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Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis

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Summary

There is a growing awareness of the impact of non-deterministic model properties on the numerical simulation of physical phenomena. These non-deterministic aspects are of great importance when there is a large amount of information to be retrieved from the numerical analysis, as for instance in a numerical reliability study or reliability based optimisation during a design process. Therefore, the non-deterministic properties form a primordial part of a trustworthy virtual prototyping environment. The implementation of such a virtual prototyping environment requires the inclusion of non-deterministic properties in the numerical finite element framework. This articel gives an overview of the emerging non-probabilistic approaches for non-deterministic numerical analysis, and compares them to the classical probabilistic methodology. Their applicability in the context in engineering design is discussed. The typical implementation strategies applied in literature are reviewed. A new concept is introduced for the calculation of envelope frequency response functions. This method is explained in detail and illustrated on a numerical example.

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References

  1. Guyader, J. and Parizet, E. (1997). Uncertainty of Vibroacoustic Behaviour of Industrially Identical Structures. A new Challenge for Structural Acoustic People.Proceedings of the Fifth International Congress on Sound and Vibration, pp. 2347–2358.

  2. Elishakoff, I. (1990). An Idea of the Uncertainty Triangle.Shock and Vibration Digest,22, No. 10, pp. 1.

    Google Scholar 

  3. Klir, G. (1999). Uncertainty and Information Measures for Imprecise Probabilities: An Overview.Proceedings of the 1 st International Symposium on Imprecise Probabilities and Their Applications.

  4. Miller, I. and Freund, J. (1985).Probability and Statistics for Engineers. Prentice Hall, Englewood Cliffs.

    Google Scholar 

  5. Haldar, A. and Mahadevan, S. (2000).Reliability Assessment Using Stochastic Finite Element Analysis. John Wiley & Sons, New York.

    Google Scholar 

  6. Soize, C. (2000). A nonparametric model of random uncertainties for reduced matrix models in structural dynamics.Probabilistic Engineering Mechanics,15, No. 3, 277–294.

    Article  Google Scholar 

  7. Ibrahim, R. (1987). Structural Dynamics with Parameter Uncertainties.ASME Applied Mechanics Reviews,40, No. 3, 309–328.

    Article  MathSciNet  Google Scholar 

  8. Manohar, C. and Ibrahim, R. (1999). Progress in structural dynamics with stochastic parameter variations: 1987–1998.ASME Applied Mechanics Reviews,52, No. 5, 177–197.

    Google Scholar 

  9. Binder, K. (1984).Application of the Monte Carlo Method in Statistical Physics Springer-Verlag, Berlin.

    Google Scholar 

  10. Rubenstein, R. (1981).Simulation and the Monte Carlo Method. John Wiley & Sons, New York.

    Google Scholar 

  11. Annis, C. (2004). Probabilistic Life Prediction Isn’t as Easy as It Looks.Journal of ASTM International,1, No. 2.

    Google Scholar 

  12. Schuëller, G. and Stix, R. (1987). A Critical Appraisal of Methods to Determine Failure Probabilities.Structural Safety,4, 293–309.

    Article  Google Scholar 

  13. Bucher, C. (1988). Adaptive Sampling—An Iterative Fast Monte Carlo Procedure.Structural Safety,5, No. 2, 119–126.

    Article  Google Scholar 

  14. Ditlevsen, O., Bjerager, P., Olesen, R. and Hasofer, A. (1988). Directional Simulation in Gaussian Processes.Probabilistic Engineering Mechanics,3, No. 4, 207–217.

    Article  Google Scholar 

  15. Kijawatworawet, W., Pradlwarter, H. and Schuëller, G. (1997). Structural Reliability Estimation by Adaptive Importance Directional Sampling.Proceedings of the 7 th International Conference on Structural Safety and Reliability. ICOSSAR ’97, pp. 891–897.

  16. Mckay, M., Beckman, R. and Conover, W. (1979). Comparison of 3 Methods for Selecting Values of Input Variables in the Analysis of Output from A Computer Code.Technometrics,21, No. 2, 239–245.

