Papers by Francesca Rapetti
J, Jan 21, 2022
We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of... more We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of high-order finite elements in general domains. (2) We focus on its connection with the question of the invertibility of the final algebraic system arising from a high-order edge finite element discretization of the magnetostatic problem formulated in terms of the magnetic vector potential. With the same purpose of invertibility, we analyse another classically used condition, the Coulomb gauge. (3) We conclude by underlying that the two gauges can be naturally considered in a high order framework without any restriction on the topology of the domain.
Calcolo, Sep 28, 2022
We propose to extend results on the interpolation theory for scalar functions to the case of diff... more We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in P - r k (T) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree r ≥ 1 , from their weights, namely their integrals on k-chains. These inte- grals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from "good" sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary numerical results for edge elements support the nonuniform choice, in agreement with the well-known nodal case.
Journal of Scientific Computing, Jun 17, 2023
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, 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.
Journal of Computational Physics, Apr 1, 2017
Existing finite element implementations for the computation of free-boundary axisymmetric plasma ... more 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. 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 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 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 outside this region. The continuity of the numerical solution in the region of overlap is weakly enforced by a mortar-like mapping Key-words: axisymmetric plasma equilibria in tokamaks; domain decomposition mortar method; overlapping meshes; linear and cubic We choose a semi-circle Γ of radius ρ Γ surrounding the iron domain Ω Fe and the coil domains Ω ci . The truncated domain, we use for the computations, is denoted Ω ⊂ Ω ∞ , with boundary ]) be the space of vector fields in L 2 (Ω × [0, 2π]) 3 with curl in L 2 (Ω × [0, 2π]) 3 . We remark that: r -1 ψe φ ∈ H(curl, Ω × [0, 2π]) if and only if ψ ∈ H 1 (Ω).
arXiv (Cornell University), May 15, 2016
This paper combines the use of high order finite element methods with parallel preconditioners of... more This paper combines the use of high order finite element methods with parallel preconditioners of domain decomposition type for solving electromagnetic problems arising from brain microwave imaging. The numerical algorithms involved in such complex imaging systems are computationally expensive since they require solving the direct problem of Maxwell's equations several times. Moreover, wave propagation problems in the high frequency regime are challenging because a sufficiently high number of unknowns is required to accurately represent the solution. In order to use these algorithms in practice for brain stroke diagnosis, running time should be reasonable. The method presented in this paper, coupling high order finite elements and parallel preconditioners, makes it possible to reduce the overall computational cost and simulation time while maintaining accuracy.
International Journal of Numerical Modelling-electronic Networks Devices and Fields, Feb 15, 2017
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, 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.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 12, 2021
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific r... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, 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.
SIAM Journal on Numerical Analysis, 2003
In this paper, we focus on the representation of a divergence-free vector field, defined, on a co... more In this paper, we focus on the representation of a divergence-free vector field, defined, on a connected nonsimply connected domain Ω ⊂ R 3 with a connected boundary Γ, by its curl and its normal component on the boundary. The considered problem is discretized with H(curl)-and H(div)-conforming finite elements. In order to ensure the uniqueness of the vector potential, we propose a spanning tree methodology to identify the independent edges. The topological features of the domain under consideration are analyzed here by means of the homology groups of first and second order.
Journal of Scientific Computing
We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak... more We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak. We adopt a non-overlapping mortar element approach, that couples C 0 piece-wise linear Lagrange finite elements in a region that does not contain the plasma and C 1 piece-wise cubic reduced Hsieh-Clough-Tocher finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal tokamak section. The inclusion of ferromagnetic parts is simplified by assuming that they fit within the axisymmetric modeling and a new formulation of the Newton algorithm for the problem solution is stated.
Calcolo
We propose to extend results on the interpolation theory for scalar functions to the case of diff... more We propose to extend results on the interpolation theory for scalar functions to the case of differential k-forms. More precisely, we consider the interpolation of fields in $${\mathcal P}^-_r \varLambda ^k(T)$$ P r - Λ k ( T ) , the finite element spaces of trimmed polynomial k-forms of arbitrary degree $$r \ge 1$$ r ≥ 1 , from their weights, namely their integrals on k-chains. These integrals have a clear physical interpretation, such as circulations along curves, fluxes across surfaces, densities in volumes, depending on the value of k. In this work, for $$k=1$$ k = 1 , we rely on the flexibility of the weights with respect to their geometrical support, to study different sets of 1-chains in T for a high order interpolation of differential 1-forms, constructed starting from “good” sets of nodes for a high order multi-variate polynomial representation of scalar fields, namely 0-forms. We analyse the growth of the generalized Lebesgue constant with the degree r and preliminary nume...
