Automatic differentiation of mechanical codes : application to sensitivity analysis of viscoelastic sandwich sheets with respect to modeling parameters

En ingénierie, pour mieux comprendre le comportement mécanique d'une structure soumise à une certaine perturbation des paramètres de conception, on procède souvent à une analyse de sensibilité. Celle-ci fournit des informations quantitatives et qualitatives sur le comportement du modèle étudié et offre un accès aux gradients utilisables dans ces méthodes d'identification et d'optimisation. Dans cette thèse, nous démontrons que ces informations peuvent être obtenues à coût de développement faible en appliquant un outil de Différentiation Automatique (DA) au code informatique qui implémente le modèle. Nous adaptons la technique DA à la méthode asymptotique numérique, dans sa version Diamant, pour le calcul de la sensibilité des solutions numériques de problèmes non-linéaires discrétisés par la méthode des éléments finis. Nous discutons de manière générique à la fois les aspects théoriques et l'implémentation de plusieurs algorithmes écrits en Matlab. Les applications concernent des poutres et des plaques sandwich dans les cas statiques et dynamique (vibration libre). Les sensibilités sont calculées par rapport aux paramètres géométriques, mécanique et par rapport à des matrices de rigidité élémentaires. La généralité de nos développements permet de prendre en compte plusieurs lois viscoélastiques sans effort supplémentaire. Trois types de modèles viscoélastiques sont étudiés : module complexe constant, faible amortissement et fort amortissement. Comparée à l'approximation par différences finis souvent utilisée en mécanique, notre approche fournit des résultats plus précis pour la sensibilité de la réponse d'une structure lorsque les paramètres de conception sont perturbés. Elle permet aussi de réduire le temps de calculIn engineering, for a better understanding of the mechanical behavior of a structure submitted to some perturbation of the modeling parameters, one often proceed to a sensitivity analysis. This provides quantitative and qualitative information on the behavior of the model under study and gives access to gradients that may be used in identification and optimization methods. In this thesis, we demonstrate that this information may be obtained at a low development effort by applying an Automatic Differentiation (AD) tool to the computer code that implements the model. We adapt the AD techniques to the Asymptotic Numerical Method (ANM), in its Diamant version for sensitivity computations of numerical solutions of nonlinear problems discretized through a finite element method. We discuss in a generic manner both the theoretical aspects and the implementation of several algorithms written in Matlab. Applications are concerned with sandwich beams and sandwich plates in both the static and dynamic (free vibration) cases. Sensitivities are computed with respect to geometric and mechanical parameters, and with respect to elementary stiffness matrix. The generality of our developments allows to take into account several viscoelastic laws with no additional effort. Three kinds of viscoelastic models are studied: constant complex modulus, low damping and higher damping. In comparison with the finite difference approximation often used in mechanics, our approach provides more accurate results for the sensitivity of the structure response to a perturbation of the modeling parameters. It also allows a reduction of the computation effor

Différentiation automatique de codes mécaniques : application à l'analyse de sensibilité des tôles sandwich aux paramètres de modélisation

In engineering, for a better understanding of the mechanical behavior of a structure submitted to some perturbation of the modeling parameters, one often proceed to a sensitivity analysis. This provides quantitative and qualitative information on the behavior of the model under study and gives access to gradients that may be used in identification and optimization methods. In this thesis, we demonstrate that this information may be obtained at a low development effort by applying an Automatic Differentiation (AD) tool to the computer code that implements the model. We adapt the AD techniques to the Asymptotic Numerical Method (ANM), in its Diamant version for sensitivity computations of numerical solutions of nonlinear problems discretized through a finite element method. We discuss in a generic manner both the theoretical aspects and the implementation of several algorithms written in Matlab. Applications are concerned with sandwich beams and sandwich plates in both the static and dynamic (free vibration) cases. Sensitivities are computed with respect to geometric and mechanical parameters, and with respect to elementary stiffness matrix. The generality of our developments allows to take into account several viscoelastic laws with no additional effort. Three kinds of viscoelastic models are studied: constant complex modulus, low damping and higher damping. In comparison with the finite difference approximation often used in mechanics, our approach provides more accurate results for the sensitivity of the structure response to a perturbation of the modeling parameters. It also allows a reduction of the computation effortEn ingénierie, pour mieux comprendre le comportement mécanique d'une structure soumise à une certaine perturbation des paramètres de conception, on procède souvent à une analyse de sensibilité. Celle-ci fournit des informations quantitatives et qualitatives sur le comportement du modèle étudié et offre un accès aux gradients utilisables dans ces méthodes d'identification et d'optimisation. Dans cette thèse, nous démontrons que ces informations peuvent être obtenues à coût de développement faible en appliquant un outil de Différentiation Automatique (DA) au code informatique qui implémente le modèle. Nous adaptons la technique DA à la méthode asymptotique numérique, dans sa version Diamant, pour le calcul de la sensibilité des solutions numériques de problèmes non-linéaires discrétisés par la méthode des éléments finis. Nous discutons de manière générique à la fois les aspects théoriques et l'implémentation de plusieurs algorithmes écrits en Matlab. Les applications concernent des poutres et des plaques sandwich dans les cas statiques et dynamique (vibration libre). Les sensibilités sont calculées par rapport aux paramètres géométriques, mécanique et par rapport à des matrices de rigidité élémentaires. La généralité de nos développements permet de prendre en compte plusieurs lois viscoélastiques sans effort supplémentaire. Trois types de modèles viscoélastiques sont étudiés : module complexe constant, faible amortissement et fort amortissement. Comparée à l'approximation par différences finis souvent utilisée en mécanique, notre approche fournit des résultats plus précis pour la sensibilité de la réponse d'une structure lorsque les paramètres de conception sont perturbés. Elle permet aussi de réduire le temps de calcu

