ANR JCJC φ-FEM (2022-2027)
Every piece of code developed in the project is open and gathered at github.com/PhiFEM: the implementations that accompany each article, and the phiFEM package, which brings the method to FEniCSx — installable from PyPI and now running in parallel.
Overview
Title
ϕ-FEM: development of a Finite Element Method for the design of real-time digital twins in surgery
Principal investigator
Michel Duprez
Budget
€320k
ANR
ANR-22-CE46-0003
Abstract
ϕ-FEM is a recently proposed finite element method for the efficient numerical solution of partial differential equations posed in domains of complex shapes, using simple regular meshes. The main goal of this project is to further develop ϕ-FEM turning it into a tool for efficient, patient-specific and real-time simulations of human organs. To reach this objective, we shall adapt ϕ-FEM to the equations appropriate to biomechanics, provide an efficient implementation for it allowing for the use of actual organ geometries, and finally combine it with convolution neural networks to make it real time after training. The ultimate, long-term, goal is thus to contribute to the construction of digital twins of organs able to guide the surgical act in real time using information acquired before the operation and to reduce the costs of a medical doctors’ training by working on visual organs. The innovation of ϕ-FEM lies in its ability to combine the ease of implementation of classical immersed boundary methods with the accuracy of more recent CutFEM/XFEM approaches. It incorporates, by its very construction, the popular description of geometry by Level Set functions, which can represent the real geometry with whatever accuracy desired which makes this approach numerically less expensive than classical finite element methods. The ϕ-FEM paradigm will also be used to develop efficient registration algorithms. Our results will be integrated into the open-source SOFA platform developed in the MIMESIS team to facilitate dissemination.
Members
- Michel Duprez Inria senior researcher, University of Strasbourg (coordinator)
- Alexei Lozinski Professor, Laboratoire de Mathématiques de Besançon, Université de Bourgogne Franche Comté
- Vanessa Lleras Associate professor, Institut de Mathématiques Alexander Grothendieck (IMAG), University of Montpellier
- Stéphane Cotin Inria senior researcher, University of Strasbourg
- Yannick Privat Professor, Institut de Recherche en Mathématiques Appliquées (IRMA), University of Strasbourg
- Stéphane Bordas Professor, University of Luxembourg
- Vincent Vigon, associate professor, Institut de Recherche en Mathématiques Appliquées (IRMA), University of Strasbourg
Preprints
- A φ-FEM approach for time-dependent domains with unfitted meshes M. Duprez, J. Marguet, A. Lozinski, I. Voulis, preprint (hal), code.
- A multigrid and neural network approach to reduce the computational cost of φ-FEM R. Bulle, M. Duprez, V. Lleras, K. Vuillemot, preprint (hal).
- A penalized φ-FEM scheme for the Poisson Dirichlet problem R. Bulle, M. Duprez, V. Lleras, K. Vuillemot, preprint (hal).
- Residual-based a posteriori error estimates with boundary correction for φ-FEM R. Becker, R. Bulle, M. Duprez, V. Lleras, preprint (hal), code.
Publications
- [8] Enriching continuous Lagrange finite element approximation spaces using neural networks H. Barucq, M. Duprez, F. Faucher, E. Franck, F. Lecourtier, V. Lleras, V. Michel-Dansac, N. Victorion, ESAIM: M2AN, 60 2 (2026) 541-603, preprint (hal), code, pdf (journal).
