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COSSAN: A Multidisciplinary Software Suite for Uncertainty Quantification and Risk Management

  • Edoardo Patelli
Living reference work entry

Abstract

Computer-aided modeling and simulation is now widely recognized as the third “leg” of scientific method, alongside theory and experimentation. Many phenomena can be studied only by using computational processes such as complex simulations or analysis of experimental data. In addition, in many engineering fields computational approaches and virtual prototypes are used to support and drive the design of new components, structures, and systems. One of the greatest challenges of virtual prototyping is to improve the fidelity of the computational analysis. This can only be achieved by explicitly including variability and uncertainties from different sources. Variability is inherent in many natural systems and therefore cannot be reduced. Uncertainty is also always present since it is not possible to perfectly model or predict future events for which no real-world data is available.Although stochastic methods offer a much more realistic approach for analysis and design, their utilization in practical applications remains quite limited. One of the reasons is that the developments of software for stochastic analysis have received considerably less attention than their deterministic counterparts. Another common limitation is that the computational cost of stochastic analysis is often by orders of magnitude higher than the deterministic analysis. Hence, robust, efficient, and scalable computational tools are necessary, i.e., by making use of the computational power of a cluster and grid computing.This chapter presents the COSSAN project: a developed multidisciplinary general-purpose software suite for uncertainty quantification and risk analysis . The computational tools satisfy the industry requirements regarding usability, numerical efficiency, flexibility, and scalability. The software can be used to solve a wide range of engineering and scientific problems. The availability of such software is particularly important for the analysis and design of resilient structures and systems. In fact, despite the different levels of uncertainty, decision makers still need to take clear choices based on the available information. They need to trust the methodology adopted to propagate the uncertainties through multidisciplinary analysis, in order to quantify the risk with the current level of information and to avoid wrong decisions due to artificial restrictions introduced by the modeling.

Keywords

Aleatory and epistemic uncertainty Computational methods High-performance computing Imprecise probability Matlab Monte Carlo simulation Open source Rare events Reliability-based optimization Risk analysis Robust optimization Sensitivity analysis Uncertainty quantification 

References

  1. 1.
    Alvarez, D.A.: Infinite random sets and applications in uncertainty analysis. PhD thesis, Arbeitsbereich für Technische Mathematik am Institut für Grundlagen der Bauingenieurwissenschaften. Leopold-Franzens-Universität Innsbruck, Innsbruck. Available at https://sites.google.com/site/Diegoandresalvarezmarin/RSthesis.pdf (2007)
  2. 2.
    Alvarez, D.A.: Reduction of uncertainty using sensitivity analysis methods for infinite random sets of indexable type. Int. J. Approx. Reason. 50(5), 750–762 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alvarez, D.A., Hurtado, J.E.: An efficient method for the estimation of structural reliability intervals with random sets, dependence modelling and uncertain inputs. Comput. Struct. 142, 54–63 (2014)CrossRefGoogle Scholar
  4. 4.
    Au, S.K., Beck, J.: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16(4), 263–277 (2001)CrossRefGoogle Scholar
  5. 5.
    Au, S.K., Patelli, E.: Subset Simulation in finite-infinite dimensional space. Reliab. Eng. Syst. Saf. 2016, 148, 66–77CrossRefGoogle Scholar
  6. 6.
    Aven, T., Zio, E.: Some considerations on the treatment of uncertainties in risk assessment for practical decision making. Reliab. Eng. Syst. Saf. 96, 64–74 (2011)CrossRefGoogle Scholar
  7. 7.
    Barber, S., Voss, J., Webster, M.: The rate of convergence for approximate Bayesian computation. arXiv preprint, arXiv:13112038 (2013)Google Scholar
  8. 8.
    Beaurepaire, P., Valdebenito, M., Schuëller, G.I., Jensen, H.: Reliability-based optimization of maintenance scheduling of mechanical components under fatigue. CMAME 221–222, 24–40 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Beck, J.L., Katafygiotis, L.S.: Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech. ASCE 124(4), 455–461 (1998)Google Scholar
  10. 10.
