FESTIM: empowering tritium transport modelling

Remi Delaporte-Mathurin

James Dark

Why FESTIM

The tritium issue

  • Radioactive substance
  • Health risks

 

→ Limit inventories

→ Detritiation

→ Accurate assessment

Safety hazards

Risk of contamination

  • Permeation to cooling systems
  • Challenges for maintenance and safety

 

→ Permeation barriers

→ Clean-up

Tritium self-sufficiency

  • Optimised fuel cycle is required for a fusion economy

 

 

→ Breeding blanket design

→ Sub-systems engineering

 

 

How to analyse this thermo-desorption spectrum?

What's the T inventory in the first wall of SPARC?

How much T will permeate to the coolant?

How long do we need to bake components for detritiation?

What is the extraction efficiency of the TES?

What is the T residence time of component X?

Hydrogen transport

Hydrogen transport

McNabb & Foster model

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) - \frac{\partial c_{\mathrm{t}, i}}{\partial t}
\frac{\partial c_{\mathrm{t}, i}}{\partial t} = k \ c_\mathrm{m} \ (n - c_{\mathrm{t}, i}) - p \ c_{\mathrm{t}, i}

Mobile H concentration

Trapped H concentration

\rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (\lambda \nabla T)

Temperature dependence

McNabb & Foster model

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) - \frac{\partial c_{\mathrm{t}, i}}{\partial t}
\frac{\partial c_{\mathrm{t}, i}}{\partial t} = k \ c_\mathrm{m} \ (n - c_{\mathrm{t}, i}) - p \ c_{\mathrm{t}, i}

Time derivatives

McNabb & Foster model

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) - \frac{\partial c_{\mathrm{t}, i}}{\partial t}
\frac{\partial c_{\mathrm{t}, i}}{\partial t} = k \ c_\mathrm{m} \ (n - c_{\mathrm{t}, i}) - p \ c_{\mathrm{t}, i}

Diffusive term

McNabb & Foster model

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) - \frac{\partial c_{\mathrm{t}, i}}{\partial t}
\frac{\partial c_{\mathrm{t}, i}}{\partial t} = k \ c_\mathrm{m} \ (n - c_{\mathrm{t}, i}) - p \ c_{\mathrm{t}, i}

Trapping term

McNabb & Foster model

\frac{\partial c_\mathrm{m}}{\partial t} = \nabla \cdot (D \nabla c_\mathrm{m}) - \frac{\partial c_{\mathrm{t}, i}}{\partial t}
\frac{\partial c_{\mathrm{t}, i}}{\partial t} = k \ c_\mathrm{m} \ (n - c_{\mathrm{t}, i}) - p \ c_{\mathrm{t}, i}

Detrapping term

Surface physics

Molecular dissociation and recombination

\varphi = K_d \ P - K_r c_\mathrm{m}^2

Sievert's law of solubility

c_\mathrm{m} = K_S \ \sqrt{P}
c_\mathrm{m} = K_H \ P

Henry's law of solubility

c_\mathrm{m} = \frac{\varphi_{\mathrm{imp}} R_p}{D} + \sqrt{\frac{\varphi_{\mathrm{imp}}+K_d P}{K_r}}

Plasma + gas exposure

Application: ITER PFCs

We need a numerical tool

W

Cu

CuCrZr

Particle and heat fluxes

Convection

14 mm

Problem: predict T retention and permeation

 

  • Multi-dimensional
  • Multi-material
  • non homogeneous temperature

What is FESTIM 

User inputs

  • Material properties
  • Trap properties
  • Geometry
  • Boundary conditions
  • Initial conditions
  • ...

Heat transfer model

T

Hydrogen transport model

  • McNabb & Foster
  • Multi-level trapping
  • Multi-isotopes
  • ...

FESTIM

Outputs

  • H concentration fields \(c(x,t)\)
  • Temperature field \(T(x,t)\)
  • surface fluxes
  • inventories
  • average concentration
  • ...

2022

2019

Start of development

  • 1D/2D/3D
  • Finite elements
  • Python
  • multi-material
  • Heat transfer

History of FESTIM

open-source

  • development continuing at MIT

Oct 2023

v1.0 release

2024

Meet the development team

Jonathan Dufour

CEA, France

+ all contributors

Remi Delaporte-Mathurin

MIT, US

@RemDelaporteMathurin

Samuele Meschini

MIT, US

@SamueleMeschini

Etienne Hodille

CEA, France

@ehodille

Gabriele Ferrero

PoliTo, Italy

@gabriele-ferrero

James Dark

CEA, France

@jhdark

Vladimir Kulagin

MEPhI, Russia

@KulaginVladimir

FESTIM is open-source

Check it out on Github

github.com/festim-dev/FESTIM

✅ More transparency

✅ More collaborations

✅ More flexibility

The code is missing a feature?