    Article  MATH  MathSciNet  Google Scholar 

  17. Schuëller, G. (2001). Computational Stochastic Mechanics—Recent Advances.Computers & Structures,79, 2225–2234.

    Article  Google Scholar 

  18. Pradlwarter, H. and Schuëller, G. (1999). Assessment of Low Probability Events of Dynamical Systems by Controlled Monte Carlo Simulation.Probabilistic Engineering Mechanics,14, 213–227.

    Article  Google Scholar 

  19. Vanmarcke, E. (1993).Random fields: analysis and synthesis. MIT Press, Cambridge.

    Google Scholar 

  20. Ghanem, R. and Spanos, P. (1991).Stochastic finite elements: a spectral approach. Springer-Verlag, New-York.

    MATH  Google Scholar 

  21. Moore, R. (1966).Interval Analysis. Prentice Hall, Englewood Cliffs.

    MATH  Google Scholar 

  22. Ben Haim, Y. and Elishakoff, I. (1990).Convex Models of Uncertainty in Applied Mechanics. Elsevier Science, Amsterdam.

    MATH  Google Scholar 

  23. Ben Haim, Y., Chen, G. and Soong, T. (1996). Maximum Structural Response Using Convex Models.Journal of Engineering Mechanics,122, No. 4, 325–333.

    Article  Google Scholar 

  24. Niki, M., Yoshikawa, N. and Nakagiri, S. (2000). Worst Case Estimation of Time-history Response Subject to Uncertain Excitation Bounded in Convex Set.Proceedings of the 5th International Conference on Probabilistic Safety Assessment and Management, 1071–1076.

  25. Elseifi, M., Gürdal, Z. and Nikolaidis, E. (1999). Convex/Probabilistic Models of Uncertainties in Geometric Imperfections of Stiffened Composite Panels.AIAA Journal,37, No. 4, 468–474.

    Google Scholar 

  26. Givoli, D. and Elishakoff, I. (1992). Stress Concentration at a Nearly Circular Hole with Uncertain Irregularities.Journal of Applied Mechanics, ASME Paper 91-WA/APM-22,59, 65–71.

    Google Scholar 

  27. Elishakoff, I. and Colombi, P. (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.

    Article  MATH  MathSciNet  Google Scholar 

  28. Zadeh, L. (1965). Fuzzy sets.Information and Control,8, 338–353.

    Article  MATH  MathSciNet  Google Scholar 

  29. Dubois, D. and Prade, H. (1980).Fuzzy Sets and Systems: Theory and Applications. Academic Press, Orlando.

    MATH  Google Scholar 

  30. Dubois, D. and Prade, H. (1986). Fuzzy Sets and Statistical Data.European Journal of Operational Research,25, 345–356.

    Article  MATH  MathSciNet  Google Scholar 

  31. Dubois, D. and Prade, H. (1988),Possibility Theory. An Approach to Computerized Processing of Uncertainty. Plenum Press, New York.

    MATH  Google Scholar 

  32. Driankov, D., Hellendoorn, H. and Reinfrank, M. (1996).An Introduction to Fuzzy Control. Springer-Verlag, Berlin.

    MATH  Google Scholar 

  33. Zadeh, L. (1978). Fuzzy Sets as a Basis for a Theory of Possibility.Fuzzy Sets and Systems,1, 3–28.

    Article  MATH  MathSciNet  Google Scholar 

  34. Wasfy, T. and Noor, A. (1998). Finite Element Analysis of Flexible Multibody Systems with Fuzzy Parameters.Computer Methods in Applied Mechanics and Engineering,160, 223–243.

    Article  MATH  Google Scholar 

  35. Schulz, K., Huwe, B. and Peiffer, S. (1999). Parameter uncertainty in chemical equilibrium calculations using fuzzy set theory.Journal of Hydrology,217, No. 1-2, 119–134.

