Uncertainty quantification in littoral erosion

International audienceWe aim at quantifying the impact of flow state uncertainties in lit-toral erosion to provide confidence bounds on deterministic predictions of bottom morphodynamics. Two constructions of the bathymetry standard deviation are discussed. The first construction involves directional quantile-based extreme scenarios using what is known on the flow state Probability Density Function (PDF) from on site observations. We compare this construction to a second cumulative one using the gradient by adjoint of a functional involving the energy of the system. These ingredients are illustrated for two models for the interaction between a soft bed and a flow in a shallow domain. Our aim is to keep the computational complexity comparable to the deterministic simulations taking advantage of what already available in our simulation toolbox

Backward uncertainty propagation in shape optimization

International audienceWe aim at quantifying the impact of state uncertainties in shape optimization. This provides confidence bounds for the optimal solution. The approach is presented for inverse designs where the target is assumed uncertain. No sampling of a large dimensional space is necessary and the approach uses what is already available in a deterministic gradient-based inversion algorithm. Our proposal is based on the introduction of directional quantile-based extreme scenarios knowing the Probability Density Function (PDF) of the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using the gradient of the functional by an adjoint method. The paper goes beyond inverse design and shows how to apply the method to general optimization problems. The ingredients of the paper are illustrated on a model problem with the Burgers equation and on the optimization of the shape of an aircraft. Overall, the computational complexity is comparable to the deterministic case

Data analytic UQ cascade

International audienceThis contribution gathers some of the ingredients presented at Erice during the third workshop on 'Variational Analysis and Aerospace Engineering'. It is a collection of several previous publications on how to set up an uncertainty quantification (UQ) cascade with ingredients of growing computational complexity for both forward and reverse uncertainty propagation. It uses data analysis ingredients in a context of existing deterministic simulation platforms. It starts with a complexity-based splitting of the independent variables and the definition of a parametric optimization problem. Geometric characterization of global sensitivity spaces through their dimensions and relative positions through principal angles between vector spaces bring a first set of information on the impact of uncertainties of the functioning parameters on the optimal solution. Joining the multi-point descent direction and Probability Density Function (PDF) quantiles of the optimization parameters permits to define the notion of Directional Extreme Scenarios (DES) without sampling of large dimension design spaces. One goes beyond DES with Ensemble Kalman Filters (EnKF) after the multi-point optimization algorithm is cast into an ensemble simulation environment. This formulation accounts for the variability in large dimension. The UQ cascade continues with the joint application of the EnKF and DES leading to the concept of Ensemble Directional Extreme Scenarios (EDES) which provides a more exhaustive description of the possible extreme scenarios. The different ingredients developed for this cascade also permits to quantify the impact of state uncertainties on the design and provide confidence bounds for the optimal solution. This is typical of inverse designs where the target should be assumed uncertain. Our proposal uses the previous DES strategy applied this time to the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using available adjoint-based gradients of the functional. Eventually , we go beyond inverse design and apply the method to general optimization problems. The ingredients of the paper have been applied to constrained aerodynamic performance analysis problems. 1 Context Our domain of interest is aerodynamic shape optimization. The questions of interest are:-can we propose an aircraft shape designed to have similar performances over a given range of some functioning parameters (to be formulated through the moments of a functional) ?-can we do that modifying as less as possible an existing mono-point optimization shape design loop ?-is it possible for the time-to-solution cost of this parametric shape design to remain comparable to the mono-point situation ? We consider a generic situation where the simulation aims at predicting a given quantity of interest j(x, α) and there are a few functioning or operating parameters α and several design parameters x involved. The ranges of the functioning parameters define the global operating/functioning conditions of a given design. This splitting of the independent variables in two sets is important for the sequel

Controlling First Four Moments for Robust Optimization

International audienceThe paper addresses the solution of robust moment-based optimization problems after a multipoint reformulation. The first four moments are considered (i.e. mean, variance, skewness and kurtosis) going beyond classical engineering optimization based on the control of the mean and variance. In particular, the impact on the design of a control of the third and fourth moments are discussed. The multipoint formulation leads to discrete expressions for the moments. linking moment-based and multipoint optimizations. The linearity of the sums in the discrete moments permits an easy evaluation of their gradients with respect to the design variables. Optimal sampling issues are analyzed and a procedure is proposed to quantify the confidence level on the robustness of the design. The proposed formulation is fully parallel and the time-to-solution is comparable to single-point situations. It is applied to three problems: an analytical least-square minimization problem, a shape optimization problem with a reduced-order model, and a full aircraft shape optimization robust over a range of transverse winds

