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Uniqueness for gradient methods in engineering optimization

Part of the Lecture Notes in Mathematics book series (LNM,volume 1086)

Abstract

This paper deals with optimization of constrained functionals with distributed parameters. The functionals may be chosen for identifying parameters that optimize performance or minimize energy forms, or for computation of eigenvalues via the Rayleigh quotient. Constraints may represent design limitations, extremes of operating conditions, or state equations for dynamical systems. The intrinsic nature of iterative solution methods for functional minimization, the functional sensitivity analysis, and state function sensitivity analysis have been the subject of extensive research. Using simple examples from engineering, this paper points out some pitfalls for gradient-type computational methods particularly in connection with computing of unstable processes or eigenvalues via minimization of a constrained Rayleigh quotient. Auxiliary conditions involving energy levels of the system for constrained problems are suggested as indicators of existence of multiple gradient directions.

Keywords

  • Trial Function
  • Optimal Design Problem
  • Lower Eigenvalue
  • Design Vector
  • Rayleigh Quotient

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Leonhardi Euleri Opera Omnia, vol.X, ser. secundae, Society for Natural Sciences of Switzerland, 1960, in particular the section of C. Truesdell, historical notes pp. 1638–1788, on the rational mechanics of flexible or elastic bodies.

    Google Scholar 

  2. J. L. Lagrange, Sur la figure des colonnes, Miscellanea Taurinensia, vol. V, 1970, (See pp. 123–125).

    Google Scholar 

  3. J. B. Keller, The shape of the strongest column, Archives of Rational Mechanics and Analysis, vol.5 (1960), pp. 275–285.

    CrossRef  MathSciNet  Google Scholar 

  4. I. Tadjbakhsh and J.B. Keller, Strongest columns and isoperimetric inequalities for eigenvalues, J. of Applied Mechanics, vol. 9, (1962), pp. 159–164.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. N. Olhoff, Optimal design against structural, vibration, and instability, Ph.D. Thesis, Technical University of Denmark, Dept. of Solid Mechanics, Lyngby, Denmark, November, 1978.

    Google Scholar 

  6. E.J. Haug, U.S. Army Material Command Pamphlet, AMC 706–902 (1972–73), Engineering Design Handbook.

    Google Scholar 

  7. F. Niordson and P. Pedersen, A review of optimal structural design, Applied Mechanics, Springer Verlag, Berlin, (1973), pp. 264–278.

    Google Scholar 

  8. N. Olhoff and S. H. Rasmussen, On single and bimodal optimum buckling loads of clamped columns, Int J. Solids & Structures, vol. B, (1979), pp. 605–614.

    MATH  Google Scholar 

  9. E. F. Masur and Z. Mróz, Nonstationary optimality conditions in structural design, Int J. Solids & Structures, vol. 15, (1979), pp. 503–512.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. E. J. Haug and J. S. Arrora, Applied Optimal Design, J. Wiley — Interscience, New York, 1977.

    Google Scholar 

  11. E. J. Haug, K.C. Pan, and T.D. Streeter, A computational method for Optimal Structural Design, Part II, Continuous problems, J. Numerical Methods in Engineering, vol. 9, 1975, pp. 649–667.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. J.E. Taylor and C.Y. Liu, Optimal Design of columns, AIAA Journal, vol. 6 (1968), pp. 1496–1502.

    CrossRef  MATH  Google Scholar 

  13. J.E. Taylor, The strongest column, An energy approach, J. Applied Mechanics, vol. 34 (1967), pp. 486–489.

    CrossRef  Google Scholar 

  14. B.N. Pshenichnii, Necessary conditions for an extremum, Marcel Dekker, New York, 1971, (Translated from Russian).

    Google Scholar 

  15. K.K. Choi and E.J. Haug, Repeated eigenvalues in mechanical optimization problems, Optimization of Distributed Parameter Structures, vol. I, Sijthoff & Noordhoff Publishers, the Netherlands, 1981, pp. 219–277.

    Google Scholar 

  16. E. Clarke, Generalized gradients and applications, Trans. Amer. Math. Society, 205, #2 (1975), pp. 247–262.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. E.F. Masur, Optimality in the presence of discreteness and discontinuity, Proceedings IUTAM Symposium on Optimization in Structural Design, A. Sawczuk and Z. Mróz editors, Springer Verlag, Berlin, 1975, pp. 441–453.

    CrossRef  Google Scholar 

  18. G. H. Knightly and D. Sather, Buckled states of a spherical shell under uniform external pressure, Arch. Rat. Mechanics Anal. 72 (1980) pp. 315–380.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. E.J. Haug and B. Rousselet, Design Sensitivity Analysis in Structural Mechanics, Part II, eigenvalue variations, J. Structural Mechanics, vol. 8, ♯2, 1980.

    Google Scholar 

  20. V. Komkov, An optimal design problem—a non-existence theorem, Arch.Mech., 33, (1981) pp. 147–151.

    MathSciNet  MATH  Google Scholar 

  21. N. Olhoff, Optimization of vibrating beams with respect to higher order natural frequencies, J. Structural Mechanics vol. 4, (1976), pp. 87–122.

    CrossRef  Google Scholar 

  22. H. Rabitz, Sensitivity methods for mathematical modelling, this issue.

    Google Scholar 

  23. J. Tilden, V. Constanza, G. McRae and J. Seinfeld, Modelling in Chemical Reaction Systems, editors, K. Ebert, P. Deuflhard and J. Jaeger, Springer Verlag, 1981.

    Google Scholar 

  24. V. Komkov, An embedding technique in problems of elastic stability, ZAMM, 60, (1980), pp. 503–507.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. E.J. Haug and J.S. Arora, Distributed Parameter Structural Optimization, vol. I, Sijthoff & Noordhoff Publishers, the Netherlands, 1981, pp. 219–277.

    CrossRef  Google Scholar 

  26. N.A. Shor, Minimization techniques for nondifferentiable functions, and their applications, Naukova Dumka, Kiev, 1979.

    MATH  Google Scholar 

  27. M.M. Vainberg, Variational Methods for Investigation of Nonlinear Operators, (English Translation) Holden Day, San Francisco, 1963.

    MATH  Google Scholar 

  28. T. Kato, Perturbution Theory for Linear Operators, Springer Verlag, Berlin and New York, Die Grundlehren der mathematischen Wissenschaften series, vol. 132, 1966.

    CrossRef  Google Scholar 

  29. V. Komkov, On formulation of variational problems in the classical continuum mechanics of solids, International Journal of Engineering Science, vol.6, 1968, pp. 695–720.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. M. M. Vainberg, On differentials and gradients of mappings, Vspekhi Matem. Nauk, 7, #49, (1952), pp. 139–143.

    MathSciNet  MATH  Google Scholar 

  31. M. Fréchet, La notion de differentielle dans l'analyse generale. Ann. Soc. de l'Ecole Norm. Super., 42, (1925), pp. 293–323.

    MathSciNet  MATH  Google Scholar 

  32. K.K. Choi, E.J. Haug, J.W. Hou and V.N. Sohoni, Pshenichnyi's Linearalization Method for Mechanical System Optimization, Trans. ASME, J. Mech. Design to appear in 1984.

    Google Scholar 

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© 1984 Springer-Verlag

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Komkov, V., Irwin, C. (1984). Uniqueness for gradient methods in engineering optimization. In: Komkov, V. (eds) Sensitivity of Functionals with Applications to Engineering Sciences. Lecture Notes in Mathematics, vol 1086. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073071

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  • DOI: https://doi.org/10.1007/BFb0073071

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