    Article  Google Scholar 

  36. Barpi, F. (2004). Fuzzy modelling of powder snow avalanches.Cold Regions Science and Technology,40, 213–227.

    Article  Google Scholar 

  37. Valliappan, S. and Pham, T. (1993). Fuzzy Finite Element Analysis of a Foundation on an Elastic Soil Medium.International Journal for Numerical and Analytical Methods in Geomechanics,17, 771–789.

    Article  MATH  MathSciNet  Google Scholar 

  38. Rao, S. and Sawyer, P. (1995). Fuzzy Finite Element Approach for the Analysis of Imprecisely Defined Systems.AIAA Journal,33, No. 12, 2364–2370.

    MATH  Google Scholar 

  39. Ross, T., Booker, J., and Parkinson, W. (2002).Fuzzy Logic and Probability Applications: Bridging the Gap. SIAM, Philadelphia, ASA, Alexandria, VA.

    MATH  Google Scholar 

  40. Zadeh, L. (1975). Concept of A Linguistic Variable and Its Application to Approximate Reasoning. 1.Information Sciences,8, No. 3, 199–249.

    Article  MathSciNet  Google Scholar 

  41. Moens, D. and Vandepitte, D. (2005). A Survey of Non-Probabilistic Uncertainty Treatment in Finite Element Analysis.Computer Methods in Applied Mechanics and Engineering,194, No. 14-16, 1527–1555.

    Article  MATH  Google Scholar 

  42. Mullen, R. and Muhanna, R. (1999). Bounds of Structural Response for All Possible Loading Combinations.Journal of Structural Engineering,125, No. 1, 98–106.

    Article  Google Scholar 

  43. Cherki, A., Plessis, G., Lallemand, B., Tison, T. and Level, P. (2000). Fuzzy Behavior of Mechanical Systems with Uncertain Boundary Conditions.Computer Methods in Applied Mechanics and Engineering,189, 863–873.

    Article  MATH  Google Scholar 

  44. Chen, L. and Rao, S. (1997). Fuzzy Finite-Element Approach for the Vibration Analysis of Imprecisely-Defined Systems.Finite Elements in Analysis and Design,27, 69–83.

    Article  MATH  Google Scholar 

  45. Wasfy, T. and Noor, A. (1998). Application of Fuzzy Sets to Transient Analysis of Space Structures.Finite Elements in Analysis and Design,29, 153–171.

    Article  MATH  Google Scholar 

  46. Valliappan, S. and Pham, T. (1995). Elasto-Plastic Finite Element Analysis with Fuzzy Parameters.International Journal for Numerical Methods in Engineering,38, 531–548.

    Article  MATH  Google Scholar 

  47. Fetz, T., Jäger, J., Köll, D., Krenn, G., Lessmann, H., Oberguggenberger, M. and Stark, R. (1999). Fuzzy Models in Geotechnical Engineering and Construction Management.Computer-Aided Civil and Infrastructure Engineering,14, 93–106.

    Article  Google Scholar 

  48. Muhanna, R. and Mullen, R. (1999). Formulation of Fuzzy Finite-Element Methods for Solid Mechanics Problems.Computer-Aided Civil and Infrastructure Engineering,14, No. 2, 107–117.

    Article  Google Scholar 

  49. Akpan, U., Koko, T., Orisamolu, I., and Gallant, B. (2001). Fuzzy Finite Element Analysis of Smart Structures.Smart Materials and Structures,10, 273–284.

    Article  Google Scholar 

  50. Muc, A. (2002). A fuzzy set approach to interlaminar cracks simulation problems.International Journal of Fatigue,24, No. 2-4, 419–427.

    Article  MATH  Google Scholar 

  51. McNeill, D. (1993).Fuzzy Logic, Simon and Schuster, New York.

    Google Scholar 

  52. Oberkampf, W., DeLand, S., Rutherford, B., Diegert, K. and Alvin, K. (1999). A New Methodology for the Estimation of Total Uncertainty in Computational Simulation.Proceedings of the 40 th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, AIAA-99-1612, 3061–3083.

