Probabilistic Failure Analysis by Importance Sampling Markov Chain Simulation
Publication: Journal of Engineering Mechanics
Volume 130, Issue 3
Abstract
A probabilistic approach for failure analysis is presented in this paper, which investigates the probable scenarios that occur in case of failure of engineering systems with uncertainties. Failure analysis can be carried out by studying the statistics of system behavior corresponding to the random samples of uncertain parameters that are distributed as the conditional distribution given that the failure event has occurred. This necessitates the efficient generation of conditional samples, which is in general a highly nontrivial task. A simulation method based on Markov Chain Monte Carlo simulation is proposed to efficiently generate the conditional samples. It makes use of the samples generated from importance sampling simulation when the performance reliability is computed. The conditional samples can be used for statistical averaging to yield unbiased and consistent estimate of conditional expectations of interest for failure analysis. Examples are given to illustrate the application of the proposed simulation method to probabilistic failure analysis of static and dynamic structural systems.
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References
Abraham, F. F.(1986). “Computational statistical mechanics: methodology, applications and supercomputing.” Adv. Phys., 35, 1–111.
Alder, B. J., and Wainwright, T. E.(1959). “Studies in molecular dynamics. I. General method.” J. Chem. Phys., 31, 459–466.
Au, S. K. (2001). “On the solution of first excursion problems by simulation with applications to probabilistic seismic performance assessment.” PhD Thesis in Civil Engineering, EERL Rep. No. 2001-02, California Institute of Technology, Pasadena. Available at URL http://www.ntu.edu.sg/home/cskau
Au, S. K., and Beck, J. L.(1999). “A new adaptive importance sampling scheme.” Struct. Safety, 21, 135–158.
Au, S. K., and Beck, J. L.(2001a). “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech., 16(4), 263–277.
Au, S. K., and Beck, J. L.(2001b). “First-excursion probabilities for linear systems by very efficient importance sampling.” Probab. Eng. Mech., 16, 193–207.
Software available at URL http://www.ntu.edu.sg/home/cskau
Au, S. K., and Beck, J. L.(2002). “Importance sampling in high dimensions.” Struct. Safety, 25(2), 139–163.
Beck, J. L., and Au, S. K.(2002). “Bayesian updating of structural models and reliability using Markov Chain Monte Carlo simulation.” J. Eng. Mech., 128(4), 380–391.
Bertsimas, D., and Tsitsiklis, J.(1993). “Simulated annealing.” Stat. Sci., 8, 10–15.
Besag, J., and Green, P. J.(1993). “Spatial statistics and Bayesian computation.” J. R. Stat. Soc. Ser. B. Methodol., 55, 25–37.
Bhanot, G.(1988). “The Metropolis algorithm.” Rep. Prog. Phys., 51, 429–457.
Chib, S., and Greenberg, E.(1994). “Bayes Inference in regression-models with ARMA(p,q) errors.” J. Econometr., 64(1-2), 183–206.
Chib, S., Greenberg, E., and Winkelmann, R.(1998). “Posterior simulation and Bayes factors in panel count data models.” J. Econometr., 86(1), 33–54.
Cornell, C. A. (1996). “Reliability-based earthquake-resistant design: the future.” Proc., 11th World Conf. on Earthquake Engineering, Acapulco, Mexico.
Cox, R. T. (1961). The algebra of probable inference, Johns Hopkins, Baltimore.
Ditlevsen, O., and Madsen, H. O. (1996). Structural reliability methods, Wiley, Chichester.
Doob, J. L. (1953). Stochastic processes, Wiley, New York.
Duane, S., Kennedy, A. D., Pendleton, B. J., and Roweth, D.(1987). “Hybrid Monte Carlo.” Phys. Lett. B, 195, 216–222.
Engelund, S., and Rackwitz, R.(1993). “A benchmark study on importance sampling techniques in structural reliability.” Struct. Safety, 12, 255–276.
Fishman, G. S. (1996). Monte Carlo: Concepts, algorithms, and applications, Springer, New York.
Freudenthal, A. M.(1947). “The safety of structures.” Trans. Am. Soc. Civ. Eng., 112, 125–180.
Freudenthal, A. M.(1956). “Safety and the probability of structural failure.” Trans. Am. Soc. Civ. Eng., 121, 1337–1397.
Freudenthal, A. M., Garrelts, J. M., and Shinozuka, M.(1966). “The analysis of structural safety.” J. Struct. Div. ASCE, 92(1), 267–325.
Geman, S., and Geman, D.(1984). “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images.” IEEE Trans. Pattern Anal. Mach. Intell., 6, 721–741.
Hajek, B.(1988). “Cooling schedules for optimal annealing.” Math. Op. Res., 13, 311–329.
Hammersley, J. M., and Handscomb, D. C. (1964). Monte-Carlo methods, Methuen, London.
Hastings, W. K.(1970). “Monte Carlo sampling methods using Markov chains and their applications.” Biometrika, 57, 97–109.
Jaynes, E. T. (1978). “Where do we stand on maximum entropy?” in The maximum entropy formalism, R. D. Levine and M. Tribus, eds., MIT, Cambridge, MA, Press.
Jaynes, E. T. (1983). E. T. Jaynes: Papers on probability, statistics, and statistical physics, Reidel, Dordrecht.
Kirkpatrick, S., Gelatt, C. D., and Vecchi, M. P.(1983). “Optimization by simulated annealing.” Science, 220, 671–680.
Melchers, R. E.(1989). “Importance sampling in structural systems.” Struct. Safety, 6, 3–10.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., and Teller, A. H.(1953). “Equations of state calculations by fast computing machines.” J. Chem. Phys., 21(6), 1087–1092.
Papoulis, A. (1965). Probability, random variables, and stochastic processes, McGraw-Hall, New York.
Rubinstein, R. Y. (1981). Simulation and the Monte-Carlo method, Wiley, New York.
Schuëller, G. I., Bucher, C. G., Bourgund, U., and Quypornprasert, W.(1989). “On efficient computational schemes to calculate structural failure probabilities.” Probab. Eng. Mech., 4(1), 10–18.
Schuëller, G. I., Pradlwarter, H. J., and Pandey, M. D. (1993). “Methods for reliability assessment of nonlinear systems under stochastic dynamic loading—A review.” Proc., of EURODYN’93, Balkema, Rotterdam, The Netherlands, 751–759.
Schuëller, G. I., and Stix, R.(1987). “A critical appraisal of methods to determine failure probabilities.” Struct. Safety, 4, 293–309.
Tierney, L.(1994). “Markov chains for exploring posterior distributions.” Ann. Stat., 22, 1701–1762.
Wen, Y. K.(2001). “Reliability and performance-based design.” Struct. Safety, 23, 407–428.
Wood, W. W., and Parker, F. R.(1957). “Monte Carlo equation state of molecules interacting with the Lennard-Jones potential. I. A supercritical isotherm at about twice the critical temperature.” J. Chem. Phys., 27(3), 720–733.
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Copyright © 2004 American Society of Civil Engineers.
History
Received: Sep 27, 2002
Accepted: May 28, 2003
Published online: Feb 19, 2004
Published in print: Mar 2004
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