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Research Papers

Design Optimization Toward Alleviating Forced Response Variation in Cyclically Periodic Structure Using Gaussian Process

[+] Author and Article Information
K. Zhou, A. Hegde, P. Cao

Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, Unit 3139,
Storrs, CT 06269

J. Tang

Professor
Department of Mechanical Engineering,
University of Connecticut,
191 Auditorium Road, Unit 3139,
Storrs, CT 06269
e-mail: jtang@engr.uconn.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received January 20, 2016; final manuscript received October 3, 2016; published online December 7, 2016. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 139(1), 011017 (Dec 07, 2016) (14 pages) Paper No: VIB-16-1038; doi: 10.1115/1.4035107 History: Received January 20, 2016; Revised October 03, 2016

Cyclically periodic structures, such as bladed disk assemblies in turbomachinery, are widely used in engineering systems. It is well known that small uncertainties exist among their substructures, which in certain situations may cause drastic change in the dynamic responses, a phenomenon known as vibration localization. Previous studies have suggested that the introduction of small, prespecified design modification, i.e., intentional mistuning, may alleviate vibration localization and reduce response variation. However, there has been no systematic methodology to facilitate the optimal design of intentional mistuning. The most significant challenge is the computational cost involved. The finite-element model of a bladed disk usually requires a very large number of degrees-of-freedom (DOFs). When uncertainties occur in a cyclically periodic structure, the response may no longer be considered as simple perturbation to that of the nominal structure. In this research, a suite of interrelated algorithms is proposed to enable the efficient design optimization of cyclically periodic structures toward alleviating their forced response variation. We first integrate model order reduction with a perturbation scheme to reduce the scale of analysis of a single run. Then, as the core of the new methodology, we incorporate Gaussian process (GP) emulation to conduct the rapid sampling-based evaluation of the design objective, which is a metric of response variation under uncertainties, in the parametric space. The optimal design modification can thus be directly identified to minimize the response variation. The efficiency and effectiveness of the proposed methodology are demonstrated by systematic case studies.

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Figures

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Fig. 1

(a) Schematic diagram of bladed disk assembly and its substructure (N substructures in total) and (b) finite-element model of the substructure (n DOFs)

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Fig. 2

Natural frequencies of first modal family: (a) validation of order reduction technique by comparison and (b) frequency bandwidth demonstration (zoom-in view)

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Fig. 3

Illustration of design optimization: the response variance is reduced while the response mean does not increase (:original; – –with design modification)

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Fig. 4

Conventional brute-force Monte Carlo-based design flow

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Fig. 5

Complete Monte Carlo analysis result: (a) design objective (variance) distribution and (b) mean summation (constraint) distribution

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Fig. 6

Enhanced design flow: (a) with single Gaussain process emulator and (b) with nested Gaussian process emulator

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Fig. 7

Gaussian process emulation result (590 training data under single GP): (a) design objective variance) distribution and (b) mean summation (constraint) distribution

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Fig. 8

Comparison of design objective evaluation versus design variable A in projection view: (a) complete Monte Carlo analysis and (b) Gaussian process emulator (590 training data under single GP)

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Fig. 9

Comparison of design objective evaluation versus design variable τ in projection view: (a) complete Monte Carlo analysis and (b) Gaussian process emulator (590 training data under single GP)

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Fig. 10

Error distribution of Gaussian process emulator (590 training data under single GP)

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Fig. 11

(a) Design objective evaluation versus design variable A in projection view and (b) design objective evaluation versus design variable τ in projection view (120 training data under single GP)

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Fig. 12

Error distribution of Gaussian process emulator (120 training data under single GP)

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Fig. 13

Error distribution of Gaussian process emulator (51 training data under single GP)

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Fig. 14

Prediction of response distribution under uncertainties at one specific excitation frequency (——: benchmark distribution by MC;·– –·–: inferred distribution by GP; *: training data)

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Fig. 15

Statistical distribution of response: (a) complete Monte Carlo analysis and (b) Gaussian process emulator (67 training data points)

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Fig. 16

(a) Comparison of prediction accuracy and (b) comparison of computational efficiency

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Fig. 17

Scenario: A (0–4%) (a) design objective evaluation versus design variable A in projection view and (b) design objective evaluation versus design variable τ in projection view (590 training data under single GP)

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Fig. 18

Scenario: A (0–5%) (a) design objective evaluation versus design variable A in projection view and (b) design objective evaluation versus design variable τ in projection view (590 training data under single GP)

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