Abstract
An algorithm for risk-based optimization (RO) of engineering systems is proposed, which couples the Cross-entropy (CE) optimization method with the Line Sampling (LS) reliability method. The CE-LS algorithm relies on the CE method to optimize the total cost of a system that is composed of the design and operation cost (e.g., production cost) and the expected failure cost (i.e., failure risk). Guided by the random search of the CE method, the algorithm proceeds iteratively to update a set of random search distributions such that the optimal or near-optimal solution is likely to occur. The LS-based failure probability estimates are required to evaluate the failure risk. Throughout the optimization process, the coupling relies on a local weighted average approximation of the probability of failure to reduce the computational demands associated with RO. As the CE-LS algorithm proceeds to locate a region of design parameters with near-optimal solutions, the local weighted average approximation of the probability of failure is refined. The adaptive refinement procedure is repeatedly applied until convergence criteria with respect to both the optimization and the approximation of the failure probability are satisfied. The performance of the proposed optimization heuristic is examined empirically on several RO problems, including the design of a monopile foundation for offshore wind turbines.
Similar content being viewed by others
References
Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscip Optim 41(2):277–294
Au S (2005) Reliability-based design sensitivity by efficient simulation. Comput Struct 83(14):1048–1061
Basudhar A, Missoum S, Sanchez AH (2008) Limit state function identification using support vector machines for discontinuous responses and disjoint failure domains. Probab Eng Mech 23(1):1–11
Beck AT, Gomes WJ, Lopez RH, Miguel LF (2015) A comparison between robust and risk-based optimization under uncertainty. Struct Multidiscip Optim 52(3):479–492
Beck AT, de Santana Gomes WJ (2012) A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty. Probab Eng Mech 28:18–29
Botev Z, Kroese DP (2004) Global likelihood optimization via the cross-entropy method with an application to mixture models. In: Proceedings of the 36th conference on winter simulation. Winter simulation conference, pp 529–535
Botev ZI, Kroese DP, Rubinstein RY, LEcuyer P, et al. (2013) The cross-entropy method for optimization. In: Govindaraju V, Rao CR (eds) Machine learning: theory and applications, vol 31. Elsevier BV, Chennai, pp 35–59
Bucher C, Bourgund U (1990) A fast and efficient response surface approach for structural reliability problems. Struct Saf 7(1):57–66
Chen X, Hasselman TK, Neill DJ, et al. (1997) Reliability based structural design optimization for practical applications. In: Proceedings of the 38th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, pp 2724–2732
Chen Z, Qiu H, Gao L, Li X, Li P (2014) A local adaptive sampling method for reliability-based design optimization using kriging model. Struct Multidiscip Optim 49(3):401–416
Cheng G, Xu L, Jiang L (2006) A sequential approximate programming strategy for reliability-based structural optimization. Comput Struct 84(21):1353–1367
Ching J, Hsieh YH (2007) Local estimation of failure probability function and its confidence interval with maximum entropy principle. Probab Eng Mech 22(1):39–49
De Angelis M, Patelli E, Beer M (2015) Advanced line sampling for efficient robust reliability analysis. Struct Saf 52:170–182
De Boer PT, Kroese DP, Mannor S, Rubinstein RY (2005) A tutorial on the cross-entropy method. Ann Oper Res 134(1):19–67
Depina I, Le TMH, Fenton G, Eiksund G (2016) Reliability analysis with metamodel line sampling. Struct Saf 60:1–15
Der Kiureghian A, Zhang Y, Li CC (1994) Inverse reliability problem. J Eng Mech 120(5):1154–1159
Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126(2):225–233
Dubourg V, Sudret B, Bourinet JM (2011) Reliability-based design optimization using kriging surrogates and subset simulation. Struct Multidiscip Optim 44(5):673–690
Enevoldsen I, Sørensen JD (1994) Reliability-based optimization in structural engineering. Struct Saf 15 (3):169–196
Fenton GA, Griffiths DV (2008) Risk assessment in geotechnical engineering. Wiley
Gomes WJ, Beck AT (2016) The design space root finding method for efficient risk optimization by simulation. Probab Eng Mech 44:99–110
