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Optimization-based inverse analysis for membership function identification in fuzzy steady-state heat transfer problem

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Abstract

Based on the optimization design technology and fuzzy uncertainty theory, this paper proposes a novel inverse analysis method for membership function identification in steady-state heat transfer problem with fuzzy modeling parameters. The system subjective uncertainties associated with expert opinions are quantified as fuzzy parameters, which can be converted into interval variables by level-cut strategy. By means of the errors between measured and calculated temperature data, the parameter identification process is executed as a nested-loop optimization model. To avoid the considerable computational cost caused by nested-loop, an interval vertex method is presented to replace the inner-loop for predicting the temperature response bounds. The eventual membership functions of input fuzzy parameters are constructed by using the fuzzy decomposition theorem. Comparing results with traditional Monte Carlo method, a numerical example about 3D air cooling system is provided to verify the feasibility of proposed method for fuzzy parameter identification in engineering.

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Acknowledgements

This work was supported by the Alexander von Humboldt Foundation, 111 Project (No. B07009), and National Natural Science Foundation of PR China (No.11432002).

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Authors and Affiliations

  1. Institute of Scientific Computing, Technische Universität Braunschweig, 38106, Braunschweig, Germany

    Chong Wang & Hermann G. Matthies

  2. Institute of Solid Mechanics, Beihang University, Beijing, 100191, People’s Republic of China

    Zhiping Qiu

Authors
  1. Chong Wang
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  2. Hermann G. Matthies
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  3. Zhiping Qiu
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Corresponding author

Correspondence to Chong Wang.

Appendix: Fuzzy set theory

Appendix: Fuzzy set theory

Definition 1 (fuzzy sets)

Given the definition domain R, a fuzzy set X can be defined as

$$ X=\left\{\left.\left(x,{\mu}_X(x)\right)\right|x\in R,{\mu}_X(x)\in \left[0,1\right]\right\} $$

where μ X (x) is the membership function representing the degree a sample x belongs to the set R.

Definition 2 (level-cut strategy)

Given a fuzzy set X, for any λ ∈ [0, 1] the normal set

$$ {X}_{\lambda }=\left\{\left.x\right|x\in X,{\mu}_X(x)\ge \lambda \right\} $$

is defined as the λ-cut set, where λ is called the cut level. Usually, the cut sets are considered as intervals of confidence, since in case of convex fuzzy sets, they are closed intervals associated with a gradation of confidence between [0,1].

Definition 3 (fuzzy decomposition theorem)

Based on the λ-cut sets, any fuzzy set X defined in the domain R can be visually expressed by the normal sets

$$ X=\underset{\lambda \in \left[0,1\right]}{\cup}\left(\lambda \times {X}_{\lambda}\right) $$

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Wang, C., Matthies, H.G. & Qiu, Z. Optimization-based inverse analysis for membership function identification in fuzzy steady-state heat transfer problem. Struct Multidisc Optim 57, 1495–1505 (2018). https://doi.org/10.1007/s00158-017-1821-2

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  • DOI: https://doi.org/10.1007/s00158-017-1821-2

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