Bayesian fatigue damage and reliability analysis using Laplace approximation and inverse reliability method

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Xuefei Guan Jingjing He Ratneshwar Jha Yongming Liu

Abstract

This paper presents an efficient analytical Bayesian method for reliability and system response estimate and update. The method includes additional data such as measurements to reduce estimation uncertainties. Laplace approximation is proposed to evaluate Bayesian posterior distributions analytically. An efficient algorithm based on inverse first-order reliability method is developed to evaluate system responses given a reliability level. Since the proposed method involves no simulations such as Monte Carlo or Markov chain Monte Carlo simulations, the overall computational efficiency improves significantly, particularly for problems with complicated performance functions. A numerical example and a practical fatigue crack propagation problem with experimental data are presented for methodology demonstration. The accuracy and computational efficiency of the proposed method is compared with simulation-based methods.

How to Cite

Guan, X. ., He, J. ., Jha, R. ., & Liu, Y. . (2011). Bayesian fatigue damage and reliability analysis using Laplace approximation and inverse reliability method. Annual Conference of the PHM Society, 3(1). https://doi.org/10.36001/phmconf.2011.v3i1.1991
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Keywords

fatigue crack growth, Bayesian updating, prognosis, inverse reliability method, Laplace approximation

References
Brauer, D., & Brauer, G. (2009). Reliability-centered maintenance. Reliability, IEEE Transactions on, 36(1), 17– 24.

Bucher, U., et al. (1990). A fast and efficient response surface approach for structural reliability problems. Structural Safety, 7(1), 57–66.

Cai, G., & Elishakoff, I. (1994). Refined second-order reliability analysis. Structural Safety, 14(4), 267–276. Cheng, J., Zhang, J., Cai, C., & Xiao, R. (2007). A new
approach for solving inverse reliability problems with implicit response functions. Engineering structures, 29(1), 71–79.

Dennis Jr, J., Gay, D., & Walsh, R. (1981). An adaptive nonlinear least-squares algorithm. ACM Transactions on Mathematical Software (TOMS), 7(3), 348–368.

Der Kiureghian, A., & Dakessian, T. (1998). Multiple design points in first and second-order reliability. Structural Safety, 20(1), 37–49.

Der Kiureghian, A., Yan, Z., & Chun-Ching, L. (1994). In- verse reliability problem. Journal of engineering mechanics, 120(5), 1154–1159.

Ditlevsen, O., & Madsen, H. (1996). Structural reliability methods (Vol. 315). Citeseer.

Du, X., Sudjianto, A., & Chen, W. (2004). An integrated framework for optimization under uncertainty using inverse reliability strategy. Journal of Mechanical Design, 126, 562.

Gelman, A., & Meng, X. (1998). Simulating normalizing constants: From importance sampling to bridge sampling to path sampling. Statistical Science, 13(2), 163– 185.

Ghahramani, Z., & Beal, M. (2000). Variational inference for Bayesian mixtures of factor analysers. Advances in neural information processing systems, 12, 449–455.

Gilks, W., Richardson, S., & Spiegelhalter, D. (1996). Markov chain Monte Carlo in practice. Chapman & Hall/CRC.

Graves, T., Hamada, M., Klamann, R., Koehler, A., & Martz, H. (2008). Using simultaneous higher-level and partial lower-level data in reliability assessments. Reliability
Engineering & System Safety, 93(8), 1273-1279. Gregory, P. (2005). Bayesian logical data analysis for the physical sciences: a comparative approach with Mathematica support. Cambridge Univ Pr.

Guan, X., Jha, R., & Liu, Y. (2009). Probabilistic fatigue damage prognosis using maximum entropy approach. Journal of Intelligent Manufacturing, 1-9. (10.1007/s10845-009-0341-3)

Hasofer, A., & Lind, N. (1974). Exact and invariant second- moment code format. Journal of the Engineering Mechanics Division, 100(1), 111–121.

Hong, H. (1997). Reliability analysis with nondestructive inspection. Structural Safety, 19(4), 383–395.

Kalos, M., & Whitlock, P. (2008). Monte carlo methods. Wiley-VCH.

Lee, I., Choi, K., Du, L., & Gorsich, D. (2008). Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems. Computer Methods in Applied Mechanics and Engineering, 198(1), 14–27.

Li, H., & Foschi, R. (1998). An inverse reliability method and its application. Structural Safety, 20(3), 257–270.

Liu, J. (1996). Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Statistics and Computing, 6(2), 113–119.

Madsen, H. (1977). Some experience with the Rackwitz- Fiessler algorithm for the calculation of structural reliability under combined loading. DIALOG-77, Danish Engineering Academy, Lyngby, Denmark, 73–98.

Madsen, H., Krenk, S., & Lind, N. (1986). Methods of structural safety. Prentice-Hall, Inc., Englewood Cliffs, NJ.

Melchers, R. (1999). Structural reliability analysis and prediction. John Wiley & Son Ltd.

Moon, T. (1996). The expectation-maximization algorithm. Signal Processing Magazine, IEEE, 13(6), 47–60.

More, J. (1978). The Levenberg-Marquardt algorithm: implementation and theory. Numerical analysis, 105–116.

Papadimitriou, C., Beck, J., & Katafygiotis, L. (2001). Updating robust reliability using structural test data. Probabilistic Engineering Mechanics, 16(2), 103–113.

Paris, P., & Erdogan, F. (1963). A critical analysis of crack propagation laws. Journal of Basic Engineering, 85(4), 528–534.

Powell, M. (1970). A FORTRAN subroutine for solving systems of nonlinear algebraic equations. Numerical methods for nonlinear algebraic equations, 115.

Rackwitz, R. (2001). Reliability analysis–a review and some perspectives. Structural Safety, 23(4), 365–395. Rackwitz, R., & Flessler, B. (1978). Structural reliability under combined random load sequences. Computers & Structures, 9(5), 489–494.

Rebba, R., & Mahadevan, S. (2008). Computational methods for model reliability assessment. Reliability Engineering & System Safety, 93(8), 1197–1207.

Saranyasoontorn, K., & Manuel, L. (2004). Efficient models for wind turbine extreme loads using inverse reliability. Journal of Wind Engineering and Industrial Aero-
dynamics, 92(10), 789–804.

Tu, J., Choi, K., & Park, Y. (1999). A new study on reliability based design optimization. Journal of Mechanical De-
sign, 121, 557.

Virkler, D., Hillberry, B., & Goel, P. (1979). The Statistical Nature of Fatigue Crack Propagation. Journal of Engineering Materials and Technology, 101, 148.
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Technical Papers

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