    Beer, M., Ferson, S.: Fuzzy probability in engineering analyses. In: Ayyub, B. (ed.) Proceedings of the First International Conference on Vulnerability and Risk Analysis and Management (ICVRAM 2011) and the Fifth International Symposium on Uncertainty Modeling and Analysis (ISUMA), pp. 53–61, 11–13 Apr 2011, University of Maryland, ASCE, Reston (2011)Google Scholar
  11. 11.
    Beer, M., Ferson, S.: Special issue of mechanical systems and signal processing “imprecise probabilities-what can they add to engineering analyses?”. Mech. Syst. Signal Process. 37(1–2), 1–3 (2013). doi:http://dx.doi.org/10.1016/j.ymssp.2013.03.018, http://www.sciencedirect.com/science/article/pii/S0888327013001180
  12. 12.
    Beer, M., Patelli, E.: Editorial: engineering analysis with vague and imprecise information. Struct. Saf. 52, Part B, 143 (2015). doi:http://dx.doi.org/10.1016/j.strusafe.2014.11.001, http://www.sciencedirect.com/science/article/pii/S0167473014001106. Special Issue: Engineering Analyses with Vague and Imprecise Information.
  13. 13.
    Beer, M., Phoon, K.K., Quek, S.T. (eds.): Special issue: Modeling and analysis of rare and imprecise information. Struct. Saf. 32 (2010)Google Scholar
  14. 14.
    Beer, M., Zhang, Y., Quek, S.T., Phoon, K.K.: Reliability analysis with scarce information: Comparing alternative approaches in a geotechnical engineering context. Struct. Saf. 41(6), 1–10 (2013). doi:http://dx.doi.org/10.1016/j.strusafe.2012.10.003, http://www.sciencedirect.com/science/article/pii/S0167473012000689
  15. 15.
    Benjamin, J., Schuëller, G., Wittmann, F. (eds.): Proceedings of the second international seminar on structural reliability of mechanical components and subassemblies of nuclear power plants, special volume. J. Nucl. Eng. Des. 59, 1–168 (1989)Google Scholar
  16. 16.
    Bratley, P., Fox, B.L.: Algorithm 659: implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Softw. 14(1), 88–100 (1988). doi:http://doi.acm.org/10.1145/42288.214372
  17. 17.
    Bucher, C., Pradlwarter, H.J., Schuëller, G.I.: Computational stochastic structural analysis (COSSAN). In: Schuëller, G.I. (ed.) Structural Dynamics – Recent Advances, pp. 301–316. Springer, Berlin/Heidelberg (1991)CrossRefGoogle Scholar
  18. 18.
    Bucher, C., Pradlwarter, H.J., Schuëller, G.I.: COSSAN – (Computational stochastic structural analysis) – Perspectives of software developments. In: Schuëller, G.I., et al. (ed.) Proceedings of the 6th International Conference on Structural Safety and Reliability (ICOSSAR’93), pp. 1733–1740. A.A. Balkema Publications, Rotterdam/Innsbruck (1994)Google Scholar
  19. 19.
    Busacca, P.G., Marseguerra, M., Zio, E.: Multiobjective optimization by genetic algorithms: application to safety systems. Reliab. Eng. Syst. Saf. 72(1), 59–74 (2001). http://www.sciencedirect.com/science/article/B6V4T-42G751J-7/2/f0bf8189c921c1d6029d1f9b56524094 CrossRefGoogle Scholar
  20. 20.
    Chiachio, M., Beck, J.L., Chiachio, J., Rus, G.: Approximate Bayesian computation by subset simulation. arXiv preprint, arXiv:14046225 (2014)Google Scholar
  21. 21.
    Ching, J., Chen, Y.: Transitional Markov Chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging. J. Eng. Mech. 133(7), 816–832 (2007). doi:10.1061/(ASCE)0733-9399(2007)133:7(816), http://ascelibrary.org/doi/abs/10.1061/%28ASCE%290733-9399%282007%29133%3A7%28816%29 CrossRefGoogle Scholar
  22. 22.
