Florent Pled - Academia.edu (original) (raw)

Papers by Florent Pled

Research paper thumbnail of On the Stochastic Modeling and Inverse Identification of a Phase-Field Fracture Model in Random Heterogeneous Elastic Materials Exhibiting Isotropic Symmetry Properties

Research paper thumbnail of Statistical inverse problem for the mesoscale model of apparent elasticity properties by training an artificial neural network

HAL (Le Centre pour la Communication Scientifique Directe), Jun 27, 2021

Research paper thumbnail of Machine learning based on neural networks for the inverse identification of probability distributions - Application to the axial transmission technique for the ultrasonic characterization of damaged cortical bone properties

HAL (Le Centre pour la Communication Scientifique Directe), Nov 23, 2020

Research paper thumbnail of Identification of the elastic properties of particle boards and stochastic simulation of wood-based furniture

HAL (Le Centre pour la Communication Scientifique Directe), Jun 11, 2018

Research paper thumbnail of Artificial neural network for a robust identification of apparent elasticity properties at mesoscale with limited experimental measurements

HAL (Le Centre pour la Communication Scientifique Directe), Jan 11, 2021

International audienceFor many materials, as for instance the biological materials, the microstru... more International audienceFor many materials, as for instance the biological materials, the microstructure is complex and highly heterogenous. An efficient approach for constructing the model of such materials consists in modeling their apparent elasticity properties at mesoscale by a tensor-valued random field [1]. Nevertheless, an important challenge is related to the identification of the hyperparameters of such a probabilistic mesoscopic model with limited experimental measurements. Some recent works [2, 3] addressed this problem and an efficient methodology has been proposed which consists in solving a multiscale and multi-objective optimization problem with limited experimental information at both macroscale and mesoscale. The multi-objective cost functions that are used rely on four experimentally measured indicators that are sensitive to the values of the hyperparameters even with a very low number of experimental specimens: good results are obtained with only one specimen. The calculation of the optimal hyperparameters is carried out in using dedicated algorithms but they cannot quantify the probability level of the solution. In order to improve the robust identification of the hyperparameters, we propose to train an artificial neural network with a multiscale computational model and a probabilistic mesoscopic model of the material, for which the output layer corresponds to the values of the hyperparameters and the input layer corresponds to the values of the experimentally measured indicators, the effective elasticity tensor at macroscale and the values of a random vector involved in the probabilistic mesoscopic model. Consequently, for given values of the indicators and the effective elasticity properties, this artificial neural network can be used to propagate the uncertainties on the vector-valued stochastic germ to the optimal hyperparameters and then, the posterior probability density function of the random hyperparameters given the experimental values of the indicators.REFERENCES[1] Soize C., Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilistic Engineering Mechanics (2008) 23(2):307–323.[2] Nguyen M-T., Desceliers C., Soize C., Allain J-M., Gharbi H., Multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations. International Journal for Multiscale Computational Engineering (2015) 13(4):281– 295.[3] Zhang T., Desceliers C., Pled F., Experimental identification of mesoscopic elasticity tensor field for heterogeneous materials with complex microstructure using multiscale experimental imaging measurements. 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Jun 2019, Hersonissos, Crete Island, Greece

Research paper thumbnail of Vers une stratégie robuste et efficace pour le contrôle des calculs par éléments finis en ingénierie mécanique

