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A robust optimization approach based on multi-fidelity metamodel

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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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Bandler JW, Cheng QS, Nikolova NK, Ismail MA (2004) Implicit space mapping optimization exploiting preassigned parameters. IEEE Transactions on Microwave Theory and Techniques 52(1):378–385

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Choi S-K, Grandhi RV, Canfield RA (2004) Structural reliability under non-gaussian stochastic behavior. Comput Struct 82(13–14):1113–1121

    Article  Google Scholar 

  • Coello CAC (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41(2):113–127

    Article  Google Scholar 

  • Gano SE, Renaud JE, Sanders B (2005) Hybrid variable fidelity optimization by using a kriging-based scaling function. AIAA J 43(11):2422–2433

    Article  Google Scholar 

  • Gano SE, Renaud JE, Agarwal H, Tovar A (2006a) Reliability-based design using variable-fidelity optimization. Struct Infrastruct Eng 2(3–4):247–260

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Haftka RT (1991) Combining global and local approximations. AIAA J 29(9):1523–1525

    Article  Google Scholar 

  • Han Z-H, Görtz S (2012) Hierarchical kriging model for variable-fidelity surrogate modeling. AIAA J 50(9):1885–1896

    Article  Google Scholar 

  • Han Z, Zimmerman R, Görtz S (2012) Alternative cokriging method for variable-fidelity surrogate modeling. AIAA J 50(5):1205–1210

    Article  Google Scholar 

  • Hu Z, Mahadevan S (2015) Global sensitivity analysis-enhanced surrogate (gsas) modeling for reliability analysis. Struct Multidiscip Optim 53(3):501–521

    Article  MathSciNet  Google Scholar 

  • Hu Z, Mahadevan S (2016) A single-loop kriging surrogate modeling for time-dependent reliability analysis. J Mech Des 138(6):061406

    Article  Google Scholar 

  • Hu Z, Mahadevan S (2017) Uncertainty quantification in prediction of material properties during additive manufacturing. Scr Mater 135:135–140

    Article  Google Scholar 

  • Hu W, Enying L, Li GY, Zhong ZH (2007) Optimization of sheet metal forming processes by the use of space mapping based metamodeling method. Int J Adv Manuf Technol 39(7–8):642–655

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Hu J, Zhou Q, Jiang P, Shao X, Xie 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Jin R, Du X, Chen W (2003) The use of metamodeling techniques for optimization under uncertainty. Struct Multidiscip Optim 25(2):99–116

    Article  Google Scholar 

  • Jin 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

    Article  MathSciNet  MATH  Google Scholar 

  • Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic press, San Diego

    Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • Kennedy MC, O'hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87(1):1–13

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • Koziel S, Bekasiewicz A (2016) Rapid design optimization of antennas using variable-fidelity em models and adjoint sensitivities. Eng Comput 33(7):2007–2018

    Article  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Lewis RM, Nash SG (2000) A multigrid approach to the optimization of systems governed by differential equations. AIAA paper 4890:2000

    Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Liu Y, Collette M (2014) Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm. Appl Soft Comput 24:482–493

    Article  Google Scholar 

  • March A, Willcox K (2012) Provably convergent multifidelity optimization algorithm not requiring high-fidelity derivatives. AIAA J 50(5):1079–1089

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Park C, Haftka RT, Kim NH (2017) Remarks on multi-fidelity surrogates. Struct Multidiscip Optim 55(3):1029–1050

    Article  MathSciNet  Google Scholar 

  • Patel J, Choi S-K (2012) Classification approach for reliability-based topology optimization using probabilistic neural networks. Struct Multidiscip Optim 45(4):529–543

    Article  MathSciNet  MATH  Google Scholar 

  • Rayas-Sanchez JE (2016) Power in simplicity with asm: tracing the aggressive space mapping algorithm over two decades of development and engineering applications. IEEE Microw Mag 17(4):64–76

