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Model Reduction and Uncertainties in Structural Dynamics

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Computational Methods in Stochastic Dynamics

Part of the book series: Computational Methods in Applied Sciences ((COMPUTMETHODS,volume 22))

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Abstract

Model reduction procedures for the purpose of reducing computational efforts in structural analysis are already well developed and widely used to compute the dynamic response of complex structural systems. For example the Guyan reduction might essentially reduce the size of the structural matrices, while component mode synthesis provides a means to consider first simpler substructures and to use its modal properties for deriving a reduced global structural model for computing the dynamic response. Moreover, component mode synthesis allows for an assembly of large finite element models consisting of substructures established by different working groups and hence has significant advantages in the design cycle. The above mentioned, frequently used finite element (FE) reduction procedures assume perfect, i.e. deterministic knowledge of all structural properties. However, the assumption of deterministically known structural properties is in most practical, real world cases, not realistic. Many items (e.g. stiffness, mass and damping parameters, etc.) incorporated in the mathematical FE model of a structure are uncertain, i.e. are either not precisely known or might reveal unpredictable random behavior. This paper will give an overview of the well established deterministic reduction techniques and approaches for the efficient uncertainty propagation. Finally, the recent advances in the combination of these two fields are reviewed and methodologies for the consideration of uncertainty when using model reduction schemes are presented. The model reduction schemes considering uncertainties, which usually involve some simplifying assumptions, similar to their deterministic counterparts, are compared with the reference solution as obtained by direct Monte Carlo simulation (MCS), where the deterministic FE reduction scheme is applied together with randomly generated structural properties.

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References

  1. Allemang, R.J., Brown, D.: A correlation coefficient for modal vector analysis. Proceedings of the 1st IMAC, pp. 110–116, Orlando (1982)

    Google Scholar 

  2. Au, S.-K. Beck, J.L.: Estimation of small failure probabilities in high dimensions by subset simulation. Probabilist. Eng. Mech. 16(4), 263–277 (2001)

    Google Scholar 

  3. Bah, M.T., Nair, P.B., Bhaskar, A., Keane, A.J.: Stochastic component mode synthesis. In: 44th AIAA/ASCE/ASME/AHS Structures, Structural Dynamics and Material Conference, AIAA-2003-1750, Norfolk (2003)

    Google Scholar 

  4. Benfield, W.A., Hruda, R.F.: Vibration analysis of structures by component mode substitution. AIAA J. 9(7), 1255–1261 (1971)

    Article  MATH  Google Scholar 

  5. Bouhaddi, N., Fillod, R.: Model reduction by a simplified variant of dynamics condensation. J. Sound Vib. 191(2), 233–250 (1996)

    Article  Google Scholar 

  6. Box, G.E.P., Wilson, K.B.: On the experimental attainment of optimum conditions. J. Roy. Statist. Soc. 13(1), 1–45 (1951)

    MathSciNet  Google Scholar 

  7. Castanier, M.P., Tan, Y.-C., Pierre, C.: Characteristic constraint modes for component mode synthesis. AIAA J. 39(6), 1182–1187 (2001)

    Article  Google Scholar 

  8. Cornell, J.A., Khuri, A.I.: Response Surfaces: Designs and Analyses, Volume 81 of Statistics: Textbooks and Monographs. Marcel Dekker, New York (1987)

    Google Scholar 

  9. Craig, R.R., Jr.: A brief tutorial on substructure analysis and testing. In: 18th International modal analysis conference, San Antonio (2000)

    Google Scholar 

  10. Craig, R.R., Jr.: Coupling of substructures for dynamic analyses: An overview. AIAA Paper No. 2000-1573, AIAA Dynamics Specialists Conference, Atlanta (2000)

    Google Scholar 

  11. Craig, R.R., Jr., Bampton, M.M.: Coupling of substructures for dynamical analysis. AIAA J. 6(7), 1313–1319 (1968)

    Article  MATH  Google Scholar 

  12. Craig, R.R., Jr., Chang, C.-J.: Free-interface methods of substructure coupling for dynamic analysis. AIAA J. 14(11), 1633–1635 (1976)

    Article  Google Scholar 

  13. Craig, R.R., Jr., Chang, C.-J.: On the use of attachment modes in substructure coupling for dynamic analysis. In: 18th AIAA/ASME Structures, Structural Dynamics and Material Conference, San Diego (1977)

    Google Scholar 

  14. De Kraker, A., Van Campen, D.A.: Rubin’s CMS reduction method for general space-state models. Comput. Struct. 58(3), 597–606 (1996)

    Article  MATH  Google Scholar 

  15. Eldred, M.S., Giunta, A.A., Castro, J.P.: Uncertainty quantification using response surface approximations. In: Proceedings of the 9th ASCE Joint Specialty Conference on Probabilistic Mechanics and Structural Reliability, Albuquerque (2007)

