Papers by Sebastien Boyaval
Numerical simulations are a highly valuable tool to evaluate the impact of the uncertainties of v... more Numerical simulations are a highly valuable tool to evaluate the impact of the uncertainties of various model parameters, and to optimize e.g. injection-production scenarios in the context of underground storage (of CO2 typically). Finite volume approximations of Darcy's parabolic model for flows in porous media are typically run many times, for many values of parameters like permeability and porosity, at costly computational efforts. We study the relevance of reduced basis methods as a way to lower the overall simulation cost of finite volume approximations to Darcy's parabolic model for flows in porous media for different values of the parameters such as permeability. In the context of underground gas storage (of CO2 typically) in saline aquifers, our aim is to evaluate quickly, for many parameter values, the flux along some interior boundaries near the well injection area-regarded as a quantity of interest-. To this end, we construct reduced bases by a standard POD-Greedy algorithm. Our POD-Greedy algorithm uses a new goal-oriented error estimator designed from a discrete spacetime energy norm independent of the parameter. We provide some numerical experiments that validate the efficiency of the proposed estimator.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 31, 2022
We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831... more We generalize a new symmetric-hyperbolic system of PDEs proposed in [ESAIM:M2AN 55 (2021) 807-831] for Maxwell fluids to a class of systems that define unequivocally multi-dimensional visco-elastic flows. Precisely, within a general setting for continuum mechanics, we specify constitutive assumptions i) that ensure the unequivocal definition of motions satisfying widely-admitted physical principles, and ii) that contain [ESAIM:M2AN 55 (2021) 807-831] as one particular realization of those assumptions. The new class can capture the mechanics of various materials, from solids to viscous fluids, possibly with temperature dependence and heat conduction.
We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional... more We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error.
arXiv (Cornell University), Feb 7, 2018
We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional... more We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error.
arXiv (Cornell University), Dec 2, 2022
Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of ... more Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021) 807-831] an upper-convected Maxwell model for compressible flows with a symmetrichyperbolic formulation. Here, that model is derived again, with new details.
arXiv (Cornell University), Dec 12, 2017
We pursue here the development of models for complex (viscoelastic) fluids in shallow free-surfac... more We pursue here the development of models for complex (viscoelastic) fluids in shallow free-surface gravity flows which was initiated by [Bouchut-Boyaval, M3AS (23) 2013] for 1D (translation invariant) cases. The models we propose are hyperbolic quasilinear systems that generalize Saint-Venant shallow-water equations to incompressible Maxwell fluids. The models are compatible with a formulation of the thermodynamics second principle. In comparison with Saint-Venant standard shallow-water model, the momentum balance includes extra-stresses associated with an elastic potential energy in addition to a hydrostatic pressure. The extra-stresses are determined by an additional tensor variable solution to a differential equation with various possible time rates. For the numerical evaluation of solutions to Cauchy problems, we also propose explicit schemes discretizing our generalized Saint-Venant systems with Finite-Volume approximations that are entropy-consistent (under a CFL constraint) in addition to satisfy exact (discrete) mass and momentum conservation laws. In comparison with most standard viscoelastic numerical models, our discrete models can be used for any retardationtime values (i.e. in the vanishing "solvent-viscosity" limit). We finally illustrate our hyperbolic viscoelastic flow models numerically using computer simulations in benchmark test cases. On extending to Maxwell fluids some free-shear flow testcases that are standard benchmarks for Newtonian fluids, we first show that our (numerical) models reproduce well the viscoelastic physics, phenomenologically at least, with zero retardation-time. Moreover, with a view to quantitative evaluations, numerical results in the lid-driven cavity testcase show that, in fact, our models can be compared with standard viscoelastic flow models in sheared-flow benchmarks on adequately choosing the physical parameters of our models. Analyzing our models asymptotics should therefore shed new light on the famous High-Weissenberg Number Problem (HWNP), which is a limit for all the existing viscoelastic numerical models.
Comptes rendus, Feb 7, 2023
A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation
New Results - Homogeneization
Flood twin experiment for estimating the potential of satellite observations in shallow-water simulations
With more than one billion people exposed to floods throughout the world, this natural hazard is ... more With more than one billion people exposed to floods throughout the world, this natural hazard is the most common and devastating one, resulting in loss of lives and damaging personal properties or sensitive infrastructures. Numerical models have become essential to forecast and to mitigate their consequences, but they remain uncertain mainly due to the lack of high-resolution data and the inherent uncertainties related to the simplified representation of natural phenomena.The growing availability of satellite observations distributed in time and space is a valuable source of information for improving flood modelling. Additional data like water level or flood extent can be extracted and used to calibrate numerical models.This study proposes to analyse the potential of remote sensing data as a complement to in-situ observations (from hydrometric stations) in the calibration process of shallow-water flood numerical models. A two-dimensional twin experiment of an extreme flood event overflowing into the floodplains is carried out on a 50 km reach on the Garonne River in France between Tonneins and La Réole. The roughness coefficients are computed as solutions to an inverse problem mixing both in-situ (pointwise and high-frequency) and satellite observations (spatially distributed but low-frequency) data. Data assimilation combining uncertain model simulations and observations has proven efficient for improving hydraulic models. However, an open question is the choice of the best information to assimilate (water level or/and flood extent maps) into the hydraulic models. We study this problem by testing different assimilation configurations. The satellite observations are not considered perfect, so the numerical solutions are compared with different noise levels.
arXiv (Cornell University), Jan 17, 2017
Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows... more Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in [Bouchut & Boyaval, 2013], which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solution to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but numerical simulations went smoothly in a practically useful range of parameters.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 7, 2016
est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés o... more 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.