Ces quatre années de formation à l'ENS Cachan et ces trois années de thèse au LMT représentent un... more Ces quatre années de formation à l'ENS Cachan et ces trois années de thèse au LMT représentent une formidable aventure humaine, riche en évènements et pleine de rencontres enrichissantes sur le plan scientifique. Je tiens tout d'abord à remercier l'ensemble des membres de mon jury pour avoir eu la gentillesse de m'accorder un peu de leur temps, si précieux à l'approche des fêtes de fin d'année : Nicolas Moës, qui m'a fait l'honneur de présider ce jury ; Pedro Díez et Martin Vohralík, pour leur relecture attentive et minutieuse, leur regard critique et leurs commentaires constructifs ; Erwin Stein, dont les questions parfois perfides, mais non moins pertinentes, ont égayé ma curiosité et ma soif de découverte ; enfin, Albert Alarcón, pour avoir apporté la dimension industrielle et la vision pratique lors ma soutenance. J'adresse de sincères remerciements à mon directeur de thèse, Pierre Ladevèze, non seulement pour m'avoir accordé sa confiance sur un sujet aussi passionnant et m'avoir guidé sur cette route semée d'embûches, mais aussi pour m'avoir transmis son amour de la recherche. Je tiens également à témoigner toute ma reconnaissance et mon amitié à Ludovic, une rencontre inestimable à mes yeux. Au-delà de l'encadrant exceptionnel, j'ai découvert un véritable ami aux qualités humaines et morales remarquables. Une pensée toute particulière aux piliers de « l'Équipe Erreur » : Éric et Ludo, pour leur joie de vivre débordante et leur grand coeur ; Valentine et Pierrounet, pour leur admirable gentillesse et toutes ces parties endiablées de « Patran Sliding Contest » ; Bibou, Christian, Gus, Julien (Long shoes) et Sylvain (La graine), pour tous ces moments inoubliables passés à vos côtés ; enfin, Frisou, pour ses soudaines attentions sous la barre des 45˚! Un gros pincement au coeur pour Manouchka, Karin, Camillou, Max, Nico, Pierrounet et PAB, qui ont partagé mes longues journées au Centre de Calcul (CdC), sans oublier Mr Zèbre et Mr Snake ! Un grand merci à toute l'équipe du CdC : Frisou, Arnaud (et son cluster), Pierre et Philippe, dont la disponibilité et la gentillesse font du CdC un lieu unique d'échanges et de convivialité ; Felipe, Hugal et Raphaël, pour leur aide indispensable et le temps consacré à répondre à toutes mes questions d'ordre numérique ! Un petit clin d'oeil à tous les collègues et amis :

Research paper thumbnail of Phase Field Model for Brittle Fracture in Random Heterogeneous Elastic Media: Forward Numerical Simulations and Sensitivity Analysis

HAL (Le Centre pour la Communication Scientifique Directe), Jun 5, 2022

Research paper thumbnail of On the Solution of Statistical Inverse Problems using Machine Learning Methods based on Artificial Neural Networks

HAL (Le Centre pour la Communication Scientifique Directe), Jul 25, 2021

Research paper thumbnail of Cyclic virtual test on wood furniture by Monte Carlo simulation: from compression behavior to connection modeling

Mechanics & Industry, 2019

Prediction of durability of wood product is a major challenge and an important goal for furniture... more Prediction of durability of wood product is a major challenge and an important goal for furniture industry. Numerical simulation based on approximation methods such as the finite element method (FEM) is an efficient and powerful tool to address this challenge while avoiding expensive experimental testing campaigns. Nevertheless, the strong heterogeneity of wood-based materials, the specific geometrical characteristics of woodbased structures (such as furniture that can often be represented as an assembly of beams, plates and/or shells) and the complex nonlinear 3D local behavior near the connections between structural parts may induce some difficulties in the numerical modeling and virtual testing of furniture for robust design purposes. Especially, when cyclic loading occurs, the behavior of junctions in furniture involves a local permanent strain that increases with the number of cycles and that can lead to an important gap potentially affecting the structural integrity of furniture. In this paper, we present an experimental campaign of cyclic compression tests carried out on spruce specimens. Theses specimens are cut out from a bunk bed and loaded under cyclic compression. The cyclic compression loading applied to the specimens leads to an evolution of the permanent strain during cycles that is modeled using a simple law describing the displacement gap as a function of the number of cycles. Considering the strong dispersion in the mechanical properties of wood-based materials and the variabilities induced by the experimental configuration, a stochastic modeling of the gap is proposed by having recourse to the maximum entropy (MaxEnt) principle in order to take into account the random uncertainties on the experimental setup and between the test specimens. The random mechanical response of a complex corner junction in a bunk bed under cyclic loading is then numerically simulated by using a Monte Carlo numerical simulation method as stochastic solver. This provides independent realizations of the random gap evolution (with respect to the number of cycles) in the bunk bed corner, allowing probabilistic quantities of interest related to the random gap, such as firstand second-order statistical moments (mean value, standard deviation) as well as confidence regions (with a given probability level), to be estimated.

Research paper thumbnail of A multiscale method for the propagation of localized uncertainties in linear parabolic equations

HAL (Le Centre pour la Communication Scientifique Directe), Aug 17, 2018

Research paper thumbnail of An artificial neural network-based identification method applied to a random elasto-acoustic wave propagation problem in computational biomechanics

HAL (Le Centre pour la Communication Scientifique Directe), Jun 27, 2021

Research paper thumbnail of Multiscale identification of apparent elastic properties at meso-scale for materials with complex microstructure using experimental imaging measurements

HAL (Le Centre pour la Communication Scientifique Directe), Jun 11, 2018

Research paper thumbnail of Identification of the Mechanical Properties of Particle Boards and Stochastic Simulation of the Behavior of Furniture

HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2017

Research paper thumbnail of Caractérisation expérimentale et simulation stochastique du comportement des meubles à base de panneaux de particules

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Research paper thumbnail of Multiscale coupling approach for solving high-dimensional stochastic problems featuring localized uncertainties and non-linearities

HAL (Le Centre pour la Communication Scientifique Directe), Sep 9, 2015

Research paper thumbnail of A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network

Computer Methods in Applied Mechanics and Engineering, 2021

Research paper thumbnail of On the Design of Artificial Neural Networks for solving Statistical Inverse Problems in Computational Biomechanics

HAL (Le Centre pour la Communication Scientifique Directe), Jul 31, 2022

Research paper thumbnail of Modélisation de la fissuration dans les matériaux bois par la méthode des champs de phase : simulation numérique d'un essai de compression surdes échantillons en bois d'épicéa

HAL (Le Centre pour la Communication Scientifique Directe), Aug 29, 2022

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Modélisation de la fissuration dans les matériaux bois par la méthode des champs de phase : simulation numérique d'un essai de compression sur des échantillons en bois d'épicéa

Research paper thumbnail of Identification of a Phase Field Model for Brittle Fracture in Random Heterogeneous Elastic Media

HAL (Le Centre pour la Communication Scientifique Directe), Jul 31, 2022

Research paper thumbnail of On the Solution of Statistical Inverse Problems using Machine Learning Methods based on Artificial Neural Networks

International audienceThis work adresses the solution of a statistical inverse problem in computa... more International audienceThis work adresses the solution of a statistical inverse problem in computational elastodynamics using machine learning based on artificial neural networks (ANNs). The stochastic computational model (SCM) corresponds to a simplified random elasto-acoustic multilayer model of a biological system that is representative of the axial transmission technique for the ultrasonic characterization of cortical bone properties from experimental velocity measurements. The three-layer biological system consists of a random heterogeneous damaged/weaken elastic solid layer (cortical bone layer) sandwiched between two deterministic homogeneous acoustic fluid layers (soft tissues and marrow bone layers) and excited by an acoustic line source [1]. Such SCM is parameterized by two geometrical parameters, corresponding to the thicknesses of the "healthy" and "damaged" elastic solid parts, a dispersion parameter controlling the level of statistical fluctuations o...

Research paper thumbnail of On the Stochastic Modeling and Inverse Identification of a Phase-Field Fracture Model in Random Heterogeneous Elastic Materials Exhibiting Isotropic Symmetry Properties

Research paper thumbnail of Statistical inverse problem for the mesoscale model of apparent elasticity properties by training an artificial neural network

HAL (Le Centre pour la Communication Scientifique Directe), Jun 27, 2021

Research paper thumbnail of Machine learning based on neural networks for the inverse identification of probability distributions - Application to the axial transmission technique for the ultrasonic characterization of damaged cortical bone properties

HAL (Le Centre pour la Communication Scientifique Directe), Nov 23, 2020

Research paper thumbnail of Identification of the elastic properties of particle boards and stochastic simulation of wood-based furniture

HAL (Le Centre pour la Communication Scientifique Directe), Jun 11, 2018

Research paper thumbnail of Artificial neural network for a robust identification of apparent elasticity properties at mesoscale with limited experimental measurements

HAL (Le Centre pour la Communication Scientifique Directe), Jan 11, 2021

International audienceFor many materials, as for instance the biological materials, the microstru... more International audienceFor many materials, as for instance the biological materials, the microstructure is complex and highly heterogenous. An efficient approach for constructing the model of such materials consists in modeling their apparent elasticity properties at mesoscale by a tensor-valued random field [1]. Nevertheless, an important challenge is related to the identification of the hyperparameters of such a probabilistic mesoscopic model with limited experimental measurements. Some recent works [2, 3] addressed this problem and an efficient methodology has been proposed which consists in solving a multiscale and multi-objective optimization problem with limited experimental information at both macroscale and mesoscale. The multi-objective cost functions that are used rely on four experimentally measured indicators that are sensitive to the values of the hyperparameters even with a very low number of experimental specimens: good results are obtained with only one specimen. The calculation of the optimal hyperparameters is carried out in using dedicated algorithms but they cannot quantify the probability level of the solution. In order to improve the robust identification of the hyperparameters, we propose to train an artificial neural network with a multiscale computational model and a probabilistic mesoscopic model of the material, for which the output layer corresponds to the values of the hyperparameters and the input layer corresponds to the values of the experimentally measured indicators, the effective elasticity tensor at macroscale and the values of a random vector involved in the probabilistic mesoscopic model. Consequently, for given values of the indicators and the effective elasticity properties, this artificial neural network can be used to propagate the uncertainties on the vector-valued stochastic germ to the optimal hyperparameters and then, the posterior probability density function of the random hyperparameters given the experimental values of the indicators.REFERENCES[1] Soize C., Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size. Probabilistic Engineering Mechanics (2008) 23(2):307–323.[2] Nguyen M-T., Desceliers C., Soize C., Allain J-M., Gharbi H., Multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations. International Journal for Multiscale Computational Engineering (2015) 13(4):281– 295.[3] Zhang T., Desceliers C., Pled F., Experimental identification of mesoscopic elasticity tensor field for heterogeneous materials with complex microstructure using multiscale experimental imaging measurements. 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP 2019), Jun 2019, Hersonissos, Crete Island, Greece

