Abstract
Metamodels have been widely used in engineering design and optimization. Sampling method plays an important role in the constructing of metamodels. This paper proposes an adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS (KMDT). In the proposed KMDT, Delaunay triangulation is employed to partition the design space according to current sample points. The area of each partitioned triangle is used to indicate the degree of dispersion of sample points, and the prediction error of Kriging metamodel at each triangle’s centroid is used to represent the local error of each triangle region. By calculating the weight of the area and prediction error for each triangle region using the entropy method and TOPSIS, the degree of dispersion of sample points and local errors of metamodel are taken into consideration to make a trade-off between global exploration and local exploitation during the sequential sampling process. As a demonstration, the proposed approach is compared to other three sampling methods using several numerical cases and the modeling of the aerodynamic coefficient for a three-dimensional aircraft. The result reveals that the proposed approach provides more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.
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Crombecq K, Laermans E, Dhaene T (2011) Efficient space-filling and non-collapsing sequential design strategies for simulation-based modeling. Eur J Oper Res 214(3):683–696
Jiang C, Han X (2007) A new uncertain optimization method based on intervals and an approximation management model. Comput Model Eng Sci 22(2):97
Eddy D C, Krishnamurty S, Grosse I R, Wileden J C, Lewis K E (2015) A predictive modelling-based material selection method for sustainable product design. J Eng Des 26(10-12):365–390
Jiang P, Wang J, Zhou Q, Zhang X (2015) An enhanced analytical target cascading and Kriging model combined approach for multidisciplinary design optimization. Math Probl Eng 2015
Kleijnen J P (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716
Zhou Q, Shao X, Jiang P, Gao Z, Wang C, Shu L (2016) An active learning metamodeling approach by sequentially exploiting difference information from variable-fidelity models. Adv Eng Inform 30(3):283–297
Chang C -J, Lin J -Y, Chang M -J (2016) Extended modeling procedure based on the projected sample for forecasting short-term electricity consumption. Adv Eng Inform 30(2):211–217
Liu J, Hu Y, Wu B, Jin C A hybrid health condition monitoring method in milling operations. Int J Adv Manuf Technol 1–12
Wang X, You M, Mao Z, Yuan P (2016) Tree-structure ensemble general regression neural networks applied to predict the molten steel temperature in ladle furnace. Adv Eng Inform 30(3):368–375
Shokri A, Dehghan M (2012) A meshless method using radial basis functions for the numerical solution of two—dimensional complex Ginzburg—Landau equation. Comput Model Eng Sci 84(4):333
Zhou Q, Shao X, Jiang P, Zhou H, Shu L (2015) An adaptive global variable fidelity metamodeling strategy using a support vector regression based scaling function. Simul Model Pract Theory 59:18–35
Zhou Q, Jiang P, Shao X, Hu J, Cao L, Wan L (2017) A variable fidelity information fusion method based on radial basis function. Adv Eng Inform 32:26–39
Mirzaei D (2015) Analysis of moving least squares approximation revisited. J Comput Appl Math 282:237–250
Kavousi-Fard A, Samet H, Marzbani F (2014) A new hybrid modified firefly algorithm and support vector regression model for accurate short term load forecasting. Exp Syst Appl 41(13):6047–6056
Zhao H, Yue Z, Liu Y, Gao Z, Zhang Y (2015) An efficient reliability method combining adaptive importance sampling and Kriging metamodel. Appl Math Model 39(7):1853–1866
Shan S, Wang G G (2010) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241
Song X, Sun G, Li G, Gao W, Li Q (2013) Crashworthiness optimization of foam-filled tapered thin-walled structure using multiple surrogate models. Struct Multidiscip Optim 47(2):221–231
Wang H, Fan T, Li G (2016) Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems. Struct Multidiscip Optim 1–15
Bursztyn D, Steinberg D M (2006) Comparison of designs for computer experiments. J Stat Plan Infer 136 (3):1103–1119