    Article  Google Scholar 

  • Raza MA, Liang W (2012) Uncertainty-based computational robust design optimisation of dual-thrust propulsion system. J Eng Des 23(8):618–634

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Taguchi G (1978) Performance analysis design. Int J Prod Res 16(6):521–530

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  MATH  Google Scholar 

  • Wang H, Fan T, Li G (2017) Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems. Struct Multidiscip Optim 55(6):2143–2157

    Article  Google Scholar 

  • Xia T, Li M, Zhou J (2016) A sequential robust optimization approach for multidisciplinary design optimization with uncertainty. J Mech Des 138(11):111406

    Article  Google Scholar 

  • Xiong S, Qian PZG, Wu CFJ (2013) Sequential design and analysis of high-accuracy and low-accuracy computer codes. Technometrics 55(1):37–46

    Article  MathSciNet  Google Scholar 

  • Zadeh PM, Toropov VV, Wood AS (2009) Metamodel-based collaborative optimization framework. Struct Multidiscip Optim 38(2):103–115

    Article  Google Scholar 

  • 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

    Article  MathSciNet  MATH  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zheng J, Shao X, Gao L, Jiang P, Qiu H (2014) Difference mapping method using least square support vector regression for variable-fidelity metamodelling. Eng Optim 47(6):719–736

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Zhou Q, Shao X, Jiang P, Cao L, Zhou H, Shu L (2015a) Differing mapping using ensemble of metamodels for global variable-fidelity metamodeling. CMES: Com Mod Eng Sci 106(5):323–355

    Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • 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

    Google Scholar 

  • Zhu J, Wang Y-J, Collette M (2013) A multi-objective variable-fidelity optimization method for genetic algorithms. Eng Optim 46(4):521–542

    Article  MathSciNet  Google Scholar 

  • Zhu P, Zhang S, Chen W (2014) Multi-point objective-oriented sequential sampling strategy for constrained robust design. Eng Optim 47(3):287–307

    Article  Google Scholar 

  • Zimmermann R, Han Z-H (2010) Simplified cross-correlation estimation for multi-fidelity surrogate cokriging models. Ad Appl Math Sci 7(2):181–202

    MathSciNet  MATH  Google Scholar 

Download references

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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Authors and Affiliations

  1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science & Technology, 430074, Wuhan, People’s Republic of China

    Qi Zhou, Ping Jiang, Xinyu Shao, Jiexiang Hu & Leshi Shu

  2. George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Qi Zhou, Yan Wang & Seung-Kyum Choi

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  1. Qi Zhou
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  2. Yan Wang
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  3. Seung-Kyum Choi
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  4. Ping Jiang
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Corresponding author

Correspondence to Ping Jiang.

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

$$ {\displaystyle \begin{array}{l}{\mu}_3\left(f\left(\boldsymbol{X}\right)\right)=E\left[Y\left(\boldsymbol{X},V\right)\right]\\ {}\kern4.199998em =E\left[E\left[Y\left(\boldsymbol{X},V\right)/V\right]\right]\\ {}\kern4.199998em =E\left[E\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)/V\right]\right]\\ {}\kern4.199998em =E\left[{\int}_wY\left(\boldsymbol{x}+\boldsymbol{w},V\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}|V\right]\\ {}\kern4.199998em ={\int}_{\boldsymbol{w}}E\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)/V\right]p\left(\boldsymbol{w}\right)d\boldsymbol{w}\end{array}} $$
(30)

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

$$ {\displaystyle \begin{array}{l}{\mu}_3\left(f\left(\boldsymbol{X}\right)\right)=E\left[Y\left(\boldsymbol{X},V\right)\right]\\ {}\kern4.199998em ={\int}_{\boldsymbol{w}}{y}_{mf}\left(\boldsymbol{x}+\boldsymbol{w}\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}\end{array}} $$
(31)