Homogenization of diffuse delamination in composite laminates

Diffuse delamination induced by transverse cracking is usually the secondary damage mode when a composite laminate experiences tensile loading. The fist damage mechanism in such a laminate is transverse cracking which has been widely investigated with both analytical methods and " mechanism-based" constitutive laws. Delamination induced by matrix cracking is already studied extensively by analytical approaches, however, a proper homogenization way has not been proposed yet. In this paper, a modification to an available cohesive constitutive law is proposed which is capable of considering the effect of diffuse delamination without the necessity of consideration of an actual discontinuity between the layers. The proposed constitutive law is then compared against its equivalent models containing interlaminar discontinuity and it is shown that the obtained results from both models are in good. Then the proposed modification is used in Double Cantilever Beam (DCB) specimen and the obtained results are found coincident with the equivalent model with diffuse discontinuities at the interface. Finally, a damaged cross-ply laminate is modeled under the boundary conditions of tensile loading and also 3-point bending with and without the proposed cohesive modification. In tensile loading, the results of both cases are similar; however, it is shown that in bending, the unmodified cohesive law predicts the lateral stiffness larger than the proposed modification. The lateral stiffness of the equivalent model with discontinuities as crack indicates that the proposed modification is able to properly consider the lateral stiffness decrease

Mathematical and numerical analysis of an alternative well-posed two-layer turbulence model

In this article, we wish to investigate the behavior of a two-layer k - ε turbulence model from the mathematical point of view, as this model is useful for the near-wall treatment in numerical simulations. First, we explain the difficulties inherent in the model. Then, we present a new variable θ that enables the mathematical study. Due to a problem of definition of the turbulent viscosity on the wall boundary, we consider an alternative version of the original equation. We show that some physical aspects of the model are preserved by the new formulation, and in particular, we show how the physicists can help us to prove the existence of a solution of our problem. Finally, we are interested in the Navier-Stokes equations coupled with the modified turbulence model and we show that the alternative model may be preferred to the original one, because of its good properties (existence of a solution of the coupled problems)

Simplifying numerical solution of constrained PDE systems through involutive completion

When analysing general systems of PDEs, it is important first to find the involutive form of the initial system. This is because the properties of the system cannot in general be determined if the system is not involutive. We show that the notion of involutivity is also interesting from the numerical point of view. The use of the involutive form of the system allows one to consider quite general situations in a unified way. We illustrate our approach on the numerical solution of several flow equations with the aim of showing the impact of the involutive form of the systems in simplifying numerical schemes

Extreme scenarios for the evolution of a soft bed interacting with a fluid using the Value at Risk of the bed characteristics

International audienceWe show how to introduce the Value at Risk (VaR) concept in the analysis of the adaptation of the shape of a soft bed to a flow knowing the Probability Density Function (PDF) of the responses of the shape to flow perturbations (bed receptivity). Our aim is to quantify our confidence on simulation scenarios by an available morphodynamic model for the shape. The approach permits to perform this task at low complexity as it does not require any sampling of the bed receptivity parameter space. The paper goes beyond stationarity for the variability of the bed receptivity by linking its dynamics to the bottom morphodynamics through an original transport equation for the local bed receptivity standard deviation. The approach has been applied to the analysis of bed morphodynamics based on minimization principles. The results show the importance of including uncertainty information during the coupling and not only eventually through simple margins on the results

A stable algorithm for the k-epsilon model for compressible flows

Projet MENUSI

Autour du contrôle - Hommage à Jean-Jacques Moreau

Un théorème de contrôlabilité en temps fini nous servira de fil conducteur. Nous verrons comment les idées de J.-J. Moreau sur la dynamique des particules et le contact peuvent servir dans des domaines aussi divers que l'optimisation multicritère, le contrôle des écoulements, la parallélisation en temps ou la quantification des incertitudes. Les ingrédients de la présentation seront illustrés par plusieurs applications industrielles.