Roy R, Hinduja S, Teti R (2008) Recent advances in engineering design optimisation: challenges and future trends. CIRP Ann-Manuf Technol 57(2):697–715
Zheng J, Li Z, Gao L, Jiang G (2016) A parameterized lower confidence bounding scheme for adaptive metamodel-based design optimization. Eng Comput 33(7):2165–2184
Zhang Y, Li W, Mao S, Zheng Z (2011) Orthogonal arrays obtained by generalized difference matrices with g levels. Sci Chin Math 54(1):133–143
Vořechovský M (2015) Hierarchical refinement of Latin hypercube samples. Comput-Aided Civil Infrastruct Eng 30(5):394–411
McKay M D, Beckman R J, Conover W J (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61
Liu H, Xu S, Wang X (2015) Sequential sampling designs based on space reduction. Eng Optim 47 (7):867–884
Younis A, Dong Z (2010) Trends, features, and tests of common and recently introduced global optimization methods. Eng Optim 42(8):691–718
Xiong F, Xiong Y, Chen W, Yang S (2009) Optimizing Latin hypercube design for sequential sampling of computer experiments. Eng Optim 41(8):793–810
Jin R, Chen W, Sudjianto A (2002) On sequential sampling for global metamodeling in engineering design. In: ASME 2002 International design engineering technical conferences and computers and information in engineering conference. American Society of Mechanical Engineers, pp 539-548
Farhang-Mehr A, Azarm S (2005) Bayesian meta-modelling of engineering design simulations: a sequential approach with adaptation to irregularities in the response behaviour. Int J Numer Methods Eng 62(15):2104–2126
Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley
Lin Y (2004) An efficient robust concept exploration method and sequential exploratory experimental design
Li G, Aute V, Azarm S (2010) An accumulative error based adaptive design of experiments for offline metamodeling. Struct Multidiscip Optim 40(1–6):137–155
Yang Q, Xue D (2015) A weighted sequential sampling method considering influences of sample qualities in input and output parameter spaces for global optimization. J Optim Theory Appl 164(2):644–665
Delaunay B (1934) Sur la sphere vide. Izv Akad Nauk SSSR. Otdelenie Matematicheskii i Estestvennyka Nauk 7(793–800):1–2
He Y, Guo H, Jin M, Ren P (2016) A linguistic entropy weight method and its application in linguistic multi-attribute group decision making. Nonlin Dyn 1–6
Hwang C-L, Yoon K (2012) Multiple attribute decision making: methods and applications a state-of-the-art survey, vol 186. Springer Science & Business Media
Kahraman C, Büyüközkan G, Ateş N Y (2007) A two phase multi-attribute decision-making approach for new product introduction. Inform Sci 177(7):1567–1582
Sacks J, Welch W J, Mitchell T J, Wynn H P (1989) Design and analysis of computer experiments. Stat Sci 409–423
Tüceryan M, Jain A K (1990) Texture segmentation using Voronoi polygons. IEEE Trans Pattern Anal Mach Intell 12(2):211–216
Tang M, Pan S (2015) A hybrid genetic algorithm for the energy-efficient virtual machine placement problem in data centers. Neural Process Lett 41(2):211–221
Rastrigin L (1974) Extremal control systems theoretical foundations of engineering cybernetics series. Nauka
Aute V, Saleh K, Abdelaziz O, Azarm S, Radermacher R (2013) Cross-validation based single response adaptive design of experiments for Kriging metamodeling of deterministic computer simulations. Struct Multidiscip Optim 48(3):581–605
Crombecq K, Couckuyt I, Gorissen D, Dhaene T (2009) Space-filling sequential design strategies for adaptive surrogate modelling. In: The first international conference on soft computing technology in civil, structural and environmental engineering
Zhao D, Xue D (2010) A comparative study of metamodeling methods considering sample quality merits. Struct Multidiscip Optim 42(6):923–938
Zhou Q, Shao X, Jiang P, Gao Z, Zhou H, Shu L (2016) 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
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. The authors also would like to thank the anonymous referees for their valuable comments.
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Jiang, P., Zhang, Y., Zhou, Q. et al. An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS. Appl Intell 48, 1644–1656 (2018). https://doi.org/10.1007/s10489-017-1031-z
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DOI: https://doi.org/10.1007/s10489-017-1031-z