Just add it!

Found a bug?

Report it and we'll fix it!

FESTIM's development workflow

  • Anyone can create their own copy (fork) of the FESTIM repository and make changes
  • PRs are a place where FESTIM's maintainers review the proposed changes
  • ~500 tests are automatically run
  • The PR is only merged when all the tests pass ✅

fork

fork

pull request

FESTIM

Jane Doe

FESTIM

John Doe

FESTIM

festim-dev

pull request

How to use FESTIM

github.com/festim-dev/festim

User-support forum

festim.discourse.group

User-support forum

Join the Slack channel

Documentation

and installation instructions

festim.readthedocs.io

FESTIM is user-friendly

  • Very easy to learn
  • Plenty of libraries (numpy, scipy, matplotlib, HTM...)
  • Inuitive interface
conda install -c conda-forge fenics
pip install festim

Two lines to install!

import festim as F
import numpy as np

my_model = F.Simulation()

my_model.mesh = F.MeshFromVertices(
    vertices=np.linspace(0, 1e-6, num=1001)
)

my_model.materials = F.Material(id=1, D_0=1.9e-7, E_D=0.2)

my_model.T = 500  # K

my_model.boundary_conditions = [
    F.DirichletBC(
        surfaces=[1, 2],
        value=1e15,  # H/m3/s
        field=0
        )
]

my_model.settings = F.Settings(
    absolute_tolerance=1e10,
    relative_tolerance=1e-10,
    final_time=100  # s
    )


my_model.dt = F.Stepsize(0.1)  # s

my_model.initialise()

my_model.run()

FESTIM workshop

The FESTIM workshop is a series of tutorials

 

from basic problems to more advanced cases

festim.discourse.group

FESTIM is user-friendly

FESTIM is fully documented

Check out the complete documentation at

festim.readthedocs.io

Installation instructions

User guide

  • Development guide
  • Tutorials
  • Theory background
  • API reference

FESTIM is verified & validated

  • Validated against TDS, permeation experiments...
  • Verified against analytical solutions in many different problems
  • New V&V online book

festim-vv-report.readthedocs.io

 

14 verification cases (4 in progress)

6 validation cases (5 in progress)

V&V

Effective-diffusion through a slab

2D multi-material

Exact

Computed concentration

Method of Exact Solution

governing equations

exact solutions

parameters (sources, BCs, ICs)

FESTIM

computed solutions

solve

run

compare

⚠️sometimes very complex!

Method of Manufactured Solutions

governing equations

manufactured solutions

source terms, BCs and ICs

FESTIM

computed solutions

compare

FESTIM is used worldwide

8 private companies

11 universities

18 research organisations

📈5 years of development

📑13+ publications

🗣️110+ citations

🧑‍💻20+ contributors

🏛️26+ institutions using the code

🧑‍💻50+ Slack members

⭐80+ stars on GitHub

FESTIM in numbers

Evolution of GitHub stars

Open source

SOFE workshop

New reference paper

Retention studies

  • Delaporte-Mathurin et al 2024 International Journal of Hydrogen Energy 63 786–802
  • Delaporte-Mathurin et al 2024 Nucl. Fusion 64 026003
  • ITER plasma facing components
  • Transient estimation of tritium retention

Detritiation studies

Delaporte-Mathurin et al 2024 Nucl. Fusion 64 026003

Influence of ELMs on retention

Kulagin, V., & Gasparyan, Y. Numerical Simulation of deuterium retention in tungsten under ELM-like conditions. J. Nucl. Mater. (under revision)

  • 1D model of a ITER monoblock
  • Transient heat transfer simulation
  • Varying surface heat flux

Breeding Blanket modelling

  • DEMO WCLL
  • Complex 3D geometry
  • Coupled to fluid dynamics
  • Tritium generation in the LiPb volume (computed from OpenMC)

James Dark et al 2021 Nucl. Fusion 61 116076

Tritium extraction system

courtesy of K. Dunnell (MIT)

  • Permeation Against Vacuum
  • Complex 3D geometry
  • Coupled with fluid dynamics
  • Tritium extraction from permeable membranes

Tritium extraction system

courtesy of K. Dunnell (MIT)

① neutrons are generated

 

② tritium is created from nuclear reactions

 

③ tritium is transported in the salt

 

④ tritium is released into the gas phase

 

⑤ tritium is collected and counted

BABY breeding experiment

👀 Presentation tomorrow...