    Crémona, C., Lukić, M.: Probability-based assessment and maintenance of welded joints damaged by fatigue. Nucl. Eng. Des. 182(3), 253–266 (1998)CrossRefGoogle Scholar
  23. 23.
    Crespo, L.G., Kenny, S.P., Giesy, D.P.: The NASA langley multidisciplinarty uncertainty quantification challenge. In: 16th AIAA Non-Deterministic Approaches Conference – AIAA SciTech, American Institute of Aeronautics and Astronautics (2014). doi:10.2514/6.2014-1347, http://dx.doi.org/10.2514/6.2014-1347 Google Scholar
  24. 24.
    de Angelis, M., Patelli, E., Beer, M.: An efficient strategy for interval computations in risk-based optimization. In: ICOSSAR, 16–20 June 2013. Columbia University, New York (2013)Google Scholar
  25. 25.
    de Angelis, M., Patelli, E., Beer, M.: Advanced line sampling for efficient robust reliability analysis. Struct. Saf. 52, 170–182 (2015). doi:10.1016/j.strusafe.2014.10.002, http://www.sciencedirect.com/science/article/pii/S0167473014000927 CrossRefGoogle Scholar
  26. 26.
    DeFinetti, B.: Theory of Probability: A Critical Introductory Treatment. Wiley, Chichester (1990)Google Scholar
  27. 27.
    Der Kiureghian, A., Dakessian, T.: Multiple design points in first and second-order reliability. Struct. Saf. 20(1), 37–49, doi:10.1016/S0167-4730(97)00026-X, http://www.sciencedirect.com/science/article/B6V54-3T2H6KD-3/2/241e203d3372ca22a2cc463c44cc98ca (1998)
  28. 28.
    Ditlevsen, O., Madsen, H.O.: Structural Reliability Methods, Internet edition. Wiley, Chichester (2005)Google Scholar
  29. 29.
    Exler, O., Schittkowski, K.: A trust region SQP algorithm for mixed-integer nonlinear programming. Optim. Lett. (2007). doi:10.1007/s11590-006-0026-1MathSciNetzbMATHGoogle Scholar
  30. 30.
    Free Software Foundation: Free software foundation, GNU lesser general public license, version 3. http://www.gnu.org/licenses/lgpl.html (2007)
  31. 31.
    Ghanem, R., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York/Berlin/Heidelberg. Revised edition 2003, Dover Publications, Mineola/New York (1991)Google Scholar
  32. 32.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization and Machine Learning. Addison Wesley, Reading (1989)zbMATHGoogle Scholar
  33. 33.
    Goller, B., Pradlwarter, H.J., Schuëller, G.I.: Robust modal updating with insufficient data. Comput. Methods Appl. Mech. Eng. 198(37–40), 3096–3104 (2009). doi:10.1016/j.cma.2009.05.009CrossRefzbMATHGoogle Scholar
  34. 34.
    Harder, R., Desmarais, R.: Interpolation using surface splines. J. Aircr. 2, 189–191 (1972)CrossRefGoogle Scholar
  35. 35.
    Hoshiya, M.: Kriging and conditional simulation of gaussian field. J. Eng. Mech. ASCE 121(2), 181–186 (1995)CrossRefGoogle Scholar
  36. 36.
    Jensen, H., Catalan, M.: On the effects of non-linear elements in the reliability-based optimal design of stochastic dynamical systems. Int. J. Nonlinear Mech. 42(5), 802–816 (2007)CrossRefzbMATHGoogle Scholar
  37. 37.
    Jensen, H., Valdebenito, M., Schuëller, G.: An efficient reliability-based optimization scheme for uncertain linear systems subject to general gaussian excitation. Comput. Methods Appl. Mech. Eng. 198(1), 72–87 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kijawatworawet, W.: Reliability of structural systems using adaptive importance directional sampling. PhD thesis, Institute of Engineering Mechanics, Leopold-Franzens University, Innsbruck, EU (1992)Google Scholar
  39. 39.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science, Number 4598, 13 May 1983 220, 4598, 671–680 (1983). citeseer.ist.psu.edu/kirkpatrick83optimization.html
  40. 40.
