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Stochastic inverse method to identify parameter random fields in a structure

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Abstract

The parameters in a structure such as geometric and material properties are generally uncertain due to manufacturing tolerance, wear, fatigue and material irregularity. Such parameters are random fields because the uncertain properties vary along the spatial domain of a structure. Since the parameter uncertainties in a structure result in the uncertainty of the structural dynamic behavior, they need to be identified accurately for structural analysis or design. In order to identify the random fields of geometric parameters, the parameters can be measured directly using a 3-dimensional coordinate measuring machine. However, it is often very expensive to measure them directly. It is even impossible to directly measure some parameters such as density and Young’s modulus. For that case, the parameter random fields should be identified from measurable response data samples. In this paper, a stochastic inverse method to identify parameter random fields in a structure using modal data is proposed. The proposed method consists of the following three steps: (i) obtaining realizations of the parameter random field from modal data samples by solving an optimization problem, (ii) obtaining the deterministic terms in the Karhunen-Loève expansion by solving an eigenvalue problem and (iii) estimating the distributions of random variables in the Karhunen-Loève expansion using a maximum likelihood estimation method with kernel density.

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Abbreviations

w(x, θ):

Random field

\( \overline{w}\left(\mathbf{x}\right) \) :

Mean function of w(x, θ)

α(x, θ):

Zero mean random field

D :

Spatial domain of a structure

x :

Spatial variable vector

θ :

Random event

C(x 1, x 2):

Covariance function of w(x, θ)

λ i :

Eigenvalue of C(x 1, x 2)

f i (x):

Eigenfunction of C(x 1, x 2)

ξ i (θ):

Random variable

a i :

Deterministic coefficient vector of a polynomial chaos expansion

ψ k (η i ):

Hermite polynomial basis function

η i :

Gaussian random variable

ω k n,j and q k j :

Natural frequency and mode vector of a structural system obtained from modal testing

\( {\widehat{\omega}}_{n,j} \) and \( {\widehat{\mathbf{q}}}_j \) :

Natural frequency and mode vector of a structural system obtained from modal analysis

[M] and [K]:

Mass and stiffness matrix of a simulation model

K(⋅):

Kernel function

h :

Smoothing parameter for kernel density

n s :

Number of realizations for kernel density

ϕ i (x):

Bending mode function of a beam

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Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2015R1A2A2A01003422).

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Authors and Affiliations

  1. School of Mechanical Engineering, Hanyang University, 222 Wangshimni-ro, Seongdong-gu, Seoul, 04763, South Korea

    Chan Kyu Choi & Hong Hee Yoo

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  1. Chan Kyu Choi
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  2. Hong Hee Yoo
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Correspondence to Hong Hee Yoo.

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Choi, C.K., Yoo, H.H. Stochastic inverse method to identify parameter random fields in a structure. Struct Multidisc Optim 54, 1557–1571 (2016). https://doi.org/10.1007/s00158-016-1534-y

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  • DOI: https://doi.org/10.1007/s00158-016-1534-y

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