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In addition to the numerical challenges of simulating the explosive dispersal of particles, validation of the simulation is often plagued with poor knowledge of the experimental conditions. The level of experimental detail required for validation is beyond what is usually included in the literature. This presentation proposes the use of forensic uncertainty quantification (UQ) to investigate validation-quality experiments to discover possible sources of uncertainty that may have been missed in initial design of experiments or under-reported. The current experience of the authors has found that by making an analogy to crime scene investigation when looking at validation experiments, valuable insights may be gained. One examines all the data and documentation provided by the validation experimentalists, corroborates evidence, and quantifies large sources of uncertainty a posteriori with empirical measurements. In addition, it is proposed that forensic UQ may benefit from an independent investigator to help remove possible implicit biases and increases the likelihood of discovering unrecognized uncertainty. Forensic UQ concepts will be discussed and then applied to a set of validation experiments performed at Eglin Air Force Base.

Colloque avec actes et comité de lecture. Internationale.; International audience; La méthode d'identification bayesienne présente l'avantage de pouvoir tenir compte de différentes sources d'incertitude présentes dans le problème et de quantifier l'incertitude avec laquelle les propriétés sont identifiées aussi bien en termes de variances que de corrélations. Son application à un problème d'identification de propriétés élastiques orthotropes à partir de mesures de champs de déplacements par Moiré interférométrique est présentée dans ce papier.

National audience; L'identification bayesienne est une des approches permettant de prendre en considération des erreurs de mesure ainsi que de modélisation. De plus, cette méthode identifie une densité de probabilité, permettant ainsi d'avoir des informations aussi bien sur la variance que sur la corrélation des propriétés identifiées. Cette procédure peut cependant être très coûteuse en temps de calcul. Pour réduire ce temps de calcul nous proposons ici une approche bayesienne basée sur la méthode des surfaces de réponse. L'approche est appliquée au problème de l'identification de constantes élastiques orthotropes à partir des mesures des fréquences propres d'une plaque vibrante libre. La procédure développée prend en compte des erreurs de mesures, de l'incertitude sur les autres paramètres d'entrée du modèle (dimensions de la plaque, densité), ainsi que des erreurs systématiques. La densité de probabilité jointe des quatre constantes élastiques du pli est identifiée et caractérisée par ses valeurs moyennes et la matrice de variance covariance. Nous trouvons que certaines paramètres, comme le coefficient de Poisson, sont identifiées avec une bien plus grande incertitude que d'autres et qu'une corrélation significative existe entre les différentes propriétés identifiées.

International audience; Some applications such as identification or Monte Carlo based uncertainty quantification often require simple analytical formulas that are fast to evaluate. Approximate closed-form solutions for the natural frequencies of free orthotropic plates have been developed and have a wide range of applicability, but they are not very accurate. For moderate ranges of plate parameters, such as those needed for identifying material properties from vibration tests, good accuracy can be achieved by using response surface methodology combined with dimensional analysis. This paper first demonstrates that such a response surface can be much more accurate then the approximate analytical solutions even for relatively large ranges of the material and geometric parameters. Second it compares the accuracy of the elastic constants identified from experiments using the two approximations. It demonstrates the advantage of high fidelity approximations in vibration-based elastic constants identification. For a least squares identification approach, the approximate analytical solution led to physically implausible properties, while the high-fidelity response surface approximation obtained reasonable estimates.

Structural health monitoring provides sensor data that follow fatigue induced damage growth in service. This information may in turn be used to improve the characterization of the material properties that govern damage propagation for the structure being monitored. These properties are often widely distributed between nominally identical structures because of differences in manufacturing and aging. The improved accuracy in damage growth characteristics allows more accurate prediction of the remaining useful life (RUL) of the structural component. In this paper, a probabilistic approach using Bayesian statistics is employed to progressively narrow the uncertainty in damage growth parameters in spite of variability and error in sensor measurements. Starting from an initial, wide distribution of damage parameters that are obtained from coupon tests, the distribution is progressively narrowed using the damage growth between consecutive measurements. Detailed discussions on how to construct the likelihood function under given variability of sensor data and how to update the distribution are presented. The approach is applied to crack growth in fuselage panels due to cycles of pressurization and depressurization. It is shown that the proposed method rapidly converges to the accurate damage parameters when the initial damage size is 20mm and the variability in sensor data is 1mm. It is observed that the distribution narrows down rapidly when the damage grows fast, while slows down when the damage grows slowly. This property works in favor because more accurate information will be obtained when the damage is dangerous. Using the identified damage parameters, the RUL is predicted with 95% confidence in order to obtain conservative prediction. The proposed approach may have the potential of turning aircraft into flying fatigue laboratories.