HAL (Le Centre pour la Communication Scientifique Directe), Nov 10, 2021
We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak... more We consider the axisymmetric formulation of the equilibrium problem for a hot plasma in a tokamak. We adopt a non-overlapping mortar element approach, that couples C 0 piece-wise linear Lagrange finite elements in a region that does not contain the plasma and C 1 piece-wise cubic reduced Hsieh-Clough-Tocher finite elements elsewhere, to approximate the magnetic flux field on a triangular mesh of the poloidal tokamak section. The inclusion of ferromagnetic parts is simplified by assuming that they fit within the axisymmetric modeling and a new formulation of the Newton algorithm for the problem solution is stated.
J, 2022
We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of... more We recall the classical tree-cotree technique in magnetostatics. (1) We extend it in the frame of high-order finite elements in general domains. (2) We focus on its connection with the question of the invertibility of the final algebraic system arising from a high-order edge finite element discretization of the magnetostatic problem formulated in terms of the magnetic vector potential. With the same purpose of invertibility, we analyse another classically used condition, the Coulomb gauge. (3) We conclude by underlying that the two gauges can be naturally considered in a high order framework without any restriction on the topology of the domain.
Mortaring the two-dimensional edge finite elements for the discretization of some electromagnetic... more Mortaring the two-dimensional edge finite elements for the discretization of some electromagnetic models∗
Journal of Computational Physics, 2021
The numerical simulation of the equilibrium of the plasma in a tokamak as well as its self-consis... more The numerical simulation of the equilibrium of the plasma in a tokamak as well as its self-consistent coupling with resistive diusion should benet from higher regularity of the approximation of the magnetic ux map. In this work, we propose a nite element approach on a triangular mesh of the poloidal section, that couples piece-wise linear nite elements in a region that does not contain the plasma and reduced Hsieh-Clough-Tocher nite elements elsewhere. This approach gives the exibility to achieve easily and at low cost higher order regularity for the approximation of the ux function in the domain covered by the plasma, while preserving accurate meshing of the geometric details in the rest of the computational domain. The continuity of the numerical solution at the coupling interface is weakly enforced by mortar projection. A new technique for the computation of the geometrical coecients is also presented.
2017 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, 2017
This paper deals with microwave tomography for brain stroke imaging using state-of-the-art numeri... more This paper deals with microwave tomography for brain stroke imaging using state-of-the-art numerical modeling and massively parallel computing. Iterative microwave tomographic imaging requires the solution of an inverse problem based on a minimization algorithm (e.g. gradient or Newton-like methods) with successive solutions of a direct problem. The solution direct requests an accurate modeling of the wholemicrowave measurement system as well as the as the whole-head. Moreover, as the system will be used for detecting brain strokes (ischemic or hemorrhagic) and for monitoring during the treatment, running times for the reconstructions should be fast. The method used is based on high-order finite elements, parallel preconditioners with the Domain Decomposition method and Domain Specific Language with open source FreeFEM++ solver.
Fekete-Gauss TSEM and application to incompressible flows
International audienc
High Order Edge Elements for Electromagnetic Waves: Remarks on Numerical Dispersion
Lecture Notes in Computational Science and Engineering, 2017
We recall one set of possible basis vector fields and two different sets of possible degrees of f... more We recall one set of possible basis vector fields and two different sets of possible degrees of freedom, those related to “small-edges” and those defined by “moments”, for the Nedelec’s first family of high order edge elements. We thus address a dispersion analysis of the resulting methods, when the time-harmonic Maxwell’s equation for the electric field is discretized on a simplicial mesh.
Spectral element method on simplicial meshes for incompressible Navier-Stokes flows
International audienc
Journal of Mathematical Study, 2018
We investigate the cubature points based triangular spectral element method and provide accuracy ... more We investigate the cubature points based triangular spectral element method and provide accuracy results for elliptic problems in non polygonal domains using various isoparametric mappings. The capabilities of the method are here again clearly confirmed.
Journal of Computational Physics, 2017
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Papers by Francesca Rapetti