Hohenbichler M, Rackwitz R (1988) Improvement of second-order reliability estimates by importance sampling. J Eng Mech 114(12):2195–2199
Jensen H, Valdebenito M, Schuëller G., Kusanovic D (2009) Reliability-based optimization of stochastic systems using line search. Comput Methods Appl Mech Eng 198(49):3915–3924
Jensen HA (2005) Design and sensitivity analysis of dynamical systems subjected to stochastic loading. Comput Struct 83(14):1062–1075
Jensen HA, Catalan MA (2007) On the effects of non-linear elements in the reliability-based optimal design of stochastic dynamical systems. Int J Non Linear Mech 42(5):802–816
Koutsourelakis P, Pradlwarter H, Schuëller G. (2004) Reliability of structures in high dimensions, part i: algorithms and applications. Probab Eng Mech 19(4):409–417
Kroese DP, Porotsky S, Rubinstein RY (2006) The cross-entropy method for continuous multi-extremal optimization. Methodol Comput Appl Probab 8(3):383–407
Kuschel N, Rackwitz R (1997) Two basic problems in reliability-based structural optimization. Math Meth Oper Res 46(3):309–333
Lee I, Choi K, Zhao L (2011) Sampling-based rbdo using the stochastic sensitivity analysis and dynamic kriging method. Struct Multidiscip Optim 44(3):299–317
Liu PL, Der Kiureghian A (1986) Multivariate distribution models with prescribed marginals and covariances. Probab Eng Mech 1(2):105–112
Matlock H (1970) Correlations for design of laterally loaded piles in soft clay. Offshore Technology in Civil Engineering Hall of Fame Papers from the Early Years:77–94
Nadaraya EA (1964) On estimating regression. Theory of Probability & Its Applications 9(1):141–142
Nikolaidis E, Burdisso R (1988) Reliability based optimization: a safety index approach. Comput Struct 28(6):781–788
Pradlwarter H, Schueller G, Koutsourelakis P, Charmpis D (2007) Application of line sampling simulation method to reliability benchmark problems. Struct Saf 29(3):208–221
Rackwitz R (2000) Optimization-the basis of code-making and reliability verification. Struct Saf 22(1):27–60
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat:470–472
Rosenblueth E, Mendoza E (1971) Reliability optimization in isostatic structures. J Eng Mech Div 97 (6):1625–1642
Royset J, Kiureghian AD, Polak E (2001) Reliability-based optimal design of series structural systems. J Eng Mech 127(6):607– 614
Royset J, Polak E (2004) Reliability-based optimal design using sample average approximations. Probab Eng Mech 19(4):331– 343
Royset JO, Der Kiureghian A, Polak E (2006) Optimal design with probabilistic objective and constraints. J Eng Mech 132(1):107–118
de Santana Gomes WJ, Beck AT (2013) Global structural optimization considering expected consequences of failure and using ann surrogates. Comput Struct 126:56–68
Schuëller G, Pradlwarter H, Koutsourelakis P (2004) A critical appraisal of reliability estimation procedures for high dimensions. Probab Eng Mech 19(4):463–474
Sørensen JD, Tarp-Johansen NJ (2005) Reliability-based optimization and optimal reliability level of offshore wind turbines. Int J Offshore Polar Eng 15(02)
Spall JC (2005) Introduction to stochastic search and optimization: estimation, simulation, and control, vol 65. Wiley
Sudret B, Der Kiureghian A (2000) Stochastic finite element methods and reliability: a state-of-the-art report. Department of Civil and Environmental Engineering University of California
Taflanidis A, Beck J (2008) Stochastic subset optimization for optimal reliability problems. Probab Eng Mech 23(2):324– 338
Valdebenito M, Schuëller G (2011) Efficient strategies for reliability-based optimization involving non-linear, dynamical structures. Comput Struct 89(19):1797–1811
Valdebenito MA, Schuëller G. I. (2010) A survey on approaches for reliability-based optimization. Struct Multidiscip Optim 42(5):645–663
Wasserman L (2006) All of nonparametric statistics. Springer Science & Business Media
Watson GS (1964) Smooth regression analysis. Sankhyā: The Indian Journal of Statistics, Series A:359–372
Yang R, Gu L (2004) Experience with approximate reliability-based optimization methods. Struct Multidiscip Optim 26(1-2):152– 159
Acknowledgments
The authors gratefully acknowledge the financial support by the Research Council of Norway and several partners through the research Centres SAMCoT and Klima 2050.