Adjoint sensitivity calculation of stress and displacement functional may be much less expensive than direct sensitivity calculation when the number of load cases is large. However, efficient implementation is problem dependent and requires strategies for reducing the number of constraints used. The study shows that for truss and plane-stress elements it is easy to implement the adjoint method. Also an example demonstrates that even an extreme form of constraint lumping can work well in optimization of a wing structure allowing us to keep only a small number of constraints.

This paper describes a design study for the structural optimization of a typical bay of a blended wing body transport. A hat stiffened laminated composite shell concept is used in the design. The geometry of the design is determined with the PANDA2 program, but due to the presence of varying axial loads, a more accurate analysis procedure is needed. This is obtained by combining the STAGS finite element analysis program with response surface approximations for the stresses and the buckling loads. This design procedure results in weight savings of more than 30 percent, albeit at the expense of a more complex design. The response surface approximations allow easy coupling of the structural analysis program with the optimization program in the easily accessible Microsoft EXCEL spreadsheet program. The response surface procedure also allows the optimization to be carried out with a reasonable number of analyses. In particular, it allows combining a large number of inexpensive beamanalysis stress calculations with a small number of the more accurate STAGS analyses.

This paper presents the optimization of FRP laminated rectangular plates under uncertainties in elastic properties. In optimization, thickness of FRP lamina and fiber angles are taken as design variables, and the maximum deflection of the plate is minimized under buckling and in-plane stiffness constraints. The uncertainties are entered by introducing fuzzy material constants, and then objectives and constraints are represented by a membership function of their own according to the Vertex Method. Thus the optimization problem is defined by two parameters, which are the membership function levels for the objective and constraints. Genetic algorithm with continuous design variables are employed in solving the optimization problem. By extensive numerical calculations the effect of uncertainties in elastic constants on the optimum design solutions are studied.

Integration of global and local analyses for structural optimization is a challenging task for designers. While computational costs are still a big hurdle for integrating local-global analysis in design optimization, additional difficulties are posed by the need to use a variety of codes or programs to perform the analysis. The paper presents a problem that required the use of two separate programs for the local and global analyses for a laminate composite liquid hydrogen tank structure optimized for minimum weight. The global and local analysis programs are integrated using constraint approximations. Both local linear approximations and more global response surface approximations are used. Optimization results of the two schemes are presented and relative merits discussed.

Response surface methodology provides a powerful tool for finding approximations to complex response functions. In many applications the evaluation of each design point required to construct the response surface approximations, is expensive and/or time consuming. The cost associated with the evaluation of design points generally motivates the use of statistical design of experiments to select a small number of 'optimal' points at which to evaluate the response quantity. Response surface approximations have two types of error; the first is random error or noise and the second is bias or modeling error. Traditionally points are selected so as to minimize the random error using minimum variance based designs, for example the Z?-Optimal design. However, when using a low order polynomial to approximate a response quantity with unknown behavior, large bias errors may be introduced. In the present work, a genetic algorithm is developed for obtaining a minimum-bias based design. Two example problems are evaluated and the results as obtained from both .D-Optimal as well as minimum-bias designs are compared.

Stacking sequence optimization is an important problem in composite laminate design. It is a combinatorial problem, which has been solved by genetic algorithms (GA). However, for thick laminates with many plies, the combinatorial problem is large, and the GA may not be reliable. The paper proposes a strategy for improving the reliability of the GA for thick laminates based on a two stage approach. In the first stage, the laminate is optimized with a coarse representation of the stacking sequence. In the second stage, instead of using a random initial population, the optimum stacking sequence of the first stage is adapted to create an initial population for the second stage. The approach is called consanguineous because the entire initial population is descended from this single optimum design. Several approaches for creating the consanguineous initial population were tested, and an approach based on expansion and interchange operators proved tp be the best. The twostage approach is shown to be much more reliable than the usual single stage GA.

The study focuses on the development of a simple and accurate global/local method for calculating the static response of stepped, simply-supported, isotropic and composite plates with circular and elliptical cutouts. The approach primarily involves two steps. In the first step a global approach, the Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function, which is accurate for uniform thickness plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cut-out boundary, nor does it accurately model the discontinuities caused by abrupt thickness changes. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. The methodology is then applied to the analysis of composite plates, and its usefulness successfully proved in such cases. The proposed approach resulted in accurate prediction of stresses, with considerable savings in CPU time and data storage for composite flat panels.

