Uncertainty quantiﬁcation and global sensitivity analysis of seismic fragility curves using kriging Clément Gauchy12 Cyril Feau1 Josselin Garnier2

2025-04-30 0 0 773.05KB 27 页 10玖币

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Uncertainty quantiﬁcation and global sensitivity

analysis of seismic fragility curves using kriging

Clément Gauchy1,2, Cyril Feau1, Josselin Garnier2

1Université Paris-Saclay, CEA, Service d’Études Mécaniques et Thermiques, 91191, Gif-sur-Yvette,

France

2CMAP, École Polytechnique, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France

Abstract

Seismic fragility curves have been introduced as key components of Seismic Proba-

bilistic Risk Assessment studies. They express the probability of failure of mechanical

structures conditional to a seismic intensity measure and must take into account the in-

herent uncertainties in such studies, the so-called epistemic uncertainties (i.e. coming

from the uncertainty on the mechanical parameters of the structure) and the aleatory

uncertainties (i.e. coming from the randomness of the seismic ground motions). For

simulation-based approaches we propose a methodology to build and calibrate a Gaus-

sian process surrogate model to estimate a family of non-parametric seismic fragility

curves for a mechanical structure by propagating both the surrogate model uncertainty

and the epistemic ones. Gaussian processes have indeed the main advantage to propose

both a predictor and an assessment of the uncertainty of its predictions. In addition, we

extend this methodology to sensitivity analysis. Global sensitivity indices such as aggre-

gated Sobol indices and kernel-based indices are proposed to know how the uncertainty

on the seismic fragility curves is apportioned according to each uncertain mechanical

parameter. This comprehensive Uncertainty Quantiﬁcation framework is ﬁnally applied

to an industrial test case consisting in a part of a piping system of a Pressurized Water

Reactor.

1 Introduction

In the 1980s, a probabilistic framework was developed to evaluate the mean annual prob-

ability of occurrence of severe damage on structures caused by seismic ground motions,

coined Seismic Probabilistic Risk Assessment (SPRA) [1, 2, 3]. One of the key elements of

this approach is the fragility curve. Such a curve expresses the probability of failure (or un-

desirable outcome) of a structure conditional to a seismic intensity measure and must take

into account the different sources of uncertainties that inevitably come into play in this type

of study and which are classiﬁed into two categories, namely: the epistemic and the aleatory

uncertainties. According to [4], distinguishing between these two types of uncertainties is a

pragmatic way of distinguishing which uncertainties engineers can reduce and which can-

not, allowing for information based design choices. For that reason, in practice, it is often

assumed that epistemic uncertainties are sources of uncertainty that can be reduced in the

short term with a reasonable budget, while aleatory uncertainties are devolved to sources

of natural hazards due to physical phenomena. Thus, a seismic fragility curve is not strictly

speaking a single curve (i.e. mean curve), but a family of fragility curves which reﬂects the

arXiv:2210.06266v1 [stat.AP] 12 Oct 2022

uncertainty on the mean seismic fragility curve due to a certain lack of knowledge of the

structure of interest and its environment (i.e including soil–structure interaction, etc.).

Since the 1980s several techniques have been developed to estimate such curves, in the

sense of mean fragility curves most of the time. When little data is available, whether experi-

mental, from post-earthquake feedback or from numerical calculations, a classic approach to

circumvent estimation difﬁculties is to use a parametric model of the fragility curve, such as

the lognormal model historically introduced in [1] (see e.g. [5, 6, 7, 8, 9, 10]). As the validity

of parametric models is questionable, non-parametric estimation techniques have also been

developed, such as kernel smoothing [8, 9] as well as other methodologies [10, 11]. Most of

these strategies are compared in [8, 9, 12] and [8] presents their advantages and disadvan-

tages. Beyond these methods, techniques based on statistical and machine learning on the

mechanical response of the structure can also be used, including: linear or generalized linear

regression [8], classiﬁcation - based techniques [13, 14], polynomial chaos expansion [15, 16]

and artiﬁcial neural networks [17, 14]. Most of these techniques take advantage of the rise

of computational power to allow estimations based on numerical simulations. They make

it possible to reduce the computational burden which remains high because such estima-

tions require a large number of numerical simulations to be precise. Nevertheless, despite

all these techniques, one of the main challenges that persists is the estimation at a lower

numerical cost (i.e. with few calls for computer codes) of non-parametric fragility curves

taking into account the two types of uncertainties.