    Koutsourelakis, P.S., Pradlwarter, H.J., Schuëller, G.I.: Reliability of structures in high dimensions, part I: algorithms and applications. Probab. Eng. Mech. 19(4), 409–417 (2004). doi:10.1016/j.probengmech.2004.05.001CrossRefGoogle Scholar
  41. 41.
    Kucherenko, S., Delpuech, B., Iooss, B., Tarantola, S.: Application of the control variate technique to estimation of total sensitivity indices. Reliab. Eng. Syst. Saf. 134, 251–259 (2015). doi:10.1016/j.ress.2014.07.008CrossRefGoogle Scholar
  42. 42.
    Laplace, P.S.: A Philosophical Essay on Probabilities. Dover Publications, New York (1814)zbMATHGoogle Scholar
  43. 43.
    Liu, J.: Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, New York (2001)zbMATHGoogle Scholar
  44. 44.
    Melchers, R.E.: Structural reliability: analysis and prediction. Wiley, Chichester (2002)Google Scholar
  45. 45.
    Melchers, R.E., Ahammed, M.: Gradient estimation for applied Monte Carlo analyses. Reliab. Eng. Syst. Saf. 78(3), 283–288 (2002). http://www.sciencedirect.com/science/article/B6V4T-475R7RS-8/2/8eaa29f83ddacc51937b7005aed69481 CrossRefGoogle Scholar
  46. 46.
    Mitseas, I., Kougioumtzoglou, I., Beer, M., Patelli, E., Mottershead, J.: Robust design optimization of structural systems under evolutionary stochastic seismic excitation. In: Vulnerability, Uncertainty, and Risk, American Society of Civil Engineers, pp. 215–224 (2014). doi:10.1061/9780784413609.022, http://dx.doi.org/10.1061/9780784413609.022
  47. 47.
    Molchanov, I.: Theory of Random Sets. Springer, London (2005)zbMATHGoogle Scholar
  48. 48.
    Möller, B., Beer, M.: Fuzzy-Randomness – Uncertainty in Civil Engineering and Computational Mechanics. Springer, Berlin/New York (2004)zbMATHGoogle Scholar
  49. 49.
    Müller, B., Graf, W., Beer, M.: Fuzzy structural analysis using alpha-level optimization. Comput. Mech. 26, 547–565 (2000)CrossRefzbMATHGoogle Scholar
  50. 50.
    NASA Standard for Models and Simulations: Tech. Rep. NASA-STD-7009, National Aeronautics and Space Administration (NASA) (2013)Google Scholar
  51. 51.
    Nelder, J., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)CrossRefzbMATHGoogle Scholar
  52. 52.
    Nissen, S.: Implementation of a fast artificial neural network library (fann). Tech. rep., Department of Computer Science University of Copenhagen (DIKU), http://fann.sf.net (2003)
  53. 53.
    Olsson, A., Sandberg, G., Dahlblom, O.: On Latin hypercube sampling for structural reliability analysis. Struct. Saf. 25, 47–68(22) (2003). doi:10.1016/S0167-4730(02)00039-5, http://www.ingentaconnect.com/content/els/01674730/2003/00000025/00000001/art00039 CrossRefGoogle Scholar
  54. 54.
    Panayirci, H.M.: Efficient solution for Galerkin based polynomial chaos expansion systems. Adv. Eng. Softw. 41(412), 1277–1286 (2010). doi:10.1016/j.advengsoft.2010.09.004CrossRefzbMATHGoogle Scholar
  55. 55.
    Paris, P., Erdogan, F.: A critical analysis of crack propagation laws. J. Basic Eng. Trans. ASME 85, 528–534 (1963)CrossRefGoogle Scholar
  56. 56.
    Patelli, E., Au, I.: Efficient Monte Carlo algorithm for rare failure event simulation. In: 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12), Vancouver, 12–15 July 2015, http://hdl.handle.net/2429/53247 (2015)
  57. 57.
    Patelli, E., Broggi, M.: On general purpose software for the efficient uncertainty management of large finite element models. In: NAFEMS World Congress, 9–12 June 2013, Salzburg, NAFEMS, http://academia.edu/attachments/31544367/download_file (2013)
  58. 58.
