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Variable-fidelity modeling of structural analysis of assemblies

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Abstract

This paper deals with the advantages of using variable-fidelity metamodeling strategies in order to develop a valid metamodel more rapidly than by using traditional methods. In our mechanical assembly design, we use the term “variable-fidelity” in reference to the convergence (or accuracy) level of the iterative solver being used. Variable-fidelity metamodeling is a way to improve the prediction of the output of a complex system by incorporating rapidly available auxiliary lower-fidelity data. This work uses two fidelity levels, but more levels can be added. The LATIN iterative algorithm is used along with a “multiparametric” strategy to calculate the various data and their different fidelity levels by means of an error indicator. Three main categories of variable-fidelity strategies are currently available. We tested at least one method from each of these categories, which comes to a total of five methods for calculating a valid metamodel using low- and high-fidelity data. Here, our objective is to compare the performances of these five methods in solving three mechanical examples.

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Abbreviations

\({{\mathcal {D}}}\) :

The design space

n :

The dimension of \({{\mathcal {D}}}\)

\(\mathbf{x } ^{i}\) :

A point \(\in {{\mathcal {D}}}\)

\(y(\mathbf{x } )\) :

The objective function at point \(\mathbf{x } \)

\(\hat{y}(\mathbf{x } )\) :

An approximation of the objective function at point \(\mathbf{x } \) obtained through a metamodel

\(\mathbf X \) :

The matrix of all the calculated points \(\mathbf X =\left[ \begin{array}{ccc} {\mathbf{x } }^{1}&\cdots&{\mathbf{x } }^{p} \end{array} \right] ^{T} \)

\(y^{i}\) :

The objective function at point \(\mathbf{x } ^{i}\)

\(\mathbf Y \) :

The vector of all the objective function \(\mathbf Y =\left[ \begin{array}{ccc} y^{1}&\cdots&y^{p} \end{array} \right] ^{T} \)

\(\bullet _{fcv}\) :

Any of the above in the case of fully converged data

\(\bullet _{pcv}\) :

Any of the above in the case of partially converged data

\(\mathbf{f } (\mathbf{x } )\) :

The regression function. \(\Big (\)If one chooses a mean regression, \(\mathbf{f } (\mathbf{x } )=1\); for a linear regression, \(\mathbf{f } (\mathbf{x } )=\left[ \begin{array}{cccc} 1&x_{1}&\cdots&x_{n} \end{array} \right] ^{T}\Big )\)

\(\varvec{\beta }\) :

The parameters of the regression function. \(\Big (\)For a linear regression, \(\varvec{\beta }=\left[ \begin{array}{ccc} \beta _{1}&\cdots&\beta _{n+1} \end{array} \right] ^{T} \Big )\)

\(\mathbf F = \left[ \begin{array}{ccc}{\mathbf{f } }({\mathbf{x } }^{1})&\cdots&{\mathbf{f } }({\mathbf{x } }^{p})\end{array}\right] ^{T}\) :

The matrix of the regression functions at the calculation points (for universal kriging case)

\(z(\mathbf{x } )\) :

A stochastic function

\(\mathbf Z \) :

The stochastic vector of the calculated approximations \(\mathbf Z =\left[ \begin{array}{ccc} z(\mathbf{x } ^{1})&\cdots&z(\mathbf{x } ^{p}) \end{array} \right] ^{T} \)

\(\mathbf{C } \) :

The covariance matrix

\(\mathbf R \) :

The correlation matrix

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Authors and Affiliations

  1. LMT Cachan (ENS Cachan/CNRS/Université Paris-Saclay), 61 Avenue du Président Wilson, 94230, Cachan, France

    Nicolas Courrier, Pierre-Alain Boucard & Bruno Soulier

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  1. Nicolas Courrier
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  2. Pierre-Alain Boucard
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Correspondence to Nicolas Courrier.

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Courrier, N., Boucard, PA. & Soulier, B. Variable-fidelity modeling of structural analysis of assemblies. J Glob Optim 64, 577–613 (2016). https://doi.org/10.1007/s10898-015-0345-9

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