Abstract
Multi-fidelity (MF) metamodeling approaches have recently attracted a significant amount of attention in simulation-based design optimization due to their ability to conduct trade-offs between high accuracy and low computational expenses by integrating the information from high-fidelity (HF) and low-fidelity (LF) models. While existing MF metamodel assisted design optimization approaches may yield an inferior or even infeasible solution since they generally treat the MF metamodel as the real HF model and ignore the interpolation uncertainties from the MF metamodel. This situation will be more serious in non-deterministic optimization. Hence, in this work, a MF metamodel assisted robust optimization approach is developed, in which the interpolation uncertainty of the MF metamodel and design variable uncertainty are quantified and taken into consideration. To demonstrate the effectiveness and merits of the proposed approach, two numerical examples and a long cylinder pressure vessel design optimization problem are tested. Results show that for the test cases the proposed approach can obtain a solution that is both optimal and within the feasible region even with perturbation of the uncertain variables.
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References
Apley DW, Liu J, Chen W (2006) Understanding the effects of model uncertainty in robust design with computer experiments. J Mech Des 128(4):945–958
Arendt PD, Apley DW, Chen W (2013) Objective-oriented sequential sampling for simulation based robust design considering multiple sources of uncertainty. J Mech Des 135(5):051005
Bahrami S, Tribes C, Devals C, Vu TC, Guibault F (2016) Multi-fidelity shape optimization of hydraulic turbine runner blades using a multi-objective mesh adaptive direct search algorithm. Appl Math Model 40(2):1650–1668
Bandler JW, Cheng QS, Nikolova NK, Ismail MA (2004) Implicit space map** optimization exploiting preassigned parameters. IEEE Transactions on Microwave Theory and Techniques 52(1):378–385
Burgee S, Giunta AA, Balabanov V, Grossman B, Mason WH, Narducci R, Haftka RT, Watson LT (1996) A coarse-grained parallel variable-complexity multidisciplinary optimization paradigm. International Journal of High Performance Computing Applications 10(4):269–299
Chen S, Jiang Z, Yang S, Apley DW, Chen W (2016) Nonhierarchical multi-model fusion using spatial random processes. Int J Numer Methods Eng 106(7):503–526
Choi S-K, Grandhi RV, Canfield RA (2004) Structural reliability under non-gaussian stochastic behavior. Comput Struct 82(13–14):1113–1121
Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127
Gano SE, Renaud JE, Sanders B (2005) Hybrid variable fidelity optimization by using a kriging-based scaling function. AIAA J 43(11):2422–2433
Gano SE, Renaud JE, Agarwal H, Tovar A (2006a) Reliability-based design using variable-fidelity optimization. Struct Infrastruct Eng 2(3–4):247–260
Gano SE, Renaud JE, Martin JD, Simpson TW (2006b) Update strategies for kriging models used in variable fidelity optimization. Struct Multidiscip Optim 32(4):287–298
Haftka RT (1991) Combining global and local approximations. AIAA J 29(9):1523–1525
Han Z-H, Görtz S (2012) Hierarchical kriging model for variable-fidelity surrogate modeling. AIAA J 50(9):1885–1896
Han Z, Zimmerman R, Görtz S (2012) Alternative cokriging method for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210
Hu Z, Mahadevan S (2015) Global sensitivity analysis-enhanced surrogate (gsas) modeling for reliability analysis. Struct Multidiscip Optim 53(3):501–521
Hu Z, Mahadevan S (2016) A single-loop kriging surrogate modeling for time-dependent reliability analysis. J Mech Des 138(6):061406
Hu Z, Mahadevan S (2017) Uncertainty quantification in prediction of material properties during additive manufacturing. Scr Mater 135:135–140
Hu W, Enying L, Li GY, Zhong ZH (2007) Optimization of sheet metal forming processes by the use of space map** based metamodeling method. Int J Adv Manuf Technol 39(7–8):642–655
Hu W, Li M, Azarm S, Almansoori A (2011) Multi-objective robust optimization under interval uncertainty using online approximation and constraint cuts. J Mech Des 133(6):061002
