FEMs on composite meshes for plasma equilibrium simulations in tokamaks

International audienceWe rely on a combination of different finite element methods on composite meshes, for the simulation of axisymmetric plasma equilibria in tokamaks.One mesh with Cartesian quadrilaterals covers the vacuum chamber and one mesh with triangles discretizes the region outside the chamber. The two meshes overlap in a narrow region around the chamber. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the area that is covered by the plasma, while preserving accurate meshing of the geometric details in the exterior. The continuity of the numerical solution across the boundary of each subdomain is enforced by a new mortar-like projection

Mortar FEs on Overlapping Meshes : Application to Magnetodynamics

Abstract The finite element (FE) method is frequently used in magnetodynamics as well suited to treat problems with complex geometries while keeping a simplicity in the implementation. However, some modelisations, as in eddy current (EC) non destructive testing (NDT), present the particularity to have moving parts. A global remeshing can be necessary which causes expensive CPU time. Domain decomposition methods allowing to take into account the movement without having to remesh the whole computational domain. The mortar element method (MEM), a variational non-conforming domain decomposition approach [1] offers attractive advantages in terms of flexibility and accuracy. In its original version for non-overlapping subdomains, the information is transferred through the skeleton of the decomposition by means of a suitable L 2 -projection of the field trace from the master to the slave subdomains. A MEM with overlapping subdomains has been proposed to coupled a global scalar potential defined everywhere in the considered domain and a local vector potential defined only in (possibly moving) conductor

FEMs on Composite Meshes for Tuning Plasma Equilibria in Tokamaks

We rely on a combination of different finite element methods on composite meshes, for the simulation of axisymmetric plasma equilibria in tokamaks. One mesh with Cartesian quadrilaterals covers the burning chamber and one mesh with triangles discretizes the region outsidethe chamber. The two meshes overlap in a narrow region around the chamber. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the area that is covered by the plasma, while preserving accurate meshing of the geometric details in the exterior. The continuity of the numerical solution across the boundary of each subdomain is enforced by a mortar-like projection. We show that higher order regularity is very beneficial to improve computational tools for tokamak research.Nous allons utiliser différentes méthodes d’éléments finis sur des maillages composite, pour la simulation des équilibres du plasma dans les tokamaks. Un maillage composé de rectangles avec des quadrilatérales cartésiennes couvre la chambre de combustion et une autre maillage des triangles discrétise la région à l’extérieur de la chambre. Les deux maillages se chevauchent dans une région étroite autour de la chambre. Cette approche a la flexibilité nécessaire pour réaliser facilement et à moindre coût une régularité plus élevé pour l’approximation du flux magnétique dans la zone couverte par le plasma, tout en préservant des détails géométriques à l’extérieur. La continuité de la solution numérique à travers la limite de chaque sous-domaine est imposée par une projection de type mortar. Nous montrons que la régularité d’ordre supérieur est très bénéfique pour affiner les outils de calcul qui sont utilisés en recherche pour mieux maîtriser les experiments dans des tokamaks

A finite element method with overlapping meshes for free-boundary axisymmetric plasma equilibria in realistic geometries

Existing finite element implementations for the computation of free-boundary axisymmetric plasma equilibria approximate the unknown poloidal flux function by standard {lowest order} continuous finite elements with discontinuous gradients. \hh{As a consequence,} the location of critical points of the poloidal flux, that are of paramount importance in tokamak engineering, is constrained to nodes of the mesh \hh{leading} to undesired jumps in transient problems. Moreover, recent numerical results for the self-consistent coupling of equilibrium with resistive diffusion and transport suggest the necessity of higher regularity when approximating the flux map.In this work we propose a mortar element method that employs two overlapping meshes. One mesh with Cartesian quadrilaterals covers the vacuum \rfA{chamber domain accessible by the plasma} and one mesh with triangles discretizes the region outside. The two meshes overlap in a narrow region. This approach gives the flexibility to achieve easily and at low cost higher order regularity for the approximation of the flux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details \hh{outside this region}. The continuity of the numerical solution in the region of overlap is weakly enforced by a mortar-like \hh{mapping}.En éléments finis, les applications existantes pour le calcul d’équilibres de plasma à frontière libre en axisymétrique approchent le flux poloidale par une fonction continue à gradient discontinu, qui est localement dans chaque élément du maillage un polynôme de degré un. La position des points critiques du flux poloidale, qui est de grande importance pour les ingénieurs des tokamaks, est par consequence limitée aux seuls nœuds du maillage, et donc à l’origine de perturbations quand on modélise le passage du plasma du transitoire vers l’équilibre. De plus, des résultats numériques recents sur le couplage à l’équilibre entre le plasma et un modèle de diffusion/transport ont montré le besoin de plus de régularité dans l’approximation du flux. Dans ce travail on propose une approche par méthode d’éléments finis avec joints sur deux maillages qui se recouvrent. Un maillage structuré composé de rectangles qui couvre la chambre à vide du tokamak et un maillage non structuré de triangles pour la partie du domaine qui se trouve à l’extérieur de la chambre à vide. Les deux maillages sont superposés sur une bande étroite qui entoure la chambre à vide. Cet approche est assez flexible pour permettre d’avoir facilement et à bas coût numérique une approximation du flux poloidale plus régulière dans la chambre à vide tout en gardant une description precise des détails géométriques dans la partie externe. La continuité de la solution numérique dans la partie de recouvrement des maillages est imposée faiblement à l’aide de projections de type mortar

International Workshop on Finite Elements for Microwave Engineering

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018

Mortar FEs on overlapping subdomains for eddy current non destructive testing

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