  53. Klir, G. and Folger, T. (1988).Fuzzy Sets, Uncertainty and Information. Prentice Hall, Englewood Cliffs.

    MATH  Google Scholar 

  54. Freudenthal, A. (1961). Fatigue Sensitivity and Reliability of Mechanical Systems, especially Aircraft Structures.WADD Technical Report 61-53.

  55. Civanlar, M. and Trussell, H. (1986). Constructing Membership Functions Using Statistical Data.Fuzzy Sets and Systems,18, 1–13.

    Article  MATH  MathSciNet  Google Scholar 

  56. Dubois, D. and Prade, H. (1991). Random Sets and Fuzzy Interval Analysis.Fuzzy Sets and Systems,42, 87–101.

    Article  MATH  MathSciNet  Google Scholar 

  57. Shafer, G. (1976).A Mathematical Theory of Evidence. Princeton University Press, Princeton.

    MATH  Google Scholar 

  58. Zang, T.A., Hemsch, M.J., Hilburger, M.W., Kenny, S.P., Luckring, J.M., Maghami, P. and Padula, S.L. (2002). Needs and Opportunities for Uncertainty-Based Multidisciplinary Design Methods for Aerospace Vehicles. Tech. Rep. NASA/TM-2002-211462. Langley Research Center, Hampton, Virginia.

    Google Scholar 

  59. Ross, T., Sellers, K. and Booker, J. (2002). Considerations for using fuzzy set theory and probability theory.Fuzzy Logic and Probability Applications: Bridging the Gap, Chap. 5, SIAM, Philadelphia, ASA, Alexandria, VA, pp. 87–104.

    Google Scholar 

  60. Casciati, F. and Roberts, J. (1996).Mathematical Models for Structural Reliability Analysis. CRC Press, New York.

    MATH  Google Scholar 

  61. Haldar, A. and Mahadevan, S. (1993). First-order/second-order reliability methods (FORM/SORM).Probabilistic Structural Mechanics Handbook, 27–52.

  62. Schuëller, G., Pradlwarter, H. and Koutsourelakis, P. (2004). A critical appraisal of reliability estimation procedures for high dimensions.Probabilistic Engineering Mechanics,19, No. 4, 463–474.

    Article  Google Scholar 

  63. Salita, M. (2004). Shuttle disasters: a common cause?Aerospace America,4, 41–43.

    Google Scholar 

  64. Ben Haim, Y. (1995). A Non-Probabilistic Measure of Reliability of Linear Systems based on Expansion of Convex Models.Structural Safety,17, 91–109.

    Article  Google Scholar 

  65. Zingales, M. and Elishakoff, I. (2000). Anti-Optimization Versus Probability in an Applied Mechanics Problem: Vector Uncertainty.Transactions of the ASME, Journal of Applied Mechanics,67, 472–483.

    Article  MATH  Google Scholar 

  66. Sawyer, J. and Rao, S. (1999). Strength-based reliability and fracture assessment of fuzzy mechanical and structural systems.AIAA Journal,37, No. 1, 84–92.

    Article  Google Scholar 

  67. Biondini, F., Bontempi, F. and Malerba, P. (2004). Fuzzy reliability analysis of concrete structures 4.Computers & Structures,82, No. 13-14, 1033–1052.

    Article  Google Scholar 

  68. Catallo, L. (2004). Genetic anti-optimization for reliability structural assessment of precast concrete structures 1.Computers & Structures,82, No. 13-14, 1053–1065.

    Article  Google Scholar 

  69. Ferrari, P. and Savoia M. (1998). Fuzzy Number Theory to Obtain Conservative Results with respect to Probability.Computer Methods in Applied Mechanics and Engineering,160, 205–222.

    Article  MATH  MathSciNet  Google Scholar 

  70. Youn, B. and Choi, K. (2004). Selecting probabilistic approaches for reliability-based design optimization.AIAA Journal,42, No. 1, 124–131.

    Google Scholar 

  71. Ross, T., Sellers, K. and Booker, J. (2002). Introduction.Fuzzy Logic and Probability Applications: Bridging the Gap, Chap. 1, SIAM, Philadelphia, ASA, Alexandria, VA, 3–27.

