現實生活中的各種不確定因素，不但會改變產品的設計結果或性能表現，更可能使原本可行的設計失敗，因此我們利用可靠度最佳化設計方法（reliability-based design optimization, RBDO），將不確定因素視為非時變之隨機變數，並將拘束條件改寫為機率型式之後，得到高可靠度的最佳設計結果。然而，現實生活中有許多不確定因素會隨時間而改變，例如產品或系統常承受動態外力，或材料特性常隨時間而改變。因此即使在設計初期就考量了不確定因素，最終結果在經過一段時間後仍可能不符合設計要求。大多數的可靠度最佳化設計研究多不考慮時間因素，不足以處理現實的時變性可靠度最佳化問題。

文獻中已有方法可處理時變可靠度問題，其中PHI2是一個可計算時變性拘束條件函數或時變性不確定因素之機率的近似方法，在利用隨機過程模擬時變性不確定因素，並利用非時變性一階可靠度方法（ first-order reliability method, FORM）與隨機過程的自相關係數（auto-correlation coefficient）代表各個時間增量之間的關聯性，PHI2可計算穿越率（outcrossing rate）進而求得時變性可靠度。雖然PHI2比直接對各個時間點隨機取點計算時變性可靠度來的快速許多，將其加入設計流程中仍需要相當昂貴的計算成本，因此本研究利用PHI2，將原本的可靠度最佳化方法延伸為時變性可靠度最佳化設計方法（time-variant reliability-based design optimization），為了減少該最佳化演算法與PHI2結合後的計算成本，我們使用MADS（mesh adaptive direct search）演算法與PHI2整合，形成可計算時變性限制條件之最佳化問題的最佳化設計流程。MADS利用網格切割進行搜索最佳值，在每次迭代之間，只有符合其他非時變性限制條件的設計點需進行PHI2時變性拘束條件的可靠度評估，我們也利用目標函數值與時變性拘束條件違反量的總合取作為懲罰函數，來進行評估，以決定收斂後的最佳值是否為設計者可接受之結果。最後我們會利用兩個工程範例，示範此設計流程的使用方法以及結果。此方法也可以延伸至生命週期與保固成本的設計。

Uncertainties in real-life could potentially alter the performance of a design or even make a feasible design unreliable. After modeling uncertainties as random variables and reformulating constraints probabilistically, reliability-based design optimization (RBDO) provides a ensuring reliability of an optimum. However, in addition to time-invariant uncertainties, engineering products are also constantly under external loadings that may vary over time. Most RBDO research does not account for time factor in reliability analysis, thus is not capable of handling these loading conditions in the current framework.

Various methods in the literature have been proposed to handle time-variant reliability, PHI2 is a popular approximation-based method that is developed based on the time-invariant first-order reliability method (FORM) in evaluating the probability of a time-variant function or a function with time-variant uncertainty. Combining the time-invariant reliability using FORM and relations between small time increments, PHI2 can estimate the out-crossing rate of a time-variant event, therefore the reliability value. Although PHI2 is faster than a traditional sampling MCS , it still needs a high computation cost . Therefore we need integret a property optimum algorithm with PHI2 to be a time-variant reliability-based design optimization routine.

To reduce the computation burden of implementing PHI2 method in optimization, we extend deterministic mesh adaptive direct search algorithm (MADS) to consider probabilistic constraints with time-variant performances. It seeks points on the deterministic feasible mesh grid and only candidates that have objective improvements are selected for reliability evaluations using PHI2. A filter containing objectives and constraint violations can be implemented to replace penalty functions to determine the acceptance of a candidate until convergence. Several examples are demonstrated to show the optimization routine and the results. This method could potentially be extended to engineering design with life cycle and warranty cost.