BABY breeding experiment

Velocity

Temperature

Tritium concentration

BABY breeding experiment

courtesy of C. Weaver (MIT)

0D model

FESTIM model

Metal Foil Pumps for DIR

  • H is implanted in the first \( 10 \ \mathrm{nm} \)
  • Super-permeation regime is attained at high recombination energy (upstream surface)
  • Source code

Benedikt & Day, (2017) Fusion Engineering and Design

Neutron-induced trap model

  • Custom trapping model
  • Dynamic evolution of trap densities including annealing
  • Parametrised on TDS data from T Schwartz Selinger

 James Dark et al 2024 Nucl. Fusion 64 086026

👀James' talk tomorrow

Permeation experiment

No barrier

with barrier

Permeation barrier

Substrate

High H pressure

Low H pressure

Conservation of chemical potential

Permeation flux

\frac{c^-}{K_S^-} = \frac{c^+}{K_S^+}

FESTIM can inform fuel cycle models

Samuele Meschini et al 2023 Nucl. Fusion 63 126005

Text

Kinetic surface model

to be submitted

New in v1.3!

Workshop time

The workshop

📖Tasks

 

Basic

  • Simple model
  • Simulating a TDS
  • Permeation experiment
  • Permeation barrier
  • Heat transfer
  • CAD-based simulation

 

Advanced

  • Advection-diffusion
  • Fitting a TDS
  • Radioactive decay
  • Soret effect
  • Modelling discontinuities
  • Non cartesian meshes
  • Kinetic Surface model
  • Custom classes

💪Challenges

  • A: Evaluate a PRF
  • B: Fit a TDS spectrum
  • C: Permeation flux through a pipe
  • D: Complex structure analysis

  • E: Compute the extraction efficiency of a PAV extractor

🚀Contribution/Hackathon

Learn how to make contributions and tackle real problems

🚀Contribution/Hackathon

You'll need a GitHub account for this

What is a contribution?

  • Bug fixes
  • New functionality
  • Enhancements
  • Documentation updates/fixes
  • Testing
  • Architecture (CI, CD...)

We curated a list of easy issues for this workshop:

  • Documentation
  • Quality of Life
  • Bug fixes

github.com/festim-dev/FESTIM/milestone/17

  • Anyone can make a copy of the repository and make changes there
     
  • Changes are reviewed by official maintainers via pull-requests
     
  • ~500 tests are run automatically
     
  • Changes are only merged when all the tests pass ✅

fork

fork

pull request

FESTIM

Jane

FESTIM

John

FESTIM

festim-dev

pull request

The contribution workflow

✅

❌

How to contribute to the documentation (easy)

festim.readthedocs.io

How to contribute to the documentation

  1. Make changes to the docs folder locally
  2. Compile the documentation locally


     
  3. Inspect the HTML files
  4. Push changes
  5. Documentation is automatically built with PR
cd docs/source
make html

We need to test our code

def add_two(x):
    return x + 2


def test_add_two():
    assert add_two(2) == 4
    assert add_two(3) == 5
    assert add_two(4) == 6
    assert add_two(5) == 7


test_add_two()

test_example.py

$ python test_example.py

We want to test this function

We write a test function

We can test more functions

def add_two(x):
    return x + 2


def add_three(x):
    return x + 3


def test_add_two():
    assert add_two(2) == 4
    assert add_two(3) == 5
    assert add_two(4) == 6
    assert add_two(5) == 7


def test_add_three():
    assert add_three(2) == 5
    assert add_three(3) == 6
    assert add_three(4) == 7
    assert add_three(5) == 8


test_add_two()
test_add_three()

test_example.py

$ python test_example.py

We use pytest instead

def add_two(x):
    return x + 2


def add_three(x):
    return x + 3


def test_add_two():
    assert add_two(2) == 4
    assert add_two(3) == 5
    assert add_two(4) == 6
    assert add_two(5) == 7


def test_add_three():
    assert add_three(2) == 5
    assert add_three(3) == 6
    assert add_three(4) == 7
    assert add_three(5) == 8

test_example.py

$ python -m pytest test_example.py
================================================================ test session starts ================================================================
platform linux -- Python 3.11.9, pytest-7.4.4, pluggy-1.0.0
rootdir: /home/remidm/FESTIM-workshop
collected 2 items                                                                                                                                   

example_test.py ..                                                                                                                            [100%]

