Bayesian parameter updating in linear structural dynamics with frequency transformed data using rational surrogate models

Schneider, Felix; Papaioannou, Iason; Straub, Daniel; Winter, Christoph; Müller, Gerhard

doi:10.1016/j.ymssp.2021.108407

“…The parameters are assumed to be independent and lognormally distributed. It should be noted that the assumption of independence does not necessarily hold in general [51], however, we consider it sufficient for the following numerical investigations. The individual mean values and coefficients of variation are given in Tab.…”

Section: Number Of Samples In Ed (Number Of Samples Divided By Number...mentioning

confidence: 99%

Sparse Bayesian Learning for Complex-Valued Rational Approximations

Schneider¹,

Papaioannou²,

Müller³

2022

Preprint

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0

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Surrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties. For models that show a strongly non-linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response. Through applying a rational approximation instead, the approximation error can be efficiently reduced for models whose non-linearity is accurately described through a rational function. Specifically, our aim is to approximate complex-valued models. A common approach to obtain the coefficients in the surrogate is to minimize the samplebased error between model and surrogate in the least-square sense. In order to obtain an accurate representation of the original model and to avoid overfitting, the sample set has be two to three times the number of polynomial terms in the expansion. For models that require a high polynomial degree or are high-dimensional in terms of their input parameters, this number often exceeds the affordable computational cost. To overcome this issue, we apply a sparse Bayesian learning approach to the rational approximation. Through a specific prior distribution structure, sparsity is induced in the coefficients of the surrogate model. This results in a significant decrease of the number of model evaluations required for an accurate representation in comparison to a least-squares approach. In the derivation, we make use of the fact that the posterior distribution of the numerator polynomial coefficients, conditional on the denominator polynomials, can be expressed analytically. The denominator polynomial coefficients as well as the hyperparameters of the problem are then determined through a type-II-maximum likelihood approach. The problem is thereby transferred to a regularized minimization problem. We apply a quasi-Newton gradient-descent algorithm in order to find the optimal denominator coefficients and derive the required gradients through application of CR-calculus. The method is applied to the frequency response functions of an algebraic frame structure model as well as that of an orthotropic plate finite element model.

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“…$$ The parameters are assumed to be independent and lognormally distributed. It should be noted that the assumption of independence does not necessarily hold in general, 51 however, we consider it sufficient for the following numerical investigations. The individual mean values and coefficients of variation are given in Table 3.…”

Section: Numerical Examplesmentioning

confidence: 99%

Sparse Bayesian learning for complex‐valued rational approximations

Schneider

¹

,

Papaioannou

²

,

Müller

³

2022

Numerical Meth Engineering

Self Cite

3

0

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Surrogate models are used to alleviate the computational burden in engineering tasks, which require the repeated evaluation of computationally demanding models of physical systems, such as the efficient propagation of uncertainties.For models that show a strongly non-linear dependence on their input parameters, standard surrogate techniques, such as polynomial chaos expansion, are not sufficient to obtain an accurate representation of the original model response.It has been shown that for models with discontinuities or rational dependencies, for example, frequency response functions of dynamic systems, the use of a rational (Padé) approximation can significantly improve the approximation accuracy. In order to avoid overfitting issues in previously proposed standard least squares approaches, we introduce a sparse Bayesian learning approach to estimate the coefficients of the rational approximation. Therein the linearity in the numerator polynomial coefficients is exploited and the denominator polynomial coefficients as well as the problem hyperparameters are determined through type-II-maximum likelihood estimation. We apply a quasi-Newton gradient-descent algorithm to find the optimal denominator coefficients and derive the required gradients through application of CR-calculus. The method is applied to the frequency response functions of an algebraic frame structure model as well as that of an orthotropic plate finite element model.

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“…But changes in the resonance frequencies are also linked to the thickness change in plates [12]. If the mode shapes are available as data, they offer a precise identification of the elastic material of structures such as plates [13]. In [14], strain mode shapes are used to calculate the sensitivity indices of defects in plates.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of inversion approaches for plates based on guided waves and modal analysis

Hoppe,

Schmid,

Kollofrath

et al. 2024

Discov Appl Sci

1

0

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The identification of defects in plate-like structures has been successfully treated using both local guided ultrasonic waves and global modal quantities. Although there are many papers on these techniques, a lack of comparability between the two methods persists. This makes it difficult for users to identify the most appropriate method for the defect in question. This paper examines the effect of different parameterizations of a circular defect in a square aluminum plate on the system response in a case study. The measured local ultrasound signal from a propagating guided wave makes up the initial data set. The first method uses the entire waveform for the objective function of the optimization problem, while the second considers the velocity of the $$A_0$$ A 0 mode. In the third and fourth methods, the first and second global natural frequencies of the plate modeled with free-free boundary conditions are investigated. The numerical models are validated experimentally through measurements with a laser Doppler vibrometer. This results in the qualitative and quantitative evaluation of the objective functions for all parameter combinations of the defect. The recommendation to sequentially employ modal analysis and then the ultrasound procedure is made for the defect type used in this study. The data gathered on the objective functions suggests that potential joint employment of the natural frequency and ultrasound methods may increase computational efficiency. For a specific case, the methods could also complement each other in terms of challenges such as local minima.

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Bayesian parameter updating in linear structural dynamics with frequency transformed data using rational surrogate models

Cited by 14 publications

References 36 publications

Sparse Bayesian Learning for Complex-Valued Rational Approximations

Sparse Bayesian Learning for Complex-Valued Rational Approximations

Sparse Bayesian learning for complex‐valued rational approximations

Evaluation of inversion approaches for plates based on guided waves and modal analysis

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