The objective of this work is to propose a methodology that meets these requirements in

a numerical simulation based framework. As we focus on approaches based on numerical

simulations that rely on real seismic signal databases enriched by means of a seismic signal

generator that well encompasses their temporal and spectral non-stationarities [18], we as-

sume that there is no epistemic uncertainty affecting the excitation which only represents the

aleatory uncertainty of the problem. Consequently, in our settings, epistemic uncertainties

only concern the mechanical parameters of the structures of interest. The physics-based ap-

proaches developed as part of Performance-Based Earthquake Engineering (PBEE) address

this problem [19]. However, they are not suitable when the use of detailed ﬁnite element

simulations is required, in order to take into account all the speciﬁcities of the structures

of interest as it can be the case nowadays for the seismic safety studies in nuclear industry

[5, 20, 21]. So, in this paper, our approach relies on the use of surrogate models of the com-

puter codes, also referenced as metamodels, based on Gaussian process regression. This

framework corresponds to a data driven approximation of the input/output relationship

of a numerical computer code based on a set of experiments (e.g. computer model calls)

at different values of the input parameters with a Gaussian process assumption on the nu-

merical computer code output values [22]. Gaussian process regression, or kriging in the

ﬁeld of geostatistics, has gained in popularity because of its predictive capabilities and its

ability to quantify the surrogate model uncertainty [23]. Gaussian process surrogates have

already been used for various applications in engineering, such as seismic risk assessment

[24, 25], thermohydraulics for safety studies of nuclear power plants [26] or hydrogeology

for radionucleide transport in groundwater [27]. In this work, we propose a methodology

to build and calibrate a Gaussian process surrogate model to estimate a family of seismic

fragility curves for mechanical structures - deﬁned here as seismic fragility quantile curves

- by propagating both the surrogate model uncertainty and the epistemic ones.

In such a context, the use of Sensitivity Analysis (SA) techniques is essential for engi-

neers. Indeed, according to [28], SA goal is to investigate how the uncertainty of the model

output can be apportioned to different sources of uncertainties of the model input. SA tech-

niques are also performed according to a range of conceptual objectives, coined as SA set-

tings, deﬁned in [28, 29]. These objectives are prioritizing the most inﬂuential inputs, thus a

possible reduction of uncertainty affecting these inputs may lead to the largest reduction of

the output uncertainty, and identifying the noninﬂuential inputs which then could be ﬁxed

at a given value without any loss of information about the model output. SA techniques

are classically applied on the model output, however it is possible to extend their ﬁelds of

application to goal-oriented quantities of interest such as seismic fragility curves. In our

case, SA techniques will help to determine which mechanical parameter uncertainties most

inﬂuence the seismic fragility curve uncertainty. Note that SA on the mechanical param-

eters of the structures is peculiarly challenging, due to the strong inﬂuence of the seismic

ground motions on their responses. However, even if the uncertainty coming from me-

chanical parameters is smaller that the one coming from the seismic ground motion, SA on

these parameters is crucial to propose information-based choices to engineers and to discuss

quantitatively the different possible designs of the mechanical structure studied, especially

in the context of nuclear industry where safety constraints imposed by regulatory agencies

are very high. In [30] CDF-based importance measures are used to address the problem of

ranking of uncertain model parameters in seismic fragility analysis. To go further, we pro-

pose to use Global Sensitivity Analysis (GSA) methods [31, 32] which take into account the

overall uncertainty ranges of the parameters. We present global sensitivity indices applied

in the particular context of seismic fragility curves as a quantity of interest. We are ﬁrst in-

terested in the estimation of the Sobol indices [33, 34] adapted to seismic fragility curves.

We also focus on recently studied global sensitivity indices based on kernel methods [35],

the βk-indices, which seem adapted to functional quantities of interest like fragility curves.

However, because the estimation of global sensitivity indices requires a large number of

simulations that is intractable using complex numerical simulations, the Gaussian process

surrogate is also used to estimate the global sensitivity indices on the seismic fragility curves.

