration using optimal slit resonators this work, these two approaches are used to which dimensions are optimized. This type of optimize the dimensions of slit resonators. Two resonators is of great interest in arquitectural different optimization algorithms are considered acoustics due to easy construction. and compared. These correspond to the genetic Slit resonators are composed by a periodic algorithm and the differential evolution structure of T-like plates. It can be described algorithm. using three physical dimensions. The height of A cubical enclosure of 5.1 m side with and its respective boundary condition can be solved without slit resonators is considered as a case of by using the finite element method [12]. study. The source and the reception point are Triangular linear lagrangian elements have been located in opposite corners. Vertically-oriented used to model the pressure in the 2D part of the slits are considered. Their length coincides with equation. The finite element formulation for the the height of the room. The sound field is equation Eq. (4) is the following eigenvalue modeled for frequencies ranging from 20 Hz to problem solved with Comsol® Multiphysics: 200 Hz using a mixture between the finite element method and a classic analytical solution. 2 (6) Kφ= kxy M φ This choice has been made to decrease the computational cost. Where, K and M are the acoustics stiffness and mass matrices, and φ is the eigenvector. 2. Theory and Governing Equations Thus, the natural frequencies can be calculated using equation Eq. (7). Finally, the sound 2.1 Formulation of the Problem and pressure at any point r inside the enclosure Application of the Method of Separation of produced by a point source located at r0 for a Variables frequency ω is the result of the combination of the solutions of the equations Eq. (3) and Eq. (4) The enclosure is excited by a flat spectrum as is shown in Eq. (8). point source. This problem is governed by the Helmholtz’s equation when considering ω =k2 + k 2 (7) harmonic solution. This is shown in Eq. 1 along nxy n z xy z with the respective boundary condition. In order to simplify the problem the stationary solution in ∞ ∞ An n (r,, r0 ω) p r,, r ω = xy z (8) the frequency domain will be studied only. ()0 ∑∑ 2 2 nxy=1 n z = 0 ω− ω nxy n z 2 2 2 ˆ (1) An n (r,, r0 ω) = jS0 ρ 0 c ω( φr, n cos(kz z)) ∇P + k P =,0 ∇P ⋅ n = 0 xy z xy ×(φr, n cos()kz z0 ) By using the method of separation of variables 0 xy the following equations and boundary conditions Where ρ is the density of the air and U is are obtained 0 0 the vibration velocity on the source surface. P x,,, y z= P x y P z (2) ( ) xy ( ) z ( ) 2.2 Determination of the Loudness Levels Using eural etworks The dependency in z is given by: The loudness may be defined as the sensation ∂ 2 P ∂P ∂P that corresponds most closely to the sound z +k2 P = ,0 z = z 2 z z (3) intensity of a stimulus [13]. An equal-loudness ∂z ∂z z=0 ∂z z= L z contour is a curve that ties up sound pressure levels having equal loudness as a function of And for the (x,y) dependency: frequency. In other words, it expresses a frequency characteristic of loudness sensation. In ∂ 2 P ∂ 2 P xy + xy +k2 P =,0 ∇ P ⋅ nˆ = 0 (4) this work a loudness model, implemented using ∂x 2 ∂y 2 xy xy xy an artificial neural network, has been developed from the equalloudness-level contours data It should be satisfied that: presented in reference [14]. The procedure described in reference [15] has been followed up. k2= k 2 + k 2 (5) The presented model aims to accurately calculate xy z loudness level at low frequencies. The artificial neural network [16] has been trained with the The equation and the boundary condition Eq. quasi Newton backpropagation algorithm (3) have a well-known solution [11]. Eq. (4) and considering 3000 epochs and an objective goal of 10-5. The final configuration corresponds to a functions, dimensional restrictions are imposed three layer feedforward neural network with 5 to the design variables (0.01m ≤ xi ≤ 0.60m, i = neurons in the hidden layer and 1 output neuron. 1, 2, 3). The posed optimization problem is The transfer function of the hidden layer is a characterized by a strong nonlinear interrelation sigmoidal hyperbolic tangent function whereas is between the variables and the fitness functions. linear for the output layer. The inputs to the The functions have many peaks and dips. This neural network are frequency and sound pressure makes the solution oversensitive to the level. The output is the respective loudness level. dimensions of the slit resonators. For this reason the frequency response curves are smoothed out 2.3 Objective Functions using the Savitzky – Golay method. Finally for both objective functions the positions of source The optimization techniques are used to and receiver will be located in opposed corners determine the best possible design in engineering of the room, because this represents the worse problems. In this case they are used to determine case. the optimal dimensions of a set of slit resonators. Since a flat room response is the goal to predict 3. umerical Simulations and Results the geometric modifications, a flattest sound frequency response could be well considered as 3.1 Comparison of Results between Genetic the best frequency response for reference, even Algorithm and Differential Evolution though a perfect flat response is practically impossible to get due to the maximums and Using the functions objectives previously minimums caused by sparsity of the room detailed. A set of simulations has been run to modes. evaluate the best possible strategy of optimal Under this consideration, the chosen design. Five simulations have been made using objective function is the square root of the sound genetic algorithm [17.18] with 100, 200, 300, frequency response deviation from a least square 400 and 500 generations, for both objective straight line drawn throughout the spectrum as functions. After that the differential evolution proposed in [6]. algorithm [19, 20] has been used with the same number of generations for both objective 1 2 functions. The results of the simulations are (9) f1 ()x = ∑[ ()()ip 1 i +− afafL 0 ] given in Tables 1 to 5. Where Gen is the number i=1 of generations. f1(x)Opt is the optimum value of T the objective function based on the sound Where x = [x1, x2, x3] , is the design vector pressure level. f2(x)Opt is the optimum value of based on the dimensions of the slit, is the the objective function based on the loudness number of points , fi is frequency , Lp(fi) the level. f1(x)Asc is the value of the first objective sound pressure level, a1 and a0 are the function when the optimization process is based coefficients of the linear regression.