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Structural and Multidisciplinary Optimization

, Volume 31, Issue 1, pp 60–75 | Cite as

Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics

  • Mahmoud I. Hussein
  • Karim Hamza
  • Gregory M. Hulbert
  • Richard A. Scott
  • Kazuhiro Saitou
Research Paper

Abstract

An important dispersion-related characteristic of wave propagation through periodic materials is the existence of frequency bands. A medium effectively attenuates all incident waves within stopbands and allows propagation within passbands. The widths and locations of these bands in the frequency domain depend on the layout of contrasting materials and the ratio of their properties. Using a multiobjective genetic algorithm, the topologies of one-dimensional periodic unit cells are designed for target frequency band structures characterizing longitudinal wave motion. The decision variables are the number of layers in the unit cell and the thickness of each layer. Binary and mixed formulations are developed for the treatment of the optimization problems. Designs are generated for the following novel objectives: (1) maximum attenuation of time harmonic waves, (2) maximum isolation of general broadband pulses, and (3) filtering signals at predetermined frequency windows. The saturation of performance with the number of unit-cell layers is shown for the first two cases. In the filtering application, the trade-off between the simultaneous realization of passband and stopband targets is analyzed. It is shown that it is more difficult to design for passbands than it is to design for stopbands. The design approach presented has potential use in the development of vibration and shock isolation structures, sound isolation pads/partitions, and multiple band frequency filters, among other applications.

Keywords

Periodic materials Phononic and photonic crystals Wave dispersion Band gap Stopband Passband Topology optimization Multiobjective genetic algorithms Vibration and shock isolation 

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References

  1. Burger M, Osher SJ, Yablonovitch E (2004) Inverse problem techniques for the design of photonic crystals. IEICE Trans Electron E87C:258–265Google Scholar
  2. Cao WW, Qi WK (1995) Plane-wave propagation in finite 2–2-composites. J Appl Phys 78:4627–4632Google Scholar
  3. Cox SJ, Dobson DC (1999) Maximizing band gaps in two dimensional photonic crystals. SIAM J Appl Math 59:2108–2120CrossRefMathSciNetGoogle Scholar
  4. Cox SJ, Dobson DC (2000) Band structure optimization of two-dimensional photonic crystals in H-polarization. J Comput Phys 158:214–224CrossRefGoogle Scholar
  5. Day NA, Zhu C, Kinra VK (1994) A study of dispersive wave propagation in periodic layered composites. Proc. Review of Progress in Quantitative Nondestructive Evaluation (held in Brunswick, Maine, 1993) vol. 13A. Plenum Press, New York, pp 243–250Google Scholar
  6. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Wiley, LondonGoogle Scholar
  7. Deb K, Argawal S, Pratab A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. In: Lecture notes in computer science no. 1917, parallel problem solving from Nature VI Conference. Springer, Paris, France, pp 849–858Google Scholar
  8. Esquivel-Sirvent R, Cocoletzi GH (1994) Band-structure for the propagation of elastic-waves in superlattices. J Acoust Soc Am 95:86–90Google Scholar
  9. Floquet G (1883) Sur les Équations Différentielles Linéaries à Coefficients Périodiques. Ann Éc Norm 12:47–88zbMATHMathSciNetGoogle Scholar
  10. Goldberg D (1989) Genetic algorithms in search optimization and machine learning. Addison-Wesley Pub. Co., Reading, MAGoogle Scholar
  11. Hussein MI (2004) Dynamics of banded materials and structures: analysis, design and computation in multiple scales. Ph.D. Thesis. University of Michigan, Ann Arbor, MIGoogle Scholar
  12. Hussein MI, Hulbert GM, Scott RA (2002) Tailoring of wave propagation characteristics in periodic structures with multilayer unit cells. Proc. 17th American Society of Composites Technical Conference (held in West Lafayette, Indiana 2002), CD ROM, pp 1–9Google Scholar
  13. Hussein MI, Hulbert GM, Scott RA (2003) Band-gap engineering of elastic wave guides using periodic materials. Proc. ASME International Mechanical Engineering Congress and R&D Expo (held in Washington, DC, 2003). ASME Publication, New York, pp 799–807Google Scholar
  14. Hussein MI, Hamza K, Hulbert GM, Scott RA, Saitou K (2004a) Design of layered structures with desired dispersion properties using a multi-objective genetic algorithm. Proc. 8th Cairo University International Conference on Mechanical Design and Production (held in Cairo, Egypt, 2004), pp 41–50Google Scholar
  15. Hussein MI, Hulbert GM, Scott RA (2004b) Effects of “finiteness” on wave propagation and vibration in elastic periodic structures. Proc. ASME International Mechanical Engineering Congress and R&D Expo (held in Anaheim, CA, 2004). ASME Publication, New York, pp 437–447Google Scholar
  16. Hussein MI, Hulbert GM, Scott RA (2005) Dispersive elastodynamics of 1D banded materials and structures: analysis. J Sound Vib (in press)Google Scholar
  17. Jensen JS (2003) Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J Sound Vib 266:1053–1078CrossRefGoogle Scholar
  18. Kushwaha MS (1996) Classical band structure of periodic elastic composites. Int J Mod Phys B 10:977–1094Google Scholar
  19. Michalewicz Z (1996) Genetic algorithms+data structures=evolution programs, 3rd edn. Springer, Berlin Heidelberg New YorkGoogle Scholar
  20. Sigmund O, Jensen JS (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans R Soc Lond, A 361:1001–1019MathSciNetGoogle Scholar
  21. Shen MR, Cao WW (2000) Acoustic bandgap formation in a periodic structure with multilayer unit cells. J Phys D Appl Phys 33:1150–1154CrossRefMathSciNetGoogle Scholar
  22. Thomson WT (1950) Transmission of elastic waves through a stratified solid medium. J Appl Phys 21:89–93zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Mahmoud I. Hussein
    • 1
  • Karim Hamza
    • 1
  • Gregory M. Hulbert
    • 1
  • Richard A. Scott
    • 1
  • Kazuhiro Saitou
    • 1
  1. 1.Department of Mechanical EngineeringThe University of MichiganAnn ArborUSA

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