    Google Scholar 

  72. Choi, K., Du, L. and Youn, B. (2004). Vertex-Domain Fuzzy Analysis Method for Possibility-Based Design Optimization.10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference.

  73. Elishakoff, I. (1998). Three Versions of the Finite Element Method Based on Concepts of either Stochasticity, Fuzziness, or Anti-Optimization.ASME Applied Mechanics Reviews,51, No. 3, 209–218.

    Google Scholar 

  74. Lima, B. and Ebecken, N. (2000). A Comparison of Models for Uncertainty Analysis by the Finite Element Method.Finite Elements in Analysis and Design,34, 211–232.

    Article  MATH  Google Scholar 

  75. Maglaras, G., Nikolaidis, E., Haftka, T and Cudney, H. (1997). Analytical-Experimental Comparison of Probabilistic Methods and Fuzzy Set Based Methods for Designing under Uncertainty.Structural Optimization,13, 69–80.

    Article  Google Scholar 

  76. Nikolaidis, E., Cudney, H. and Chen, Q. (1999). Bayesian and Fuzzy Set Methods for Design Under Uncertainty.Proceedings of the 13th ASCE Engineering Mechanics Division Specialty Conference (CDROM).

  77. Langley, R. (2000). Unified Approach to Probabilistic and Possibilistic Analysis of Uncertain Systems.Journal of Engineering Mechanics,126, No. 11, 1163–1172.

    Article  Google Scholar 

  78. Rao, S., Chen, L. and Mulkay, E. (1998). Unified Finite Element Method for Engineering Systems with Hybrid Uncertainties.AIAA Journal,36, No. 7, 1291–1299.

    Google Scholar 

  79. Köylüoĝlu, U., Çakmak, A. and Nielsen, S. (1995). Interval Algebra to Deal with Pattern Loading and Structural Uncertainties.Journal of Engineering Methanics,121, No. 11, 1149–1157.

    Article  Google Scholar 

  80. Möller, B., Graf, W. and Beer, M. (2000). Fuzzy Structural Analysis Using α-level Optimization.Computational Mechanics,26, 547–565.

    Article  MATH  Google Scholar 

  81. Dong, W. and Shah, H. (1987). Vertex Method for Computing Functions of Fuzzy Variables.Fuzzy Sets and Systems,24, 65–78.

    Article  MATH  MathSciNet  Google Scholar 

  82. Hanss, M. and Willner, K. (2000). A Fuzzy Arithmetical Approach to the Solution of Finite Element Problems with Uncertain Parameters.Mechanics Research Communications,27, No. 3, 257–272.

    Article  MATH  Google Scholar 

  83. Hanss, M. (2003). The extended transformation method for the simulation and analysis of fuzzy-parameterized models.International Journal of Uncertainty Fuzziness and Knowledge-Based Systems,11, No. 6, 711–727.

    Article  MATH  MathSciNet  Google Scholar 

  84. Shary, S. (1997). Algebraic Approach in the ‘Outer Problem’ for Interval Linear Equations.Reliable Computing,3, 103–135.

    Article  MATH  MathSciNet  Google Scholar 

  85. Oettli, W. and Prager, W. (1964). Compatibility of Approximate Solution of Linear Equations with given Error Bounds for Coefficients and Right-hand Sides.Numerische Mathematik,6, 405–409.

    Article  MATH  MathSciNet  Google Scholar 

  86. Neumaier, A. (1990).Interval Methods for Systems of Equations. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  87. Alefeld, G. and Herzberger, J. (1983).Introduction to Interval Computations. Academic Press, Inc., New York.

    MATH  Google Scholar 

  88. Shary, S. (1995). On Optimal Solution of Interval Linear Equations.SIAM Journal on Numerical Analysis,32, No. 2, 610–630.

    Article  MATH  MathSciNet  Google Scholar 

  89. Rao, S. and Berke, L. (1997). Analysis of Uncertain Structural Systems Using Interval Analysis.AIAA Journal,34, No. 4, 727–735.

