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International audience

This paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p-adaptive algorithms for discontinuous Galerkin (DG) methods. Residual error, discretization-error and feature-based indicators are compared by studying their effect on convergence history, computational gain and refinement regions selected by the adaptive algorithm. The analysis is initially carried out on steady flow configurations. Static p-adaptive simulations of the periodic flow past a cylinder at Re = 100 are then performed. Compared to uniform p-refinement, a reduction between 50% and 75% in the number of degrees of freedom is obtained for all test cases considered. The accuracy and efficiency observed for the small-scale energy density and spectral decay indicators demonstrate their great potential for the adaptive simulation of unsteady flows.

International audience; A-priori analyses to study the effect of the large-scale space definition in the variational multiscale simulation (VMS) approach to LES are carried out in the context of a modal discontinuous Galerkin (DG) method. A numerically consistent framework is introduced to derive the expression of the ideal subgrid-scale (SGS) energy transfer, which takes into account the different terms involved in the DG discretization. Based on the insight gained from this study, a locally adaptive modeling strategy able to adjust the definition of the large-scale space to the local resolution requirements of the flow is proposed.

This paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p-adaptive algorithms for discontinuous Galerkin (DG) methods. Residual error, discretization-error and feature-based indicators are compared by studying their e ect on convergence history, computational gain and refinement regions selected by the adaptive algorithm. The analysis is initially carried out on steady flow configurations. Static p-adaptive simulations of the periodic flow past a cylinder at Re = 100 are then performed. Compared to uniform p-refinement, a reduction between 50% and 75% in the number of degrees of freedom is obtained for all test cases considered. The accuracy and eciency observed for the VMS and spectral decay indicators demonstrate their great potential for the adaptive simulation of unsteady flows.

International audience; This paper presents the stability, convergence and optimization characteristics of interface treatments for steady conjugate heat transfer problems. The Dirichlet-Robin and Neumann-Robin procedures are presented in detail and compared on the basis of the Godunov-Ryabenkii normal mode analysis theory applied to a canonical aero-thermal coupling prototype. Two fundamental parameters are introduced, a "numerical" Biot number that controls the stability process and an optimal coupling coefficient that ensures unconditional stability. This coefficient is derived from a transition of the amplification factor. A comparative study of these two treatments is made in order to implement numerical schemes based on adaptive and local coupling coefficients, with no arbitrary relaxation parameters, and with no assumptions on the temporal advancement of the fluid domain. The coupled numerical test case illustrates that the optimal Dirichlet-Robin interface conditions provide effective and oscillation-free solutions for low and moderate fluid-structure interactions. Moreover, the computation time is slightly shorter than the time required for a CFD computation only. However, for higher fluid-structure interactions, a Neumann interface condition on the fluid side presents good numerical properties so that no relaxation coefficients are required.; Cet article présente les caractéristiques de stabilité, de convergence et d'optimisation des traitements d'interface des problèmes couplés aérothermiques. Les procédures de Dirichlet-Robin et Neumann-Robin sont présentées en détail et comparées sur la base de la décomposition en modes normaux suivant la théorie de Godunov-Ryabenkii appliquée à un modèle de couplage aéro-thermique simplifié. Deux paramètres fondamentaux sont introduits, un nombre Biot "numérique" qui contrôle le processus de stabilité et un coefficient de couplage optimal qui assure une stabilité inconditionnelle. Ce coefficient est obtenu à partir d'une transition du facteur d'amplification du modèle couplé. Une étude comparative de ces deux traitements est réalisée afin de mettre en œuvre des schémas numériques basés sur des coefficients de couplage adaptatifs et locaux, sans avoir recours à des paramètres de relaxation arbitraire et sans hypothèse sur l'avancement temporel du domaine fluide. Le cas test numérique couplé que nous présentons montrent que les conditions optimales d'interface Dirichlet-Robin fournissent des solutions performantes et sans oscillation pour les interactions fluide-structure faibles et modérées. De plus, le temps de calcul est légèrement inférieur au temps requis pour un calcul CFD seul. Cependant, pour des interactions fluide-structure plus fortes, une condition d'interface de Neumann du côté fluide présente de bonnes propriétés numériques, de sorte qu'aucun coefficient de relaxation n'est requis.

We studied the construction of subgrid-scale models for large-eddy simulationof incompressible turbulent flows, focusing on consistency with importantmathematical and physical properties. In particular, we considered the symmetriesof the Navier-Stokes equations, and the near-wall scaling and dissipation behaviorof the turbulent stresses. After showing that existing models do not all satisfy thedesired properties, we discussed a general class of subgrid-scale models based onthe local filtered velocity gradient. We provided examples of models from this classthat preserve several of the symmetries of the Navier-Stokes equations and exhibitthe same near-wall scaling behavior as the turbulent stresses. Furthermore, thesemodels are capable of describing nondissipative effects.

Goal-oriented error estimation [1] has seen increasing interest for adaptive discretizations of engineering problems, as these methods aim to minimize the error in the quantity of interest that motivates the engineering analysis in the first place. However, the main disadvantage of deriving refinement indicators based on goal-oriented error estimators is that an additional “dual problem” needs to be solved, often in higher resolution than the primal problem, comparing badly with the simpler estimators requiring only local operations in terms of the computational cost.In this work, we aim to reduce the cost of approximating the dual solution. Taking inspiration from the fact that often its interpolation on the primal mesh is subtracted from the approximate dual solution in obtaining localized refinement indicators, we propose to discretize the dual problem using a space orthogonal to the primal discretization space. We assume to start with a discrete dual space that is a strict superset of the primal discretization space and decompose it into coarse and fine parts and only use the fine-scale solution, discarding the coarse problem. We then apply the ideas originating from the variational multiscale framework [2] to take care of the influence of the discarded coarse scales on the finer resolved scales. We prove that under certain conditions the fine-coarse coupling can be chosen to disappear altogether, resulting in a cheap error estimator identical to the conventional and more expensive estimator based on dual discretizations using a refinement of the discrete primal spaces.We choose to apply these concepts to the discontinuous Galerkin methods [3], leveraging the simplicity of constructing orthogonal bases for discontinuous polynomial spaces. We numerically investigate the quality of the resulting estimator for a 1D convection-diffusion problem with varying Peclet numbers. Our preliminary results indicate that the proposed estimator performs exceptionally in the diffusion-dominated range; however, the dependence on the choice of the projection in splitting the scales and whether or not a coarse scale model is employed appears to be very significant in the advection-dominated regime.