FESTIM v2.0: Upgraded framework for multi-species hydrogen transport
Remi Delaporte-Mathurin, James Dark, Jorgen Dokken, Huihua Yang, Kaelyn Dunnell, Chirag Khurana, Ethan Peterson
FEniCS 2026 | June, 2026 | Paris, France
Nuclear Fusion
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+
Deuterium
Tritium
+
+
Helium
neutron
Fusion
The tritium issue
The tritium issue
The maths behind FESTIM
\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}
\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t}) - p_i c_\mathrm{t}
Diffusion-reaction
- D_1 \nabla c_\mathrm{m,1} = - D_2 \nabla c_\mathrm{m,2}
\frac{c_1}{K_{1}} =\frac{c_2}{K_{2}}
\frac{c_1}{K_{1}} =\left( \frac{c_2}{K_{2}} \right) ^ {2}
or
At interfaces
Hard
Very hard!
Flux continuity
Conservation of chemical potential
FESTIM Evolution Timeline
2019 - 2024
FESTIM ≤ 1
- Based on Legacy FEniCS
- Initial development and validation
- Established core H-transport models
2024 - Today
FESTIM 2
- Migration to DOLFINx
- New multi-species paradigm
- Improved performance & scalability
- Modern FEniCS ecosystem integration
FESTIM2 in a nutshell github.com/festim-dev/festim
User friendly - Intuitive, Python API easy to use
Apache 2.0 license - Suitable for academic and commercial
Easy install - conda install -c conda-forge festim
Online tutorials - Dozens of tutorials
Online documentation - Auto-generated
8,000+ forum views/month - Public, searchable Q&A
27+ institutions - Adopted by universities, labs and industry
30+ contributors - Open to new developers and users
Monthly dev meetings - Community-driven roadmap
130+ Slack users - Active developer and user support
Built on FEniCSx - Proven FEM backend, 1.5M+ downloads
5 team members at MIT - more around the world
On GitHub - Transparent, version-controlled development
NumFOCUS affiliation - Backed by a sustainability non-profit
CI/CD workflows - Rigorous testing and streamlined releases
Performance
FEniCSx - JIT to fast C++ rapid assembly
High-performance solvers - PETSc back-end
Built for HPC - Native MPI, runs on 1000’s of cores
Tuned control - Custom meshes, solvers, finite elements
Fast I/O - Parallel mesh loading, low memory footprint
Usability
Sustainability
Community
Examples
The FESTIM team at MIT and beyond
James Dark
Postdoctoral associate
Huihua Yang
Postdoctoral associate
Rémi Delaporte-Mathurin
Research Scientist
Kaelyn Dunnell
Graduate Student
Chirag Khurana
Graduate Student
MIT Team
Joe Dean
Research Associate
University of Cambridge
University of Cambridge
Jørgen Dokken
Senior Research Scientist
Simula Research Laboratory
Simula Research Laboratory
FEniCS collaborators
FESTIM Community
31+ contributors
FESTIM is used worldwide
Docs users
Fusion machine developers
Research organisations
Average person in this room
- knows maths
- knows how to code
Us
- can't read greek
- Not FEM experts
\nu \nabla \tau \eta \xi \delta
import festim as F
import numpy as np
my_model = F.HydrogenTransportProblem()
my_model.mesh = F.Mesh1D(vertices=np.linspace(0, 1, num=1001))
mat = F.Material(D_0=1, E_D=0.0)
volume_subdomain = F.VolumeSubdomain1D(id=1, borders=[0, 1], material=mat)
boundary_left = F.SurfaceSubdomain1D(id=1, x=0)
boundary_right = F.SurfaceSubdomain1D(id=2, x=1)
my_model.subdomains = [volume_subdomain, boundary_left, boundary_right]
H = F.Species("H")
my_model.species = [H]
my_model.temperature = 300
my_model.boundary_conditions = [
F.FixedConcentrationBC(subdomain=boundary_left, value=1, species=H),
F.FixedConcentrationBC(subdomain=boundary_right, value=0, species=H),
]
my_model.settings = F.Settings(atol=1e-10, rtol=1e-10, final_time=2)
my_model.settings.stepsize = F.Stepsize(0.05, milestones=[0.05, 0.1, 0.2, 0.5, 1]) # s
profile = F.Profile1DExport(field=H, subdomain=volume_subdomain, times=my_model.settings.stepsize.milestones)
my_model.exports = [
F.VTXSpeciesExport(
field=H,
filename="hydrogen_concentration.bp",
),
profile,
]
my_model.initialise()
my_model.run()Before FEniCSx....
\frac{c_1}{K_{1}} =\frac{c_2}{K_{2}}
\theta =\frac{c}{K}
\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) + S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}
\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ c_\mathrm{m} \ (n_\mathrm{trap,i} - c_\mathrm{t}) - p_i c_\mathrm{t}
\theta_1 =\theta_2
\frac{\partial (\theta K)}{\partial t} = \nabla \cdot (D \nabla (\theta K) )+ S - \sum \frac{\partial c_\mathrm{t,i}}{\partial t}
\frac{\partial c_\mathrm{t, i}}{\partial t} = k_i \ (\theta K) \ (n_\mathrm{trap,i} - c_\mathrm{t}) - p_i c_\mathrm{t}
Change of variable method
With FEniCSx
Mixed formulation
With FEniCSx
Mixed formulation
Submesh 1
Submesh 2
With FEniCSx
Mixed formulation
\partial_t c_1 = \nabla \cdot (D \nabla c_1) + S
\partial_t c_2 = \nabla \cdot (D \nabla c_2) + S
\frac{c_1}{K_{1}} =\frac{c_2}{K_{2}}
1. Penalty method
- \int_{\Gamma} \langle D \nabla c^n\rangle \cdot \mathbf{n} \llbracket v \rrbracket - \langle D \nabla v \rangle\cdot\mathbf{n} \llbracket \frac{c^n}{K}\rrbracket + \frac{\gamma}{\langle h\rangle}\llbracket \frac{c^n}{K} \rrbracket\llbracket v \rrbracket ~\mathrm{d}s
2. Nitsche method
- \int_{\Gamma} \gamma \left( \frac{c_1}{K_1} - \frac{c_2}{K_2} \right) \llbracket v \rrbracket ~\mathrm{d}s
Dark, James, Rémi Delaporte-Mathurin, Jørgen S. Dokken, et al. “FESTIM v2.0: Upgraded Framework for Multi-Species Hydrogen Transport and Enhanced Performance.” International Journal of Hydrogen Energy 220 (March 2026): 153987. https://doi.org/10.1016/j.ijhydene.2026.153987.
Please redirect complex maths questions to this guy!
With FEniCSx
Mixed formulation
c_1
c_2
With FEniCSx
Mixed formulation
- No need for change of variable
- but requires a penalty term
- Provides flexibility on interface physics
- Flux interface conditions
- Multi-species reactions
The mixed formulation is great for multiphysics
velocity, temperature, turbulent viscosity...
tritium production
Heat source
Temperature
Neutronics
Thermo-hydraulics
Tritium transport
Using FESTIM for estimating tritium contamination in a heat exchanger
Thank you for your attention!
Any questions?
FEniCS 2026
By Remi Delaporte-Mathurin
