Reference prior for Bayesian estimation of seismic fragility curves

One of the crucial quantities of probabilistic seismic risk assessment studies is the fragility curve, which represents the probability of failure of a mechanical structure conditional to a scalar measure derived from the seismic ground motion. Estimating such curves is a difficult task because for most structures of interest, few data are available, whether they come from complex numerical simulations or experimental campaigns. For this reason, a wide range of the methods of the literature rely on a parametric log-normal model. Bayesian approaches allow for efficient learning of the model parameters. However, for small data set sizes, the choice of the prior distribution has a non-negligible influence on the posterior distribution, and therefore on any resulting estimate. We propose a thorough study of this parametric Bayesian estimation problem when the data are binary (i.e. data indicate the state of the structure, failure or non-failure). Using the reference prior theory as a support, we suggest an objective approach for the prior choice to simulate a posteriori fragility curves. This approach leads to the Jeffreys prior and we prove that this prior depends only of the ground motion characteristics, making its calculation suitable for any equipment in an industrial installation subjected to the same seismic hazard. Our proposal is theoretically and numerically compared to those classically proposed in the literature by considering three different case studies. The results show the robustness and advantages of the Jeffreys prior in terms of regularization (no degenerate estimations) and stability (no outliers of the parameters) for fragility curves estimation

Finding Single Copy Genes Out of Sequenced Genomes for Multilocus Phylogenetics in Non-Model Fungi

Historically, fungal multigene phylogenies have been reconstructed based on a small number of commonly used genes. The availability of complete fungal genomes has given rise to a new wave of model organisms that provide large number of genes potentially useful for building robust gene genealogies. Unfortunately, cross-utilization of these resources to study phylogenetic relationships in the vast majority of non-model fungi (i.e. “orphan” species) remains an unexamined question. To address this problem, we developed a method coupled with a program named “PHYLORPH” (PHYLogenetic markers for ORPHans). The method screens fungal genomic databases (107 fungal genomes fully sequenced) for single copy genes that might be easily transferable and well suited for studies at low taxonomic levels (for example, in species complexes) in non-model fungal species. To maximize the chance to target genes with informative regions, PHYLORPH displays a graphical evaluation system based on the estimation of nucleotide divergence relative to substitution type. The usefulness of this approach was tested by developing markers in four non-model groups of fungal pathogens. For each pathogen considered, 7 to 40% of the 10–15 best candidate genes proposed by PHYLORPH yielded sequencing success. Levels of polymorphism of these genes were compared with those obtained for some genes traditionally used to build fungal phylogenies (e.g. nuclear rDNA, β-tubulin, γ-actin, Elongation factor EF-1α). These genes were ranked among the best-performing ones and resolved accurately taxa relationships in each of the four non-model groups of fungi considered. We envision that PHYLORPH will constitute a useful tool for obtaining new and accurate phylogenetic markers to resolve relationships between closely related non-model fungal species

Probabilistic response of an elastic perfectly plastic oscillator under Gaussian white noise

International audienceThe response of an elastic perfectly plastic oscillator under zero mean Gaussian white noise excitation is studied in this paper. Considering the work of Vanmarcke and Veneziano, a closed form expression of the mean largest maximum of the plastic drift is given assuming that the plastic process is equivalent to a Brownian motion. In order to better describe the plastic drift a probabilistic model is proposed for the yield increments which occur in clumps. To estimate the input parameters of this model, three theoretical models, based on numerical computations of some relevant integrals, are presented. Alternatively, these parameters can be estimated, more conveniently, according to the results obtained by Ditlevsen with the Slepian model approach. The results of numerical simulations show a quite satisfactory agreement with theoretical predictions

An empirical study on plastic deformations of an elasto-plastic problem with noise

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Some aspects of floor spectra of 1DOF nonlinear primary structures

International audienceIn this paper the influence of the nonlinear behaviour of the primary structure on floor spectra is investigated by means of simple models. The general trends of floor spectra for different types of nonlinear behaviour of one degree of freedom (1DOF) primary structure are shown and we point out their common futures and their differences. A special attention is given to the cases of elastoplastic and nonlinear elastic behaviours and methods to determine an equivalent linear oscillator are proposed. The properties (frequency and damping) of this equivalent linear oscillator are quite different from the properties of equivalent linear oscillators commonly considered in practice. In particular, in the case of elastoplastic behaviour, there is no frequency shift and damping is smaller than assumed by other methods commonly used. In the case of nonlinear elastic behaviour, the concept of an equivalent frequency which is a random variable is used. Finally, a design floor spectrum of primary structures, exhibiting energy dissipating nonlinear behaviour is proposed

Influence of Input Motion's Control Point Location in Nonlinear SSI Analysis of Equipment Seismic Fragilities: Case Study on the Kashiwazaki-Kariwa NPP