Sensitivity computations in higher order continuation methods

International audienceSensitivity analysis is a key tool in the study of the relationships between the input parameters of a model and the output solution. Although sensitivity analysis is extensively addressed in the literature, little attention has been brought to the methodological aspects of the sensitivity of nonlinear parametric solutions computed through a continuation technique. This paper proposes four combinations of sensitivity analysis with continuation and homotopy methods, including sensitivity analysis along solution branches or at a particular point. Theoretical aspects are discussed in the higher order continuation framework Diamant. The sensitivity methods are applied to a thermal ignition problem and some free vibration problems. Remarkable eigenvalue maps are produced for the complex nonlinear eigenvalue problems

Hyper-reduced predictions for lifetime assessment of elasto-plastic structures

International audienceFinite element (FE) elasto-plastic or elasto-viscoplastic simulations of complex components can still be prohibitive for lifetime predictions. There is a need for fast estimation methods of plasticity in a given region of interest, where a crack could be initiated. Some rules are already available for fast predictions of elasto-plastic stress and strain, by using elastic simulations. Furthermore, as shown by the Herbland’s model, inclusion theory can be incorporated in simplified rules to improve their accuracy. Recent advances in model reduction methods for nonlinear mechanical models give access to fast elasto-plastic or elasto-viscoplastic predictions having both accuracy and computational complexity in between usual FE predictions and these simplified rules. Hyper-reduction performs quite well in the simplification of elasto-plastic or elasto-viscoplastic models. Similarly to Herbland’s model, we show in this paper a first attempt to improve hyper-reduced models by the recourse to a virtual inclusion placed in the region of interest. In the proposed numerical example, a finite element model involving 5000 degrees of freedom is reduced to 12 variables. The mesh is also reduced to 550 elements over a total of 900 elements for the original mesh. The approximation error on the predicted plastic strains and stresses is lower than 3 % and the computational time is reduced up to a factor 5

Différentiation automatique de codes mécaniques (application à l'analyse de sensibilité des tôles sandwich aux paramètres de modélisation)

