Acoustical topology optimization of Zwicker's loudness with Padé approximation

Junghwan Kook, Jakob Søndergaard Jensen, Semyung Wang

Research output: Contribution to journalJournal articleResearchpeer-review


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. © 2012 Elsevier B.V.
Original languageEnglish
JournalComputer Methods in Applied Mechanics and Engineering
Pages (from-to)40-66
Publication statusPublished - 2013


  • Acoustics
  • Design
  • Finite element method
  • Frequency response
  • Iterative methods
  • Optimization
  • Acoustic signal processing
  • Padé approximation
  • Acoustical topology optimization
  • Adjoint variable method
  • Zwicker’s loudness
  • Main specific loudness


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