Erkenntnis

, Volume 41, Issue 2, pp 139–156 | Cite as

Convex models of uncertainty: Applications and implications

  • Yakov Ben-Haim
Article

Abstract

Modern engineering has included the basic sciences and their accompanying mathematical theories among its primary tools. The theory of probability is one of the more recent entries into standard engineering practice in various technological disciplines. Probability and statistics serve useful functions in the solution of many engineering problems. However, not all technological manifestations of uncertainty are amenable to probabilistic representation. In this paper we identify the conceptual limitations of probabilistic and related theories as they occur in a wide range of engineering tasks. We discuss the structure and properties of an alternative, non-probabilistic, method — convex modelling — for quantitatively representing uncertain phenomena.

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References

  1. [1]
    J. Arbocz and J. G. Williams: 1977, ‘Imperfection Surveys on a 10 ft. Diameter Shell Structure’,Amer. Inst. of Aeronautics and Astronautics Journal 15, 949–956.Google Scholar
  2. [2]
    Z. Artstein: 1974, ‘On the Calculus of Closed Set-Valued Functions’,Indiana University Mathematics Journal 24, 433–441.Google Scholar
  3. [3]
    Z. Artstein and J. C. Hansen: 1985, ‘Convexification in Limit Laws of Random Sets in Banach Spaces’,Annals of Probability 13, 307–309.Google Scholar
  4. [4]
    R. J. Aumann: 1965, ‘Integrals of Set-Valued Functions’,Journal of Mathematical Analysis and Applications 12, 1–12.Google Scholar
  5. [5]
    Y. Ben-Haim: 1985,The Assay of Spatially Random Material, Kluwer Academic Publishers, Dordrecht, Holland.Google Scholar
  6. [6]
    Y. Ben-Haim: 1990, ‘Detecting Unknown Lateral Forces on a Bar by Vibration Measurement’,Journal of Sound and Vibration 140, 13–29.Google Scholar
  7. [7]
    Y. Ben-Haim: 1992, ‘Convex Models for Optimizing Diagnosis of Uncertain Slender Obstacles on Surfaces’,Journal of Sound and Vibration 152, 327–341.Google Scholar
  8. [8]
    Y. Ben-Haim: 1993, ‘Convex Models of Uncertainty in Radial Pulse Buckling of Shells’,ASME Journal of Applied Mechanics 60, 683–688.Google Scholar
  9. [9]
    Y. Ben-Haim: 1993, ‘Failure of an Axially Compressed Beam with Uncertain Initial Deflection of Bounded Strain Energy’,International Journal of Engineering Science 31, 989–1001.Google Scholar
  10. [10]
    Y. Ben-Haim: ‘A Non-Probabilistic Concept of Reliability’, to appear inStructural Safety. Google Scholar
  11. [11]
    Y. Ben-Haim and E. Elias: 1987, ‘Indirect Measurement of Surface Temperature and Heat Flux: Optimal Design Using Convexity Analysis,International Journal of Heat and Mass Transfer 30, 1673–1683.Google Scholar
  12. [12]
    Y. Ben-Haim and I. Elishakoff: 1989, ‘Non-Probabilistic Models of Uncertainty in the Non-Linear Buckling of Shells with General Imperfections: Theoretical Estimates of the Knockdown Factor’,ASME Journal of Applied Mechanics 56, 403–410.Google Scholar
  13. [13]
    Y. Ben-Haim and I. Elishakoff: 1990,Convex Models of Uncertainty in Applied Mechanics, Elsevier, Amsterdam.Google Scholar
  14. [14]
    Y. Ben-Haim and I. Elishakoff: 1991, ‘Convex Models of Vehicle Response to Uncertain but Bounded Terrain’,ASME Journal of Applied Mechanics 58, 354–361.Google Scholar
  15. [15]
    Y. Ben-Haim and N. Shenhav: 1984, ‘The Measurement of Spatially Random Material’,SIAM Journal of Applied Mathematics 44, 1150–1163.Google Scholar
  16. [16]
    R. F. Drenick: 1968, ‘Functional Analysis of Effects of Earthquakes’,2nd Joint United States—Japan Seminar on Applied Stochastics, Washington, D.C., Sept. 19–24.Google Scholar
  17. [17]
    R. F. Drenick: 1970, ‘Model-Free Design of Aseismic Structures’,Journal of the Engineering Mechanics Division’, Proceedings of the ASCE 96, 483–493.Google Scholar
  18. [18]
    D. Dubois and H. Prade: 1988,Possibility Theory: An Approach to Computerized Processing of Uncertainty, Plenum Press, New York.Google Scholar
  19. [19]
    I. Elishakoff: 1983,Probabilistic Methods in the Theory of Structures, Wiley, New York.Google Scholar
  20. [20]
    I. Elishakoff and Y. Ben-Haim: 1990, ‘Dynamics of a Thin Shell Under Impact with Limited Deterministic Information on its Initial Imperfections’,International Journal of Structural Safety 8, 103–112.Google Scholar
  21. [21]
    I. Elishakoff and P. Colombi: 1993, ‘Combination of Probabilistic and Convex Models of Uncertainty when Scarce Knowledge is Present on Acoustic Excitation Parameters’,Computer Methods in Applied Mechanics and Engineering,104, 187–209.Google Scholar
  22. [22]