En ingénierie, pour mieux comprendre le comportement mécanique d'une structure soumise à une certaine perturbation des paramètres de conception, on procède souvent à une analyse de sensibilité. Celle-ci fournit des informations quantitatives et qualitatives sur le comportement du modèle étudié et offre un accès aux gradients utilisables dans ces méthodes d'identification et d'optimisation. Dans cette thèse, nous démontrons que ces informations peuvent être obtenues à coût de développement faible en appliquant un outil de Différentiation Automatique (DA) au code informatique qui implémente le modèle. Nous adaptons la technique DA à la méthode asymptotique numérique, dans sa version Diamant, pour le calcul de la sensibilité des solutions numériques de problèmes non-linéaires discrétisés par la méthode des éléments finis. Nous discutons de manière générique à la fois les aspects théoriques et l'implémentation de plusieurs algorithmes écrits en Matlab. Les applications concernent des poutres et des plaques sandwich dans les cas statiques et dynamique (vibration libre). Les sensibilités sont calculées par rapport aux paramètres géométriques, mécanique et par rapport à des matrices de rigidité élémentaires. La généralité de nos développements permet de prendre en compte plusieurs lois viscoélastiques sans effort supplémentaire. Trois types de modèles viscoélastiques sont étudiés : module complexe constant, faible amortissement et fort amortissement. Comparée à l'approximation par différences finis souvent utilisée en mécanique, notre approche fournit des résultats plus précis pour la sensibilité de la réponse d'une structure lorsque les paramètres de conception sont perturbés. Elle permet aussi de réduire le temps de calculIn engineering, for a better understanding of the mechanical behavior of a structure submitted to some perturbation of the modeling parameters, one often proceed to a sensitivity analysis. This provides quantitative and qualitative information on the behavior of the model under study and gives access to gradients that may be used in identification and optimization methods. In this thesis, we demonstrate that this information may be obtained at a low development effort by applying an Automatic Differentiation (AD) tool to the computer code that implements the model. We adapt the AD techniques to the Asymptotic Numerical Method (ANM), in its Diamant version for sensitivity computations of numerical solutions of nonlinear problems discretized through a finite element method. We discuss in a generic manner both the theoretical aspects and the implementation of several algorithms written in Matlab. Applications are concerned with sandwich beams and sandwich plates in both the static and dynamic (free vibration) cases. Sensitivities are computed with respect to geometric and mechanical parameters, and with respect to elementary stiffness matrix. The generality of our developments allows to take into account several viscoelastic laws with no additional effort. Three kinds of viscoelastic models are studied: constant complex modulus, low damping and higher damping. In comparison with the finite difference approximation often used in mechanics, our approach provides more accurate results for the sensitivity of the structure response to a perturbation of the modeling parameters. It also allows a reduction of the computation effortMETZ-SCD (574632105) / SudocNANCY1-Bib. numérique (543959902) / SudocNANCY2-Bibliotheque electronique (543959901) / SudocNANCY-INPL-Bib. électronique (545479901) / SudocSudocFranceF

DIAMANLAB – AN INTERACTIVE TAYLOR-BASED CONTINUATION TOOL IN MATLAB

Abstract. With the interactive continuation tool Diamanlab, solution branches of a parametric nonlinear problem are computed as sets of Taylor-based solutions stored in checkpoints. Theoretical aspects and implementation are generic, taking advantage of the efficient higher-order asymptotic numerical method in its Diamant form that interprets the generic nonlinear problem as a sequence of linear ones, of Automatic Differentiation (AD) for Taylor coefficient computations, of object-oriented programming and graphical user interface capabilities of MATLAB. The implementation involves four classes devoted to the interactive management of the continuation, to the manipulation of a generic system of nonlinear equations, to the checkpoint management and to higher-order AD, respectively. In practice, any analytical nonlinear system of equation may be implemented in a natural way as a subclass of the generic system class, then solved in an easy manner using the graphical user interface. A benchmark of classical nonlinear problems is provided to serve as a basis for the implementation userdefined problems. Diamanlab usage and bifurcation detection are discussed on the Brusselator problem whose solution involves three interconnected loops. Additional user-defined graphics are presented for the Bratu problem. Asymptotic numerical method, automatic differentiation, MANLAB, Diamant, graphical user interface 1

Eigenmode sensitivity of damped sandwich structures

International audienceThe modeling of the linear free vibration of a sandwich structure including viscoelastic layers yields a complex nonlinear eigenvalue problem. In this paper, the sensitivity of eigensolutions is computed using a homotopy-based asymptotic numerical method, then a first-order automatic differentiation. The generality of the proposed method enables us to consider any analytical frequency-dependent viscoelastic law in the modeling and the sensitivity computation. Its application potential is demonstrated by computing the sensitivity of eigenmodes, eigenfrequencies and modal loss factors of sandwich beams and plates to various perturbations

Numerical modeling of triboelectric separation: application to vegetal powders

International audienceElectrostatic separation processes rely on the triboelectric properties of particles to sort them in an electrical field. One major benefit of these processes is that they produce no effluent and allow separation for various materials as for example separation of plastic grains from waste or for the removal of unburned carbon from fly ash. More recently the triboelectric separation was used for vegetal powders using and has proved to be well adapted for the purification of targeted compounds as Peeling and Gluten [1] or lignin in wheat straw [2].To better understand the triboelectric separation, we developed a numerical model based on the Discrete Element Method (DEM). This model takes into account 1) the triboelectric charging during particles contacts and collisions, 2) the exchange of electric charges and 3) the collective effect of the electric fields generated by all particles and by electrodes. In this study, we first investigate the dynamics of charging of a vibrated packing of particles. The simulated evolution compares well with experimental data from the literature. This evolution allows to determine the so-called work function which is an intrinsic physical property characterizing charge transmission during collisions. Assuming that the effect of the air results only in a drag force applied to the center of mass of the grains, we were able to take into account the aerodynamic transport of particles. Finally, the real geometry of the experimental setup is considered and we highlight the capabilities of the model to simulate complex features as electrodes clogging, flow eddies resulting in dead zones or particles agglomeration