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. Youn
  • Byung C. Jung
  • Joung Taek Yoon
RESEARCH PAPER

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.

Keywords

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

Nomenclature

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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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Zhimin Xi
    • 1
  • Byeng D. Youn
    • 2
  • 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

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