Efficient surrogate construction by combining response surface methodology and reduced order modeling

  • Christian GoguEmail author
  • Jean-Charles Passieux
Research Paper


Response surface methodology is an efficient method for approximating the output of complex, computationally expensive codes. Challenges remain however in decreasing their construction cost as well as in approximating high dimensional output instead of scalar values. We propose a novel approach addressing both these challenges simultaneously for cases where the expensive code solves partial differential equations involving the resolution of a large system of equations, such as by finite element. Our method is based on the combination of response surface methodology and reduced order modeling by projection, also known as reduced basis modeling. The novel idea is to carry out the full resolution of the system only at a small, appropriately chosen, number of points. At the other points only the inexpensive reduced basis solution is computed while controlling the quality of the approximation being sought. A first application of the proposed surrogate modeling approach is presented for the problem of identification of orthotropic elastic constants from full field displacement measurements based on a tensile test on a plate with a hole. A surrogate of the entire displacement field was constructed using the proposed method. A second application involves the construction of a surrogate for the temperature field in a rocket engine combustion chamber wall. Compared to traditional response surface methodology a reduction by about an order of magnitude in the total system resolution time was achieved using the proposed sequential surrogate construction strategy.


Response surface methodology Surrogate modeling Reduced basis modeling Proper orthogonal decomposition Key points 