- [7] φ-FEM-FNO: a new approach to train a Neural Operator as a fast PDE solver for variable geometries M. Duprez, V. Lleras, A. Lozinski, V. Vigon and K. Vuillemot. Communications in Nonlinear Science and Numerical Simulation, Volume 152, Part A, January 2026, 10913, preprint (hal), code, pdf (journal)
- [6] φ-FD : A well-conditioned finite difference method inspired by φ-FEM for general geometries on elliptic PDEs M. Duprez, V. Lleras, A. Lozinski, V. Vigon and K. Vuillemot, Journal of Scientific Computing, Volume 104, article number 23, (2025), preprint (hal), code, pdf (journal)
- [5] φ-FEM for the heat equation: optimal convergence on unfitted meshes in space M. Duprez, V. Lleras, A. Lozinski and K. Vuillemot. Comptes Rendus. Mathématique, Volume 361 (2023) p. 1699-1710, preprint (hal), code (github), pdf (journal)
- [4] φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations M. Duprez, V. Lleras and A. Lozinski. ESAIM: M2AN 57 (2023), 1111-1142, preprint (hal), pdf (journal), code (github)
- [3] A new φ-FEM approach for problems with natural boundary conditions M. Duprez, V. Lleras and A. Lozinski. Numer. Methods Partial Differ. Eq. 39 (2023) n°1, 281-303., preprint (hal), pdf (journal), code (github).
- [2] φ-FEM: an efficient simulation tool using simple meshes for problems in structure mechanics and heat transfer S. Cotin, M. Duprez, V. Lleras, A. Lozinski, K. Vuillemot. Partition of Unity Methods (Wiley Series in Computational Mechanics) 2023,preprint (hal), code (github), pdf (journal).
- [1] φ-FEM: a finite element method on domains defined by level-sets M. Duprez and A. Lozinski. SIAM J. Numer. Anal. 58 (2020), no. 2, pp. 1008-1028,preprint (hal), pdf (journal), code (github).
Code
All the project’s code is openly available at github.com/PhiFEM under a free licence: every published article has its implementation repository there, and the phiFEM package provides the method ready to use in FEniCSx, installable from PyPI and usable in parallel.
Events organised
4-8 Sept. 2023
Mini-symposium at the ENUMATH conference
https://enumath2023.com/
Location: Lisbon
Organiser: Vanessa Lleras
Number of speakers: 6
Title: Approximated boundary methods: modelling, mathematical analysis and simulations
Talks at conferences and seminars
10 Sept. 2026
DMV-Jahrestagung
https://www.dmv-jahrestagung.de/
Location: Konstanz (Germany)
Speaker: Michel Duprez
Title: Recent advances in φ-FEM: an immersed finite element method for domains defined by level sets
19 July 2026
WCCM conference
Location: Munich (Germany)
Speaker: Michel Duprez
Title: Recent advances in φ-FEM: an immersed finite element method for domains defined by level sets
25 Feb. 2026
FEniCS conference
Location: Paris
Speaker: Michel Duprez
Title: φ-FEM: an immersed boundary finite element method for geometries defined by a level-set
Poster: Shape optimization using φ-FEM — R. Bulle, S. Cotin, L. Ducongé, M. Duprez, J. Diaz-Avalos, A. Laurain, hal
1 Sept. 2025
ENUMATH conference
Location: Heidelberg (Germany)
Speaker: Michel Duprez
Title: φ-FEM: a fictitious domain method for FEM on domains defined by level-sets
24 June 2025
« The 30th Biennial Numerical Analysis Conference »
Location: Glasgow (UK)
Speaker: Michel Duprez
Title: φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations
26 May 2025
« Coupled Problems »
Location: Villasimius (Sardinia, Italy)
Speaker: Michel Duprez
Title: φ-FD: a finite difference scheme with an optimal convergence for elliptic PDEs on domains defined by a level-set function
26 Feb. 2025
Séminaire Calcul Scientifique et Modélisation, IMB
https://www.math.u-bordeaux.fr/imb/seminaire-calcul-scientifique-et-modelisation
Location: Bordeaux
Speaker: Michel Duprez
Title: φ-FEM: a fictitious domain method for finite element methods on domains defined by level-sets