[1] P. Papalambros and D. Wilde, Principles of Optimal Design. New York, USA: Cambridge University Press, 2nd ed., 2000.
[2] J. S. Arora, Introduction to Optimum Design. Elsevier Academic Press, 2004.
[3] M. Allen and K. Maute, “Reliability-based shape optimization of structures undergoing fluid-structure interaction phenomena,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 30-33, pp. 3472 – 3495, 2005.
[4] L. Gu, R. Yang, C. Tho, M. Makowski, O. Faruque, and Y. Li, “Optimization and robustness for crashworthiness of side impact,” International Journal of Vehicle Design, vol. 26, no. 4, pp. 348–360, 2001.
[5] A. Haldar and S. Mahadevan, Probability, Reliability and Statistical Methods in Engineering Design. John Wiley & Sons Inc, 1999.
[6] L. Huyse, S. Padula, R. Lewis, andW. Li,“Probabilistic approach to free-form airfoil shape optimization under uncertainty,” AIAA Journal, vol. 40, no. 9, pp. 764–1772, 2002.
[7] P. Kales, Reliability For Technology Engineering and Management. Upper Saddle River, New Jersey Columbus, Ohio: Prentice Hall, 1998.
[8] J. Tu, K. Choi, and Y. Park, “A new study on reliability-based design optimization,” Journal of Mechanical Design, Transactions of the ASME, vol. 121, no. 4, pp. 557–564,1999.
[9] W. Gao, D. Wu, C. Song, F. Tin-Loi, and X. Li, “Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation monte-carlo method,” Finite Elements in Analysis and Design, vol. 47, no. 7, pp. 643 – 652, 2011.
[10] L. A. Zadeh, “Fuzzy sets as a basis for a theory of possibility,” Fuzzy Sets and Systems, vol. 100, no. Supplement 1, pp. 9 – 34, 1999.
[11] X. Du, A. Sudjianto, and B. Huang, “Reliability-based design with the mixture of random and interval variables,” Journal of Mechanical Design, Transactions of the ASME, vol. 127, no. 6, pp. 1068–1076, 2005.
[12] D. Athow and J. Law, “Development and applications of a random variable model for cold load pickup,” IEEE Transactions on Power Delivery, vol. 9, no. 3, pp. 1647–1653, 1994.
[13] G. Feng, “Eye movements as time-series random variables: A stochastic model of eye movement control in reading,” Cognitive Systems Research, vol. 7, no. 1, pp. 70–95, 2006.
[14] H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing. Prentice Hall, 2001.
[15] H. Stark and J. W. Woods, Probability, Random Processes, and Estimation Theory for Engineers. Prentice Hall, 1994.
[16] R. Rackwitz, “Reliability analysis-a review and some perspective,” Structural Safety, vol. 23, no. 4, pp. 365–395, 2001.
[17] C. Cornell, “Bounds on the reliability of structural systems,” American Society of Civil Engineers Proceeding , Journal of the Structural Division, vol. 93, no. ST1, pp. 171–200, 1967.
[18] N. C. Hasofer, Abraham M.and Lind, “Exact and invariant second-moment code format,” ASCE Journal of the Engineering Mechanics Division, vol. 100, no. EM1, pp. 111–121,1974.
[19] M. Hohenbichler and R. Rackwitz, “First-order concepts in system reliability,” Structural Safety, vol. 1, no. 3, pp. 177 – 188, 1983.
[20] M. Hohenbichler and R. Rackwitz, “Non-normal dependent vectors in structural safety,” ASCE Journal of Engineering Mechanics Division, vol. 107, no. 6, pp. 1227–1238, 1981.
[21] X. Chen and N. Lind, “Fast probability integration by three-parameter normal tail approximation,”Structural Safety, vol. 1, no. 4, pp. 269 – 276, 1983.
[22] R. Rackwitz and B. Flessler, “Structural reliability under combined random load sequences,”Computers and Structures, vol. 9, no. 5, pp. 489–494, 1978.
[23] B. Fiessler, H.-J. Neumann, and R. Rackwitz, “Quadratic limit states in structural reliability,”
ASCE Journal Engineering Mechanics Division, vol. 105, no. 4, pp. 661–676,1979.
[24] K. Breitung, “Asymptotic approximations for multinormal integrals,” Journal of Engineering
Mechanics, vol. 110, no. 3, pp. 357–366, 1984.
[25] F. James, “Monte carlo theory and practice,” Reports on Progress in Physics, vol. 43, no. 9, pp. 1145–1189, 1980.
[26] A. Harbitz, “An efficient sampling method for probability of failure calculation,” Structural Safety, vol. 3, no. 2, pp. 109–115, 1986.
[27] A. Karamchandani, P. Bjerager, and C. Cornell, “Adaptive importance sampling,” in Proceedings International Conference on Structural Safety and Reliability , San Francisco, pp. 855–862, 1989.
[28] C.-C. Li and A. Der Kiureghian, “Optimal discretization of random fields,” Journal of Engineering Mechanics, vol. 119, no. 6, pp. 1136–1154, 1993.
[29] B. Sudret and A. D. Kiureghian, “Stochastic finite element method and reliability,” Tech. Rep. UCB/SEMM-2000/08, Department of Civil and Environmental Engineering, University of California at Berkeley, 2000.
[30] R. Griffiths and S. Tavar, “Monte carlo inference methods in population genetics,” Mathematical and Computer Modelling, vol. 23, no. 8-9, pp. 141–158, 1996.
[31] K. Zapp, “Monte carlo simulations of jet quenching in heavy ion collisions,” Nuclear Physics A, vol. 855, no. 1, pp. 60–66, 2011.