  1. Allen DM (1971) Mean square error of prediction as a criterion for selecting variables. Technometrics 13:469–475zbMATHCrossRefGoogle Scholar
  2. Chen VCP, Tsui K-L, Barton RR, Meckesheimer M (2006) A review on design, modeling and applications of computer experiments. IIE Transactions 38(4):273–291CrossRefGoogle Scholar
  3. Cheng B, Titterington DM (1994) Neural networks: a review from a statistical perspective. Stat Sci 9(1):2–54MathSciNetzbMATHCrossRefGoogle Scholar
  4. Coelho RF, Breitkopf P, Vayssade CK-L (2008) Model reduction for multidisciplinary optimization-application to a 2D wing. Struct Multidisc Optim 37(1):29–48CrossRefGoogle Scholar
  5. Coelho RF, Breitkopf P, Vayssade CK-L, Villon P (2009) Bi-level model reduction for coupled problems. Struct Multidisc Optim 39(4):401–418CrossRefGoogle Scholar
  6. Craig RR, Bampton MCC (1968) Coupling of substructures for dynamic analyses. AIAA J 6(7):1313–1319zbMATHCrossRefGoogle Scholar
  7. Daniel C (1973) One-at-a-time plans. J Am Stat Assoc 68:353–360CrossRefGoogle Scholar
  8. De Lucas S, Vega JM, Velazquez A (2011) Aeronautic conceptual design optimization method based on high-order singular value decomposition. AIAA J 49(12):2713–2725CrossRefGoogle Scholar
  9. Forrester AIJ, Keane AJ (2009) Recent advances in surrogate-based optimization. Prog Aerosp Sci 45(1–3):50–79CrossRefGoogle Scholar
  10. Goel T, Stander N (2009) Comparing three error criteria for selecting radial basis function network topology. Comput Methods Appl Mech Eng 198(27–29):2137–2150MathSciNetzbMATHCrossRefGoogle Scholar
  11. Gogu C, Haftka RT, Bapanapalli SK, Sankar BV (2009) Dimensionality reduction approach for response surface approximations: application to thermal design. AIAA J 47(7):1700–1708CrossRefGoogle Scholar
  12. Gogu C, Yin W, Haftka RT, Ifju PG, Molimard J, Le Riche R, Vautrin A (2012) Bayesian identification of elastic constants in multi-directional laminate from moiré interferometry displacement fields. Exp Mech doi: 10.1007/s11340-012-9671-8 Google Scholar
  13. Gosselet P (2003) Méthode de décomposition de domaines et méthodes d’accélération pour les problemes multichamps en mécanique non-linéaire, 2003. PhD thesis, Université Paris 6Google Scholar
  14. Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math Model Numer Anal 41:575–605. doi: 10.1051/m2an:2007031 MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hotteling H (1933) Analysis of Complex of statistical variables into principal component. J Educ Psychol 24:417–441CrossRefGoogle Scholar
  16. Karhunen K (1943) Uber lineare methoden für wahrscheinigkeitsrechnung. Ann Acad Sci Fenn, A1 Math-Phys 37:3–79Google Scholar
  17. Kerfriden P, Gosselet P, Adhikaric S, Bordas SPA (2011) Bridging proper orthogonal decomposition methods and augmented Newton–Krylov algorithms: an adaptive model order reduction for highly nonlinear mechanical problems. Comput Methods Appl Mech Eng 200(5–8):850–866zbMATHCrossRefGoogle Scholar
  18. Kerfiden P, Passieux J-C, Bordas SPA (2011) Local/global model order reduction strategy for the simulation of quasi-brittle fracture. Int J Numer Methods Eng 89(2):154–179CrossRefGoogle Scholar
  19. Khuri AI, Cornell JA (1996) 7Response surfaces: designs and analyses, 2nd edn. Dekker, New YorkzbMATHGoogle Scholar
  20. Kleijnen JPC (2009) Kriging metamodeling in simulation: a review. Eur J Oper Res 192(3):707–716MathSciNetzbMATHCrossRefGoogle Scholar
  21. Kleijnen JPC, Sanchez SM, Lucas TW, Cioppa TM (2005) A user’s guide to the brave new world of designing simulation experiments. INFORMS J Comput 17(3):263–289zbMATHCrossRefGoogle Scholar
  22. Krysl P, Lall S, Marsen JE (2001) Dimensional model reduction in non-linear finite element dynamics of solids and structures. Int J Numer Methods Eng 51(479):504Google Scholar
  23. Kunisch K, Volkwein S (2002) Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J Numer Anal 40(2):492–515MathSciNetzbMATHCrossRefGoogle Scholar
  24. Ladeveze P, Passieux J-C, Neron D (2009) The LATIN multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng 199(21):1287–1296MathSciNetCrossRefGoogle Scholar
  25. Loeve MM (1955) Probability theory. Van Nostrand, Princeton, NYzbMATHGoogle Scholar
  26. Mika S, Tsuda K (2001) An introduction to kernel-based learning algorithms. IEEE Trans Neural Netw 12(2):181–201CrossRefGoogle Scholar
  27. Myers DE (1982) Matrix formulation of co-kriging. Math Geol 14(3):249–257MathSciNetCrossRefGoogle Scholar
  28. Myers RH, Montgomery DC (2002) Response surface methodology: process and product in optimization using designed experiments. Wiley, New YorkzbMATHGoogle Scholar
  29. Park J, Sandberg IW (1991) Universal approximation using radial-basis-function networks. Neural Comput 3(2):246–257CrossRefGoogle Scholar
  30. Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41:1–28CrossRefGoogle Scholar
  31. Riccius JR, Haidn OJ, Zametaev EB (2004) Influence of time dependent effects on the estimated life time of liquid rocket combustion chamber walls. In: AIAA 2004-3670, 40th AIAA/ASME/SAE/ASEE joint propulsion conference, Fort Lauderdale, FloridaGoogle Scholar
  32. Riccius J, Gogu C, Zametaev E, Haidn O (2006) Liquid rocket engine chamber wall optimization using plane strain and generalized plane strain models. In: AIAA paper 2006-4366, 42nd AIAA/ASME/SAE/ASEE joint propulsion conference, Sacramento, CAGoogle Scholar
  33. Ryckelynck D (2005) A priori hypereduction method: an adaptive approach. J Comput Phys 202(1):346–366MathSciNetzbMATHCrossRefGoogle Scholar
  34. Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4):409–435MathSciNetzbMATHCrossRefGoogle Scholar
  35. Scholkopf B, Smola AJ (2002) Learning with Kernels. CambridgeGoogle Scholar
  36. Silva GHC (2009) Identification of material properties using finite elements and full-field measurements with focus on the characterization of deterministic experimental errors, PhD thesis, Ecole des Mines de Saint Etienne, FranceGoogle Scholar
  37. Simpson TW, Mauery TM, Korte JJ, Mistree F (2001) Kriging models for global approximation in simulation-based multidisciplinary design optimization. AIAA J 39(12):2233–2241CrossRefGoogle Scholar
  38. Simpson TW, Toropov V, Balabanov V, Viana FAC (2010) Design and analysis of computer experiments in multidisciplinary design optimization: a review of how we have come or not. In: 12th AIAA/ISSMO multidisciplinary analysis and optimization conference, Victoria, British Colombia, 10–12 September 2008. doi: 10.2514/6.2008-5802
  39. Singh G, Grandhi RV (2009) Propagation of structural random field uncertainty using improved dimension reduction method. In: 50th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference, Palm Springs, California, AIAA paper 2009-2254; 2009Google Scholar
  40. Smola AJ, Scholkopf B (2004) A tutorial on support vector regression. Stat Comput 14(3):199–222MathSciNetCrossRefGoogle Scholar
  41. Sobol I (1993) Sensitivity estimates for non-linear mathematical models. Math Model Comput Exper 4:407–414MathSciNetGoogle Scholar
  42. Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkzbMATHCrossRefGoogle Scholar
  43. Viana FAC (2010) SURROGATES toolbox user’s guide, version 2.1.
  44. Viana FAC, Gogu C, Haftka RT (2010) Making the most out of surrogate models: tricks of the trade. In ASME 2010 international design engineering technical conferences & computers and information in engineering conference, Montreal, Canada, 15–18 August 2010Google Scholar
  45. Wehrwein D, Mourelatos ZP (2006) Reliability based design optimization of dynamic vehicle performance using bond graphs and time dependent metamodels, 2006. SAE Technical Paper 2006-01-0109. doi: 10.4271/2006-01-0109
  46. Yee TW (2000) Vector splines and other vector smoothers. In: Bethlehem JG, Van der Heijden PGM (eds) Proc. computational statistics COMPSTAT 2000. Physica-Verlag, HeidelbergGoogle Scholar
  47. Yin X, Lee S, Chen W, Liu WK, Horstemeyer MF (2009) Efficient random field uncertainty propagation in design using multiscale analysis. J Mech Des 131(2):021006 (10 pages)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Université de Toulouse; UPS, INSA, Mines Albi, ISAE; ICA (Institut Clément Ader)ToulouseFrance

Personalised recommendations