Acoustical topology optimization of Zwicker’s loudness with Pade-approximation
JungHwan Kook(Technical University of Denmark)
Nederland | Computer Methods in Applied Mechanics and Engineer
2012-07-10 | 바로가기
Cited by 13
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Computer Methods in Applied Mechanics and Engineering
Received 10 July 2012, Revised 12 October 2012, Accepted 31 October 2012, Available online 20 November 2012.
Junghwan Kooka, Jakob S.Jensenb, Semyung Wanga
a School of Mechatronics, Gwangju Institute of Science and Technology
b Department of Mechanical Engineering, Technical University of Denmark
Zwicker’s loudness is a conventional standard index for measuring human hearing annoyance and has been widely considered in many industrial fields for noise evaluations. The calculation of Zwicker’s loudness, which is needed for a wide range of frequency responses with a fine frequency resolution, using the finite element procedure usually requires significant computation time, since a numerical solution must be obtained for each considered frequency. Furthermore, if the analysis is the basis for an iterative optimization procedure such as a gradient-based acoustical topology optimization, this approach imposes prohibitively high computational costs. In this research, we propose a computationally-efficient approach to resolve the computational issue in the computation and optimization of Zwicker’s loudness. We present an efficient approach which combines the finite element method (FEM) with the Padé approximation (PA) procedure for obtaining Zwicker’s loudness and for applying it in a gradient-based acoustical topology optimization procedure applied to the design of acoustic devices to minimize Zwicker’s loudness. In this respect, the calculation of Zwicker’s loudness is represented by the PA. The main specific loudness considered as an objective function is evaluated using the PA procedure with a sufficient number of subintervals and expansion terms. An adjoint variable formulation for the design sensitivities that uses the advantages of the PA is used. We compare the performance of the proposed algorithm with the standard FEM in terms of accuracy and the CPU-time required for the calculation of Zwicker’s loudness. In addition, we also compare the optimized designs obtained by the proposed method to optimized designs obtained by the standard method in terms of objective values, optimized topology, and iterations or CPU-times needed for the optimization. Through several examples including 2-D and 3-D acoustics, the efficiency and reliability of using PA for computation and acoustical topology optimization of Zwicker’s loudness are compared and validated.
In this paper, it was shown that the PA procedure can be employed as an alternative approach for obtaining Zwicker’s loudness as the objective function, and as a formulation for analytical design sensitivities that uses the advantages of the PA in order to apply to a gradient-based ATO procedure.
A finite element procedure combined with the PA was developed for computation of Zwicker’s loudness. Two numerical examples of 2-D and 3-D acoustic systems were presented in order to demonstrate the proposed numerical procedure with different numbers of expansion terms and subintervals. It was demonstrated that the numerical procedure yields an accurate approximation of Zwicker’s loudness in general. Above all, it was observed that for the 3-D car cavity problem, which has a complex frequency response with many frequency peaks in the whole frequency interval, the accuracy of the PA problems is excellent. In all examples considered in the computation, it was confirmed that the PA schemes can be used for rapid and accurate computation of Zwicker’s loudness.
A finite element procedure combined with the PA was extended to deal with ATO problems considering Zwicker’s loudness and it was used to compute the main specific loudness and the corresponding design sensitivities. The analytical sensitivities of the main specific loudness were obtained using the adjoint variable method with the PA, which is one of the main contributions of this study. This makes the formulation particularly suited for ATO of Zwicker’s loudness, which typically involves thousands of design variables and a wide ranging frequency response with a fine frequency resolution.
Two examples of ATO of Zwicker’s loudness were presented to demonstrate the proposed optimization procedure. The optimized designs obtained using the PAs with different employed expansion terms and subintervals were compared to designs obtained by using the standard method with respect to the objective values, optimized topology, and iterations or CPU-time needed for the optimization. In the investigation of the reliability of the PA, it was shown that the optimized results obtained by using the PA are identical to the optimized results obtained by using the standard method. In the investigation on the speedup of the PA, the proposed method has been proven to be significantly faster than the standard method. For speedup while ensuring accurate approximations, a reduction of the number of subintervals is more efficient than a reduction of the number of expansion terms. This tendency becomes more obvious for larger acoustic problems for which the LU-factorization cost becomes dominant. It should be noticed that since numerical characteristics such as the CPU-time and solution accuracy are highly dependent on the number of expansion terms and subintervals, a compromise between the accuracy and efficiency of these methods should be considered when the PA parameters are chosen.
In all examples considered in the optimization study, the proposed method results in a significant reduction of computational effort, and is expected to provide an alternative approach, while the optimized designs are identical to the optimized designs obtained by using the standard method. In conclusion, it is confirmed that the proposed optimization procedure combined with the PA can effectively solve the ATO for Zwicker’s loudness problems, where a computational issue inevitably arises. The present research will be useful in addressing real engineering design problems involving large-scale FE models with a wide range of frequency.
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