    Patelli, E., de Angelis, M.: Line sampling approach for extreme case analysis in presence of aleatory and epistemic uncertainties. In: European Safety and Reliability Conference – ESREL – 7–10 Sept 2015. CRC Press/Balkema (2015)Google Scholar
  59. 59.
    Patelli, E., Pradlwarter, H.: Monte Carlo gradient estimation in high dimensions. Int. J. Numer. Methods Eng. 81(2), 172–188 (2010). doi:10.1002/nme.2687MathSciNetzbMATHGoogle Scholar
  60. 60.
    Patelli, E., Schuëller, G.I.: Computational optimization strategies for the simulation of random media and components. Comput. Optim. Appl. 1–29 (2012). doi:10.1007/s10589-012-9463-1, http://dx.medra.org/10.1007/s10589-012-9463-1
  61. 61.
    Patelli, E., Pradlwarter, H.J., Schuëller, G.I.: Global sensitivity of structural variability by random sampling. Comput. Phys. Commun. 181, 2072–2081 (2010). doi:10.1016/j.cpc.2010.08.007MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Patelli, E., Pradlwarter, H., Schuëller, G.: On multinormal integrals by importance sampling for parallel system reliability. Struct. Saf. 33, 1–7 (2011). doi:10.1016/j.strusafe.2010.04.002CrossRefGoogle Scholar
  63. 63.
    Patelli, E., Pradlwarter, H.J., Schuëller, G.I.: On multinormal integrals by importance sampling for parallel system reliability. Struct. Saf. 33, 1–7 (2011). doi:10.1016/j.strusafe.2010.04.002CrossRefGoogle Scholar
  64. 64.
    Patelli, E., Valdebenito, M.A., Schuëller, G.I.: General purpose stochastic analysis software for optimal maintenance scheduling: application to a fatigue-prone structural component. Int. J. Reliab. Saf. 5, 211–228 (2011). Special Issue on: “Robust Design – Coping with Hazards Risk and Uncertainty”Google Scholar
  65. 65.
    Patelli, E., Panayirci, H.M., Broggi, M., Goller, B., Pradlwarter, P.B.H.J., Schuëller, G.I.: General purpose software for efficient uncertainty management of large finite element models. Finite Elem. Anal. Des. 51, 31–48 (2012). doi:10.1016/j.finel.2011.11.003, http://dx.medra.org/10.1016/j.finel.2011.11.003 CrossRefGoogle Scholar
  66. 66.
    Patelli, E., Alvarez, D.A., Broggi, M., de Angelis, M.: An integrated and efficient numerical framework for uncertainty quantification: application to the NASA Langley multidisciplinary uncertainty quantification challenge. In: 16th AIAA Non-Deterministic Approaches Conference (SciTech 2014), American Institute of Aeronautics and Astronautics, AIAA SciTech (2014). doi:10.2514/6.2014-1501Google Scholar
  67. 67.
    Patelli, E., Broggi, M., de Angelis, M., Beer, M.: Opencossan: an efficient open tool for dealing with epistemic and aleatory uncertainties. In: Vulnerability, Uncertainty, and Risk, American Society of Civil Engineers, pp. 2564–2573 (2014). doi:10.1061/9780784413609.258, http://dx.doi.org/10.1061/9780784413609.258
  68. 68.
    Patelli, E., Alvarez, D.A., Broggi, M., de Angelis, M.: Uncertainty management in multidisciplinary design of critical safety systems. J. Aerosp. Inf. Syst. 12, 140–169 (2015). doi:10.2514/1.I010273Google Scholar
  69. 69.
    Pedroni, N., Zio, E., Ferrario, E., Pasanisi, A., Couplet, M.: Propagation of aleatory and epistemic uncertainties in the model for the design of a food protection dike. In: PSAM 11 & ESREL, Jun 2012, Helsinki, pp. 1–10 (2012)Google Scholar
  70. 70.