Research paper thumbnail of Vers une stratégie robuste et efficace pour le contrôle des calculs par éléments finis en ingénierie mécanique

Ces quatre années de formation à l'ENS Cachan et ces trois années de thèse au LMT représentent un... more Ces quatre années de formation à l'ENS Cachan et ces trois années de thèse au LMT représentent une formidable aventure humaine, riche en évènements et pleine de rencontres enrichissantes sur le plan scientifique. Je tiens tout d'abord à remercier l'ensemble des membres de mon jury pour avoir eu la gentillesse de m'accorder un peu de leur temps, si précieux à l'approche des fêtes de fin d'année : Nicolas Moës, qui m'a fait l'honneur de présider ce jury ; Pedro Díez et Martin Vohralík, pour leur relecture attentive et minutieuse, leur regard critique et leurs commentaires constructifs ; Erwin Stein, dont les questions parfois perfides, mais non moins pertinentes, ont égayé ma curiosité et ma soif de découverte ; enfin, Albert Alarcón, pour avoir apporté la dimension industrielle et la vision pratique lors ma soutenance. J'adresse de sincères remerciements à mon directeur de thèse, Pierre Ladevèze, non seulement pour m'avoir accordé sa confiance sur un sujet aussi passionnant et m'avoir guidé sur cette route semée d'embûches, mais aussi pour m'avoir transmis son amour de la recherche. Je tiens également à témoigner toute ma reconnaissance et mon amitié à Ludovic, une rencontre inestimable à mes yeux. Au-delà de l'encadrant exceptionnel, j'ai découvert un véritable ami aux qualités humaines et morales remarquables. Une pensée toute particulière aux piliers de « l'Équipe Erreur » : Éric et Ludo, pour leur joie de vivre débordante et leur grand coeur ; Valentine et Pierrounet, pour leur admirable gentillesse et toutes ces parties endiablées de « Patran Sliding Contest » ; Bibou, Christian, Gus, Julien (Long shoes) et Sylvain (La graine), pour tous ces moments inoubliables passés à vos côtés ; enfin, Frisou, pour ses soudaines attentions sous la barre des 45˚! Un gros pincement au coeur pour Manouchka, Karin, Camillou, Max, Nico, Pierrounet et PAB, qui ont partagé mes longues journées au Centre de Calcul (CdC), sans oublier Mr Zèbre et Mr Snake ! Un grand merci à toute l'équipe du CdC : Frisou, Arnaud (et son cluster), Pierre et Philippe, dont la disponibilité et la gentillesse font du CdC un lieu unique d'échanges et de convivialité ; Felipe, Hugal et Raphaël, pour leur aide indispensable et le temps consacré à répondre à toutes mes questions d'ordre numérique ! Un petit clin d'oeil à tous les collègues et amis :

Research paper thumbnail of Phase Field Model for Brittle Fracture in Random Heterogeneous Elastic Media: Forward Numerical Simulations and Sensitivity Analysis

HAL (Le Centre pour la Communication Scientifique Directe), Jun 5, 2022

Research paper thumbnail of On the Solution of Statistical Inverse Problems using Machine Learning Methods based on Artificial Neural Networks

HAL (Le Centre pour la Communication Scientifique Directe), Jul 25, 2021

Research paper thumbnail of Cyclic virtual test on wood furniture by Monte Carlo simulation: from compression behavior to connection modeling