In the same manner, the variance value of the objective function can be expressed as

$$ {\displaystyle \begin{array}{l}{\sigma}_3^2\left(f\left(\boldsymbol{X}\right)\right)= Var\left[Y\left(\boldsymbol{X},V\right)\right]\\ {}\kern3.9em = Var\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)\right]\\ {}\kern3.9em =E\left[{Y}^2\left(\boldsymbol{x}+\boldsymbol{w},V\right)\right]-E{\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)\right]}^2\\ {}\kern3.9em =E\left[E\left[{Y}^2\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]-E{\left[E\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]}^2\end{array}} $$
(32)

The first term in the fourth line of (32) can be further expanded by the law of total expectation as

$$ {\displaystyle \begin{array}{l}E\left[E\left[{Y}^2\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]\hfill \\ {}=E\left[ Var\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]+E{\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]}^2\right]\hfill \\ {}=E\left[ Var\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]+E\left[E{\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]}^2\right]\hfill \end{array}} $$
(33)

Substituting (33) to (32), one can rewrite (32) as

$$ {\displaystyle \begin{array}{l}{\sigma}_3^2\left(f\left(\boldsymbol{X}\right)\right)= Var\left[Y\left(\boldsymbol{X},V\right)\right]\\ {}\kern4.199998em =E\left[ Var\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]+E\left[E{\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]}^2\right]\\ {}\kern5.099998em -E{\left[E\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]\right]}^2\\ {}\kern4.299998em ={\int}_{\boldsymbol{w}} Var\left[Y\left(\boldsymbol{x}+\boldsymbol{w},V\right)|V\right]p\left(\boldsymbol{w}\right)d\boldsymbol{w}+{\int}_{\boldsymbol{w}}\left({y}_{mf}^2\left(\boldsymbol{x}+\boldsymbol{w}\right)\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}\\ {}\kern5.099998em -{\left[{\int}_{\boldsymbol{w}}\left({y}_{mf}\left(\boldsymbol{x}+\boldsymbol{w}\right)\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}\right]}^2\end{array}} $$
(34)

Because Var[Y(x + w, V)| V] = s 2(y mf (x + w)) is the MSE for the Hierarchical Kriging prediction, recalling (18)

$$ Var\left({y}_{mf}\left(\boldsymbol{X}\right)\right)={\int}_{\boldsymbol{w}}{y}_{mf}^2\left(\boldsymbol{x}+\boldsymbol{w}\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}-{\left[{\int}_{\boldsymbol{w}}{y}_{mf}\left(\boldsymbol{x}+\boldsymbol{w}\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}\right]}^2 $$
(35)

The variance value of the objective function can be expressed as

$$ {\displaystyle \begin{array}{l}{\sigma}_3^2\left(f\left(\boldsymbol{X}\right)\right)= Var\left[Y\left(\boldsymbol{X},V\right)\right]\\ {}\kern3.9em ={\int}_{\boldsymbol{w}}{s}^2\left({y}_{mf}\left(\boldsymbol{x}+\boldsymbol{w}\right)\right)p\left(\boldsymbol{w}\right)d\boldsymbol{w}+ Var\left[{y}_{mf}\left(\boldsymbol{x}+\boldsymbol{w}\right)\right]\end{array}} $$
(36)

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.

Table 10 Monte Carlo Samples and corresponding responses for the solution only considering design variable uncertainty
Full size table
Table 11 Monte Carlo Samples and corresponding responses for solution considering both design variable and MF metamodel uncertainties
Full size table

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“+”.

Fig. 20
figure 20

The boxplots of the objective functions for different approaches under all examples in 15 runs

Full size image

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.

Fig. 21
figure 21

Plots of the constructed MF metamodels in additional case 3

Full size image

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.

Table 12 Comparison of result in the additional case 3
Full size table

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.

Fig. 22
figure 22

Plots of the constructed MF metamodels in additional case 4

Full size image

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.

Table 13 Comparison of result in the additional case 4
Full size table

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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