Response surface methodology is used to construct response surface approximations for the fatigue life of structures with complex geometry and/or loading conditions. The smooth nature of the response surface approximations eliminates numerical noise, which is inherent to the numerical analysis procedures used to estimate the fatigue life. Procedures for constructing the required response surface approximations, as well as for implementing the resulting approximations in the design process, are discussed. The durability of a structure is introduced into the design process by including the minimum fatigue life as a constraint in a minimum weight optimization of the structure. The smoothness of the response surface approximations allows the use of a derivative based optimization in the design process. Two example problems, typical of the automotive industry, are evaluated to illustrate the proposed methodology. Both example problems have structures modeled by plate elements and the objective of the optimization problem is to find the optimum thickness distribution for a minimum specified fatigue life. It is shown that the thickness

http://218.196.244.90/comp_meeting/IACM-ECCOMAS08/pdfs/a1601.pdf; International audience; Identifying parameters of a model using experimental data has been extensively studied in various areas, including for determining elastic material properties from strain measurements or modal vibration data. In order to find the parameters that make the model agree best with the experiments, the most widely used method is based on minimizing the least squares error between the experimental data and the model predictions. The least squares method leads however to suboptimal results under a variety of conditions. In a previous study by the same authors [1] it has been shown on two simple test problems that the least squares method can be negatively affected by different magnitude of the experimental data, different uncertainty and correlation among the experimental data. Multiple sophistications to the least squares method exist in the literature, addressing these issues. Another natural way to address the previous shortcomings is through statistical frameworks, based on maximum likelihood or on Bayes' rule. The Bayesian framework is more general since it can include prior knowledge. Isenberg proposed the application of the Bayesian framework for parameter estimation in 1979 [2] and several articles have been published since on the application of this approach to frequency or modal identification in particular, i.e. identifying material properties from vibration test data [3],[4]. The statistical nature of the Bayesian approach implies that it can intrinsically handle the shortcomings discussed earlier on the classical least squares method. However it also implies numerical and computational cost issues since we usually have to carry out large numbers of simulations. Different types of uncertainty can be considered in a Bayesian framework, with error and variability representing two major categories. By error we mean either measurement error, which includes systematic as well as random error (or noise) or modeling error, that is, the error induced by using a certain model to represent the actual experiment. Variability on the other hand relates to uncertainty of fixed input parameters of our model, such as dimensions of the plate or plies, exact fiber volume fraction or fiber orientation. The uncertainty in the knowledge of these fixed parameters, unlike the error uncertainty, can be accurately propagated through the model to determine its effects on the identified parameters. However this propagation process is particularly computationally expensive. Previous studies such as [4] that applied the Bayesian approach for material properties identification from vibration test data of a beam considered error but not the more expensive variability. In our previous study [1] we did consider both error and variability however the Bayesian approach was applied on a simple vibration test problem for which we used an analytical model. The aim of this previous study was to illustrate on an example where computational issues do not cloud the process how the Bayesian approach can handle variable uncertainties as well as correlation in the measurements. In the current study we are interested in what happens when we apply a Bayesian updating procedure, accounting for both error and variability, to an actual identification problem, where inevitably we have to deal with numerical and computational cost issues. The chosen application consists in identifying orthotropic elastic properties from frequency response data of a free vibrating composite plate. The focus of the study is on the numerical side so we chose to use experimental results already available in the literature [5]. The final contribution will discuss the difficulties of accounting for both error and variability in a Bayesian approach and the solutions retained to tackle this problem, applied on the identification of orthotropic material properties from vibration test data.

http://www.iccm-central.org/Proceedings/ICCM17proceedings/Themes/Behaviour/FULL%20FIELD%20MEASU%20METH/F14.7%20Gogu.pdf; International audience; The principal components analysis is applied to simulated full field displacement maps of 4569 measurement points thus allowing representation of any field within a certain domain as a linear combination of five basis vectors without losing significant information. A dimensionality reduction from 4569 to only 5 is thus achieved.

This dissertation addresses the issue of dealing with uncertainties when surrogate models are used to approximate costly numerical simulators. First, we propose alternatives to compensate for the surrogate model errors in order to obtain safe predictions with minimal impact on the accuracy (conservative strategies). The efficiency of the different methods are analyzed with the help of engineering problems, and are applied to the optimization of a laminate composite under reliability constraints. We also propose two contributions to the field of design of experiments (DoE) in order to minimize the uncertainty of surrogate models. Firstly, we developed a sequential method to build DoEs that minimize the error in a target region of the design space. Secondly, we proposed optimal sampling strategies when simulators with noisy responses are considered., Cette thèse est consacrée à la planification et l'exploitation d'expériences numériques à l'aide de modèles de remplacement (métamodèles), et plus particulièrement à la prise en compte des incertitudes induites par l'utilisation de ces métamodèles. Dans un premier temps, différentes méthodes sont proposées pour compenser ces incertitudes en biaisant les modèles afin de limiter le risque d'erreur 'défavorable' (méthodes conservatives). Cette étude s'appuie sur des applications en mécanique des structures, et en particulier, l'optimisation d'un système soumis a des contraintes de fiabilité. Cette thèse propose également deux contributions au domaine de la planification d'expériences numériques. D'une part, une méthode a été développée pour construire des plans permettant de minimiser l'erreur du modèle dans une région cible de l'espace de conception. Enfin, nous avons proposé des résultats pour la planification optimale des calculs dans le cas de simulateurs à réponse bruitées.