================================================================= 2 passed in 0.01s =================================================================

Let's split the scripts

from example import *

def test_add_two():
    assert add_two(2) == 4
    assert add_two(3) == 5
    assert add_two(4) == 6
    assert add_two(5) == 7


def test_add_three():
    assert add_three(2) == 5
    assert add_three(3) == 6
    assert add_three(4) == 7
    assert add_three(5) == 8

test_example.py

$ python -m pytest test_example.py
def add_two(x):
    return x + 2


def add_three(x):
    return x + 3

example.py

================================================================ test session starts ================================================================
platform linux -- Python 3.11.9, pytest-7.4.4, pluggy-1.0.0
rootdir: /home/remidm/FESTIM-workshop
collected 2 items                                                                                                                                   

example_test.py ..                                                                                                                            [100%]

================================================================= 2 passed in 0.01s =================================================================

This is basically FESTIM

from example import *

def test_add_two():
    assert add_two(2) == 4
    assert add_two(3) == 5
    assert add_two(4) == 6
    assert add_two(5) == 7


def test_add_three():
    assert add_three(2) == 5
    assert add_three(3) == 6
    assert add_three(4) == 7
    assert add_three(5) == 8

test_example.py

def add_two(x):
    return x + 2


def add_three(x):
    return x + 3

example.py

festim/festim

festim/test

github.com/festim-dev/FESTIM/tree/main/festim

github.com/festim-dev/FESTIM/tree/main/test

Testing is automated with GitHub Actions

name: CI
on: [pull_request, push]

jobs:
  build:
    runs-on: ubuntu-latest

    steps:
    - name: Checkout code
      uses: actions/checkout@v2

    - name: Set up Conda
      uses: conda-incubator/setup-miniconda@v2
      with:
        activate-environment: myenv
        channels: conda-forge, defaults

    - name: Create Conda environment
      shell: bash -l {0}
      run: |
        conda install -c conda-forge fenics numpy=1.24

    - name: Install dependencies
      shell: bash -l {0}
      run: |
        pip install pytest pytest-cov

    - name: Run tests
      shell: bash -l {0}
      run: |
        pytest test/ --cov festim --cov-report xml --cov-report term
    
    - name: Upload coverage to Codecov
      uses: codecov/codecov-action@v4
      with:
        token: ${{ secrets.CODECOV_TOKEN }}

Triggered by pushes and PRs

Runs-on cloud ubuntu machine

Clone FESTIM repository

Install dependencies

Run the tests

Upload code coverage

Workflow file stored in .github/workflows/

It shows up during PRs

Now you know

✅Continuous Integration

We also have automation for

  • pip distribution
  • building the documentation
  • testing the documentation
  • computing coverage
  • ...

Code coverage tells you how much of the code is tested

app.codecov.io/gh/festim-dev/FESTIM/tree/main/festim

Live demo 👩‍💻

FESTIM development plans

FESTIM limitations

Parallel scaling issues

Performance issues

 Technical debt

Legacy FEniCS deprecated

Current interface condition  does not scale well

Many hard coded foudations

More complex implementationof new features

Bugs not being fixed

Can't take advantage of   new features

Dolfinx

FESTIM 2.0

Next steps...

New computational engine

Community-driven

New Features

New computational engine

FEniCS deprecated in 2019

FEniCSx

  • Designed from the ground up for parallelisation
  • Better performance
  • More cell types and control over the domain

Community-driven

Mailing list

User-support forum

Slack channel

Features wanted by the community?

New Features

Multi-isotope transport

New mesh types

Validation on PFCs

Chemical reactions

External physics integration

Anisotropic diffusion

Multilevel trapping

Massively parallel

Towards the future: FESTIM 2

Chemical reactions

  • More suitable for molten salts
  • Even more flexibility

Multi-species transport

  • Isotopic exchange
  • Multi-level trapping
  • Transport of impurities, He, interstitials...
  • Validation on WEST PFCs

Finite element engine upgrade

  • Better performances (including in parallel)
  • Bigger models
  • Mixed topology meshes

Graphical user interface?

