A Copula-Based and Monte Carlo Sampling Approach for Structural Dynamics Model Updating with Interval Uncertainty
A Copula-Based and Monte Carlo Sampling Approach for Structural Dynamics Model Updating with Interval Uncertainty
Xueqian Chen,1,2 Zhanpeng Shen,1,2 and Xin’en Liu1,2
1Institute of Systems Engineering, China Academy of Engineering Physics (CAEP), Mianyang Sichuan 621999, China
2Shock and Vibration of Engineering Materials and Structures Key Laboratory of Sichuan Province, Mianyang Sichuan 621999, China
Correspondence should be addressed to Xueqian Chen; moc.uhos@721ddqxc
Received 23 March 2018; Revised 29 May 2018; Accepted 4 June 2018; Published 9 July 2018
Academic Editor: Aly Mousaad Aly
Copyright © 2018 Xueqian Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
As the uncertainty is widely existent in the engineering structure, it is necessary to study the finite element (FE) modeling and updating in consideration of the uncertainty. A FE model updating approach in structural dynamics with interval uncertain parameters is proposed in this work. Firstly, the mathematical relationship between the updating parameters and the output interesting qualities is created based on the copula approach and the vast samples of inputs and outputs are obtained by the Monte Carlo (MC) sampling technology according to the copula model. Secondly, the samples of updating parameters are rechosen by combining the copula model and the experiment intervals of the interesting qualities. Next, 95% confidence intervals of updating parameters are calculated by the nonparameter kernel density estimation (KDE) approach, which is regarded as the intervals of updating parameters. Lastly, the proposed approach is validated in a two degree-of-freedom mass-spring system, simple plates, and the transport mirror system. The updating results evidently demonstrate the feasibility and reliability of this approach.
1. Introduction
Finite element (FE) models that numerically solve various engineering problems can aid virtual prototyping, reduce product development cycle, and cut down the cost of performing the physical tests. However, the reliability of the simulation results by finite element modeling is not always guaranteed since FE models are the approximations of real world phenomena based on various assumptions. These assumptions may detract from the quality and accuracy of simulation results. In order to improve the accuracy of FE simulating to serve the structural design better, the FE model updating techniques are needed to develop. In the past few decades, various kinds of FE model updating approaches have been widely investigated based on the actually observed behaviors of the system. Additionally, experimental modal and vibration data are often used in FE model updating in the field of structural dynamics [1–6].
In most model updating approaches, the simulations are usually deterministic where each of the updating parameters is considered to have one “true” value and the purpose of the updating procedure is to provide a deterministic estimation. In reality, there are always uncertainties in nominally identical structures, such as the structural parameter uncertainty (physical material properties, geometric parameters), the assembly joints uncertainty, and the experiment uncertainty (measurement noise, modal identification techniques, etc.). As a result, the FE model updating approaches with uncertainty have received great attentions recently. Studies have shown that the simulation results are more reliable when the uncertainties are taken into account [7], suggesting that it is necessary to consider the uncertainties during modeling and simulating [8].
In FE model updating approaches with uncertainty, the updated parameters are no longer deterministic and are described as random variables. Usually, the FE model updating approaches with uncertainty can be classified into two major categories: probabilistic and nonprobabilistic approaches. In the earlier works, a probabilistic approach proposed incorporated the measurement noise into model updating [9]. Subsequently, Bayesian statistical frameworks were adopted to estimate the posterior probabilities of uncertain parameters [10–12]. However, high computational costs due to a large amount of samples required for a satisfactory estimation greatly restrain the applications of Bayesian updating approaches. As a result, surrogate models such as the Gaussian process model with the perturbation approaches and sensitivity analysis approaches have been employed in stochastic model updating to improve the efficiency [13–16]. Though, the surrogate model approaches own the superiority of computational efficiency over Monte Carlo (MC) based methods. Nevertheless, the prerequisite of small uncertainties, together with the Gaussian distribution assumption, also limits the applications to complex problems. Mo (...truncated)