    Google Scholar 

  90. Rao, S. and Chen, L. (1998). Numerical Solution of Fuzzy Linear Equations in Engineering Analysis.International Journal for Numerical Methods in Engineering,43, 391–408.

    Article  MATH  MathSciNet  Google Scholar 

  91. Dessombz, O., Thouverez, F., Laîné, J.-P. and Jézéquel, L. (2001). Analysis of Mechanical Systems Using Interval Computations Applied to Finite Element Methods.Journal of Sound and Vibration,239, No. 5, 949–968.

    Article  MathSciNet  Google Scholar 

  92. Qiu, Z. and Elishakoff, I. (1998). Antioptimization of Structures with Large Uncertain-but-non-random Parameters via Interval Analysis.Computer Methods in Applied Mechanics and Engineering,152, 361–372.

    Article  MATH  Google Scholar 

  93. McWilliam, S. (2001). Anti-Optimization of Uncertain Structures Using Interval Analysis.Computers & Structures,79, 421–430.

    Article  Google Scholar 

  94. Moens, D. (2002).A Non-Probabilistic Finite Element Approach for Structural Dynamic Analysis with Uncertain Parameters, PhD thesis, K.U. Leuven, Leuven.

    Google Scholar 

  95. Chen, S., Qiu, Z. and Song, D. (1995). A New Method for Computing the Upper and Lower Bounds on Frequencies of Structures with Interval Parameters.Mechanics. Research Communications,22, No. 5, 431–439.

    Article  MATH  Google Scholar 

  96. El Gebeily, M., Abu-Baker, Y. and Elgindi, M. (1999). The Generalized Eigenvalue Problem for Tridiagonal Symmetric Interval Matrices.International Journal on Control,72, No. 6, 531–535.

    Article  MATH  Google Scholar 

  97. Chen, S., Lian, H. and Yang, X. (2003). Interval eigenvalue analysis for structures with interval parameters 1.Finite Elements in Analysis and Design,39, No. 5-6, 419–431.

    Article  Google Scholar 

  98. Qiu, Z., Wang, X. and Friswell, M. (2005). Eigenvalue bounds of structures with uncertain-but-bounded parameters 2.Journal of Sound and Vibration,282, No. 1-2, 297–312.

    Article  MathSciNet  Google Scholar 

  99. Fox, R. and Kapoor, M. (1968). Rates of Change of Eigenvalues and Eigenvectors.AIAA Journal,6, No. 12, 2426–2429.

    MATH  Google Scholar 

  100. Balmès, E. (1998). Predicted Variability and Differences between Tests of a Single Structure.Proceedings of the 16th International Modal Analysis Conference, IMAC XVI, 558–564.

    Google Scholar 

  101. Hasselman, T. and Chrostowski, J. (1997). Effects of Product and Experimental Variability on Model Verification of Automobile Structures.Proceedings of the 15th International Modal Analysis Conference, IMAC XV, 612–620.

    Google Scholar 

  102. Worden, K. (1998). Confidence Bounds for Frequency Response Functions from Time Series Models.Mechanical Systems and Signal Processing,12, No. 4, 559–569.

    Article  MathSciNet  Google Scholar 

  103. Enling, L. (1997). A New Solution to Structural Fuzzy Finite Element Equilibrium Equations (SFFEEE).Applied Mathematics and Mechanics,18, No. 4, 385–391.

    Article  MATH  Google Scholar 

  104. Laveuve, S. (1975). Definition einer Kahan-Arithmetik und ihre Implementierung.Interval Methematics 1975, Lecture Notes in Computer Science, 29, 236–245.

    Google Scholar 

  105. Hitchings, D. (1992). A Finite Element Dynamics Primer.A Finite Element Dynamics Primer, 170–196.

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  1. Department of Mechanical Engineering, K.U. Leuven, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

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Moens, D., Vandepitte, D. Recent advances in non-probabilistic approaches for non-deterministic dynamic finite element analysis. ARCO 13, 389–464 (2006). https://doi.org/10.1007/BF02736398

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