Moreover, as in [36], the Gaussian process surrogate uncertainty will be propagated into the

global sensitivity indices estimates.

This paper, which presents a comprehensive Uncertainty Quantiﬁcation (UQ) framework

for seismic fragility curves of mechanical structures, taking into account metamodel and

mechanical parameter uncertainties, is organized as follows: Section 2 is devoted to the esti-

mation of seismic fragility curves using Gaussian process regression, Section 3 concerns the

deﬁnition of global sensitivity indices tailored for seismic fragility curves, the aggregated

Sobol indices and the βkindices. Section 4 presents an illustration of the methodology de-

veloped in this article to an industrial test case consisting in a mock-up of a piping system

of a French Pressurized Water Reactor (PWR).

2 Estimation of seismic fragility curves using Gaussian Pro-

cess surrogates

As discussed in the introduction, the sources of uncertainties are in this work divided into

two categories, the aleatory and epistemic uncertainties.

Aleatory uncertainties are related to the stochastic ground motions. To account for them,

we use a synthetic generator of ground motions to enrich a set of real seismic signals selected

in a database for a given magnitude (M) - source-to-site distance (R) scenario. This generator

is based on a ﬁltered modulated white-noise process [18]. It is common in SPRA studies

to sum up the seismic hazard by a so-called Intensity Measure (IM), which is the variable

against which the fragility curves are conditioned. This is often a scalar value obtained from

the seismic signals such as the Peak Ground Acceleration (PGA) or the Pseudo Spectral

Acceleration (PSA). In [37], the author recalls the main assumptions according to which it is

possible to reduce the seismic hazard to the IM values (see also [38]). In the following, we

denote by athe scalar value corresponding to the IM.

Epistemic uncertainties are related to the mechanical properties of the model of the struc-

ture. These parameters are denoted by the vector xPXĂRd. Furthermore, we denote by z

the Engineering Demand Parameter (EDP) of interest, which can be the peak inter story drift

for a multistoreys building or a rotation angle of a speciﬁc elbow of a piping system of a nu-

clear power plant. A very common statistical model between the EDP and the combination

of structural and seismic uncertainty is the log-normal model:

logpzpa, xqq “ gpa, xq ` εpa, xq,(1)

where xis the vector of the mechanical properties of the structure, ais the IM, gpa, xqis the

regression function, and ε„Np0, σεpa, xq2qis a centered Gaussian noise. Note that this log-

normal assumption for the EDP distribution is not necessary for the proposed methodology,

any functional transformation of z(such as Box-Cox transformation [39]) is possible as long

as it is normally distributed after this transformation. For the sake of notation simplicity, we

denote ypa, xq “ logpzpa, xqq. The fragility curve is then deﬁned by:

Ψpa, xq “ PpzpA, Xq ą C|A“a, X“xq,(2)

where Ais the real-valued random variable of the seismic intensity measure and Xthe ran-

dom vector of the mechanical parameters of the structure. Ccorresponds to a deterministic

threshold of acceptable robustness of the structure. Substituting the model Equation (1) into

Equation (2) we get the form of the fragility curve

Ψpa, xq “ Φˆgpa, xq ´ logpCq

σεpa, xq˙,(3)

where Φis the cumulative distribution function (cdf) of the standard Gaussian distribution.

In this framework the numerical simulations of the structure are made by a computer model.

The computer model is considered of high-ﬁdelity with respect to the mechanical problem

studied and therefore it may involve a chain of multi-physics simulation codes (involving

ﬁnite elements or ﬁnite volumes, computational ﬂuid dynamics...) and thus it is considered

as a black-box. This means that the different strategies described throughout this paper are

non-intrusive with respect to this black-box computer model.

2.1 Gaussian process surrogate with homoskedastic nugget noise

In this section, we suppose that the regression function gis a realization of a Gaussian pro-

cess Gand the Gaussian noise εpa, xqis homoskedastic and will be denoted by εsuch that

ε„Np0, σ2

εq. We thus deﬁne the random observation by:

Ypa, xq “ Gpa, xq ` ε . (4)

Remark in Equation (4) that thanks to the Gaussian noise assumption on the noise ε, the

random observations Ypa, xqis also a Gaussian process. We make the assumption that Gis a

zero mean Gaussian process with a tensorized anisotropic stationary Matérn 5{2covariance

function parametrized by its intensity σand its lengthscales pρiq1ďiďd`1. This covariance

function is motivated by is popularity in the machine learning community as it covers a

large number of applications. Note also that with such a covariance function the Gaussian

process Gis two times mean-square differentiable, which is a good compromise between

the regularity of the regression function gand the potential sparsity of the data.