Optimization and Engineering, Jun 4, 2018
We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional... more We propose a certified reduced basis approach for the strong-and weak-constraint four-dimensional variational (4D-Var) data assimilation problem for a parametrized PDE model. While the standard strong-constraint 4D-Var approach uses the given observational data to estimate only the unknown initial condition of the model, the weak-constraint 4D-Var formulation additionally provides an estimate for the model error and thus can deal with imperfect models. Since the model error is a distributed function in both space and time, the 4D-Var formulation leads to a large-scale optimization problem for every given parameter instance of the PDE model. To solve the problem efficiently, various reduced order approaches have therefore been proposed in the recent past. Here, we employ the reduced basis method to generate reduced order approximations for the state, adjoint, initial condition, and model error.
Environmental Modeling & Assessment, Oct 24, 2017
Assessing epistemic uncertainties is considered as a milestone for improving numerical prediction... more Assessing epistemic uncertainties is considered as a milestone for improving numerical predictions of a dynamical system. In hydrodynamics, uncertainties in input parameters translate into uncertainties in simulated water levels through the shallow water equations. We investigate the ability of generalized polynomial chaos (gPC) surrogate to evaluate the probabilistic features of water level simulated by a 1-D hydraulic model (MASCARET) with the same accuracy as a classical Monte Carlo method but at a reduced computational cost. This study highlights that the water level probability density function and covariance matrix are better estimated with the polynomial surrogate model than with a Monte Carlo approach on the forward model given a limited budget of MASCARET evaluations. The gPC-surrogate performance is first assessed on an idealized channel with uniform geometry and then applied on
Alternative mixed finite element space for (u h ,p h) with inf-sup condition. .. . 2-IV-Cc Altern... more Alternative mixed finite element space for (u h ,p h) with inf-sup condition. .. . 2-IV-Cc Alternative mixed finite element space for (u h ,p h) without inf-sup. .. .. .. . 2-V Positivity, free energy estimate and the long-time issue .
arXiv (Cornell University), Dec 14, 2022
Maxwell's models for viscoelastic flows are famous for their potential to unify elastic motions o... more Maxwell's models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But rigorous proofs are lacking. The present note is a contribution toward well-defined viscoelastic flows proved to encompass both solid and (liquid) fluid regimes. In a first part, we consider the structural stability of particular viscoelastic flows: 1D shear waves solutions to damped wave equations. We show the convergence toward purely elastic 1D shear waves solutions to standard wave equations, as the relaxation time and the viscosity grow unboundedly ≡ 1 → ∞ in Maxwell's constitutive equation + = 2 () for the stress of viscoelastic fluids with velocity. In a second part, we consider the structural stability of general multi-dimensional viscoelastic flows. To that aim, we embed Maxwell's constitutive equation in a symmetric-hyperbolic system of PDEs which we proposed in our previous publication [ESAIM:M2AN 55 (2021) 807-831] so as to define multi-dimensional viscoelastic flows unequivocally. Next, we show the continuous dependence of multi-dimensional viscoelastic flows on ≡ 1 using the relative-entropy tool developped for symmetric-hyperbolic systems after C. M. Dafermos. It implies convergence of the viscoelastic flows defined in [ESAIM:M2AN 55 (2021) 807-831] toward compressible neo-Hookean elastodynamics when → ∞.
New Results - Application of greedy algorithms
Ima Journal of Numerical Analysis, Jul 13, 2018
We state corrections to the proofs of Proposition 4.2 and Proposition 6.2, which were correct in ... more We state corrections to the proofs of Proposition 4.2 and Proposition 6.2, which were correct in the original submitted version; see Barrett & Boyaval (2017). The proofs are correct as stated, except we replace "=" by " " in the definitions of B γ and B h γ , and we do not appeal to Proposition 2.8 in Zeidler (1986), due to the choice of test functions. To correct the latter, we make the following additions. Proof of Proposition 4.2. On assuming, for any γ ∈ R >0 , that the continuous mapping F has no zero in B γ , we define the continuous mapping G γ : B γ → B γ such that
Ima Journal of Numerical Analysis, Oct 20, 2017
We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite e... more We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain D ⊂ R d , d = 2 or 3, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conformation tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics (d = 2) or a reduced version, where the tangential component on each simplicial edge (d = 2) or face (d = 3) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretization, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case d = 2, as the spatial and temporal discretization parameters tend to zero, towards global-intime weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.
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Papers by Sebastien Boyaval