2022-04741 - Post-Doctoral Research Visit F/M High-order polyhedral methods for eddy current testing simulation
Le descriptif de l’offre ci-dessous est en Anglais

Type de contrat : CDD

Niveau de diplôme exigé : Thèse ou équivalent

Fonction : Post-Doctorant

A propos du centre ou de la direction fonctionnelle

The Inria Lille - Nord Europe research center, created in 2008, employs 360 people including 305 scientists in 15 research teams. Recognized for its strong involvement in the socio-economic development of the Hauts-De-France region, the Inria Lille - Nord Europe research center pursues a close relationship with large companies and SMEs. By promoting synergies between researchers and industrialists, Inria participates in the transfer of skills and expertise in digital technologies and provides access to the best European and international research for the benefit of innovation and companies, particularly in the region.

For more than 10 years, the Inria Lille - Nord Europe center has been located at the heart of the university and scientific ecosystem in Lille, as well as at the heart of Frenchtech, with a technology showroom based on avenue de Bretagne in Lille, on the site of economic excellence dedicated to information and communication technologies (ICT) that is EuraTechnologies.

Contexte et atouts du poste


This two-year post-doctoral position falls into the framework of a partnership between Inria, RAPSODI team (Lille) and EDF, ERMES department (Palaiseau). The successful candidate will spend one half of his/her time at Inria and the other half at EDF.

Supervision: Simon LEMAIRE (Inria) & Jean-Pierre DUCREUX and Jérémy DALPHIN (EDF)


Scientific context:

Eddy current testing (ECT) is a nondestructive assessment method based on electromagnetic induction that enables to detect and characterize flaws in conductive materials. In its most basic form, an ECT device consists in a coil of conductive wire excited with an alternating electrical current. This device is brought close to the conductive material to assess. The wire coil produces an alternating magnetic field in the material, which gives raise to currents (the so-called eddy currents) that are opposite to those in the coil (Lenz–Faraday’s law). The resulting impedance changes in the coil directly correlate to the local structure of the material under assessment. They can therefore be used to detect defects in the material. At EDF, ECT is used (among other applications) to assess the mechanical integrity of heat exchanger tubes in nuclear plants. In practice, in order to calibrate and validate the ECT devices that are employed, engineers rely on modeling and numerical simulation [1]. For a given configuration, i.e., for a given ECT device and a given conductive material to assess (featuring given defects), one has to solve numerically potential formulations of the (linear) 3D Maxwell’s equations in the time-harmonic regime. Prior to this, defects must be accounted for in the 3D meshing process. At EDF, these computations are currently performed using code_Carmel, a Fortran 90 platform conjointly developed by EDF and the L2EP of the University of Lille (in the framework of the common laboratory LAMEL). code_Carmel is essentially based on lowest-order (nodal-based) H^1-conforming and (edge-based) H(curl)-conforming finite elements on matching tetrahedral meshes.

When numerically simulating ECT with code_Carmel, engineers at EDF face two main difficulties:

• even on relatively fine grids, the magnitude of the control signal (based on which the presence of a defect is inferred) is comparable with the magnitude of the noise stemming from the discretization (numerical error);

• when the flaw inside the material under assessment has a complex topology (think, e.g., of a network of cracks), the 3D meshing process using matching tetrahedra is challenging.

In practice, to attenuate the noise on the control signal, a lock-step technique is employed. It consists in performing a pre-simulation without the defect, in order to generate a control signal subsequently taken as a reference for the simulation with defect. Even though this trick seems to work quite well in practice, there is no theoretical guarantee that such a technique is indeed able to reduce the noise on the quantity of interest. As of the meshing issue, this does restrict in practice the simulations to quite simple flaw topologies.


[1] T. Henneron, Y. Le Menach, F. Piriou, O. Moreau, S. Clénet, J.-P. Ducreux, and J.-C. Vérité, Source Field Computation in NDT Applications, IEEE Transactions on Magnetics, 43(4):1785–1788, 2007.