Mechanics & Industry, 2019

Prediction of durability of wood product is a major challenge and an important goal for furniture... more Prediction of durability of wood product is a major challenge and an important goal for furniture industry. Numerical simulation based on approximation methods such as the finite element method (FEM) is an efficient and powerful tool to address this challenge while avoiding expensive experimental testing campaigns. Nevertheless, the strong heterogeneity of wood-based materials, the specific geometrical characteristics of woodbased structures (such as furniture that can often be represented as an assembly of beams, plates and/or shells) and the complex nonlinear 3D local behavior near the connections between structural parts may induce some difficulties in the numerical modeling and virtual testing of furniture for robust design purposes. Especially, when cyclic loading occurs, the behavior of junctions in furniture involves a local permanent strain that increases with the number of cycles and that can lead to an important gap potentially affecting the structural integrity of furniture. In this paper, we present an experimental campaign of cyclic compression tests carried out on spruce specimens. Theses specimens are cut out from a bunk bed and loaded under cyclic compression. The cyclic compression loading applied to the specimens leads to an evolution of the permanent strain during cycles that is modeled using a simple law describing the displacement gap as a function of the number of cycles. Considering the strong dispersion in the mechanical properties of wood-based materials and the variabilities induced by the experimental configuration, a stochastic modeling of the gap is proposed by having recourse to the maximum entropy (MaxEnt) principle in order to take into account the random uncertainties on the experimental setup and between the test specimens. The random mechanical response of a complex corner junction in a bunk bed under cyclic loading is then numerically simulated by using a Monte Carlo numerical simulation method as stochastic solver. This provides independent realizations of the random gap evolution (with respect to the number of cycles) in the bunk bed corner, allowing probabilistic quantities of interest related to the random gap, such as firstand second-order statistical moments (mean value, standard deviation) as well as confidence regions (with a given probability level), to be estimated.

Research paper thumbnail of A multiscale method for the propagation of localized uncertainties in linear parabolic equations

HAL (Le Centre pour la Communication Scientifique Directe), Aug 17, 2018

Research paper thumbnail of An artificial neural network-based identification method applied to a random elasto-acoustic wave propagation problem in computational biomechanics

HAL (Le Centre pour la Communication Scientifique Directe), Jun 27, 2021

Research paper thumbnail of Multiscale identification of apparent elastic properties at meso-scale for materials with complex microstructure using experimental imaging measurements

HAL (Le Centre pour la Communication Scientifique Directe), Jun 11, 2018

Research paper thumbnail of Identification of the Mechanical Properties of Particle Boards and Stochastic Simulation of the Behavior of Furniture

HAL (Le Centre pour la Communication Scientifique Directe), Jun 7, 2017

Research paper thumbnail of Caractérisation expérimentale et simulation stochastique du comportement des meubles à base de panneaux de particules

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Research paper thumbnail of Multiscale coupling approach for solving high-dimensional stochastic problems featuring localized uncertainties and non-linearities

HAL (Le Centre pour la Communication Scientifique Directe), Sep 9, 2015

Research paper thumbnail of A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network

Computer Methods in Applied Mechanics and Engineering, 2021

Research paper thumbnail of On the Design of Artificial Neural Networks for solving Statistical Inverse Problems in Computational Biomechanics

HAL (Le Centre pour la Communication Scientifique Directe), Jul 31, 2022

Research paper thumbnail of Modélisation de la fissuration dans les matériaux bois par la méthode des champs de phase : simulation numérique d'un essai de compression surdes échantillons en bois d'épicéa

HAL (Le Centre pour la Communication Scientifique Directe), Aug 29, 2022

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Modélisation de la fissuration dans les matériaux bois par la méthode des champs de phase : simulation numérique d'un essai de compression sur des échantillons en bois d'épicéa

Research paper thumbnail of Identification of a Phase Field Model for Brittle Fracture in Random Heterogeneous Elastic Media

HAL (Le Centre pour la Communication Scientifique Directe), Jul 31, 2022

Research paper thumbnail of On the Solution of Statistical Inverse Problems using Machine Learning Methods based on Artificial Neural Networks

International audienceThis work adresses the solution of a statistical inverse problem in computa... more International audienceThis work adresses the solution of a statistical inverse problem in computational elastodynamics using machine learning based on artificial neural networks (ANNs). The stochastic computational model (SCM) corresponds to a simplified random elasto-acoustic multilayer model of a biological system that is representative of the axial transmission technique for the ultrasonic characterization of cortical bone properties from experimental velocity measurements. The three-layer biological system consists of a random heterogeneous damaged/weaken elastic solid layer (cortical bone layer) sandwiched between two deterministic homogeneous acoustic fluid layers (soft tissues and marrow bone layers) and excited by an acoustic line source [1]. Such SCM is parameterized by two geometrical parameters, corresponding to the thicknesses of the "healthy" and "damaged" elastic solid parts, a dispersion parameter controlling the level of statistical fluctuations o...