An improved Bayesian collocation method for steady-state response analysis of structural dynamic systems with large interval uncertainties

doi:10.1016/j.amc.2021.126523

Applied Mathematics and Computation

Volume 411, 15 December 2021, 126523

https://doi.org/10.1016/j.amc.2021.126523 Get rights and content

Highlights

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An improved Bayesian collocation method (IBCM) is developed for structural steady-state response analysis with large interval uncertainties.
•
Sequential Gaussian process surrogate model is applied for interval analysis.
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A bi-directional sampling strategy is proposed to guide to search the extrema.
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A decayed weight function is presented to balance exploration and exploitation in highly nonlinear cases.

Abstract

This paper presents an improved Bayesian collocation method (IBCM) for steady-state response analysis of structural dynamic systems with large interval uncertainties. The main task of interval analysis is to search the extrema of steady-state response within the parametric intervals, so that the response bounds can be obtained. However, interval analysis problems with large parametric uncertainties are usually highly nonlinear. Thus, to improve efficiency and accuracy for nonlinear interval analysis, the IBCM executes a bi-directional global optimization process by using a sequential Gaussian process surrogate model. In this method, IBCM constructs crude surrogate models based on Gaussian process. Then a bi-directional sampling strategy is proposed to guide to search the extrema within the parametric interval. Meanwhile, the surrogate model will also be refined. A decayed weight function is presented to balance exploration and exploitation in highly nonlinear cases. The above process repeats until it converges. The interval of steady-state response can be calculated with low computational cost according to the refined Gaussian process surrogate model. The feasibility and validity of the IBCM are demonstrated by numerical examples and engineering applications.

Introduction

Steady-state response analysis is an indispensable part in structural engineering. However, in practical engineering, uncertainties such as modelling error, fluctuation of surrounding environment and variation of material properties, are everywhere. When it comes to structural dynamic systems, the results of steady-state response analysis may be affected severely. Therefore, uncertainty quantification (UQ) is necessary in order to obtain a more credible result.

Aleatory and epistemic uncertainty are two major categories [1]. Aleatory uncertainty depicts natural randomness, in which probability methods are used. Many distinguished researches have been published in this field [2, 3]. An accurate probability density function (PDF) is necessary to obtain a credible result. However, enough data are required to obtain the correct prior, which, in many cases, is intractable, especially in the field of marine and aerospace engineering. Under the circumstance, epistemic uncertainty, in which non-probabilistic methods are applied, can be used. Non-probabilistic methods includes fuzzy set [4], interval model [5, 6], evidence theory [7, 8] et al. Compared with other methods, interval model is applicable when the data are very scarce and only the bounds of a parameter is available [9].

Many researches have been conducted into the field of interval uncertainty quantification (IUQ) and some algorithms have been presented to solve the problem. The perturbation method (PM) [10], [11], [12], [13] as well as the vertex method (VM) [14], [15], [16] are widely used in previous researches. The efficiency of the PM is high. But its accuracy will deteriorate severely as the interval increase. To alleviate the problem, subinterval perturbation methods (SIPM) are introduced [17, 18]. VM can obtain precise results in monotonic cases. Otherwise, catastrophic results may be derived.

In the case of large parametric interval and high nonlinearity, the response bounds obtained by above methods may not meet the precision requirement. Recently, the surrogate based method (SBM) for interval analysis draws a lot of interest and many researches have been conducted. The main works concentrate on constructing static surrogate model with polynomial chaos expansion (PCE) to represent the structural response. Then, the structural response bounds can be calculated computational-cheaply from the surrogate model. Compared with PM and VM, more computation will be cost when using SBM for interval analysis. Nevertheless, the accuracy will be increased greatly in high nonlinear cases. Wu et al. [19, 20] applied Chebyshev polynomial series in nonlinear interval analysis for dynamic systems. It shows that compared with PM, this method can control overestimation. Arbitrary polynomial chaos is applied by Yin et al. [21, 22] for IUQ in structural-acoustic systems. Besides, Gegenbauer polynomials [23, 24], Jacobi polynomials [25] and Legendre polynomials [26] are also researched in nonlinear IUQ. Apart from PCE, radial basis functions [27] and Kriging surrogate model [28] are also investigated in IUQ. In order to ease the computation, recently, a dimension wise method [29], [30], [31], [32] is proposed. It constructs surrogate model dimension by dimension and search the extrema sequentially. Naturally, the precision of this method is lower while in highly nonlinear cases compared with above methods.

Although SBM overcomes the drawbacks of VM and PM, there are two main limitations when implementing it for IUQ: (1) High order polynomial expansion is needed in highly nonlinear and non-monotonic cases; (2) SBM is a universal approximation method. The surrogate models are constructed base on criteria such as minimum mean squared error (MSE). However, the core task of IUQ is to obtain the response extrema within parametric intervals. Therefore, too much calculation is used in reduce overall MSE, instead of searching the extrema.

