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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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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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
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
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
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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