Comparaison de différentes techniques d'approximation de séries temporelles incertaines issues d'écoulements océaniques

International audienceThe analysis of time series is a fundamental task in many flow simulations such as oceanic and atmospheric flows. A major challenge is the construction of a faithful and accurate time-dependent surrogate with a manageable number of samples. Several techniques have been tested to handle the time-dependent aspects of the surrogate including a direct approach, low-rank decomposition, auto-regressive model and global Bayesian emulators. These techniques rely on two popular methods for uncertainty quantification, namely Polynomial chaos expansion and Gaussian processes regression. The different techniques were tested and compared on the uncertain evolution of the sea surface height forecast at two location exhibiting different levels of variance. Two ensembles sizes were considered as well as two versions of polynomial chaos (ordinary least squares or ridge regression) and Gaussian processes (exponential or Matern covariance function) to assess their impact on the results. Our conclusions focus on the advantages and the drawbacks, in terms of accuracy, flexibility and computational costs of the different techniques

A Hybrid High-Order method for nonlinear elasticity

In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.Comment: 29 pages, 7 figures, 4 table

Mass conservative BDF-discontinuous Galerkin/explicit finite volume schemes for coupling subsurface and overland flows

Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem.Comment: 30 pages, 15 Figure

Study of overland flow with uncertain infiltration using stochastic tools

The saturated hydraulic conductivity is one of the key parameters in the modelling of overland flow water fluxes. In this study, this parameter is defined as a stochastic parameter, idealized as a piecewise constant random field with uniform distribution. This paper aims at investigating the effects of the spatial and temporal scales in uncertainty propagation within overland flow models, and at identifying the localization of the most influential saturated hydraulic conductivity using sensitivity analysis. The results show that the influence of saturated hydraulic conductivity depends on the soil saturation and its spatial localization. For instance, in case of low saturated soils, the most influent parameter is the one located downslope, whereas in case of high saturated soils, the most influent one is either the most infiltrating or the intermediate one. The results indicate where efforts should be concentrate when collecting input parameters to reduce modelling uncertainties

Numerical approximation of poroelasticity with random coefficients using Polynomial Chaos and Hybrid High-Order methods

In this work, we consider the Biot problem with uncertain poroelastic coefficients. The uncertainty is modelled using a finite set of parameters with prescribed probability distribution. We present the variational formulation of the stochastic partial differential system and establish its well-posedness. We then discuss the approximation of the parameter-dependent problem by non-intrusive techniques based on Polynomial Chaos decompositions. We specifically focus on sparse spectral projection methods, which essentially amount to performing an ensemble of deterministic model simulations to estimate the expansion coefficients. The deterministic solver is based on a Hybrid High-Order discretization supporting general polyhedral meshes and arbitrary approximation orders. We numerically investigate the convergence of the probability error of the Polynomial Chaos approximation with respect to the level of the sparse grid. Finally, we assess the propagation of the input uncertainty onto the solution considering an injection-extraction problem

A review of physically based models for soil erosion by water

International audiencePhysically-based models rely on fundamental physical equations describing stream flow and sediment and associated nutrient generation in a catchment. This paper reviews several existing erosion and sediment transport approaches. The process of erosion include soil detachment, transport and deposition, we present various forms of equations and empirical formulas used when modelling and quantifying each of these processes. In particular, we detail models describing rainfall and infiltration effects and the system of equations to describe the overland flow and the evolution of the topography. We also present the formulas for the flow transport capacity and the erodibility functions. Finally, we present some recent numerical schemes to approach the shallow water equations and it's coupling with infiltration and erosion source terms

State Of the Art Report in the fields of numerical analysis and scientific computing. Final version as of 16/02/2020 deliverable D4.1 of the HORIZON 2020 project EURAD.: European Joint Programme on Radioactive Waste Management

Document information Project Acronym EURAD Project Title European Joint Programme on Radioactive Waste Management Project Type European Joint Programme (EJP) EC grant agreement No. 847593 Project starting / end date 1 st June 2019-30 May 2024 Work Package No. 4 Work Package Title Development and Improvement Of NUmerical methods and Tools for modelling coupled processes Work Package Acronym DONUT Deliverable No. 4.

Méthodes numériques pour les écoulements souterrains et couplage avec le ruissellement

Accurate and robust numerical schemes are proposed to simulate subsurface flows and their coupling with surface runoff. Subsurface flows are modelled by the (unsteady) Richards' equation discretized by a BDF in time and a discontinuous Galerkin method with symmetric interior penalty in space. Infiltration column test cases confirm the robustness of our schemes. Firstly, we consider Signorini conditions for the Richards' equation to model buried drains or the water table reaching the ground surface ; thus we assume that exfiltrated water immediately exits the system. Secondly, runoff flow is taken into account through coupling conditions enforcing water flux equality and pressure continuity at the interface. Overland flows are modelled by the kinematic wave approximation of the shallow water equations, which is discretized by a Godunov method. Both schemes, that for the subsurface flow and that for the overland flow, are mass conservative and can be coupled within a multiple time step procedure. To ensure total water mass conservation for the whole system, interface fluxes must be carefully designed. We specify the form of these fluxes for BDF1 and BDF2. Accuracy and robustness of our schemes are assessed on drainage, exfiltration and hortonian runoff test cases. Finally, we apply the methology to investigate the hydrological behavior of a small-scale drained watershed. keywords : Richards' equation, kinematic wave approximation, discontinuous finite elements, coupling algorithms, mass conservation.Des schémas numériques précis et robustes sont proposés pour modéliser les écoulements souterrains et leur couplage avec le ruissellement surfacique. Les écoulements souterrains sont d´écrits par l'équation de Richards (instationnaire) qui est discrétisée par une méthode BDF en temps et une méthode de Galerkine discontinue à pénalisation intérieure symétrique en espace. Des cas tests sur des colonnes d'infiltration confirment la robustesse des schémas choisis. Dans un premier temps, nous considérons des conditions de Signorini pour l'équation de Richards afin de modéliser la présence de drains en fond d'aquifère ou l'affleurement de la nappe en négligeant le ruissellement, c'est-à-dire en supposant que l'eau exfiltrée est immédiatement évacuée du système. Dans un second temps, nous prenons en compte le ruissellement par le biais de conditions de couplage qui imposent l'égalité des flux d'eau échangés et la continuité de la pression à l'interface. Les écoulements superficiels sont d´écrits par l'équation de l'onde cinématique qui constitue une approximation des équations de Saint-Venant. L'équation de l'onde cinématique est discrétisée par une méthode de Godunov. Les deux schémas, pour l'écoulement souterrain et pour l'écoulement superficiel, sont conservatifs et peuvent être utilisés dans des algorithmes de couplage faisant intervenir un ou plusieurs pas de temps. Pour assurer la conservation de la masse d'eau totale du système couplé, les flux à l'interface doivent être convenablement choisis. Nous donnons en particulier la construction de ces flux pour les schémas BDF1 et BDF2. La précision et la robustesse de nos schémas sont évaluées sur plusieurs cas tests dont le drainage d'une lame d'eau, deux cas d'exfiltration de nappe (l'un provoqué par la pluie et l'autre par une injection en fond d'aquifère) et un ruissellement hortonien. Enfin, nous présentons une application concrète portant sur le fonctionnement hydrologique d'un petit bassin versant drainé