Author information
Authors and Affiliations
Corresponding author
Appendices
Appendix: A: Nadaraya-Watson kernel estimator
The Nadaraya-Watson kernel estimator (Nadaraya 1964; Watson 1964; Wasserman 2006) is constructed on N pairs of observations (t 1,Y 1),...,(t N ,Y N ), where a response variable Y is related to the covariate t=[t 1,...,t n ]T with the following model:
such that r is the regression function, while σ(t i )κ i is the residual with location dependent variance, σ 2(t i ).
The Nadaraya-Watson kernel estimator of r(t) is defined by:
where w i (t) is a weight:
with kernel function:
K is a function defined to provide higher weights to observations closer to v=0, while H is a nonsingular positive definite bandwith matrix. Often the covariates are scaled so that a one-dimensional kernel with bandwidth, h>0, can be employed:
The variance of the estimator in (34) is (Wasserman 2006):
An estimate of σ 2(t) is evaluated based on a vector of fitted values, \( \hat {\textbf {r}}_{N}=\left \lbrack \hat {r}_{N}(\textbf {t}_{1}),...,\hat {r}_{N}(\textbf {t}_{N}) \right \rbrack ^{T} \), which is calculated as:
where Y=[Y 1,...,Y N ]T is the vector of observed response variables, while W is a N×N ’hat’ or ’smoothing’ matrix with entries W i j = w j (t i ). Starting from the expression in (33), a second regression model is introduced to estimate σ 2(t) (Wasserman 2006):
From (40) it can be observed that an estimate of ln(σ 2(t)) can be obtained by regressing Z i ’s on t i ’s. For example, a non-parametric regression model can be employed to obtain an estimate \(\hat {\nu }(\textbf {t})\) of logσ 2(t). The estimate of the variance then becomes:
A value of h for the kernel function in (37) is commonly selected by minimizing the leave-one-out cross-validation score (e.g., Wasserman2006):
where W i i = w i (t i ) is the ith diagonal element of the smoothing matrix.
Appendix: B: Analytical solution to the linear optimization problem
The RO problem in (24a–24d) can be solved analytically, based on the fact that g(u,t) is a linear combination of independent standard normally distributed random variables. Consequently, this leads to g(u,t) being a normally distributed random variable with mean, μ g , and standard deviation, σ g . The mean and the standard deviation are calculated to be, respectively, \( \mu _{g}=\sum \limits _{i=1}^{n}t_{i}\) and \(\sigma _{g}=\sqrt {m}\). The failure probability is calculated as:
where Φ is the standard normal cumulative density function. With the analytical solution of the reliability problem in (43), the total cost is formulated as:
The minimum of the total cost is located by differentiating the cost function with respect to the design parameters, setting it equal to zero, and solving for the design parameters. The derivative of the cost function with respect to t i is:
where ϕ is the standard normal probability density function. After setting the derivative equal to zero, the following expression is obtained:
Since (46) contains t i on both sides, the ith component of the minimizer is defined by specifying a desired reliability index, β min at the minimizer.
where all the design parameters have the same value at the minimum, t min. From (47) it follows:
In order for (46) to be consistent, the values of the design cost parameters, C i , are defined based on the values of β min, t min, and C F :
An additional requirement for the results in (47) to (49) is that \({\Phi }(-\beta _{\min }) \le P_{F}^{\lim } \). Otherwise, the minimum is found at the reliability constraint.
Rights and permissions
About this article
Cite this article
Depina, I., Papaioannou, I., Straub, D. et al. Coupling the cross-entropy with the line sampling method for risk-based design optimization. Struct Multidisc Optim 55, 1589–1612 (2017). https://doi.org/10.1007/s00158-016-1596-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-016-1596-x