Recently, a Bayesian collocation method (BCM) [33] is proposed for IUQ of static analysis. In this paper, the BCM is further developed and an improved Bayesian collocation method (IBCM) is proposed for steady-state response analysis of structural dynamic systems with large interval uncertainties. Problems under the circumstance are usually highly nonlinear and non-monotonous. Compared with the BCM, there are two improvements of the IBCM: (1). The BCM is sample-based method. The samples are updated based on the Gaussian process. The response bounds equal the extrema of chosen samples. The samples should be placed right at the extrema in order to achieve high accuracy. The IBCM is surrogate-based method. The surrogate model is refined by new samples and the response bounds are obtained by the final surrogate model. That means a high accuracy can be achieved even if samples are collocated at the vicinity of the extrema. Therefore, the efficiency of the IBCM can be improved. (2). In IBCM, a bi-directional global optimization algorithm for interval analysis is proposed. A new search function is presented to balance exploration and exploitation in extrema searching.

Section snippets

Steady-state response analysis with interval parameters

The governing equation for steady-state response analysis of structural dynamic systems can be expressed as follows: $(K - ω^{2} M + i ω C) U = F$ where K ∈ R^N × N, M ∈ R^N × N, C ∈ R^N × N denote stiffness matrix, mass matrix and damping matrix of the structural dynamic system, respectively; U ∈ R^N × 1 represents the general structural response; F ∈ R^N × 1 is the external load vector; ω is a scalar which denotes vibration frequency of the structural dynamic system; i stands for unit imaginary number. Eq.(1) is

Interval normalization

For simplicity, U(α; ω) will be simplified as U(α) in this section. In order to apply the IBCM, the interval parameter α^I should be first transformed into a normalized interval parameter $ξ^{I} = [ξ_{1}^{I}, \dots, ξ_{n}^{I}], ξ_{i}^{I} = [- 1, 1]$ : $α_{i}^{I} = α_{i}^{c} + α_{i}^{r} ξ_{i}^{I}$ where $α_{i}^{c} = \frac{{\bar{α}}_{i} + α_{i}}{2}, α_{i}^{r} = \frac{{\bar{α}}_{i} - α_{i}}{2}$

Then U(α) can be replaced by U(ξ), ξ ∈ ξ^I. This step transforms parametric interval at each dimension into the same scale.

Construction of crude surrogate model with Gaussian process

Gaussian process is used here to construct surrogate model to approximate steady-state response. The structural response

Procedure of applying IBCM into interval analysis

The procedure of applying IBCM into IUQ of steady-state response analysis is described by the following steps. The corresponding flowchart is presented in Fig. 3. Among these steps, Step 4-7 are the core steps, and sequential Gaussian process surrogate models are constructed.

Step 1. Choose frequency range [ω₀,ω₁] and frequency interval dω.

Step 2. Choose interval parameters and determine their interval α^I according to experience in practical engineering. Then, normalize α^I to ξ^I.

Step 3. Obtain

Numerical and engineering examples

First, it's worth to clarify how the size of interval uncertainty can influence the IUQ problem. For the problems of steady-state response analysis with uncertainties, in many cases, there exist inherent high nonlinearity. When the uncertainty intervals are small, the problem can be linearized and dealt with methods such as PM with high efficiency. However, when the uncertainty intervals are large, the nonlinear effect should be seriously considered, or the results will be unacceptable. Here is

Conclusion

In this paper, an improved Bayesian collocation method (IBCM) is presented for steady-state response analysis of structural dynamic systems with large interval uncertainties. The IBCM executes a bi-directional global optimization process by using a sequential Gaussian process surrogate model. It constructs crude surrogate models based on Gaussian process. Then a bi-directional sampling strategy is proposed to guide to search the extrema as well as refine the surrogate model. A decayed weight

Acknowledgements

This study was supported by the National Nature Science Foundation of China (Nos. 11872089, 12072006) and the Defense Industrial Technology Development Program (Nos. JCKY2017601B001, JCKY2018601B001, JCKY2019203A003, JCKY2019209C004). The authors also wish to thank the reviewers for useful and constructive comments.

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View full text

An improved Bayesian collocation method for steady-state response analysis of structural dynamic systems with large interval uncertainties

Highlights

Abstract

Introduction

Section snippets

Steady-state response analysis with interval parameters

Interval normalization

Construction of crude surrogate model with Gaussian process

Procedure of applying IBCM into interval analysis

Numerical and engineering examples

Conclusion

Acknowledgements

Comput. Meth. Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Struct.

Comput. Struct.

Comput. Struct.

Comput. Meth. Appl. Mech. Eng.

Aerospace Sci. Technol.

Int. J. Mech. Sci.

Int. J. Solids Struct.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

J. Sound Vib.

Appl. Math. Modell.

Mech. Syst. Sig. Process.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.