Hu Z, Du X, Kolekar NS, Banerjee A (2013) Robust design with imprecise random variables and its application in hydrokinetic turbine optimization. Eng Optim 46(3):393–419
Hu J, Zhou Q, Jiang P, Shao X, **e T (2017) An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging. Eng Optim http://dx.doi.org/10.1080/0305215X.2017.1296435
Huang D, Allen TT, Notz WI, Miller RA (2006) Sequential kriging optimization using multiple-fidelity evaluations. Struct Multidiscip Optim 32(5):369–382
Huang T, Song X, Liu X (2015) The multi-objective robust optimization of the loading path in the t-shape tube hydroforming based on dual response surface model. Int J Adv Manuf Technol 82(9–12):1595–1605
Jiang P, Zhou Q, Shao X, Long R, Zhou H (2016) A modified blisco method and its combination with variable fidelity metamodel for engineering design. Eng Comput 33(5):1353–1377
** R, Du X, Chen W (2003) The use of metamodeling techniques for optimization under uncertainty. Struct Multidiscip Optim 25(2):99–116
** R, Chen W, Sudjianto A (2005) An efficient algorithm for constructing optimal design of computer experiments. Journal of Statistical Planning and Inference 134(1):268–287
Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic press, San Diego
Keane AJ, Sóbester A, Forrester AIJ (2007) Multi-fidelity optimization via surrogate modelling. Proc Royal Soc A: Math Phys Eng Sci 463(2088):3251–3269
Kennedy MC, O'hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13
Kim N-K, Kim D-H, Kim D-W, Kim H-G, Lowther D, Sykulski JK (2010) Robust optimization utilizing the second-order design sensitivity information. Mag IEEE Trans on 46(8):3117–3120
Koziel S, Bekasiewicz A (2016) Rapid design optimization of antennas using variable-fidelity em models and adjoint sensitivities. Eng Comput 33(7):2007–2018
Le Gratiet L, Garnier J (2014) Recursive co-kriging model for design of computer experiments with multiple levels of fidelity. Int J Uncertain Quantif 4(5):365–386
Lewis RM, Nash SG (2000) A multigrid approach to the optimization of systems governed by differential equations. AIAA paper 4890:2000
Li M, Gabriel SA, Shim Y, Azarm S (2011) Interval uncertainty-based robust optimization for convex and non-convex quadratic programs with applications in network infrastructure planning. Netw Spat Econ 11(1):159–191
Li X, Qiu H, Jiang Z, Gao L, Shao X (2017) A vf-slp framework using least squares hybrid scaling for rbdo. Struct Multidiscip Optim 55(5):1629–1640
Liu Y, Collette M (2014) Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm. Appl Soft Comput 24:482–493
March A, Willcox K (2012) Provably convergent multifidelity optimization algorithm not requiring high-fidelity derivatives. AIAA J 50(5):1079–1089
Nguyen NV, Tyan M, Lee J-W (2014) A modified variable complexity modeling for efficient multidisciplinary aircraft conceptual design. Optim Eng 16(2):483–505
Papadimitriou DI, Giannakoglou KC (2013) Third-order sensitivity analysis for robust aerodynamic design using continuous adjoint. Int J Numer Methods Fluids 71(5):652–670
Park C, Haftka RT, Kim NH (2017) Remarks on multi-fidelity surrogates. Struct Multidiscip Optim 55(3):1029–1050
Patel J, Choi S-K (2012) Classification approach for reliability-based topology optimization using probabilistic neural networks. Struct Multidiscip Optim 45(4):529–543
Rayas-Sanchez JE (2016) Power in simplicity with asm: tracing the aggressive space map** algorithm over two decades of development and engineering applications. IEEE Microw Mag 17(4):64–76
Raza MA, Liang W (2012) Uncertainty-based computational robust design optimisation of dual-thrust propulsion system. J Eng Des 23(8):618–634
Song X, Sun G, Li G, Gao W, Li Q (2012) Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models. Struct Multidiscip Optim 47(2):221–231
Sun G, Li G, Zhou S, Li H, Hou S, Li Q (2010) Crashworthiness design of vehicle by using multiobjective robust optimization. Struct Multidiscip Optim 44(1):99–110
Sun G, Song X, Baek S, Li Q (2013) Robust optimization of foam-filled thin-walled structure based on sequential kriging metamodel. Struct Multidiscip Optim 49(6):897–913
Taguchi G (1978) Performance analysis design. Int J Prod Res 16(6):521–530