    Powell, M.: Direct search algorithms for optimization calculations. Acta Numer. 7, 287–336 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    Powell, M.J.D.: The BOBYQA algorithm for bound constrained optimization without derivatives. Tech. rep., Department of Applied Mathematics and Theoretical Physics, Cambridge, http://fann.sf.net (2009)
  72. 72.
    Pradlwarter, H., Schuëller, G.: Reliability assessment of uncertain linear systems in structural dynamics. In: Belyaev, A.K., Langley, R.S. (eds.) IUTAM Symposium on the Vibration Analysis of Structures with Uncertainties, Saint Petersburg, pp. 363–378 (2011)CrossRefGoogle Scholar
  73. 73.
    Romero, V., Mullins, J., Swiler, L., Urbina, A.: A comparison of methods for representing and aggregating uncertainties involving sparsely sampled random variables – more results. SAE Int. J. Mater. Manuf. 6(3) (2013). http://www.scopus.com/inward/record.url?eid=2-s2.0-84876425264&partnerID=40&md5=72ea116c4e8d25c856e55d3d07afd890
  74. 74.
    Roux, W.J., Stander, N., Haftka, R.T.: Response surface approximation for structural optimization. Int. J. Numer. Methods Eng. 42, 517–534 (1998)CrossRefzbMATHGoogle Scholar
  75. 75.
    Rubinstein, R.: Simulation and the Monte Carlo Method. John Wiley & Sons, New York/Chichester/Brisbane/Toronto (1981)CrossRefzbMATHGoogle Scholar
  76. 76.
    Saltelli, A., Bolado, R.: An alternative way to compute fourier amplitude sensitivity test (fast). Comput. Stat. Data Anal. 26(4), 445–460 (1998). doi:10.1016/S0167-9473(97)00043-1, http://www.sciencedirect.com/science/article/B6V8V-3SX829Y-5/2/1147936f52dcb9461d1f69aa319bb117 CrossRefzbMATHGoogle Scholar
  77. 77.
    Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Salsana, M., Tarantola, S.: Global Sensitivity Analysis: The Primer. Wiley, Chichester (2008)zbMATHGoogle Scholar
  78. 78.
    Schenk, C.A., Schuëller, G.I.: Uncertainty Assessment of Large Finite Element Systems, Lecture Notes in Applied and Computational Mechanics, vol 24. Springer, Berlin/Heidelberg/New York (2005). http://www.springer.com/materials/mechanics/book/978-3-540-25343-3, ISBN:978-3-540-25343-3
  79. 79.
    Schuëller, G.: Efficient Monte Carlo simulation procedures in structural uncertainty and reliability analysis – recent advances. J. Struct. Eng. Mech. 32(1), 1–20 (2009)CrossRefGoogle Scholar
  80. 80.
    Schuëller, G.I.: On procedures for reliability assessment of mechanical systems and structures. J. Struct. Eng. Mech. 25(3), 275–289 (2007)CrossRefGoogle Scholar
  81. 81.
    Schuëller, G.I., Pradlwarter, H.J.: Computational stochastic structural analysis(COSSAN) – a software tool. Struct. Saf. 28(1–2), 68–82 (2006). doi:10.1016/j.strusafe.2005.03.005CrossRefGoogle Scholar
  82. 82.
    Schuëller, G.I., Pradlwarter, H.J.: Uncertainty analysis of complex structural systems. Int. J. Numer. Methods Eng. 80(6–7), 881–913 (2009). doi:10.1002/nme.2549MathSciNetCrossRefzbMATHGoogle Scholar
  83. 83.
    Schuëller, G.I., Stix, R.: A critical appraisal of methods to determine failure probabilities. J. Struct. Saf. 4(4), 293–309 (1987)CrossRefGoogle Scholar
  84. 84.
    Schuëller, G.I. (ed.): GI Uncertainties in structural mechanics and analysis – computational methods. Comput. Struct. – Special Issue 83(14), 1031–1149 (2005). doi:10.1016/j.compstruc.2005.01.004Google Scholar
  85. 85.
    Schuëller, G.I. (ed.): GI Structural reliability software. Struct. Saf. – Special Issue 28(1–2), 1–216 (2006). doi:10.1016/j.strusafe.2005.03.001Google Scholar
  86. 86.
