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]

Reduced Order modeling of high magnetic field magnets

International audienceWe present applications of the reduced basis method (RBM) to large-scale non-linear multi-physics problems connected to real industrial applications arising from the High Field Resistive Magnets development at the Laboratoire National des Champs Magnétiques Intenses

Reduced basis methods and high performance computing. Applications to non-linear multi-physics problems

International audienceWe present an open-source framework for the reduced basis methods implemented in the library Feel++ [3,4] and we consider in particular multi-physics, possibly non-linear, applications [1,2] which require high performance computing. We present how the mathematical methodology and technology scale with respect to complexity and the gain obtained in industrial context [1]. We present also briefly our first developments on low-rank methods within our framework with our colleagues from ECN. One of the main application presented is developed with the Laboratoire National des Champs Magnétiques Intenses (LNCMI), a large french equipment, allowing researchers to do experiments with magnetic fields up to 35T provided by water cooled resistive electro-magnet. Existing technologies (material properties,...) are pushed to the limits and users require now specific magnetic field profiles or homogeneous fields. These constraints and the international race for higher magnetic fields demand conception tools which are reliable and robust. The reduced basis methodology is now part of this tool chain. Another domain of application we will consider in the talk is fluid flows, both Stokes and Navier-Stokes.[1] Cécile Daversin, Stéphane Veys, Christophe Trophime, Christophe Prud'Homme. A Reduced Basis Framework: Application to large scale non-linear multi-physics problems http://hal.archives-ouvertes.fr/hal-00786557 [2] Elisa Schenone, Stéphane Veys, Christophe Prud'Homme. High Performance Computing for the Reduced Basis Method. Application to Natural Convection http://hal.archives-ouvertes.fr/hal-00786560[3] http://www.feelpp.org [4] C. Prudhomme, V. Chabannes, V. Doyeux, M. Ismail, A. Samake, G. Pena. Feel++ :A Computational Framework for Galerkin Methods and Advanced NumericalMethods, ESAIM Proc., Multiscale Coupling of Complex Models in Scientific Computing, 38 (2012), 429–455

Rôle de la nociceptine dans le système nerveux périphérique et central et perspectives dans le traitement de la douleur

PARIS-BIUP (751062107) / SudocSudocFranceF

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]

Bases réduites certifiées pour des problèmes multi-physiques non-linéaires de grande taille. Application au design d'aimants à haut champ

Le champ magnétique agit sur la matière, permettant ainsi de la sonder et de déterminer ses propriétés. C'est pourquoi de nombreuses expériences scientifiques utilisent les champs magnétiques, dans des domaines très variés parmi lesquels la physique du solide, la (bio)chimie ou encore la caractérisation et/ou l'élaboration de nouveaux matériaux. Le Laboratoire National des Champs Magnétiques Intenses (LNCMI) met à la disposition de la communauté scientifique internationale des aimants dits à haut champ, c'est-à-dire capables de produire un champ magnétique supérieur à celui des supraconducteurs (25 Tesla).La conception d'aimants pouvant générer des champs magnétiques de plus en plus intenses et/ou de plus en plus homogènes est un défi en terme de design, notamment en raison des contraintes mécaniques et thermiques. La modélisation des phénomènes physiques (non linéaires et couplés) mis en jeu pour ces aimants est donc essentielle à leur optimisation.L'objectif est de développer une gamme de modèles multi-physiques représentatifs de ces phénomènes, et prenant en compte les incertitudes liées aux propriétés des matériaux utilisés et aux mesures expérimentales donnant les paramètres de fonctionnement des aimants.Ces développements s'appuient et s'inscrivent dans la synergie autour de la librairie éléments finis Feel++.Concrètement, ces modèles sont à la base d'études paramétriques et d'analyses de sensibilité sur des géométries réelles d'aimants. La complexité de ces géométries entraîne des coûts de calcul très importants, qui sont souvent non compatibles avec la réalisation de telles études. L'utilisation d'une méthode de réduction d'ordre est la solution envisagée pour faire face à cette complexité, se basant sur le framework bases réduites certifiées disponible dans Feel++.Le calcul haute performance est également un ingrédient important de ces développements. En effet, l'utilisation de super ordinateurs donnant accès à des milliers de cœurs, et permettant ainsi de distribuer la complexité des simulations, conduit également à un gain notable en terme de coût de calcul

Full 3D MultiPhysics Model of High Field PolyHelices Magnets

International audienceHigh field Resistive magnets for static field de-veloped at the Laboratoire National des Champs Magn ́etiquesIntenses (LNCMI) are based on the so-called polyhelix technique.Their design relies on non-linear 3D multi-physic models. As theuser demands for higher magnetic field or specific field profileare growing we have to revisit our numerical models. They needto include more physics and more precise geometry. In thiscontext we have rewritten our numerical model in the frameof a collaboration with Institut de Recherche en Math ́ematiqueAvanc ́ee (IRMA). New models have been implemented with thefinite element library Feel++. This paper gives a status ofthese developments and the new features available. Results arepresented for a 14 polyhelices insert targeting 36 Tesla in a 34mm bore.High field Resistive magnets for static field de-veloped at the Laboratoire National des Champs Magn ́etiquesIntenses (LNCMI) are based on the so-called polyhelix technique.Their design relies on non-linear 3D multi-physic models. As theuser demands for higher magnetic field or specific field profileare growing we have to revisit our numerical models. They needto include more physics and more precise geometry. In thiscontext we have rewritten our numerical model in the frameof a collaboration with Institut de Recherche en Math ́ematiqueAvanc ́ee (IRMA). New models have been implemented with thefinite element library Feel++. This paper gives a status ofthese developments and the new features available. Results arepresented for a 14 polyhelices insert targeting 36 Tesla in a 34mm bore

Reduced Basis method applied to large scale non linear multiphysics problems

International audienceThe Laboratoire National des Champs Magnétiques Intenses (LNCMI) is a French large scalefacility enabling researchers to perform experiments in the highest possible magnetic field. Thedesign and optimization of such magnets require the prediction of performance metrics which can bethe magnetic field in the center, maximum stresses, or maximum and average temperatures. Theseoutputs are expressed as functionals of field variables associated with a set of coupled parametrizedPDEs involving materials properties as well as magnet operating conditions. These inputs are notexactly known and form uncertainties that are essential to consider, since existing magnet technologiesare pushed to the limits.Solutions of a multi-physics model involving electro-thermal, magnetostatics and mechanics arerequested to evaluate these implicit input-output relationships, but represent a huge computationaltime when applied on real geometries. The models typically include mesh (resp. finite element approximations)with (tens of) millions of elements (resp. degrees of freedom) requiring high performancecomputing solutions. Moreover, the non affine dependance of materials properties on temperaturerender these models non linear and non affinely parametrized.The reduced basis (RB) method offers a rapid and reliable evaluation of this input-output relationshipin a real-time or many-query context for a large class of problems among which non linearand non affinely parametrized ones. This methodology is well adapted to this context of many modelevaluations for parametric studies, inverse problems and uncertainty quantification.In this talk, we will present the RB method applied to the 3D non-linear and non affinely parametrizedmulti-physics model used in a real magnet design context. This reduced model enjoys features of reducedbasis framework available with opensource library Feel++. (Finite Element methodEmbedded Language in C++, http://www.feelpp.org). Validations and examples will be presentedfor small to large magnet models, involving parametric studies and uncertainty quantifications