[32] H. Rief, E. Gelbard, R. Schaefer, and K. Smith, Review of monte carlo techniques for analyzing reactor perturbations,” Nuclear Science and Engineering, vol. 92, no. 2, pp. 289–297, 1986.
[33] J. Turner, H. Wright, and R. Hamm, “A monte carlo primer for health physicists,” Health Physics, vol. 48, no. 6, pp. 717–733, 1985.
[34] B. Ayyub and C.-Y. Chia, “Generalized conditional expectation for structural reliability assessment,” Structural Safety, vol. 11, no. 2, pp. 131–146, 1992.
[35] Y. K. Wen and H.-C. Chen, “On fast integration for time variant structural reliability,” Probabilistic Engineering Mechanics, vol. 2, no. 3, pp. 156–162, 1987.
[36] H. Madsen and L. Tvedt, “Methods for time-dependent reliability and sensitivity analysis,”Journal of Engineering Mechanics, vol. 116, no. 10, 1990.
[37] A. Der Kiureghian, “The geometry of random vibrations and solutions by form and sorm,” Probabilistic Engineering Mechanics, vol. 15, no. 1, pp. 81–90, 2000.
[38] O. Hagen and L. Tvedt, “Vector process out-crossing as parallel system sensitivity measure,”Journal of Engineering Mechanics, vol. 117, no. 10, pp. 2201–2220, 1991.
[39] C. Andrieu-Renaud, B. Sudret, and M. Lemaire, “The phi2 method: A way to compute time-variant reliability,” Reliability Engineering and System Safety, vol. 84, no. 1, pp. 75–86, 2004.
[40] Y. Son and G. Savage, “Set theoretic formulation of performance reliability of multiple response time-variant systems due to degradations in system components,” Quality and Reliability Engineering International, vol. 23, no. 2, 2007.
[41] Y. Son and G. Savage, “A new sample-based approach to predict system performance reliability,” IEEE Transactions on Reliability, vol. 57, no. 2, pp. 322–330, 2008.
[42] A. Singh, Z. P. Mourelatos, and E. Nikolaidis, “A simulation-based random process method for time-dependent reliability of dynamic systems,” in ASME 2010 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference,
vol. 1, pp. 927–940, 2010.
[43] A. Singh, Z. P. Mourelatos, and E. Nikolaidis, “An importance sampling approach for time-dependent reliability,” in ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2011.
[44] S. Rao and P. Bhatti, “Probabilistic approach to manipulator kinematics and dynamics,”Reliability Engineering and System Safety, vol. 72, no. 1, pp. 47–58, 2001.
[45] L. Burgazzi, “About time-variant reliability analysis with reference to passive systems assessment,” Reliability Engineering & System Safety, vol. 93, no. 11, pp. 1682 – 1688,2008.
[46] B. Sudret, “Probabilistic models for the extent of damage in degrading reinforced concrete structures,” Reliability Engineering & System Safety, vol. 93, no. 3, pp. 410 – 422, 2008.
[47] J. Zhang and X. Du, “Time-dependent reliability analysis for function generator mechanisms,”Journal of Mechanical Design, Transactions of the ASME, vol. 133, no. 3, 2011.
[48] X. Chen, T. K. Hasselman, and D. J. Neill, “Reliability based structural design optimization for practical applications,” in Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, vol. 4, pp. 2724–2732, 1997.
[49] R. Yang, C. Chuang, and L. G. Gu, L., “Numerical experiments of reliability-based optimization methods,” in Collection of Technical Papers - AIAA/ASME/ASCE/AHS/ASC
Structures, Structural Dynamics and Materials Conference, vol. 7, pp. 5393–5405, 2004.
[50] J. Royset and E. Polak, “Implementable algorithm for stochastic optimization using sample average approximations,” Journal of Optimization Theory and Applications, vol. 122, no. 1, pp. 157–184, 2004.
[51] J. Royset and E. Polak, “Reliability-based optimal design using sample average approximations,” Probabilistic Engineering Mechanics, vol. 19, no. 4, pp. 331–343, 2004.
[52] N. Kuschel and R. Rackwitz, “Optimal design under time-variant reliability constraints,” Structural Safety, vol. 22, no. 2, pp. 113–127, 2000.
[53] H. Streicher and R. Rackwitz, “Time-variant reliability-oriented structural optimization and a renewal model for life-cycle costing,” Probabilistic Engineering Mechanics, vol. 19, no. 1, pp. 171–183, 2004.
[54] Y. Son and G. Savage, “Optimal probabilistic design of the dynamic performance of a vibration absorber,” Journal of Sound and Vibration, vol. 307, no. 1-2, pp. 20–37, 2007.
[55] G. Savage and Y. Son, “Dependability-based design optimization of degrading engineering systems,” Journal of Mechanical Design, Transactions of the ASME, vol. 131, no. 1, pp. 0110021–01100210, 2009.
[56] C. Audet and J. Dennis Jr., “Mesh adaptive direct search algorithms for constrained optimization,” SIAM Journal on Optimization, vol. 17, no. 1, pp. 188–217, 2007.
[57] C. Audet and J. Dennis Jr., “Analysis of generalized pattern searches,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 889–903, 2003.
[58] X.-P. Lu, H.-L. Li, and P. Papaalambros, “Design procedure for the optimization of vehicle suspensions,” International Journal of Vehicle Design, vol. 5, no. 1-2, pp. 129–142, 1984.