Follow the development roadmap

github.com/orgs/festim-dev/projects/2

New examples

github.com/festim-dev/FESTIM-workshop-fenicsx

Isotope swapping

\mathrm{H} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{H}]_\mathrm{trap}
\mathrm{D} + [\ \ \ ]_\mathrm{trap} \ \substack{p \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k} \ [\mathrm{D}]_\mathrm{trap}
\mathrm{D} + [\mathrm{H}]_\mathrm{trap} \ \substack{k_\mathrm{swap} \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1.2em] k_\mathrm{swap}} \ [\mathrm{D}]_\mathrm{trap} + \mathrm{H}

same underlying equations!

Can be represented by festim.Reaction

Isotope swapping

my_model.species = [
    mobile_H,
    mobile_D,
    trapped_H,
    trapped_D,
]

my_model.reactions = [
    F.Reaction(
        k_0=k_0,
        E_k=0.39,
        p_0=1e13,
        E_p=1.2,
        reactant1=mobile_H,
        reactant2=empty_trap,
        product=trapped_H,
        volume=my_subdomain,
    ),
    F.Reaction(
        k_0=k_0,
        E_k=0.39,
        p_0=1e13,
        E_p=1.2,
        reactant1=mobile_D,
        reactant2=empty_trap,
        product=trapped_D,
        volume=my_subdomain,
    ),
    F.Reaction(
        k_0=k_0,
        E_k=0.1,
        p_0=k_0,
        E_p=0.1,
        reactant1=mobile_H,
        reactant2=trapped_D,
        product=[mobile_D, trapped_H],
        volume=my_subdomain,
    ),
]

Usual trapping reactions

Swapping reaction

4 species are defined

Multi-isotope, multi-level trapping

  • 7 different species
  • 6 reactions
  • \( c_H = 10^{20} \ \mathrm{m^{-3}} \) on the left
  • \( c_D = 10^{19} \ \mathrm{m^{-3}} \) on the right
  • Source code available here

1 trap, 2 levels, 2 isotopes

\mathrm{H} + [\ \ \ ] \ \substack{p_1 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_1} \ [\mathrm{H}_1\mathrm{D}_0]
\mathrm{H} + [\mathrm{H}_1\mathrm{D}_0] \ \substack{p_2 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_2} \ [\mathrm{H}_2\mathrm{D}_0]
\mathrm{D} + [\ \ \ ] \ \substack{p_3 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_3} \ [\mathrm{H}_0\mathrm{D}_1]
\mathrm{D} + [\mathrm{H}_0\mathrm{D}_1] \ \substack{p_4 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_4} \ [\mathrm{H}_0\mathrm{D}_2]
\mathrm{H} + [\mathrm{H}_0\mathrm{D}_1] \ \substack{p_5 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_5} \ [\mathrm{H}_1\mathrm{D}_1]
\mathrm{D} + [\mathrm{H}_1\mathrm{D}_0] \ \substack{p_6 \\[-1em] \longleftarrow\\[-1em] \longrightarrow \\[-1em] k_6} \ [\mathrm{H}_1\mathrm{D}_1]

\( \mathrm{H} \) = mobile H

\( \mathrm{D} \) = mobile D

\( [\mathrm{H}_x\mathrm{D}_y] \) = \(x\) H and \(y\) D in trap

Spherical cavity trapping

see Zibrov and Schmid, NME, 2024

for complete description

  • Implementation in FESTIM of the spherical cavity trapping model developed by Zibrov and Schmid
  • Custom trapping equations
  • Smooth implementation in FESTIM

Anisotropy

  • Anisotropic materials can be simulated with very few modifications
  • Composites
  • Anisotropic microstructures
D = \begin{bmatrix} D_{xx} & 0\\ 0 & D_{yy} \end{bmatrix}

Interface discontinuity 

  • 2 materials
  • 1 mobile + 1 trap

Example dolfinx code available in issue #719

But this changes a lot of things!

FESTIM 1

with dolfinx

1 mesh for the whole domain

1 mesh for the whole domain

Submesh 1

Submesh 2

VectorFunctionSpace \(V\) (eg. 3 components)

Function \(u\) (eg. 3 components)

VectorFunctionSpace \(V_1\)

\(V_2\)

Function \(u_1\) (eg. 2 comps.)

\(u_2\) (eg. 3 comps.)

1 function for the whole domain

1 function per subdomain