The aim of this case study is to evaluate the influence of input seismic motion control point on the fragility curves of some nuclear power plant’s equipment considering (1) a strong soil–structure interaction problem and (2) the variability of the input seismic signals. To this end, a current engineering methodology was implemented for computation efficiency, based on a simplified model representing the largely embedded Unit 7 reactor building of the Kashiwazaki-Kariwa NPP (Japan). Seismic signals were generated with NGA GMPE according to the July 2007 NCOE scenario. In the process, soil non-linearity caused by each seismic signal was taken into account using the equivalent linear method. A study on the uncertainty propagation through the simplified soil–structure system is also presented. It allowed the implementation of an empirical sensitivity analysis to confirm the main results, which suggest that defining the control point of the input motion at the soil surface as prescribed in the French nuclear practice is not appropriate and may lead to biased outcomes when performing non-linear soil–structure fragility analysis. Instead, it should be defined at the “outcropping bedrock” level

Estimation de courbes de fragilité sismique par planification séquentielle d'expériences

Seismic probabilistic risk assessment studies consist in evaluating the probabilities of failure of mechanical structures when submitted to seismic ground motions. These studies are often concentrated on fragility curve estimation. The fragility curve is the probability of failure of the structure conditionally to a seismic intensity measure. However, its estimation requires computer experiments involving huge computation time. Such a computational burden makes crude Monte Carlo methods untractable, fragility curves estimation must then be economical in terms of sample size. We propose an algorithm of sequential planning of experiments by supposing a Gaussian process prior on the output of the mechanical computer model.Les études probabilistes de sûreté sismique consistent à évaluer les probabilités de défaillance de structures mécaniques soumises à des excitations sismiques. Ces études nécessitent l'estimation de courbes de fragilité sismique, qui sont la probabilité de défaillance de la structure conditionnellement à une mesure d'intensité du signal sismique. Cependant, leur estimation requiert de nombreuses expériences numériques qui peuvent etre très coûteuses en temps de calcul, ce qui rend l'estimation par une méthode Monte Carlo inappropriée. Nous proposons dans ce papier de construire un algorithme de planification séquentielle d'expériences en supposant un a priori de processus Gaussien sur la réponse du code de calcul mécanique

Asymptotic formulae for the risk of failure related to an elasto-plastic problem with noise

The risk of failure of mechanical structures under random forcing is an important concern in earthquake engineering. For a class of simple structures that can be modeled by an elasto-plastic oscillator, the risk of failure can be expressed in terms of the probability that, on a certain interval of time, the plastic deformation goes beyond a threshold related to a failure zone. In this note, asymptotic formulae for the risk of failure of an elasto-perfectly-plastic oscillator excited by a white noise are proposed. Our approach exploits the long cycle (repeating pattern) property of the aforementioned oscillator as introduced in A. Bensoussan, L. Mertz, S.C.P. Yam, Long cycle behaviour of the plastic deformation of an elasto-perfectly-plastic oscillator with noise, C. R. Acad. Sci. Paris Ser. I, 2012. We show that asymptotically the plastic deformation behaves like a Wiener process for which analytical formulae are available. Our result is a consequence of the Anscombe–Donsker Invariance Principle. Numerical experiments and comments are carried out

Importance sampling based active learning for parametric seismic fragility curve estimation

The key elements of seismic probabilistic risk assessment studies are the fragility curves which express the probabilities of failure of structures conditional to a seismic intensity measure. A multitude of procedures is currently available to estimate these curves. For modeling-based approaches which may involve complex and expensive numerical models, the main challenge is to optimize the calls to the numerical codes to reduce the estimation costs. Adaptive techniques can be used for this purpose, but in doing so, taking into account the uncertainties of the estimates (via confidence intervals or ellipsoids related to the size of the samples used) is an arduous task because the samples are no longer independent and possibly not identically distributed. The main contribution of this work is to deal with this question in a mathematical and rigorous way. To this end, we propose and implement an active learning methodology based on adaptive importance sampling for parametric estimations of fragility curves. We prove some theoretical properties (consistency and asymptotic normality) for the estimator of interest. Moreover, we give a convergence criterion in order to use asymptotic confidence ellipsoids. Finally, the performances of the methodology are evaluated on analytical and industrial test cases of increasing complexity

Importance sampling based active learning for parametric seismic fragility curve estimation

The key elements of seismic probabilistic risk assessment studies are the fragility curves which express the probabilities of failure of structures conditional to a seismic intensity measure. A multitude of procedures is currently available to estimate these curves. For modeling-based approaches which may involve complex and expensive numerical models, the main challenge is to optimize the calls to the numerical codes to reduce the estimation costs. Adaptive techniques can be used for this purpose, but in doing so, taking into account the uncertainties of the estimates (via confidence intervals or ellipsoids related to the size of the samples used) is an arduous task because the samples are no longer independent and possibly not identically distributed. The main contribution of this work is to deal with this question in a mathematical and rigorous way. To this end, we propose and implement an active learning methodology based on adaptive importance sampling for parametric estimations of fragility curves. We prove some theoretical properties (consistency and asymptotic normality) for the estimator of interest. Moreover, we give a convergence criterion in order to use asymptotic confidence ellipsoids. Finally, the performances of the methodology are evaluated on analytical and industrial test cases of increasing complexity