    Google Scholar 

  16. Fishman, G.S.: Monte Carlo: Concepts, Algorithms, Applications. Springer, New York (1996)

    MATH  Google Scholar 

  17. Ghanem, R., Kruger, R.: Numerical solution of spectral stochastic finite element systems. Comput. Method. Appl. Mech. Eng. 129, 289–303 (1996)

    Article  MATH  Google Scholar 

  18. Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)

    MATH  Google Scholar 

  19. Ghanem, R., Ghosh, D., Red-Horse, J.: Analysis of eigenvalues and modal interaction of stochastic systems. AIAA J. 43(10), 2196–2201 (2005)

    Article  Google Scholar 

  20. Giunta, A., Watson, L., Koehler, J.: A comparison of approximation modeling techniques: Polynomial versus interpolating models. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis (1998)

    Google Scholar 

  21. Guyan, R.: Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965)

    Article  Google Scholar 

  22. Hintz, R.M.: Analytical methods in component modal synthesis. AIAA J. 13(8), 1007–1016 (1975)

    Article  MATH  Google Scholar 

  23. Hurty, W.C.: Dynamic analysis of structural systems using component modes. AIAA J. 3(4), 678–685 (1965)

    Article  Google Scholar 

  24. Jin, R., Chen, W., Simpson, T.W.: Comparative studies of metamodelling techniques under multiple modelling criteria. Struct. Multidiscip. Optim. 23(1), 1–13 (2001)

    Article  Google Scholar 

  25. Johnson, E.A., Wojtkiewicz, S.F., Bergman, L.A., Spencer, B.F., Jr.: Observations with regard to massively parallel computation for Monte Carlo simulation of stochastic dynamical systems. Int. J. Nonlinear Mech. 32(4), 721–734 (1997)

    Article  MATH  Google Scholar 

  26. Johnson, E.A., Proppe, C., Spencer, B.F. Jr., Bergman, L., Székely, G.S., Schuëller, G.I.: Parallel processing in computational stochastic dynamics. Probabilist. Eng. Mech. 18(1), 37–60 (2003)

    Article  Google Scholar 

  27. Keane, A.J., Nair, P.B.: Computational Approaches for Aerospace Design. Wiley, New York (2005)

    Book  Google Scholar 

  28. Kleiber, M., Hien, T.D.: The stochastic finite element method: basic perturbation technique and computer implementation. Wiley, New York (1992)

    MATH  Google Scholar 

  29. Koutsourelakis, P.S., Pradlwarter, H.J., Schuëller, G.I.: Reliability of structures in high dimensions, part I: Algorithms and applications. Probabilist. Eng. Mech. 19(4), 409–417 (2004)

    Article  Google Scholar 

  30. Mace, B.R., Shorter, P.J.: A local modal/perturbational method for estimating frequency response statistics of built-up structures with uncertain properties. J. Sound Vib. 242(5), 793–811 (2001)

    Article  Google Scholar 

  31. MacNeal, R.H.: A hybrid method of component mode synthesis. Comput. Struct. 1(4), 581–601 (1971)

    Article  Google Scholar 

  32. Mayer, M.: Die Sicherheit der Bauwerke und ihre Berechnung nach Grenzkräften anstatt nach zulässigen Spannungen. Springer, Berlin (1926)

    MATH  Google Scholar 

  33. MCS Software Corporation: MCS Nastran 2001: Superelement User’s Guide, Santa Ana, CA 92707 USA (2004)

    Google Scholar 

  34. Myers, R.H., Montgomery, D.C.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley, New York (2002)

    MATH  Google Scholar 

  35. Nair, P.B., Keane, A.J.: Stochastic reduced basis methods. AIAA J. 40(8), 1653–1664 (2002)

    Article  Google Scholar 

  36. Nair, P.B., Keane, A.J.: An approximate solution scheme for the algebraic random eigenvalue problem. J. Sound Vib. 260(1), 45–65 (2003)

    Article  Google Scholar 

  37. Ohayon, R., Sampaio, R., Soize, C.: Dynamic substructuring of damped structures using singular value decomposition. J. Appl. Mech. 64, 292–298 (1997)

    Article  MATH  Google Scholar 

  38. Pellissetti, M.: Parallel processing in structural reliability. J. Struct. Eng. Mech. 32, 95–126 (2009)

    Google Scholar 

  39. Pichler, L., Pradlwarter, H.J., Schuëller, G.I.: A mode-based meta-model for the frequency response functions of uncertain structural systems. Comput. Struct. 87(5–6), 332–341 (2009)

    Article  Google Scholar 

  40. Pradlwarter, H.J., Schuëller, G.I.: Stochastic response evaluation of large FE-models with non-linear hysteretic elements by complex mode synthesis. In: Spanos, P.D. (ed.) Proceedings of the Third International Conference on Computational Stochastic Mechanics, pp. 551–558, Santorini, Greece. A.A. Balkema, Rotterdam (1999)

