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An optimality criterion for shape optimization in eigenfrequency problems

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Abstract

For a broad class of static problems an optimality criterion of constant energy density at the designed boundary is known. In the present paper we prove a similar criterion for eigenfrequency problems. This optimality criterion serves as the tool for more basic understanding and for idealized reference cases as well as the basis for recursive procedures. Eigenfrequencies for in-plane vibrations as well as for out-of-plane vibrations of plates are optimized. The focus is on simplicity and multiple eigenfrequencies are not considered.

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References

  1. KJB96 Bathe KJ (1996) Finite element procedures. 2nd edn. Prentice-Hall, NJ, USA

  2. KDZM78 Dems K, Mroz Z (1978) Multiparameter structural shape optimization by finite element method. Int J Numer Methods Eng 13:247–263

  3. RG93 Grandhi R (1993) Structural optimization with frequency constraints—a rewiev. AIAA J 31(12):2296–2303

  4. JHPN96 Haslinger J, Neittaanmaki P (1996) Finite element approximation for optimal shape, material and topology design. 2nd edn. Wiley, Chichester, UK

  5. EJHKKCVK86 Haug EJ, Choi KK, Komkov V (1986) Design sensitivity analysis of structural systems. Academic, Orland, FL, USA

  6. JSJNLP03 Jensen JS, Pedersen NL (2003) On maximal eigenfrequency separation in two-material structures: the 1d and 2d scalar cases. DCAMM (687):1–24 (submitted)

  7. FNPP73-74 Niordson F, Pedersen P (1973–74). A review of optimal structural design. In Thirteenth Int. Congress of Theoretical and Applied Mechanics, 264–278, IUTAM, Moscow, Russia, 148 references

  8. FIN65 Niordson FI (1965) On the optimal design of a vibrating beam. Q Appl Math 23:47–53

  9. NO80 Olhoff N (1980) Optimal design with respect to structural eigenvalues. In: Rimrott FPJ, Tabarrot B (eds.) IUTAM Congress, 133–149, North-Holland, Canada

  10. PP82-83 Pedersen P (1982–83). Design with several eigenvalue constraints by finite elements and linear programming. J Struct Mech 10(3):243–271

    Google Scholar 

  11. PPb98 Pedersen P (1998) Some general optimal design results using anisotropic power law non-linear elasticity. Struct Optim 15:73–80

  12. PPa00 Pedersen P (2000) On optimal shapes in materials and structures. Struct Multidisc Optim 19:169–182

  13. NLP04 Pedersen NL (2004) Optimization of holes in plates for control of eigenfrequencies. Struct Multidisc Optim 28(1):1–10

  14. WPJET68 Prager W, Taylor JE (1968) Problems of optimal structural design. J Appl Mech Trans ASME 35(1):102–106

  15. GINR89 Rozvany GNI (1989) Structural Design via Optimality Criteria. Kluwer, Dordrecht, The Netherlands

  16. KS87 Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373

  17. ZW60 Wasiutynski Z (1960) On the congruency of the forming according to the minimum potential energy with that according to equal strength. Bull. de l’Academie Polonaise des Sciences, Serie des Sciences Techniques 8(6):259–268

  18. WHW62 Wittrick WH (1962) Rates of change of eigenvalues, with reference to buckling and vibration problems. J R Aeronaut Soc 66:590–591

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Authors and Affiliations

  1. Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Nils Koppels Allè, B 404, DK-2800, Kgs. Lyngby, Denmark

    P. Pedersen

  2. Institute of Mechanical Engineering, Aalborg University, Pontoppidanstraede 101, DK-9220, Aalborg East, Denmark

    N.L. Pedersen

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  1. P. Pedersen
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  2. N.L. Pedersen
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Correspondence to P. Pedersen.

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Pedersen, P., Pedersen, N. An optimality criterion for shape optimization in eigenfrequency problems. Struct Multidisc Optim 29, 457–469 (2005). https://doi.org/10.1007/s00158-004-0499-4

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  • DOI: https://doi.org/10.1007/s00158-004-0499-4

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