    Schuëller, G., Jensen, H.: Computational methods in optimization considering uncertainties – an overview. Comput. Methods Appl. Mech. Eng. 198(1), 2–13 (2008)CrossRefzbMATHGoogle Scholar
  87. 87.
    Sobol’, I.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1(4), 407–414 (1993)Google Scholar
  88. 88.
    Sobol’, I.: Global sensitivity indices for nonlinear mathematical modes and their Monte Carlo estimates. Math. Comput. Simul. 55, 217–280 (2001)Google Scholar
  89. 89.
    Sudret, B.: Meta-models for structural reliability and uncertainty quantification. ArXiv e-prints 1203.2062 (2012)Google Scholar
  90. 90.
    Sudret, B., Der Kiureghian, A.: Stochastic finite element methods and reliability a state-of-the-art report. Tech. rep., Department of Civil and Environmental Engineering, University of California, Berkeley (2000)Google Scholar
  91. 91.
    Thomas, B.: Evolutionary algorithms in theory and practice : evolution strategies, evolutionary programming, genetic algorithms. Oxford University Press, New York (1996). doi:0-19-509971-0Google Scholar
  92. 92.
    Valdebenito, M.: Reliability-based optimization: Efficient strategies for high dimensional reliability problems. PhD thesis, Institute of Engineering Mechanics, University of Innsbruck, Innsbruck (2010)Google Scholar
  93. 93.
    Valdebenito, M., Schuëller, G.: Design of maintenance schedules for fatigue-prone metallic components using reliability-based optimization. Comput. Methods Appl. Mech. Eng. 199, 2305–2318 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  94. 94.
    Valdebenito, M., Patelli, E., Schuëller, G.: A general purpose software for reliability-based optimal design. In: Muhanna, M.B.R., Mullen, R. (eds.) 4th International Workshop on Reliable Engineering Computing: Robust Design – Coping with Hazards, Risk and Uncertainty, Research Publishing Services, Singapore, pp. 3–22 (2010). doi:10.3850/978-981-08-5118-7_plenary-1Google Scholar
  95. 95.
    Valdebenito, M., Pradlwarter, H., Schuëller, G.: The role of the design point for calculating failure probabilities in view of dimensionality and structural non linearities. Struct. Saf. 32(2), 101–111 (2010). doi:10.1016/j.strusafe.2009.08.004CrossRefGoogle Scholar
  96. 96.
    Vanmarcke, E.: Random fields: analysis and synthesis. Published by MIT Press, Cambridge, MA (1983); Web Edition by Rare Book Services, Princeton University. Princeton, Cambridge, MA (1998)Google Scholar
  97. 97.
    Wang, P., Lu, Z., Tang, Z.: A derivative based sensitivity measure of failure probability in the presence of epistemic and aleatory uncertainties. Comput. & Math. Appl. 65(1), 89–101 (2013). doi:10.1016/j.camwa.2012.08.017, http://www.sciencedirect.com/science/article/pii/S0898122112006438 MathSciNetCrossRefzbMATHGoogle Scholar
  98. 98.
    Youssef, H., Sait, S.M., Adiche, H.: Evolutionary algorithms, simulated annealing and tabu search: a comparative study. Eng. Appl. Artif. Intell. 14(2), 167–181 (2001). doi:10.1016/S0952-1976(00)00065-8, http://www.sciencedirect.com/science/article/B6V2M-42JRD52-6/2/a02150bf476eeff0d9f64652698ddea7 CrossRefGoogle Scholar
  99. 99.
    Zhang, H., Mullen, R.L., Muhanna, R.L.: Interval Monte Carlo methods for structural reliability. Struct. Saf. 32(3), 183–190 (2010)CrossRefGoogle Scholar
  100. 100.
    Zhang, M., Beer, M., Quek, S.T., Choo, Y.S.: Comparison of uncertainty models in reliability analysis of offshore structures under marine corrosion. Struct. Saf. 32(6), 425–432 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institute for Risk and UncertaintyUniversity of LiverpoolLiverpoolUK

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