ANR JCJC φ-FEM (2022-2026)

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 junior 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

Publications

  • [9] Residual-based a posteriori error estimates with boundary correction for φ-FEM R. Becker, R. Bulle, M. Duprez, V. Lleras, preprint (hal), code.
  • [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, preprint (hal), code.
  • [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

The code developed in this project is openly available at:

https://github.com/PhiFEM

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

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