Feel++ : A Versatile High Performance Finite Element Embedded Library into C++

International audienceA Domain Specific Language for PDEs embedded in C++ providing a syntax very close to the mathematical language in order to describe the variational formulation: Supports generalized arbitrary order Galerkin methods (cG, dG) in 1D, 2D, 3D Supports simplex, hypercube and high order meshes Supports finite elements: Lagrange, Hermite, Nedelec, Raviart-Thomas Supports seamless parallel computing Supports seamless interpolation between grids/function spaces Supports symbolic computation thanks to GiNaC Supports large scale parallel linear and non-linear solvers (PETSc/SLEPc) Supports hybrid computing: MPI, multi-thread, GPU (HARTS) Supports in-situ visualization with ParaView</ul

Dynamique de la construction topographique et toponymique à l'aven d'Orgnac. Ardèche, France

In August 1935, Robert de Joly and his fellows discovered and explored the Orgnac pot hole. In the following weeks, a first survey (plan and section) is published in Spelunca and in La Nature with a toponymy applied to the most remarkable speleothems and chambers. From 1965, Jean Trébuchon and his team discovered new cave networks beyond the North chambers; they undertook a synthetic survey of the whole cave. Following this work, several partial surveys have still been published for tourism or science. This paper deals only with the chambers discovered in 1935, actually known as Orgnac I. StranGély, the existing surveys show many lakes, and no complete survey of this network exists. So, this paper complements the first survey synthesis of Orgnac I (Jaillet et al. in this issue of Collection EDYTEM). During this surveying, we had the opportunity of a reflexion about the ways of drawing a great show cave: some places are concealed or neglected (Upper chambers for instance), or the North chambers are systematically undersized. This survey has been completed by a cave toponymy study: why all the places have not been named? What were the criteria for the choice to name or not to name? What are the links between toponymy and tourist arrangement? The comparison between texts, pictures captions and names written on the surveys and between successive documents show the evolution of toponymy (some names appear or disappear) and the spatial slip of several names. Then, this paper crosses surveys, toponymy and historic texts about aven d'Orgnac to show the evolving perception of this major show cave for 75 years.En août 1935, Robert de Joly et ses coéquipiers explorent l'aven d'Orgnac. Dans les semaines qui suivent, une première topographie, coupe et plan, est publiée dans Spelunca et dans La Nature : plusieurs concrétions remarquables, plusieurs salles sont identifiées par des locutions descriptives ou par des noms dont certains s'inscriront progressivement dans la toponymie. À partir de 1965, l'équipe constituée autour de Jean Trébuchon découvre les nouveaux réseaux au-delà des salles Nord et entreprend une synthèse topographique de l'aven. Par la suite, différents documents topographiques partiels seront encore publiés, souvent en lien avec l'activité touristique ou scientifique. L'article s'intéressera exclusivement à la partie découverte en 1935, aujourd'hui connue comme Orgnac I. Curieusement, aucune topographie complète n'existait, chaque document présentant un certain nombre de lacunes ; l'article accompagne donc la première synthèse topographique d'Orgnac I (Jaillet et al., 2012). Ce travail topographique a fourni l'occasion de réfléchir aux modalités de représentation, d'identifier des secteurs de la cavité occultés ou délaissés (salles Hautes par exemple), ou dont le développement était visiblement minoré (salles Nord en particulier). Il a également été complété par une étude de la toponymie souterraine : quels sont les lieux et les objets nommés ? Quelles ont été les logiques de dénomination ? Dans quelle mesure cette toponymie rend-elle compte des impératifs liés à la mise en tourisme ? La comparaison entre les textes descriptifs, les légendes des photos et les indications portées sur les topographies, et entre les documents successifs qui forment le corpus, rend compte de l'évolution toponymique (apparition/disparition de noms) et du glissement spatial de certains noms associés à des lieux mal identifiés. Au final, qu'il s'agisse de représentation imagée (topographie), verbale (toponymie) ou de leur combinaison, Orgnac I révèle, sur 75 années, les modes de représentation, d'appropriation et de médiation de ce monde souterrain par les divers acteurs impliqués

Model order reduction and sensitivity analysis for complex heat transfer simulations inside the human eyeball

Heat transfer in the human eyeball, a complex organ, is significantly influenced by various pathophysiological and external parameters. Particularly, heat transfer critically affects fluid behavior within the eye and ocular drug delivery processes. Overcoming the challenges of experimental analysis, this study introduces a comprehensive three-dimensional mathematical and computational model to simulate the heat transfer in a realistic geometry. Our work includes an extensive sensitivity analysis to address uncertainties and delineate the impact of different variables on heat distribution in ocular tissues. To manage the model's complexity, we employed a very fast model reduction technique with certified sharp error bounds, ensuring computational efficiency without compromising accuracy. Our results demonstrate remarkable consistency with experimental observations and align closely with existing numerical findings in the literature. Crucially, our findings underscore the significant role of blood flowand environmental conditions, particularly in the eye's internal tissues. Clinically, this model offers a promising tool for examining the temperature-related effects of various therapeutic interventions on the eye. Such insights are invaluable for optimizing treatment strategies in ophthalmology

Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++

In this paper, we propose a mathematical model to describe the functioning of a bioreactor landfill, that is a waste management facility in which biodegradable waste is used to generate methane. The simulation of a bioreactor landfill is a very complex multiphysics problem in which bacteria catalyze a chemical reaction that starting from organic carbon leads to the production of methane, carbon dioxide and water. The resulting model features a heat equation coupled with a non-linear reaction equation describing the chemical phenomena under analysis and several advection and advection-diffusion equations modeling multiphase flows inside a porous environment representing the biodegradable waste. A framework for the approximation of the model is implemented using Feel++, a C++ open-source library to solve Partial Differential Equations. Some heuristic considerations on the quantitative values of the parameters in the model are discussed and preliminary numerical simulations are presented

Overlapping Domain Decomposition Methods with FreeFem++

International audienceIn this note, the performances of a framework for two-level overlapping domain decomposition methods are assessed. Numerical experiments are run on Curie, a Tier-0 system for PRACE, for two second order elliptic PDE with highly heterogeneous coefficients: a scalar equation of diffusivity and the system of linear elasticity. Those experiments yield systems with up to ten billion unknowns in 2D and one billion unknowns in 3D, solved on few thousands cores

Scalable Domain Decomposition Preconditioners for Heterogeneous Elliptic Problems

International audienceDomain decomposition methods are, alongside multigrid methods, one of the dominant paradigms in contemporary large-scale partial differential equation simulation. In this paper, a lightweight implementation of a theoretically and numerically scalable preconditioner is presented in the context of overlapping methods. The performance of this work is assessed by numerical simulations executed on thousands of cores, for solving various highly heterogeneous elliptic problems in both 2D and 3D with billions of degrees of freedom. Such problems arise in computational science and engineering, in solid and fluid mechanics. While focusing on overlapping domain decomposition methods might seem too restrictive, it will be shown how this work can be applied to a variety of other methods, such as non-overlapping methods and abstract deflation based preconditioners. It is also presented how multilevel preconditioners can be used to avoid communication during an iterative process such as a Krylov method

Efficient solving strategies for incompressible Navier-Stokes equations for large scale simulations using the open source software Feel++

International audienceOver the past few decades, the computational fluid dynamics has evolved to become an important tool for the description of the complex multi-physics, multi-scale phenomena characterizing blood flow. Its reliability depends both on the verification of the numerical methods and on the validation of the mathematical models. The aim of the first part of the talk is to present a preconditioning framework for the linear system arising from the finite element discretizations and time advancing finite difference schemes of the 3D steady and unsteady Navier-Stokes equations. In particular, we are interested in preconditioners based on an algebraic factorization of the system's matrix which exploit its block structure, such as the Pressure Convection-Diffusion (PCD) preconditioner , the SIMPLE preconditioner or the LSC preconditioner, see [Elman et al. (2014)]. A comparison between the efficiency of these preconditioners is ascertain by testing them over the 3D backward facing step benchmark. The iteration counts using the PCD preconditioner are independent of mesh size and high order finite elements and mildly dependent on Reynolds numbers for steady flow problems which is not the case for the other preconditioners. In the second part of the talk we describe a framework for the solution of flow problems relevant to biomechanics strongly supported by the aforementioned solving strategies. We assess the efficiency of this framework through experimental data for fluid flow in a nozzle model with rigid boundaries, a device designed to reproduce acceleration, deceleration and recirculation, features commonly encountered in medical devices [Stewart et al. (2012)]. The flow rates were chosen to cover laminar (Re = 500), transient (Re = 2000) and turbulent (Re = 3500) regimes. The numerical results displayed during the presentation are obtained using the open-source library Feel++ (Finite Element method Embedded Language in C++, www.feelpp.org). Figure 1: Computational Fluid Dynamics FDA Benchmark at Re = 2000

Substructuring Preconditioners for h-p Mortar FEM

International audienceWe build and analyze a substructuring preconditioner for the mortar method in the h-p finite element framework. Particular attention is given to the construction of the coarse component of the preconditioner in this framework, in which continuity at the cross points is not required. Two variants are proposed: the first one is an improved version of a coarse preconditioner already presented in [12]. The second is new and is built by using a Discontinuous Galerkin interior penalty method as coarse problem. A bound of the condition number is proven for both variants and their efficiency and scalability is illustrated by numerical experiments

Hemodynamic simulations in the cerebral venous network: A study on the influence of different modeling assumptions

International audienceBlood flow computations in complex geometries are of major interest in various cardio-vascular applications. However, deriving an appropriate computational model is still an open issue and a central question is how to incorporate and quantify uncertainties due to different modeling assumptions. The present work is intended as a first step in this direction, in the particular case of blood flow in the cerebral venous system. After a careful evaluation of the influence of the computational methodology, the study investigates the impact on the velocity field and the wall shear stress of three inflow boundary conditions, two strategies for treating the outflow boundary condition and two different viscosity models. The results demonstrate that the effect of setting the inflow boundary condition on the forces created by blood flow, is likely greater than for other modeling assumptions, the other important factor being the blood viscosity model, especially in wall shear stress computations. They suggest that improvements on the one hand on the mathematical and computational approach, and on the other hand on available data for their treatment are needed

Simultaneous empirical interpolation and reduced basis method for non-linear problems

In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [4, 3]. To deal with this issue, it is now standard to apply the EIM methodology [8, 9] before deploying the Reduced Basis (RB) methodology. However the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [9, 1]. We propose a simultaneous EIM Reduced basis algorithm, named SER, that provides a huge computational gain and requires as little as N + 1 finite element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [9]