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

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Abstract

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.

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Abbreviations

Θ :

random field

μ ν :

mean and variation of the random field

ϕ :

signature of the random field

Σ :

covariance matrix

Δ :

distribution parameter vector

d :

design vector of random parameter variables

γ :

design vector of random field variables

α:

coefficient of the random field signature

λ :

eigenvalue of the covariance matrix

τ :

Kendall’s tau

C c :

cumulative distribution function and probability density function of the Copula

D :

bivariate data

F f :

cumulative distribution function and probability density function

M :

number of random fields

m n :

number of random field data and number of measurement locations

Q :

number of test Copulas

V :

random field variable

x :

measurement location

MD :

number of random field design variables

ND :

number of design variables

NC :

number of probabilistic constraints

NP :

number of random parameters

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Acknowledgments

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.

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Authors and Affiliations

  1. Department of Industrial and Manufacturing Systems Engineering, University of Michigan – Dearborn, Dearborn, MI, 48168, USA

    Zhimin Xi

  2. Department of Mechanical and Aerospace Engineering, Seoul National University, Gwanak-Ro 1, Gwanak-Gu, Seoul, 151-742, Korea

    Byeng D. Youn & Joung Taek Yoon

  3. Korea Institute of Machinery and Materials, 156 Gajeongbuk-Ro, Yuseong-Gu, Daejeon, 305-343, Korea

    Byung C. Jung

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  1. Zhimin Xi
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  2. Byeng D. Youn
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Correspondence to Byeng D. Youn.

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Xi, Z., Youn, B.D., Jung, B.C. et al. Random field modeling with insufficient field data for probability analysis and design. Struct Multidisc Optim 51, 599–611 (2015). https://doi.org/10.1007/s00158-014-1165-0

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  • DOI: https://doi.org/10.1007/s00158-014-1165-0

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