    I. Elishakoff, G. Q. Cai and J. H. Starnes, jr.: 1993, ‘Non-Linear Buckling of a Column with Initial Imperfection via Stochastic and Non-Stochastic Convex Models’,International Journal of Non-Linear Mechanics 27.Google Scholar
  23. [23]
    I. Ekeland: 1988,Mathematics and the Unexpected, University of Chicago Press, Chicago.Google Scholar
  24. [24]
    B. R. Gaines: 1978, ‘Fuzzy and Probability Uncertainty Logics’,Information and Control 38, 154–169.Google Scholar
  25. [25]
    D. Givoli and I. Elishakoff: 1992, ‘Stress Concentration at a Nearly Circular Hole with Uncertain Irregularities’,ASME Journal of Applied Mechanics 59, 65–71.Google Scholar
  26. [26]
    W. H. Hartt and N. K. Lin: 1986, ‘A Proposed Stress History for Fatigue Testing Applicable to Offshore Structures’,International Journal of Fatigue 8: 91–93.Google Scholar
  27. [27]
    S. W. Kirkpatrick and B. S. Holmes: 1989, ‘Effect of Initial Imperfections on Dynamic Buckling of Shells’,ASCE Journal of Engineering Mechanics 115, 1075–1093.Google Scholar
  28. [28]
    H. E. Lindberg: 1992, ‘An Evaluation of Convex Modelling for Multimode Dynamic Buckling’,ASME Journal of Applied Mechanics 59, 929–936.Google Scholar
  29. [29]
    H. E. Lindberg: 1992, ‘Convex Models for Uncertain Imperfection Control in Multimode Dynamic Buckling’,ASME Journal of Applied Mechanics 59, 937–945.Google Scholar
  30. [30]
    K. Murota and K. Ikeda: 1992, ‘On Random Imperfections for Structures of Regular-Polygonal Symmetry’,SIAM J. Appl. Math. 52, 1780–1803.Google Scholar
  31. [31]
    H. G. Natke and T. T. Soong: 1993, ‘Topological Structural Optimization Under Dynamic Loads’, in S. Hernandez and C. A. Brebbia (eds.),Optimization of Structural Systems and Applications, Proceedings of 3rd Intl. Conf. on Computer Aided Optimum Design of Structures, Elsevier Applied Science, London.Google Scholar
  32. [32]
    D. E. Newland: 1986, ‘General Linear Theory of Vehicle Response to Random Road Roughness’, in I. Elishakoff and R. H. Lyons (eds.),Random Vibrations — Status and Recent Developments, Elsevier Science Publishers, Amsterdam, pp. 303–326.Google Scholar
  33. [33]
    J. Pearl: 1988,Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufman Publishers, San Mateo.Google Scholar
  34. [34]
    F. C. Schweppe: 1968, ‘Recursive State Estimation: Unknown but Bounded Errors and System Inputs,IEEE Transactions on Automatic Control AC-13, 22–28.Google Scholar
  35. [35]
    F. C. Schweppe: 1973,Uncertain Dynamic Systems, Prentice-Hall, Englewood Cliffs.Google Scholar
  36. [36]
    N. Shenhav and Y. Ben-Haim: 1984, ‘A General Method for Optimal Design of Nondestructive Assay Systems’,Nuclear Science and Engineering 88, 173–183.Google Scholar
  37. [37]
    M. Shinozuka: 1970, ‘Maximum Structural Response to Seismic Excitations’,Journal of the Engineering Mechanics Division’, Proceedings of the ASCE 96, 729–738.Google Scholar
  38. [38]
    K. Sobczyk and B. F. Spencer: 1992,Random Fatigue: From Data to Theory, Academic Press, Boston.Google Scholar
  39. [39]
    T. T. Soong: 1981,Probabilistic Modelling in Science and Engineering, Wiley, New York.Google Scholar
  40. [40]
    T. T. Soong: 1990,Active Structural Control: Theory and Practice, Wiley, New York.Google Scholar
  41. [41]
    S. M. Stigler: 1986,The History of Statistics: The Measurement of Uncertainty Before 1900, Belnap Press, Cambridge.Google Scholar
  42. [42]
    P. Suppes and M. Zanotti: 1977, ‘On Using Random Relations to Generate Upper and Lower Probabilities’,Synthese 36, 427–440.Google Scholar
  43. [43]
    A. Talmor, Y. Leichter, Y. Ben-Haim and A. Kushlevsky: 1990, ‘Adaptive Assay of Radioactive Pulmonary Aerosol with an External Detector’,International Journal of Applied Radiation and Isotopes, Part A,41, 989–993.Google Scholar
  44. [44]
    U. Vulkan and Y. Ben-Haim: 1989, ‘Global Optimization in the Adaptive Borehole Assay of Uranium’,International Journal of Applied Radiation and Isotopes, Part E: Nuclear Geophysics 3, 97–105.Google Scholar
  45. [45]
    H. S. Witsenhausen: 1968, ‘A Minimax Control Problem for Sampled Linear Systems’,IEEE Transactions on Automatic Control AC-13, 5–21.Google Scholar
  46. [46]
    H. S. Witsenhausen: 1968, ‘Sets of Possible States of a Linear System Given Perturbed Observations’,IEEE Transactions on Automatic Control AC-13, 556–558.Google Scholar
  47. [47]
    J. T. P. Yao: 1985,Safety and Reliability of Existing Structures, Pitman, Boston.Google Scholar

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Yakov Ben-Haim
    • 1
  1. 1.Faculty of Mechanical EngineeringTechnion — Israel Institute of TechnologyHaifaIsrael

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