Mission confiée


The goal of the present project is to address at once both of the issues raised above, by taking advantage of the potential of Hybrid High-Order (HHO) methods [2]. HHO methods provide a novel framework for the discretization of models based on PDEs. Their construction hinges on polynomial unknowns attached to both the mesh cells and the mesh faces. HHO methods have several assets: (i) their capability of handling general polyhedral meshes, (ii) their arbitrary approximation order, and (iii) their reduced computational cost (after static condensation). They have already been applied in a large variety of contexts [3, 4], including recently to magnetostatics [5]. In practice, the possibility of resorting to general polyhedral cells in the meshing process turns out to be very useful when it comes to meshing domains with embedded interfaces, or to perform local mesh refinement/coarsening. Hanging nodes, that are not supported by standard conforming finite element methods, are indeed seamlessly handled in the polyhedral context. This geometrical flexibility will here be leveraged both to ease the meshing process (also for local mesh refinement) and to allow for the treatment of flaws with complex topologies. High-order methods also provide a natural way to increase the accuracy of the approximation for a given mesh size. This is essentially true in those regions of the simulation domain where the solution is regular. In the vicinity of geometric singularities, local mesh refinement and/or local ad hoc space enrichment shall be considered instead. Such strategies will here be leveraged to efficiently reduce the noise on the quantities of interest.


[2] D. A. Di Pietro, A. Ern, and S. Lemaire, An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators, Comput. Methods Appl. Math., 14(4):461–472, 2014.

[3] D. A. Di Pietro and J. Droniou, The Hybrid High-Order Method for Polytopal Meshes, vol. 19 of Modeling, Simulation and Applications, Springer International Publishing, 2020.

[4] M. Cicuttin, A. Ern, and N. Pignet, Hybrid High-Order methods. A primer with applications to solid mechanics, SpringerBriefs in Mathematics, Springer, Cham, 2021.

[5] F. Chave, D. A. Di Pietro, and S. Lemaire, A discrete Weber inequality on three-dimensional hybrid spaces with application to the HHO approximation of magnetostatics, Math. Models Methods Appl. Sci., 32(1):175–207, 2022.

Principales activités

Planned workflow:

The work will be devided into three main tasks.

1) In a first time, HHO methods will be devised and analyzed for simple ECT configurations, i.e., for configurations featuring flaws that do not induce topological changes. The design of methods in the time-harmonic regime will rely on the preliminary work [5] addressing magnetostatics.

2) In a second time, the case of crack-type flaws, and more broadly of non-trivial topologies, will be tackled. In this case, the well-posedness of HHO methods hinges on discrete Weber inequalities. Their proof is difficult owing to the non-compatibility of the HHO setting. It may be necessary in that case to resort to the Discrete De Rham (DDR) approach [6]. Introduced very recently, DDR methods, at the price of being computationally more expensive than HHO methods, advantageously offer a compatible setting.

3) In a third time, and if time allows, local ad hoc space enrichment strategies will be investigated to increase the accuracy of the approximation in the vicinity of singularities (crack tips for example). The ideas introduced in [7, 8] on a toy problem will have to be extended to the electomagnetics context.

The implementation of the methods will be performed in a C++ prototype platform developed by the academic members of the project. The methods and algorithms will be validated on representative applications of interest for EDF from the nondestructive testing community.


[6] D. A. Di Pietro and J. Droniou, An arbitrary-order discrete de Rham complex on polyhedral meshes: Exactness, Poincaré inequalities, and consistency, Found. Comput. Math., in press.

[7] E. Artioli and L. Mascotto, Enrichment of the nonconforming virtual element method with singular functions, Comput. Methods Appl. Mech. Eng., 385:114024, 2021.

[8] L. Yemm, Design and analysis of the Extended Hybrid High-Order method for the Poisson problem, preprint arXiv:2104.14843, 2021.


The successful candidate will hold a PhD in applied mathematics (numerical analysis/scientific computing) or in computational electromagnetics, and have a solid knowledge of C++.


  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • Access to vocational training
  • Social security coverage


Grossly salary per month : 2653 €