Tyan M, Nguyen NV, Lee J-W (2014) Improving variable-fidelity modelling by exploring global design space and radial basis function networks for aerofoil design. Eng Optim 47(7):885–908
Viana FA, Steffen V Jr, Butkewitsch S, De Freitas Leal M (2009) Optimization of aircraft structural components by using nature-inspired algorithms and multi-fidelity approximations. J Glob Optim 45(3):427–449
Viana FA, Simpson TW, Balabanov V, Toropov V (2014) Special section on multidisciplinary design optimization: metamodeling in multidisciplinary design optimization: how far have we really come? AIAA J 52(4):670–690
Wang H, Li GY, Li E (2010) Time-based metamodeling technique for vehicle crashworthiness optimization. Comput Methods Appl Mech Eng 199(37–40):2497–2509
Wang H, Fan T, Li G (2017) Reanalysis-based space map** method, an alternative optimization way for expensive simulation-based problems. Struct Multidiscip Optim 55(6):2143–2157
**a T, Li M, Zhou J (2016) A sequential robust optimization approach for multidisciplinary design optimization with uncertainty. J Mech Des 138(11):111406
**ong S, Qian PZG, Wu CFJ (2013) Sequential design and analysis of high-accuracy and low-accuracy computer codes. Technometrics 55(1):37–46
Zadeh PM, Toropov VV, Wood AS (2009) Metamodel-based collaborative optimization framework. Struct Multidiscip Optim 38(2):103–115
Zhang S, Zhu P, Chen W, Arendt P (2012) Concurrent treatment of parametric uncertainty and metamodeling uncertainty in robust design. Struct Multidiscip Optim 47(1):63–76
Zhang Y, Li M, Zhang J, Li G (2016) Robust optimization with parameter and model uncertainties using gaussian processes. J Mech Des 138(11):111405
Zheng J, Shao X, Gao L, Jiang P, Li Z (2013) A hybrid variable-fidelity global approximation modelling method combining tuned radial basis function base and kriging correction. J Eng Des 24(8):604–622
Zheng J, Shao X, Gao L, Jiang P, Qiu H (2014) Difference map** method using least square support vector regression for variable-fidelity metamodelling. Eng Optim 47(6):719–736
Zhou X, Ma Y, Tu Y, Feng Y (2013) Ensemble of surrogates for dual response surface modeling in robust parameter design. Qual Reliab Eng Int 29(2):173–197
Zhou Q, Shao X, Jiang P, Cao L, Zhou H, Shu L (2015a) Differing map** using ensemble of metamodels for global variable-fidelity metamodeling. CMES: Com Mod Eng Sci 106(5):323–355
Zhou Q, Shao XY, Jiang P, Zhou H, Cao LC, Zhang L (2015b) A deterministic robust optimisation method under interval uncertainty based on the reverse model. J Eng Des 26(10–12):416–444
Zhou Q, Shao X, Jiang P, Gao Z, Wang C, Shu L (2016a) An active learning metamodeling approach by sequentially exploiting difference information from variable-fidelity models. Adv Eng Inform 30(3):283–297
Zhou Q, Shao X, Jiang P, Gao Z, Zhou H, Shu L (2016b) An active learning variable-fidelity metamodelling approach based on ensemble of metamodels and objective-oriented sequential sampling. J Eng Des 27(4–6):205–231
Zhou Q, Wang Y, Jiang P, Shao X, Choi S-K, Hu J, Cao L, Meng X (2017) An active learning radial basis function modeling method based on self-organization maps for simulation-based design problems. Knowl-Based Syst 131:10–27
Zhu P, Zhang Y, Chen GL (2009) Metamodel-based lightweight design of an automotive front-body structure using robust optimization. Proceedings of the Institution of Mechanical Engineers, Part D: J Automobile Eng 223(9):1133–1147
Zhu J, Wang Y-J, Collette M (2013) A multi-objective variable-fidelity optimization method for genetic algorithms. Eng Optim 46(4):521–542
Zhu P, Zhang S, Chen W (2014) Multi-point objective-oriented sequential sampling strategy for constrained robust design. Eng Optim 47(3):287–307
Zimmermann R, Han Z-H (2010) Simplified cross-correlation estimation for multi-fidelity surrogate cokriging models. Ad Appl Math Sci 7(2):181–202
Acknowledgements
This research has been supported by the National Natural Science Foundation of China (NSFC) under Grant No. 51505163, No. 51421062 and No. 51323009, National Basic Research Program (973 Program) of China under Grant No. 2014CB046703, and the Fundamental Research Funds for the Central Universities, HUST: Grant No. 2016YXMS272.