Given an experimental design made of nsimulations of the mechanical computer model,

we obtain the dataset Dn“ ppai,xiq, ypai,xiqq1ďiďn. By the maximum likelihood method, we

can provide estimates for the unknown covariance function hyperparameters σ, pρiq1ďiďd`1

and also the Gaussian noise variance σε(see [27] for a practical implementation of the

method). The dataset Dncan then be used to derive the conditional distribution of the Gaus-

sian process Yfor any pa, xq:

pYpa, xq|Dnq „ N`mnpa, xq, σnpa, xq2˘,(5)

where mnpa, xqand σnpa, xq2are obtained from the kriging equations [23, p.16 - 17]. In the

same fashion, we can derive the conditional distribution of the Gaussian process Gon the

regression function for any pa, xq:

pGpa, xq|Dnq „ N`mnpa, xq, snpa, xq2˘,(6)

where σnpa, xq2“snpa, xq2`σ2

ε. The fragility curve is then obtained by replacing the com-

puter model output yby a Gaussian process Ynwhich follows the distribution of the Gaus-

sian process Yconditioned to Dndetailed in Equation (5). Hence for any vector pa, xqwe

derive the estimator of the fragility curve Ψp1q:

Ψp1qpa, xq “ PpYnpa, xq ą logpCq|A“a, X“xq.(7)

We can then use the distribution of Ynto estimate the fragility curve:

Ψp1qpa, xq “ Φˆmnpa, xq ´ logpCq

σnpa, xq˙.(8)

Moreover, the Gaussian process surrogate allows us to propagate the surrogate model un-

certainty into the fragility curve, thanks to the conditional distribution of the regression

function pGpa, xq|Dnq. We introduce Gna Gaussian process with the same distribution as the

Gaussian process pG|Dnq, then the fragility curve tainted by the uncertainty of the Gaussian

process surrogate writes:

Ψp2qpa, xq “ ΦˆGnpa, xq ´ logpCq

σε˙,(9)

where Gnpa, xq „ Npmnpa, xq, snpa, xq2q. Remark that Ψp1qis the mean of Ψp2qwith respect

to the distribution of Gn. In order to estimate the distribution of Ψp2q, we simulate Prealiza-

tions pGn,ppa, xqq1ďpďPwith the distribution of pGpa, xq|Dnqto estimate a sample of Ψp2q:

Ψp2q

ppa, xq “ ΦˆGn,ppa, xq ´ logpCq

σε˙.(10)

However, some mechanical structures have nonlinear behavior that can inﬂuence the

local variability of the log-EDP ypa, xq. Thus, a varying nugget with respect to pa, xqis nec-

essary to capture the form of ypa, xq. This comes with a cost in terms of dataset size, due

to the increase in the numbers of parameters to estimate. We deal with this case in the next

section.

2.2 Gaussian process surrogate with heteroskedastic nugget noise

In this section, the log-EDP ypa, xqis now supposed to follow the statistical model described

by Equation (1) where εpa, xq „ Np0, σεpa, xq2q. There are two ways of estimating σεpa, xq

described in [25]. The ﬁrst one, called Stochastic Kriging (SK), is to consider several replica-

tions at the same value of the input parameters pa, xqand to provide an empirical estimation

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UncertaintyquanticationandglobalsensitivityanalysisofseismicfragilitycurvesusingkrigingClémentGauchy1;2,CyrilFeau1,JosselinGarnier21UniversitéParis-Saclay,CEA,Serviced'ÉtudesMécaniquesetThermiques,91191,Gif-sur-Yvette,France2CMAP,ÉcolePolytechnique,InstitutPolytechniquedeParis,91128PalaiseauCedex,F...

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Uncertainty quantiﬁcation and global sensitivity analysis of seismic fragility curves using kriging Clément Gauchy12 Cyril Feau1 Josselin Garnier2

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