In order to verify the promising theoretically performance of the optimisation, a ... Genetic algorithms (GA) is a robust search scheme based on organic evolution.
High degree of freedom muffler optimisation using genetic algorithms: experimental verification S. Pottie* and D. Botteldooren*
University of Gent, INTEC, acoustics group. St-Pietersnieuwstraat
41 B-9000 Gent, Belgium
Abstract: This paper reports on an experimentat
verification of a non conventional muffler design. The optimal design is searched using a genetic algorithm (GA). The degrees of freedom used to describe the structure is far beyond the usual set. The performance of the mufflers is measured by the theoretically achieved sound reduction and by the ease of construction of the resulting muffler. In order to verify the promising theoretically performance of the optimisation, a good performing design is built and tested in the laboratory. General agreement between the calculations and the measurements is obtained.
INTRODUCTION This paper describes the design of a muffler on a duct to create a high insertion loss for a given specific sound power spectrum. Two additional conditions are imposed: firstly, the impedance seen at the inlet and outlet side of the source and the end impedance are assumed to be finite and are located close to the muffler, secondly, the available space for optimisation is limited to a region at two sides of the duct. The duct area is not available for optimisation to ensure a minimal pressure drop. In the simulation an inlet duct of 30 cm is modelled. The outlet duct is 20 cm long. The regions at both sides of the duct are 90 cm long and 10 cm wide.
THE OPTIMISATION
PROCESS
Genetic algorithms (GA) is a robust search scheme based on organic evolution. It simultaneously evaluates many points in the search space and is more likely to converge towards the global solution. The GA can be seen as a randomised search technique which uses a random choice as a tool to guide a highly exploitative search trough a coding of a parameter space. Given a set of points in a search space and a value or cost for each point, the GA selects from that set points with a probability proportional to their value or cost. It then uses genetically inspired operators of cross-over and mutation to generate a new set of points to test. Details of the method can be found in Goldberg (I). Prior to the final optimisation different coding alternatives were analysed (2). Best results were obtained with a coding that describes the size and the location of holes and plates in the interior of the muffler. The space available for optimisation is assumed to acoustically hard by default. The holes defined in the chromosomes are cut out of the structure, next the plates are added to the resulting structure. GA theory contains several parameters that must be tuned for optimal performance: the initial population size n, the mutation probability Pm, the fitness scaling parameter Cs, the crossover and mutation type and the replacement mechanism. For the application studied in this work, a good set of algorithm parameters was searched. The set of algorithm parameters used in the optimisation below is: n=200, Pm=OS, Cs=1.3, diagonal crossover (a specific crossover type for structural optimisation which uses geometrical information) and fitter replacement. A source signal containing a Gaussian wavelet with central frequency of 800 Hz and a bandwidth (3 dB reduction) of 460 Hz was used to optimise the muffler within the constrains given above. The impedance at both ends of the duct were realised with a priori characterised absorbent material. To evaluate the acoustical performance (sound reduction at the outlet of the duct) of a particular muffler design a finite difference time domain mode1 is used [3]. The use of direct time domain simulation has the advantage that the test signals can he included directly, that transient requirements can he taken in account and that the usual performance evaluating quantities can be calculated directly. In addition to the acoustical performance a construction cost was introduced which reflects the ease of construction of the muffler.
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EXPERIMENTAL The best muffler design found after 250 generations man
bDEL;
VERIFICATION (i.e. 50000 function evaluations)
is shown in figure 1.
*x:5
FIGURE 1. Sound Pressure Level (dB) in a cross section of tKe optimised muffler
Figure 1 also shows the sound pressure level in a cross section of the duct with the optimum muffler fitted. The transfer function from inlet to outlet was calculated using the FDTD-model. The results are shown in figure 2.
FIGURE 2. Calculated (-)
and measured (- - - -) transfer functions
In order to verify the good theoretical performance , the optimised muffler was build and the noise reduction was measured in the laboratory. The casing and the blocks are made out of polymethylmetacrylate. The plates are made out of steel with a thickness of Imm. The impedance at both ends of the duct is realised with an absorbent. The input signal is generated with a sound generator and the sound pressure is measured with a spectral sound analyser. Figure 2 shows the calculated and the measured transfer functions in the frequency range of interest. There is a good general agreement between the both results. However, the measured transfer function in the lower frequency region is much smoother that the calculated transfer function. This could be due to an absorption phenomenon which is not taken into account in the theoretically simulations.
CONCLUSIONS The experimental results prove that it is possible to design noise control devices for the reduction of specific sound spectra using the high degree of freedom (HDOF) optimisation that is proposed. GA optimisation combined with FDTD-simulations seems to be successful for the implementation of the HDOF-optimisation. It is possible that in addition to the good results, the new approach possibly can lead to new insights that could be used in conventional muffler design.
REFERENCES 1. Goldberg, D.E., Genetic Algorithms in Search, optimisation and Machine Learning, New York: AddisonWesIy, 1989. 2. Medlund, M., Toepassing van genetische algoritmes bij vormoptimalisatie, eindwerk, 1997. 3. Botteldooren, D., Acoustical Finite Diflerence Time Domain Simulation in a Quasi-Cartesian Grid, J. Acoust. Sot. Am. 9523 13-2319, 1994
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