Structural and Multidisciplinary Optimization

, Volume 51, Issue 3, pp 599–611 | Cite as

Random field modeling with insufficient field data for probability analysis and design

  • Zhimin Xi
  • Byeng D. YounEmail author
  • Byung C. Jung
  • Joung Taek Yoon


Often engineered systems entail randomness as a function of spatial (or temporal) variables. The random field can be found in the form of geometry, material property, and/or loading in engineering products and processes. In some applications, consideration of the random field is a key to accurately predict variability in system performances. However, existing methods for random field modeling are limited for practical use because they require sufficient field data. This paper thus proposes a new random field modeling method using a Bayesian Copula that facilitates the random field modeling with insufficient field data and applies this method for engineering probability analysis and robust design optimization. The proposed method is composed of three key ideas: (i) determining the marginal distribution of random field realizations at each measurement location, (ii) determining optimal Copulas to model statistical dependence of the field realizations at different measurement locations, and (iii) modeling a joint probability density function of the random field. A mathematical problem was first employed for the purpose of demonstrating the accuracy of the random field modeling with insufficient field data. The second case study deals with the assembly process of a two-door refrigerator that challenges predicting the door assembly tolerance and minimizing the tolerance by designing the random field and parameter variables in the assembly process with insufficient random field data. It is concluded that the proposed random field modeling can be used to successfully conduct the probability analysis and robust design optimization with insufficient random field data.


Random field Copula Proper orthogonal decomposition (POD) Robust design optimization Bayes 



random field

μ ν

mean and variation of the random field


signature of the random field


covariance matrix


distribution parameter vector


design vector of random parameter variables


design vector of random field variables


coefficient of the random field signature


eigenvalue of the covariance matrix


Kendall’s tau

C c

cumulative distribution function and probability density function of the Copula


bivariate data

F f

cumulative distribution function and probability density function


number of random fields

m n

number of random field data and number of measurement locations


number of test Copulas


random field variable


measurement location


number of random field design variables


number of design variables


number of probabilistic constraints


number of random parameters



Research was supported by the Faculty Research Initiation and Seed Grant at University of Michigan Dearborn, by the Basic Research Project of Korea Institute of Machinery and Materials which is originally supported by Korea Research Council for Industrial Science & Technology, by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A2A2A01068627), by the Brain Korea 21 Plus Project in 2013, and by the Institute of Advanced Machinery and Design at Seoul National University (SNU-IAMD). In addition, the authors appreciate Dr. Hui Wang at University of Michigan – Ann Arbor for providing the random field realization of a V-8 engine head.


