Abstract
Reliability-based design optimization (RBDO) is powerful for probabilistic constraint problems. Metamodeling is usually used in RBDO to reduce the computational cost. Kriging model-based RBDO is very suitable to solve engineering problems with implicit constraint functions. However, the efficiency and accuracy of the kriging model constrain its use in RBDO. In this research, the importance boundary sampling (IBS) method is enhanced by the probability feasible region (PFR) method to fit kriging model with high accuracy. The proposed probability feasible region enhanced importance boundary sampling (PFRE-IBS) method selects sample points for inactive constraint functions only in its important region, thus reducing the number of sample points to improve the efficiency of sampling method. In order to verify the efficiency and accuracy of the proposed PFRE-IBS method, three RBDO problems are used in this paper. The comparison results with other sampling methods show that the proposed PFRE-IBS method is very efficient and accurate.
Similar content being viewed by others
References
Aoues Y, Chateauneuf A (2010) Benchmark study of numerical methods for reliability-based design optimization. Struct Multidiscipl Optim 41:277–294. https://doi.org/10.1007/s00158-009-0412-2
Azad S, Alexander-Ramos MJ (2020) Robust mdsdo for co-design of stochastic dynamic systems. J Mech Des 142:8. https://doi.org/10.1115/1.4044430
Bichon BJ, Eldred MS, Swiler LP, Mahadevan S, McFarland JM (2008) Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J 46:2459–2468. https://doi.org/10.2514/1.34321
Chen ZZ, Qiu HB, Gao L, Li PG (2013a) An optimal shifting vector approach for efficient probabilistic design. Struct Multidiscipl Optim 47:905–920. https://doi.org/10.1007/s00158-012-0873-6
Chen ZZ, Qiu HB, Gao L, Su L, Li PG (2013b) An adaptive decoupling approach for reliability-based design optimization. Comput Struct 117:58–66. https://doi.org/10.1016/j.compstruc.2012.12.001
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 Multidiscipl Optim 49:401–416. https://doi.org/10.1007/s00158-013-0988-4
Chen ZZ, Peng SP, Li XK, Qiu HB, Xiong HD, Gao L, Li PG (2015) An important boundary sampling method for reliability-based design optimization using kriging model. Struct Multidiscipl Optim 52:55–70. https://doi.org/10.1007/s00158-014-1173-0
Chen Z, Li X, Chen G, Gao L, Qiu H, Wang S (2017) A probabilistic feasible region approach for reliability-based design optimization. Struct Multidiscipl Optim 57:359–372. https://doi.org/10.1007/s00158-017-1759-4
Chen ZZ, Wu ZH, Li XK, Chen G, Chen GF, Gao L, Qiu HB (2019a) An accuracy analysis method for first-order reliability method. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science 233:4319–4327. https://doi.org/10.1177/0954406218813389
Chen ZZ et al (2019b) A multiple-design-point approach for reliability-based design optimization. Eng Optim 51:875–895. https://doi.org/10.1080/0305215x.2018.1500561
Du X, Chen W (2004) Sequential optimization and reliability assessment method for efficient probabilistic design. J Mech Des 126:225–233. https://doi.org/10.1115/1.1649968
Echard B, Gayton N, Lemaire M (2011) AK-MCS: an active learning reliability method combining Kriging and Monte Carlo simulation. Struct Saf 33:145–154. https://doi.org/10.1016/j.strusafe.2011.01.002
Gano SE, Renaud JE, Martin JD, Simpson TW (2006) Update strategies for kriging models used in variable fidelity optimization. Struct Multidiscipl Optim 32:287–298. https://doi.org/10.1007/s00158-006-0025-y
Goel T, Hafkta RT, Shyy W (2008) Comparing error estimation measures for polynomial and kriging approximation of noise-free functions. Struct Multidiscipl Optim 38:429–442. https://doi.org/10.1007/s00158-008-0290-z
Hu Z, Mahadevan S (2016) A single-loop Kriging surrogate modeling for time-dependent reliability analysis. J Mech Des 138:061406. https://doi.org/10.1115/1.4033428
Jiang C, Qiu H, Gao L, Cai X, Li P (2017) An adaptive hybrid single-loop method for reliability-based design optimization using iterative control strategy. Struct Multidiscipl Optim 56:1271–1286. https://doi.org/10.1007/s00158-017-1719-z
Ju BH, Lee BC (2008) Reliability-based design optimization using a moment method and a kriging metamodel. Engineering Optimization 40:421–438. https://doi.org/10.1080/03052150701743795
Kim C, Choi KK (2008) Reliability-based design optimization using response surface method with prediction interval estimation. J Mech Des 130:121401. https://doi.org/10.1115/1.2988476
Kim D-W, Choi N-S, Choi KK, Kim D-H (2015) A single-loop strategy for efficient reliability-based electromagnetic design optimization. Ieee Trans Magnet:51. https://doi.org/10.1109/tmag.2014.2357996
Kleijnen JPC (2009) Kriging metamodeling in simulation: a review. Eur J Operat Res 192:707–716. https://doi.org/10.1016/j.ejor.2007.10.013
Lee TH, Jung JJ (2008) A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: constraint boundary sampling. Comput Struct 86:1463–1476. https://doi.org/10.1016/j.compstruc.2007.05.023
Li X, Qiu H, Chen Z, Gao L, Shao X (2016) A local Kriging approximation method using MPP for reliability-based design optimization. Comput Struct 162:102–115. https://doi.org/10.1016/j.compstruc.2015.09.004
Li X, Chen Z, Ming W, Qiu H, Ma J, He W (2017) An efficient moving optimal radial sampling method for reliability-based design optimization. Int J Performabil Eng 13:864