    Google Scholar 

  41. Pradlwarter, H.J., Schuëller, G.I., Székely, G.S.: Random eigenvalue problems for large systems. Comput. Struct. 80, 2415–2424 (2002)

    Article  Google Scholar 

  42. Rahman, S.: Stochastic dynamic systems with complex-valued eigensolutions. Int. J. Num. Meth. Eng. 71, 963–986 (2007)

    Article  MATH  Google Scholar 

  43. Rubin, S.: Improved component-mode representation for structural dynamic analysis. AIAA J. 13(8), 995–1006 (1975)

    Article  MATH  Google Scholar 

  44. Sachdeva, S.K., Nair, P.B., Keane, A.J.: Comparative study of projection schemes for stochastic finite element analysis. Comput. Method. Appl. Mech. Eng. 195(19–22): 2371–2392 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Sacks, J., Schiller, S.B., Welch, W.J.: Design for computer experiment. Technometrics 31(1), 41–47 (1989)

    MathSciNet  Google Scholar 

  46. Schuëller, G.I.: Structural reliability – recent advances – Freudenthal Lecture. In: Shiraishi, N., Shinozuka, M., Wen, Y.K. (eds.) Proceedings of the 7th International Conference on Structural Safety and Reliability (ICOSSAR’97), pp. 3–35, Kyoto, Japan. A.A. Balkema, Rotterdam (1998)

    Google Scholar 

  47. Schuëller, G.I.: Developments in stochastic structural mechanics. Arch. Appl. Mech. 75, 755–773 (2006)

    Article  MATH  Google Scholar 

  48. Schuëller, G.I.: On the treatment of uncertainties in structural mechanics and analysis (based on a Plenary Keynote Lecture presented at the Third M.I.T. Conference on Computational Fluid and Solid Mechanics), Boston, USA. Comput. Struct. 85(5–6): 235–243 (2007)

    Google Scholar 

  49. Schuëller, G.I. (ed.): A state-of-the-art report on computational stochastic mechanics. J. Probabilist. Eng. Mech. 12(4), 197–321 (1997)

    Google Scholar 

  50. Schuëller, G.I., Stix, R.: A critical appraisal of methods to determine failure probabilities. J. Struct. Safety, 4, 293–309 (1987)

    Article  Google Scholar 

  51. Soize, C.: A nonparametric model of random uncertainties for reduced matrix models in structural dynamics. Probabilist. Eng. Mech. 15, 277–294 (2000)

    Article  Google Scholar 

  52. Soize, C.: Maximum entropy approach for modeling random uncertainties in transient elastodynamics. J. Acoust. Soc. Am. 109, 1979–1996 (2001)

    Article  Google Scholar 

  53. Székely, G.S., Schuëller, G.I.: Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties. Comput. Method. Appl. Mech. Eng. 191(8–10): 799–816 (2001)

    Article  Google Scholar 

  54. Székely, G.S., Teichert, W.H., Brenner, C.E., Pradlwarter, H.J., Klein, M., Schuëller, G.I.: Practical procedures for reliability estimation of spacecraft structures and their components. AIAA J. 36(8), 1509–1515 (1998)

    Article  Google Scholar 

  55. Vasta, M., Schuëller, G.I.: Phase space reduction in stochastic dynamics. J. Eng. Mech. 126(6), 626–632 (2000)

    Article  Google Scholar 

  56. Verhoosel, C.A., Gutièrrez, M.A., Hulsdoff, S.J.: Iterative solution of the random eigenvalue problem with application to spectral stochastic finite element analysis. Int. J. Numer. Method. Eng. 68, 401–424 (2006)

    Article  MATH  Google Scholar 

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Acknowledgements

This research was partially supported by the Austrian Research Council (FWF) under Project No. P19781-N13 which is gratefully acknowledged by the author.

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Authors and Affiliations

  1. Institute of Engineering Mechanics, University of Innsbruck, Technikerstr. 13, A-6020, Innsbruck, Austria

    Gerhart I. Schuëller

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  1. Gerhart I. Schuëller
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Correspondence to Gerhart I. Schuëller .

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Editors and Affiliations

  1. Inst. Structural Analysis &, Seismic Research, National Technical Univ. Athen, Iroon Polytechniou Str. 9, Athens, 157 80, Greece

    Manolis Papadrakakis

  2. , Intitute of Structural Analysis and Seis, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    George Stefanou

  3. , Institute of Structural Analysis and Se, National Technical University of Athens, Iroon Polytechniou, Zografou Campus 9, Athens, 15780, Greece

    Vissarion Papadopoulos

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Schuëller, G.I. (2011). Model Reduction and Uncertainties in Structural Dynamics. In: Papadrakakis, M., Stefanou, G., Papadopoulos, V. (eds) Computational Methods in Stochastic Dynamics. Computational Methods in Applied Sciences, vol 22. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9987-7_1

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  • DOI: https://doi.org/10.1007/978-90-481-9987-7_1

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