  1. Basudhar A, Missoum S (2009) A sampling-based approach for probabilistic design with random fields. Comput Methods Appl Mech Eng 198(47–48):3647–3655CrossRefzbMATHGoogle Scholar
  2. Berg, BA (2004) “Markov Chain Monte Carlo Simulations and Their Statistical Analysis”, World Scientific Publishing Co. Pte. Ltd. ISBN, 981-238-935-0Google Scholar
  3. Berkooz, G, Holmes, P, and Lumley, JL (1996) “Turbulence, Coherent Structuress, Dynamical Systems and Symmetry.” Cambridge University Press: Cambridge Monographs on MechanicsGoogle Scholar
  4. Bhushan RK, Kumar S, Das S (2012) GA approach for optimization of surface roughness parameters in machining of Al alloy SiC particle composite. J Mater Eng Perform 21(8):1676–1686CrossRefGoogle Scholar
  5. Boone EL, Ye K, Smith EP (2005) Assessment of two approximation methods for computing posterior model probabilities. Comput Stat Data Anal 48(2):221–234CrossRefzbMATHMathSciNetGoogle Scholar
  6. Chen X, Fan Y (2005) Pseudo-likelihood ratio tests for semiparametric multivariate copula model selection. La Rev Can Stat 33(3):389–414CrossRefzbMATHMathSciNetGoogle Scholar
  7. Chen S, Chen W, Lee S (2010) Level Set based robust shape and topology optimization under random field uncertainties. Struct Multidiscip Optim 41(4):507–524CrossRefzbMATHMathSciNetGoogle Scholar
  8. Choi SK, Canfield RA, Grandhi RV (2006) Estimation of structural reliability for gaussian random fields. Struct Infrastruct Eng 2:161–173CrossRefGoogle Scholar
  9. Der Kiureghian A, Ke JB (1988) The stochastic finite element method in structural reliability. Probab Eng Mech 3(2):83–91CrossRefGoogle Scholar
  10. Fermanian JD (2005) Goodness-of-Fit tests for copulas. J Multivar Anal 95:119–152CrossRefzbMATHMathSciNetGoogle Scholar
  11. Fukunaga K (1990) Introduction to statistical recognition. Academic, NewYorkzbMATHGoogle Scholar
  12. Guan X, Giffin A, Jha R, Liu Y (2012) Maximum relative entropy-based probabilistic inference in fatigue crack damage prognostics. Probab Eng Mech 29:157–166CrossRefGoogle Scholar
  13. Haran, M (2012) “Gaussian random field models for spatial data”, Markov Chain Monte Carlo Handbook, CRC PressGoogle Scholar
  14. Hu C, Youn BD (2009) Adaptive-sparse polynomial chaos expansion for reliability analysis and design of complex engineering systems. Struct Multidiscip Optim 43(3):419–442CrossRefMathSciNetGoogle Scholar
  15. Huang B, Du X (2006) Uncertainty analysis by dimension reduction integration and saddlepoint approximations. J Mech Des 128:26–33CrossRefGoogle Scholar
  16. Huard D, Evin G, Favre AC (2006) Bayesian copula selection. Comput Stat Data Anal 51(2):809–822CrossRefzbMATHMathSciNetGoogle Scholar
  17. Jaynes ET, Bretthorst GL (2003) Probability theory: the logic of science. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  18. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22(1):79–86CrossRefzbMATHMathSciNetGoogle Scholar
  19. Lee SH, Chen W (2009) A comparative study of uncertainty propagation methods for black-box type problems. Struct Multidiscip Optim 37(3):239–253CrossRefMathSciNetGoogle Scholar
  20. Li C, Der Kiureghian A (1993) Optimal discretization of random fields. J Eng Mech 119(6):1136–1154CrossRefGoogle Scholar
  21. Liu WK, Belytschko T, Mani A (1986) Random field finite elements. Int J Numer Methods Eng 23(10):1831–1845CrossRefzbMATHMathSciNetGoogle Scholar
  22. Missoum S (2008) Probabilistic optimal design in the presence of random fields. Struct Multidiscip Optim 35:523–530CrossRefzbMATHMathSciNetGoogle Scholar
  23. Orfanidis SJ (1996) Introduction to signal processing. Prentice-Hall, EnglewoodGoogle Scholar
  24. Panchenko V (2005) Goodness-of-fit test for copulas. Phys A: Stat Mech Appl 355(1):176–182CrossRefMathSciNetGoogle Scholar
  25. Rabitz H, Alis OF (1999) General foundations of high-dimensional model representations. J Math Chem 25:197–233CrossRefzbMATHMathSciNetGoogle Scholar
  26. Rabitz H, Alis OF, Shorter J, Shim K (1999) Efficient input–output model representations. Comput Phys Commun 117:11–20CrossRefzbMATHGoogle Scholar
  27. Rajaee M, Karlsson SKF, Sirorich L (1994) Low-dimensional description of free-shear-flow coherent structures and their dynamical behavior. J Fluid Mech 258:1–29CrossRefzbMATHGoogle Scholar
  28. Roser BN (1999) An introduction to copulas. Springer, New YorkGoogle Scholar
  29. Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publ Inst Stat’Univer Paris 8:229–231MathSciNetGoogle Scholar
  30. Sudret, B and Der Kiureghian A (2000) “Stochastic Finite Element Methods and Reliability: A State-Of-The-Art Report,” Technical Report UCB/SEMM-2000/08, Department of Civil & Environmental Engineering, University of California, Berkeley, CAGoogle Scholar
  31. Tamura Y, Suganuma S, Kikuchi H, Hibi K (1999) Proper orthogonal decomposition of random wind pressure field. J Fluids Struct 13:1069–1095CrossRefGoogle Scholar
  32. Turk M, Pentland A (1991) Eigenfaces for recognition. J Cogn Neurosci 3(1):71–86CrossRefGoogle Scholar
  33. Vanmarcke EH, Grigoriu M (1983) Stochastic finite element analysis of simple beams. J Eng Mech, ASCE 109(5):1203–1214CrossRefGoogle Scholar
  34. Wang C, Blei DM (2013) Variational inference in nonconjugate models. J Mach Learn Res 14(1):1005–1031zbMATHMathSciNetGoogle Scholar
  35. Wang H, Suriano S, Zhou L, Hu SJ (2009) High-definition metrology based spatial variation pattern analysis for machining process monitoring and diagnosis. In: Proceedings of ASME 2009 International Manufacturing Science and Engineering Conference, MSEC2009-84154, West Lafayette, Indiana, USAGoogle Scholar
  36. Xi Z, Youn BD, Hu C (2010) Random field characterization considering statistical dependence for probability analysis and design. J Mech Des 132(10):101008(12)CrossRefGoogle Scholar
  37. Xu H, Rahman S (2004) A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. Int J Numer Method Eng 61:1992–2019CrossRefzbMATHGoogle Scholar
  38. Yin X, Lee S, Chen W, Liu WK (2009) Efficient random field uncertainty propagation in design using multiscale analysis. J Mech Des 131(2):021006(10)CrossRefGoogle Scholar
  39. Youn BD, Xi Z (2009) Reliability-based robust design optimization using the eigenvector dimension reduction (EDR) method. Struct Multidiscip Optim 37(5):475–492CrossRefMathSciNetGoogle Scholar
  40. Youn BD, Choi KK, Yi K (2005) Performance moment integration (PMI) method for quality assessment in reliability-based robust design optimization. Mech Based Des Struct Mach 33:185–213CrossRefGoogle Scholar
  41. Zhan Z, Fu Y, Yang R-J, Xi Z, Shi L (2012) A Bayesian inference based model interpolation and extrapolation. SAE Int J Mater Manuf 5(2):357–364CrossRefGoogle Scholar
  42. Zhu J, Xing EP (2009) Maximum entropy discrimination markov networks. J Mach Learn Res 10:2531–2569zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Zhimin Xi
    • 1
  • Byeng D. Youn
    • 2
    Email author
  • Byung C. Jung
    • 3
  • Joung Taek Yoon
    • 2
  1. 1.Department of Industrial and Manufacturing Systems EngineeringUniversity of Michigan – DearbornDearbornUSA
  2. 2.Department of Mechanical and Aerospace EngineeringSeoul National UniversitySeoulKorea
  3. 3.Korea Institute of Machinery and MaterialsDaejeonKorea

Personalised recommendations