Liu X, Wu Y, Wang B, Ding J, Jie H (2016) An adaptive local range sampling method for reliability-based design optimization using support vector machine and Kriging model. Struct Multidiscipl Optim. https://doi.org/10.1007/s00158-016-1641-9
Liu X, Yin L, Hu L, Zhang Z (2017) An efficient reliability analysis approach for structure based on probability and probability box models. Struct Multidiscipl Optim. https://doi.org/10.1007/s00158-017-1659-7
Lopez RH, Beck AT (2012) Reliability-based design optimization strategies based on FORM: a review. J Braz Soc Mech Sci Eng 34:506–514
Lophaven SN, Nielsen HB, Søndergaard J (2002) A MATLAB Kriging toolbox. In: Technical University of Denmark, Kongens Lyngby. Technical Report No. IMM-TR-2002-12
Luo X, Li X, Zhou J, Cheng T (2012) A Kriging-based hybrid optimization algorithm for slope reliability analysis. Structural Safety 34:401–406. https://doi.org/10.1016/j.strusafe.2011.09.004
Mansour R, Olsson M (2016) Response surface single loop reliability-based design optimization with higher-order reliability assessment. Struct Multidiscipl Optim 54:63–79. https://doi.org/10.1007/s00158-015-1386-x
Matheron G (1963) Principles of geostatistics. Econ Geol 58:1246–1266
Mattson CA, Messac A (2003) Handling equality constraints in robust design optimization. In: 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 2003, April 7, 2003 - April 10, 2003, Norfolk, VA, United states, 2003. 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics Inc.,
Meng Z, Zhang DQ, Li G, Yu B (2019) An importance learning method for non-probabilistic reliability analysis and optimization. Struct Multidiscipl Optim 59:1255–1271. https://doi.org/10.1007/s00158-018-2128-7
Messac A, Ismail-Yahaya A (2002) Multiobjective robust design using physical programming. Struct Multidiscipl Optim 23:357–371. https://doi.org/10.1007/s00158-002-0196-0
Rangavajhala S, Mullur A, Messac A (2007) The challenge of equality constraints in robust design optimization: examination and new approach. Struct Multidiscipl Optim 34:381–401. https://doi.org/10.1007/s00158-007-0104-8
Singh P, Herten JVD, Deschrijver D, Couckuyt I, Dhaene T (2016) A sequential sampling strategy for adaptive classification of computationally expensive data. Struct Multidiscipl Optim. https://doi.org/10.1007/s00158-016-1584-1
Sun Z, Wang J, Li R, Tong C (2017) LIF: a new Kriging based learning function and its application to structural reliability analysis. Reliabil Eng Syst Saf 157:152–165. https://doi.org/10.1016/j.ress.2016.09.003
Vitali R, Haftka RT, Sankar BV (2002) Multi-fidelity design of stiffened composite panel with a crack. Struct Multidiscipl Optim 23:347–356. https://doi.org/10.1007/s00158-002-0195-1
Wang L, Wang X, Xia Y (2013) Hybrid reliability analysis of structures with multi-source uncertainties. Acta Mech 225:413–430. https://doi.org/10.1007/s00707-013-0969-0
Wu YT, Shin Y, Sues RH, Cesare MA (2001) Safety-factor based approach for probability-based design optimization. In: 19th AIAA Applied Aerodynamics Conference 2001, June 11, 2001 - June 14, 2001, Anaheim, CA, United states, 2001. 19th AIAA Applied Aerodynamics Conference. American Institute of Aeronautics and Astronautics Inc https://doi.org/10.2514/6.2001-1522
Xia B, Ren Z, Koh CS (2017) A novel reliability-based optimal design of electromagnetic devices based on adaptive dynamic Taylor Kriging. IEEE transactions on Magnetics:1-1. https://doi.org/10.1109/tmag.2017.2654261
Youn BD, Choi KK (2004) Selecting probabilistic approaches for realiability-based design optimization. AIAA J 42:124–131. https://doi.org/10.2514/1.9036
Zhang JF, Du XP (2010) A second-order reliability method with first-order efficiency. J Mech Des 132. https://doi.org/10.1115/1.4002459
Zhang D, Han X, Jiang C, Liu J, Li Q (2017) Time-dependent reliability analysis through response surface method. J Mech Des 139:041404. https://doi.org/10.1115/1.4035860
Funding
Financial supports from the Fundamental Research Funds for the Central Universities of China (grant number 18D110316); Key Scientific and Technological Research Projects in Henan Province (grant number 192102210069); and Natural Science Foundation of Shanghai (grant number 19ZR1401600) are gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Replication of results
For facilitate replication of the results of this paper, some important MATLAB code are shown in Fig. 17.
Additional information
Responsible Editor: Byeng D Youn
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Highlights
1. In the proposed probability feasible region enhanced important boundary sampling (PFRE-IBS) method, sample points selected for the constraint functions are executed separately. PFRE-IBS method can build metamodel efficiently and accurately.
2. Probabilistic feasible region (PFR) method is used to solve reliability-based design optimization (RBDO) problems. PFR method can also classify functions and distinguish whether they are active or inactive.
3. PFRE-IBS method only selects sample points for active constraint functions.
Rights and permissions
About this article
Cite this article
Wu, Z., Chen, Z., Chen, G. et al. A probability feasible region enhanced important boundary sampling method for reliability-based design optimization. Struct Multidisc Optim 63, 341–355 (2021). https://doi.org/10.1007/s00158-020-02702-4
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-020-02702-4