3-7 June 2024
ECCOMAS conference
https://eccomas2024.org/
Location: Lisbon
Speaker 1: Michel Duprez
Title 1: A finite difference scheme with an optimal convergence for elliptic PDEs on domains defined by a level-set function
Speaker 2: Vanessa Lleras
Title 2: A new immersed boundary method (φ-FEM) to treat Stokes equations and particulate flows
Speaker 3: Killian Vuillemot
Title 3: A φ-FEM approach to train a FNO for variable geometries
Speaker 4: Alexei Lozinski
Title 4: φ-FEM: a fictitious domain finite element method on geometries defined by level-sets (keynote)
5-6 Feb. 2024
“MACARON seminar”
https://sites.google.com/view/jnb2024
Location: Belmont (France)
Speaker: Michel Duprez
Title: A phi-FEM approach to train a Neural Operator as a fast PDE solver for variable geometries
29-30 Jan. 2024
“Journées Numériques de Besançon”
https://sites.google.com/view/jnb2024
Location: Besançon
Speaker: Michel Duprez
Title: Phi-FEM : an immersed boundary finite element method for domains defined by level-set functions
15-19 Jan. 2024
“WONAPDE”
https://www.ci2ma.udec.cl/wonapde2024/
Location: Concepción (Chile)
Speaker 1: Vanessa Lleras
Title 1: An unfitted method (φ-FEM) combined with deep learning: variable geometries and correction
Speaker 2: Michel Duprez
Title 2: φ-FEM: an optimally convergent and easily implementable immersed boundary method for particulate flows and Stokes equations
9 Nov. 2023
“CSM seminar”
https://www.math.u-bordeaux.fr/imb/seminaire-calcul-scientifique-et-modelisation
Location: Bordeaux
Speaker: Vanessa Lleras
Title: phiFEM, a new non-conforming finite element method
4-8 Sept. 2023
ENUMATH conference
https://enumath2023.com/
Location: Lisbon
Speaker 1: Michel Duprez
Title 1: ϕ-FEM: a immersed finite element method on domains defined by level-sets to solve elliptic PDEs
Speaker 2: Vanessa Lleras
Title 2: A ϕ-FEM approach with deep learning and varying geometry
15 June 2023
“PDE seminar”
https://www-ljk.imag.fr/spip.php?article10
Location: Grenoble
Speaker: Michel Duprez
Title: phiFEM, a new non-conforming finite element method
13 June 2023
“MACS seminar”
https://www.umpa.ens-lyon.fr/seminaires/
Location: Lyon
Speaker: Vanessa Lleras
Title: phiFEM, a new non-conforming finite element method
5-7 June 2023
“COUPLED 2023”
https://coupled2023.cimne.com/
Location: Crete
Speaker: Alexei Lozinski
Title: phi-FEM for creeping flows around moving rigid particles
22-26 May 2023
“SMAI 2023”
https://smai2023.math.cnrs.fr/fr/
Location: Le Gosier (Guadeloupe)
Speaker: Vanessa Lleras
Title: Combining the non-conforming finite element method phi-FEM with neural networks
3 May 2023
“Seminario de EDP e matemática aplicada”
https://edpemaplicada.uff.br/
Location: online
Video: https://www.youtube.com/watch?v=fqnyAprIWCk
Speaker: Michel Duprez
Title: A fictitious domain method for finite element methods on domains defined by level-sets
25-28 April 2023
“22nd IACM Computational Fluids Conference – CFC 2023”
https://cfc2023.iacm.info
Location: Cannes
Speaker: Michel Duprez
Title: phi-FEM: a finite element method on domains defined by level-sets
18 April 2023
“ACSIOM seminar”
https://imag.umontpellier.fr/?page_id=625&idsem=8
Location: Montpellier
Speaker: Vanessa Lleras
Title: phiFEM, a new non-conforming finite element method
26 February – 3 March 2023
“SIAM Conference on Computational Science and Engineering”
https://www.siam.org/conferences/cm/conference/cse23
Location: Amsterdam
Speaker: Alexei Lozinski
Title: phi-FEM for creeping flows around moving rigid particles
15 Nov. 2022
“Numerical methods for fluid, structure and interactions problems”
https://indico.math.cnrs.fr/event/7591/
Location: Toulouse
Speaker: Alexei Lozinski
Title: phi-FEM, an optimally convergent fictitious domain method and its applications to flows around moving rigid particles