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Appendices
Appendix A: Derivations of (19) and (20)
In this appendix, the derivation of the (19) and (20) are presented. For simplicity, Y(X, V) is used to denote the responses considering the MF metamodel uncertainty, where X represents the uncertainty of design variables and V represents the MF metamodel uncertainty. The mean value of the objective function can be expressed as
Note that the integral ∫ w Y(x + w, V)p(w)d w∣V in the fourth line is random because of its functional dependence on V and not because of any dependence on w. The random effects of w are integrated out in the function. Because of E[Y(x + w, V)/V] = y mf (x + w), one can obtain the mean value of the objective function as
In the same manner, the variance value of the objective function can be expressed as
The first term in the fourth line of (32) can be further expanded by the law of total expectation as
Substituting (33) to (32), one can rewrite (32) as
Because Var[Y(x + w, V)| V] = s 2(y mf (x + w)) is the MSE for the Hierarchical Kriging prediction, recalling (18)
The variance value of the objective function can be expressed as
Appendix B: The Monte Carlo samples involved in the engineering problem
The Monte Carlo samples and corresponding values of maximum von Mises stress involved in the engineering problem are listed in Tables 10 and 11.
Appendix C: The boxplots of the objective functions for different approaches under all examples in 15 runs
In this appendix, the boxplots of the objective functions for different approaches under all examples in 15 runs are provided. In Fig. 20, the symbol “A” represents the robust optimization only considering design variable uncertainty, the symbol “B” represents the robust optimization considering both design variable and MF metamodel uncertainties, and the symbol “T” represents the robust optimization without using the MF metamodel. The dotted lines extending from the bottom and top of the box represents 1.5 times of the inter-quartile range. Data that lie out of this range are considered as outliers and are marked by the symbol“+”.
Appendix D: Discussion of Additional case 3 and Additional case 4 in the Example 1
In this appendix, additional case 3 and Additional case 4 in the Example 1 are discussed. For the additional case 3, the HF sample point x h = 7.5 is replaced by the HF sample point x h = 7.87 (the true robust optimum), it means the sampling will be lucky in that it captured the actual minimum of the function. For the additional case 4, the HF sample point x h = 7.5 is replaced by the HF sample point x h = 8.5 (which is located on the right side of the true robust optimum), it means the sampling cannot capture the actual minimum of the function.
In the additional case 3, the constructed MF metamodel (with HF sample point x h = 7.87), together with the sampling points, are plotted in Fig. 21.
The comparison results are listed in Table 12 For this case, the robust optimization considering the MF metamodel uncertainty can find a more accurate optimum, while the optimum A3 from the robust optimization without considering the MF metamodel uncertainty is still higher than those of robust optima B3 and T.
In the additional case 4, the constructed MF metamodel (with HF sample point x h = 8.5), together with the sampling points, are plotted in Fig. 22.
The comparison results are listed in Table 13 For this situation, the robust optimum B4 from the robust optimization considering the MF metamodel uncertainty is a little worse than that in the additional case 3, while, it is still better than that from the robust optimization without considering the MF metamodel uncertainty.
Notice that if a MF metamodel cannot capture the trend of the function, the MF metamodel based on robust optimization may result in an inferior solution. Therefore, we assume that when a MF metamodel is used, it can capture the general trend of the function.
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Zhou, Q., Wang, Y., Choi, SK. et al. A robust optimization approach based on multi-fidelity metamodel. Struct Multidisc Optim 57, 775–797 (2018). https://doi.org/10.1007/s00158-017-1783-4
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DOI: https://doi.org/10.1007/s00158-017-1783-4