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Acoustic design by simulated annealing algorithm Nicolae Cretu*,[1], Mihail-Ioan Pop*, Ioan-Calin Rosca**[1] *Physics Department, Transilvania University, Eroilor nr. 29, Brasov, , Romania **Department of the Strength of Materials and Vibrations, Transilvania University, Eroilor nr. 29, Brasov, , Romania [1][1] Corresponding author address:

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Abstract The work extends the matrix method formalism, by using a supplementary computational method based on a simulated annealing algorithm, with the aim to design acoustical structures, especially acoustic filters. The algorithm introduces a cost function, which is minimized by the simulated annealing algorithm. Also, some numerical computations have been carried out to design some special acoustic filters and an experimental analysis of the designed acoustic filters is provided to test the validity of the method.

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Simulated annealing algorithm Simulated annealing (SA) algorithm represents an intelligent random search useful to approach by computational way the global optimization of a function depending of many parameters. Other methods: -genetic algorithms -neural networks

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Simulated annealing algorithm The name of algorithm comes from the method of metal annealing, which assume small steps in temperature and long time spending in the vicinity of the phase transition temperature of a solid.

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Simulated annealing algorithm It is possible to consider the evolution of the solid to the equilibrium state as a succession of states, for which the difference between the corresponding energies satisfy the condition,each energy corresponding to an admitted spatial configuration. Whole process can be considered as an iterative process in which are accepted only the states which satisfy the condition of the lowering energy. If the condition for two successive states is valid, the new configuration is accepted and will be used as starting point of the next step in the transition process of the sample to the final equilibrium state. The case will be approached probabilistically by introducing of the transition probability between two configuration states. This probability will be compared with a random generated number uniformly distributed in the interval (0,1). If it is less than P( E), the new configuration is accepted and retained, if not the original configuration is used to start the next step. This step confers to the algorithm the stochastic property.

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Simulated annealing algorithm In general case, in place of the energy the simulated annealing algorithm uses a cost function, which depends on the defined configuration parameters and starting from a given configuration, looks for a final configuration, following the global optimization procedure at a given temperature parameter. The temperature parameter in the new context become a control parameter, which has the same units as the cost function, will be also considered in the optimization process, so by lowering the temperature parameter by slow stages until the system “freezes” and no further changes occur.

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Simulated annealing algorithm The traveling salesman problem. Given a list of N cities and a means of calculating the cost of traveling between any two cities one must plan the salesman’s route which will pass through each city once and return finally to the starting point, minimizing the total cost. Problems with this flavor arise in all areas of scheduling and design. The problem is to predict the expected cost of the salesman’s optimal route averaged over some class of typical arrangements of cities, and estimating or obtaining bounds for the computing effort necessary to determine that route. G:\CHILE\salesman_applet\start.html,

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Simulated annealing algorithm Initialize: X,T Modify X=X+dX X: better => accept X; X: worse => accept X with P=exp(- C/T) Decrease T=fT, f=0.995 Stop if condition met

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The Simulated Annealing algorithm

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Transfer matrix formalism

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Transfer matrix T=D*P

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Transfer matrix formalism

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We consider the multilayered medium with layers, each layer being characterized by an acoustic impedance, a thickness and a speed of sound. All the parameters characterizing this medium are arranged as vectors Z, a and c,. The multilayered medium will be fully described by the vector X=X(Z,a,c), formed by the concatenation of the three vectors given above. It follows that the transfer matrix is a function of the parameters of the multilayered medium: and also depends on the angular frequency.

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Algorithm implementation Problem: Given a target filter F(f), where f is the frequency in a given interval [f min, f max ] and a multilayered medium with n layers, (n fixed), determine the parameters X(Z i,a i c i ) i=1…n of the multilayered medium such that |A out (f)| is as close as possible to F(f) for f Є [f min, f max ] Cost function: computed over a set of frequencies f 1,f 2,…f N in the given interval Minimisation problem: The multilayered medium is coupled to an input medium with impedance Z in and an output medium with impedance Z out.

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Algorithm implementation Layer parameters are chosen in typical intervals Typically n=10 or n=20 ; bigger n gives better solutions with slower computation The base algorithm is modified to an elitist form: it returns the best solution ever found

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Computer simulations High-pass filter

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Computer simulations Linear filter Low-pass filter

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Algorithm analysis Two different results obtained for n=10 layers (above) and the time variation of the corresponding cost function during a run of the algorithm

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Experimental analysis A designed low-pass filter: cutoff frequency Hz

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Experimental analysis Puls injection from the impact hammer-designed filter

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Experimental analysis Pulse injection from the impact hammer in aluminum bar

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Experimental analysis Direct transfer measurements for designed filter Direct transfer measurements for the aluminum bar

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Conclusions This work is an extension of the matrix method application and presents an implementation of simulated annealing to design one-dimensional acoustical structures based on the simulated annealing probabilistic meta-algorithm. Starting from an imposed transfer characteristic, the implemented method finds the optimum design of a multilayered arrangement to obtain the best fit of the acoustic behavior. The algorithm used appears to be quite good for designing acoustic filters if the parameters of the layers making up the filter are well chosen. One advantage of the method is the relatively short time (or low number of iterations) it takes to find a filter structure, given a target filter. The method is adaptable to the design of almost any filter, specifically smooth filters or filters with few discontinuities.

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Thanks for your attention

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