Applied Intelligence

, Volume 48, Issue 6, pp 1644–1656 | Cite as

An adaptive sampling strategy for Kriging metamodel based on Delaunay triangulation and TOPSIS

  • Ping Jiang
  • Yahui Zhang
  • Qi ZhouEmail author
  • Xinyu Shao
  • Jiexiang Hu
  • Leshi Shu


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.


Delaunay triangulation TOPSIS Kriging Sequential sampling Metamodel 



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.


  1. 1.
    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):683–696CrossRefGoogle Scholar
  2. 2.
    Jiang C, Han X (2007) A new uncertain optimization method based on intervals and an approximation management model. Comput Model Eng Sci 22(2):97MathSciNetzbMATHGoogle Scholar
  3. 3.
    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):365–390CrossRefGoogle Scholar
  4. 4.
    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 2015Google Scholar
  5. 5.
    Kleijnen J P (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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):283–297CrossRefGoogle Scholar
  7. 7.
    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):211–217CrossRefGoogle Scholar
  8. 8.
    Liu J, Hu Y, Wu B, Jin C A hybrid health condition monitoring method in milling operations. Int J Adv Manuf Technol 1–12Google Scholar
  9. 9.
    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):368–375CrossRefGoogle Scholar
  10. 10.
    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):333MathSciNetzbMATHGoogle Scholar
  11. 11.
    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 59:18–35CrossRefGoogle Scholar
  12. 12.
    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 32:26–39CrossRefGoogle Scholar
  13. 13.
    Mirzaei D (2015) Analysis of moving least squares approximation revisited. J Comput Appl Math 282:237–250MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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):6047–6056CrossRefGoogle Scholar
  15. 15.
    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):1853–1866MathSciNetCrossRefGoogle Scholar
  16. 16.
    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):219–241MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    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):221–231MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang H, Fan T, Li G (2016) Reanalysis-based space mapping method, an alternative optimization way for expensive simulation-based problems. Struct Multidiscip Optim 1–15Google Scholar
  19. 19.
    Bursztyn D, Steinberg D M (2006) Comparison of designs for computer experiments. J Stat Plan Infer 136 (3):1103–1119MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Roy R, Hinduja S, Teti R (2008) Recent advances in engineering design optimisation: challenges and future trends. CIRP Ann-Manuf Technol 57(2):697–715CrossRefGoogle Scholar
  21. 21.
    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):2165–2184CrossRefGoogle Scholar
  22. 22.
    Zhang Y, Li W, Mao S, Zheng Z (2011) Orthogonal arrays obtained by generalized difference matrices with g levels. Sci Chin Math 54(1):133–143MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Vořechovský M (2015) Hierarchical refinement of Latin hypercube samples. Comput-Aided Civil Infrastruct Eng 30(5):394–411CrossRefGoogle Scholar
  24. 24.
    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):55–61CrossRefzbMATHGoogle Scholar
  25. 25.
    Liu H, Xu S, Wang X (2015) Sequential sampling designs based on space reduction. Eng Optim 47 (7):867–884CrossRefGoogle Scholar
  26. 26.
    Younis A, Dong Z (2010) Trends, features, and tests of common and recently introduced global optimization methods. Eng Optim 42(8):691–718MathSciNetCrossRefGoogle Scholar
  27. 27.
    Xiong F, Xiong Y, Chen W, Yang S (2009) Optimizing Latin hypercube design for sequential sampling of computer experiments. Eng Optim 41(8):793–810CrossRefGoogle Scholar
  28. 28.
    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 539-548Google Scholar
  29. 29.
    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):2104–2126CrossRefzbMATHGoogle Scholar
  30. 30.
    Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. WileyGoogle Scholar
  31. 31.
    Lin Y (2004) An efficient robust concept exploration method and sequential exploratory experimental designGoogle Scholar
  32. 32.
    Li G, Aute V, Azarm S (2010) An accumulative error based adaptive design of experiments for offline metamodeling. Struct Multidiscip Optim 40(1–6):137–155CrossRefGoogle Scholar
  33. 33.
    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):644–665MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Delaunay B (1934) Sur la sphere vide. Izv Akad Nauk SSSR. Otdelenie Matematicheskii i Estestvennyka Nauk 7(793–800):1–2Google Scholar
  35. 35.
    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 1–6Google Scholar
  36. 36.
    Hwang C-L, Yoon K (2012) Multiple attribute decision making: methods and applications a state-of-the-art survey, vol 186. Springer Science & Business MediaGoogle Scholar
  37. 37.
    Kahraman C, Büyüközkan G, Ateş N Y (2007) A two phase multi-attribute decision-making approach for new product introduction. Inform Sci 177(7):1567–1582CrossRefGoogle Scholar
  38. 38.
    Sacks J, Welch W J, Mitchell T J, Wynn H P (1989) Design and analysis of computer experiments. Stat Sci 409–423Google Scholar
  39. 39.
    Tüceryan M, Jain A K (1990) Texture segmentation using Voronoi polygons. IEEE Trans Pattern Anal Mach Intell 12(2):211–216CrossRefGoogle Scholar
  40. 40.
    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):211–221CrossRefGoogle Scholar
  41. 41.
    Rastrigin L (1974) Extremal control systems theoretical foundations of engineering cybernetics series. NaukaGoogle Scholar
  42. 42.
    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):581–605CrossRefGoogle Scholar
  43. 43.
    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 engineeringGoogle Scholar
  44. 44.
    Zhao D, Xue D (2010) A comparative study of metamodeling methods considering sample quality merits. Struct Multidiscip Optim 42(6):923–938CrossRefGoogle Scholar
  45. 45.
    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):205–231CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Ping Jiang
    • 1
  • Yahui Zhang
    • 1
  • Qi Zhou
    • 1
    • 2
    Email author
  • Xinyu Shao
    • 1
  • Jiexiang Hu
    • 1
  • Leshi Shu
    • 1
  1. 1.The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and EngineeringHuazhong University of Science & TechnologyWuhanPeople’s Republic of China
  2. 2.George W. Woodruff School of Mechanical EngineeringGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations