diff --git a/book/_toc.yml b/book/_toc.yml index 0105206..842b596 100644 --- a/book/_toc.yml +++ b/book/_toc.yml @@ -57,6 +57,11 @@ parts: - file: content/misc/settings - file: content/misc/stepsize + - caption: Multiphysics + chapters: + - file: content/multiphysics/cfd + - file: content/multiphysics/neutronics + - caption: Misc chapters: diff --git a/book/content/multiphysics/cfd.md b/book/content/multiphysics/cfd.md new file mode 100644 index 0000000..4fbf2f1 --- /dev/null +++ b/book/content/multiphysics/cfd.md @@ -0,0 +1,437 @@ +--- +jupytext: + formats: ipynb,md:myst + text_representation: + extension: .md + format_name: myst + format_version: 0.13 + jupytext_version: 1.19.1 +kernelspec: + display_name: festim-workshop + language: python + name: python3 +--- + +# Coupling to computational fluid dynamics # + +This tutorial discusses how to couple FESTIM to computational fluid dynamics (CFD) solvers, which is relevant in modeling advective-assisted and turbulent diffusion. + + +```{admonition} Objectives +:class: objectives +* Understanding the importance of fluid flow in hydrogen diffusion +* Utilizing external CFD solvers to generate velocity fields +* Reading fields into FESTIM for advective-assisted diffusion +``` + ++++ + +## Importance of fluid flow in hydrogen diffusion + +Let's review the governing equation for FESTIM, neglecting reactions and source terms: + +$$ +\frac{\partial c_m}{\partial t} = \nabla \cdot (D \nabla c_m) + \mathbf{u} \cdot \nabla c_m , +$$ + +where $\mathbf{u}$ is the velocity field in units of $\mathrm{ms^{-1}}$. This velocity term, which represents the species' advection, requires solving for the $\textbf{u}$ field. This can be done using custom (DOLFINx) or external (OpenFOAM) CFD solvers. + +Additionally, turbulence is a diffusion-enhancing property. This can be incorporated in FESTIM by adding a turbulent diffusion term $D_t$: + +$$ D_t = \frac{\nu_t}{Sc} $$ + +where $\nu_t$ is the turbulent viscosity and $Sc$ is the Schmidt number (measures the relative rates of mass and momentum diffusion). This turbulent diffusion term is simply added to the Fickian diffusivity in FESTIM. + ++++ + +### When do I need advection in my diffusion model? + +Advection can add complexity to your FESTIM model, and sometimes may not be needed. Users can choose whether or not they need advection by referring to the Péclet number $Pe$: + +$$ Pe = \frac{\text{advection transport rate}}{\text{diffusive transport rate}} = \frac{\mathrm{Lu}}{D} = \mathrm{Re_L}\mathrm{Sc} $$ + +where $L$ is the characteristic length, $u$ the local flow velocity, $D$ the mass diffusion coefficient, $Re_L$ the Reynolds number, $Sc$ the Schmidt number. If $Pe << 1$, diffusion dominates and advection is not needed. + ++++ + +## Solving a lid-driven cavity problem + +This section discusses how to setup and solve a coupled problem between OpenFOAM and FESTIM, as outlined in the FESTIM 2 review paper. Specifically, we'll solve a lid-driven cavity problem by calculating the velocity field in OpenFOAM and exporting it to FESTIM to solve our diffusion model. `.foam` files from OpenFOAM are converted to `.vtk` files for DOLFINx using `foam2dolfinx`. + +```{seealso} +Check out the [FESTIM 2 review paper](https://arxiv.org/abs/2509.24760) or [foam2dolfinx](https://github.com/festim-dev/foam2dolfinx) to learn more. +``` + ++++ + +Our rectangular geometry will have the following velocity boundary conditions: + +$$ +\mathbf{u}(x=0, y) = (0, 0), \quad +\mathbf{u}(x=L, y) = (0, 0), \\ +\mathbf{u}(x, y=0) = (0, 0), \quad +\mathbf{u}(x, y=L) = (U, 0) +$$ + ++++ + +### Reading OpenFOAM data + +Let us first import our OpenFOAM data (which has been stored on the [FESTIM 2 review paper repo](https://github.com/festim-dev/FESTIM-v2-review/tree/main/coupling/coupling_cfd/data).) + +```{seealso} +The OpenFOAM case for this problem can be found [here](https://github.com/festim-dev/FESTIM-v2-review/blob/main/coupling/coupling_cfd/ldc_coupling.py). This problem uses the *icoFOAM* solver, take a look at the [OpenFOAM documentation](https://www.openfoam.com/documentation/tutorial-guide/2-incompressible-flow/2.1-lid-driven-cavity-flow#x6-60002.1) for more information about running an OpenFOAM case. +``` + +```{code-cell} ipython3 +from foam2dolfinx import OpenFOAMReader +from pathlib import Path +import zipfile + +export_field = True +zip_path = Path("cfd_data/cavity.zip") +extract_path = Path("cfd_data/cavity") + +# Extract if needed +if not extract_path.exists(): + with zipfile.ZipFile(zip_path, "r") as zip_ref: + zip_ref.extractall(extract_path) + +# Ensure .foam file exists +foam_file = extract_path / "cavity" / "cavity.foam" +foam_file.touch(exist_ok=True) + +# Read OpenFOAM case +ldc_reader = OpenFOAMReader( + filename=str(foam_file), + cell_type=12 +) + +velocity_field = ldc_reader.create_dolfinx_function(t=2.5, name="U") +``` + +We can visualize the velocity field using PyVista: + +```{code-cell} ipython3 +:tags: [hide-input] + +from dolfinx import plot +import pyvista +import numpy as np + +pyvista.set_jupyter_backend("html") + +V = velocity_field.function_space +topology, cell_types, geometry = plot.vtk_mesh(V) + +grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) + +num_dofs = V.dofmap.index_map.size_local +value_size = V.dofmap.index_map_bs # number of components (e.g. 2 or 3) + +u_array = velocity_field.x.array.real.reshape(num_dofs, value_size) + +grid.point_data["U"] = u_array +grid.set_active_vectors("U") + +grid["U_mag"] = np.linalg.norm(u_array, axis=1) + +plotter = pyvista.Plotter() +plotter.add_mesh(grid, scalars="U_mag", cmap="coolwarm") +glyphs = grid.glyph(orient="U", factor=0.02) +plotter.add_mesh(glyphs, cmap="coolwarm") +plotter.view_xy() +plotter.show() +``` + +````{tip} +To save the OpenFOAM velocity field, you can export it to `.bp` and view the results in ParaView (as shown in the [post-processing section](../post_process/paraview.md)). +``` +writer = VTXWriter( + MPI.COMM_WORLD, + "results/velocity_field.bp", + velocity_field, + "BP5" +) +writer.write(t=0) +``` +```` + ++++ + +### Setting up our FESTIM model + +Now, let's initiate our hydrogen transport problem. We have insulated boundary conditions ($c=0$ on all boundaries): + +```{code-cell} ipython3 +import festim as F +from mpi4py import MPI +from dolfinx.mesh import create_rectangle +import numpy as np + +my_model = F.HydrogenTransportProblem() +fenics_mesh = create_rectangle(MPI.COMM_WORLD, [[0, 0], [0.1, 0.1]], [50, 50]) +my_model.mesh = F.Mesh(fenics_mesh) + +# define species +H = F.Species("H", mobile=True) +my_model.species = [H] + +# define subdomains +my_mat = F.Material(D_0=5e-04, E_D=0) + +fluid = F.VolumeSubdomain(id=1, material=my_mat) +left = F.SurfaceSubdomain(id=2, locator=lambda x: np.isclose(x[0], 0.0)) +right = F.SurfaceSubdomain(id=3, locator=lambda x: np.isclose(x[0], 0.1)) +bottom = F.SurfaceSubdomain(id=4, locator=lambda x: np.isclose(x[1], 0.0)) +top = F.SurfaceSubdomain(id=5, locator=lambda x: np.isclose(x[1], 0.1)) +my_model.subdomains = [fluid, left, right, bottom, top] + +# define boundary conditions +my_model.boundary_conditions = [ + F.FixedConcentrationBC(subdomain=left, value=0, species=H), + F.FixedConcentrationBC(subdomain=right, value=0, species=H), + F.FixedConcentrationBC(subdomain=bottom, value=0, species=H), + F.FixedConcentrationBC(subdomain=top, value=0, species=H), +] + +# define temperature +my_model.temperature = 500 +``` + +In this example, we add a particle source using `ParticleSource': + +```{code-cell} ipython3 +my_model.sources = [ + F.ParticleSource(volume=fluid, species=H, value=10), +] +``` + +### Adding advection into FESTIM + +To add the velocity field to our FESTIM model, we need to create an `AdvectionTerm`, which requires the velocity field (in our case the exported field from OpenFOAM), which species the velocity field is acting on, and which volume subdomain is this velocity field on: + +```{code-cell} ipython3 +advection = F.AdvectionTerm(velocity=lambda t: velocity_field(t), species=H, subdomain=fluid) +``` + +```{note} +We use a `lambda` function to utilize transient velocity fields, although our example here is steady-state and therefore only uses data at $t=2.5$ seconds. +``` + ++++ + +We add this to our problem's `advection_terms` attribute: + +```{code-cell} ipython3 +my_model.advection_terms = [advection] +``` + +Let us solve and visualize the results: + +```{code-cell} ipython3 +my_model.settings = F.Settings(atol=1e-10, rtol=1e-10, transient=False) +my_model.initialise() +my_model.run() +``` + +```{code-cell} ipython3 +:tags: [hide-input] + +pyvista.set_jupyter_backend("html") + +c = H.post_processing_solution + +topology, cell_types, geometry = plot.vtk_mesh(c.function_space) +u_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) +u_grid.point_data["c"] = c.x.array.real +u_grid.set_active_scalars("c") +u_plotter = pyvista.Plotter() + +u_plotter.add_mesh(u_grid, cmap="viridis", show_edges=False) +u_plotter.view_xy() + +if not pyvista.OFF_SCREEN: + u_plotter.show() +else: + figure = u_plotter.screenshot("concentration.png") +``` + +Compare this to the results without advection: + +```{code-cell} ipython3 +my_model.advection_terms = [] +my_model.initialise() +my_model.run() +``` + +```{code-cell} ipython3 +:tags: [hide-input] + +c = H.post_processing_solution + +topology, cell_types, geometry = plot.vtk_mesh(c.function_space) +u_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) +u_grid.point_data["c"] = c.x.array.real +u_grid.set_active_scalars("c") +u_plotter = pyvista.Plotter() + +u_plotter.add_mesh(u_grid, cmap="viridis", show_edges=False) +u_plotter.view_xy() + +if not pyvista.OFF_SCREEN: + u_plotter.show() +else: + figure = u_plotter.screenshot("concentration.png") +``` + +### Getting velocity field from custom DOLFINx script + +For simple velocity fields (such as our 2D, steady-state, incompressible flow), users can write their own DOLFINx script that outputs a `dolfinx.fem.Function`. This can then be used in FESTIM the same exact was as shown above. To create a simi + +```{code-cell} ipython3 +import numpy as np + +from dolfinx import fem, mesh, io +from dolfinx.fem import Function, dirichletbc, locate_dofs_topological +from dolfinx.fem.petsc import NonlinearProblem + +from ufl import TestFunctions, split, inner, grad, div, dx, nabla_grad, dot +from basix.ufl import element, mixed_element + +Re = 100.0 +nx = ny = 50 + +msh = fenics_mesh +gdim = msh.geometry.dim + +P2 = element("Lagrange", msh.basix_cell(), 2, shape=(gdim,)) +P1 = element("Lagrange", msh.basix_cell(), 1) +W = fem.functionspace(msh, mixed_element([P2, P1])) + +fdim = msh.topology.dim - 1 +msh.topology.create_connectivity(fdim, msh.topology.dim) + +def on_lid(x): + return np.isclose(x[1], 0.1) + +def on_walls(x): + return np.isclose(x[0], 0.0) | np.isclose(x[0], 0.1) | np.isclose(x[1], 0.0) + +W0 = W.sub(0) +V_collapsed, _ = W0.collapse() + +lid_facets = mesh.locate_entities_boundary(msh, fdim, on_lid) +lid_dofs = locate_dofs_topological((W0, V_collapsed), fdim, lid_facets) +u_lid = Function(V_collapsed) +u_lid.interpolate(lambda x: np.vstack([np.ones(x.shape[1]), np.zeros(x.shape[1])])) +bc_lid = dirichletbc(u_lid, lid_dofs, W0) + +wall_facets = mesh.locate_entities_boundary(msh, fdim, on_walls) +wall_dofs = locate_dofs_topological((W0, V_collapsed), fdim, wall_facets) +u_wall = Function(V_collapsed) +u_wall.x.array[:] = 0.0 +bc_wall = dirichletbc(u_wall, wall_dofs, W0) + +bcs = [bc_lid, bc_wall] + +w = Function(W) +u, p = split(w) +v, q = TestFunctions(W) + +F = ( + inner(grad(u), grad(v)) * dx + + inner(dot(u, nabla_grad(u)), v) * dx + - inner(p, div(v)) * dx + + inner(div(u), q) * dx +) + +petsc_options = { + "snes_type": "newtonls", + "snes_rtol": 1e-10, + "snes_atol": 1e-10, + "snes_max_it": 50, + "snes_monitor": None, + "ksp_type": "preonly", + "pc_type": "lu", + "pc_factor_mat_solver_type": "mumps", +} + +problem = NonlinearProblem( + F, w, bcs=bcs, + petsc_options_prefix="cavity_", + petsc_options=petsc_options, +) + +result = problem.solve() + +u_h = w.sub(0).collapse() +p_h = w.sub(1).collapse() +u_h.name = "velocity" +p_h.name = "pressure" +``` + +The resulting velocity field `u_h` will be used for FESTIM.: + +```{seealso} +Check out the [the DOLFINx tutorial](https://docs.fenicsproject.org/dolfinx/v0.10.0.post5/python/demos/demo_stokes.html#implementation) to learn more about solving the Stokes equations using Taylor-Hood elements. +``` + ++++ + +Let's take a look at the velocity field generated from our DOLFINx script: + +```{code-cell} ipython3 +:tags: [hide-input] + +from dolfinx import plot +import pyvista +import numpy as np + +V = u_h.function_space +topology, cell_types, geometry = plot.vtk_mesh(V) + +grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) + +num_dofs = V.dofmap.index_map.size_local +value_size = V.dofmap.index_map_bs # number of components (e.g. 2 or 3) + +u_array = u_h.x.array.real.reshape(num_dofs, value_size) +grid["U_mag"] = np.linalg.norm(u_array, axis=1) + +plotter = pyvista.Plotter() +plotter.add_mesh(grid, scalars="U_mag", cmap="coolwarm") +plotter.view_xy() +plotter.show() +``` + +To add this field to FESTIM, we create an `AdvectionTerm` and add it to our problem's `advection_terms`, just as above: + +```{code-cell} ipython3 +import festim as F + +dolfinx_velocity = F.AdvectionTerm(velocity=u_h, species=H, subdomain=fluid) +my_model.advection_terms = [dolfinx_velocity] +my_model.initialise() +my_model.run() +``` + +```{code-cell} ipython3 +:tags: [hide-input] + +c = H.post_processing_solution + +topology, cell_types, geometry = plot.vtk_mesh(c.function_space) +u_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) +u_grid.point_data["c"] = c.x.array.real +u_grid.set_active_scalars("c") +u_plotter = pyvista.Plotter() + +u_plotter.add_mesh(u_grid, cmap="viridis", show_edges=False) +u_plotter.view_xy() + +if not pyvista.OFF_SCREEN: + u_plotter.show() +else: + figure = u_plotter.screenshot("concentration.png") +``` diff --git a/book/content/multiphysics/cfd_data/cavity.zip b/book/content/multiphysics/cfd_data/cavity.zip new file mode 100644 index 0000000..041b56a Binary files /dev/null and b/book/content/multiphysics/cfd_data/cavity.zip differ diff --git a/book/content/multiphysics/neutronics.md b/book/content/multiphysics/neutronics.md new file mode 100644 index 0000000..c955e62 --- /dev/null +++ b/book/content/multiphysics/neutronics.md @@ -0,0 +1,288 @@ +--- +jupytext: + formats: ipynb,md:myst + text_representation: + extension: .md + format_name: myst + format_version: 0.13 + jupytext_version: 1.19.1 +kernelspec: + display_name: festim-workshop + language: python + name: python3 +--- + +# Coupling to neutronics + +This tutorial discusses how to couple FESTIM to neutronics solvers such as OpenMC. + +```{admonition} Objectives +:class: objectives +* Understanding the importance of neutronics in hydrogen transport +* Utilizing external neutronics solvers to generate tritium source terms +* Reading OpenMC results into FESTIM for accurate particle sources +``` + ++++ + +## Where does neutronics couple into hydrogen transport? ## + +Let us take a look at the governing equation for FESTIM, neglecting reactions and advection: + +$$ \frac{\partial c_i}{\partial t} = \nabla\cdot (D_i \nabla c_i) + S_i(\mathbf{x}, t, T) $$ + +Neutronics solvers can provide a value for the $S_i$, which represents a volumetric source of mobile hydrogen isotope that can depend on space, time and temperature. In particular, solvers such as OpenMC can provide an accurate source of of tritium generation. + +### Converting data from OpenMC to FESTIM + +OpenMC can tally spatial distributions of reaction rates and export the results in VTK format. Users can then use the `openmc2dolfinx` package to convert them into DOLFINx functions. + +```{seealso} +Check out the [OpenMC documentation](https://docs.openmc.org/en/stable/) and [openmc2dolfinx repo](https://github.com/festim-dev/foam2dolfinx) to learn more. +``` + ++++ + +## Solving a hydrogen transport problem in an neutron-irradiated lithium cube + +This section discusses how to setup and solve a coupled problem between OpenMC and FESTIM, as outlined in the FESTIM 2 review paper. + +Specifically, we'll calculate the tritium generation in a neutron-irradiated lithium cube using OpenMC. Then, we will export the source term to FESTIM to solve our diffusion model. + +```{seealso} +Check out the [FESTIM 2 review paper](https://arxiv.org/abs/2509.24760) to learn more. +``` + ++++ + +### Getting tritium source from OpenMC + +First, run the OpenMC model: + +``` +import openmc +import openmc_data_downloader as odd + +dim = 60 +lithium = openmc.Material(name="lithium") +lithium.set_density("g/cc", 0.534) +lithium.add_element("Li", 1.0) + +mats = openmc.Materials([lithium]) + +odd.download_cross_section_data( + mats, + libraries=["FENDL-3.1d"], + set_OPENMC_CROSS_SECTIONS=True, + particles=["neutron"], +) + +# Geometry +cube_surface = openmc.model.RectangularParallelepiped( + -dim, dim, -dim, dim, -dim, dim +) +region = -cube_surface +cell = openmc.Cell(region=region, fill=lithium) + +vacuum_surf = openmc.Sphere(r=dim * 2, boundary_type="vacuum") +vacuum_region = +cube_surface & -vacuum_surf +vacuum = openmc.Cell(region=vacuum_region, fill=None) + +geometry = openmc.Geometry([cell, vacuum]) + +# Tallies +tally = openmc.Tally(name="tritium_production") +tally.scores = ["(n,Xt)"] +mesh = openmc.RegularMesh() +mesh.dimension = [30, 30, 15] +mesh.lower_left = [-dim, -dim, -dim] +mesh.upper_right = [dim, dim, dim] +tally.filters = [openmc.MeshFilter(mesh)] + +tallies = openmc.Tallies([tally]) + +# Settings +source = openmc.IndependentSource() +source_pos_z = dim + 10 +source.space = openmc.stats.Point((0, 0, source_pos_z)) +source.angle = openmc.stats.Isotropic() +source.energy = openmc.stats.Discrete([14.1e6], [1.0]) + +settings = openmc.Settings() +settings.run_mode = "fixed source" +settings.source = source +settings.batches = 10 +settings.particles = 10000 + +my_model = openmc.Model( + geometry=geometry, settings=settings, materials=mats, tallies=tallies +) + +my_model.run(apply_tally_results=True) + +mesh.write_data_to_vtk( + "tritium_production_mesh.vtk", {"tritium_production": tally.mean} +) +``` + +This should output a file named `tritium_production_mesh.vtk` which can then be read into FESTIM. + ++++ + +````{tip} +We provide the code to run the OpenMC model, but don't run it this tutorial. To run this code, create a new environment and install OpenMC with the following lines of code: + +``` +conda config --add channels conda-forge +conda config --set channel_priority strict +conda create --name openmc-env openmc +conda activate openmc-env +``` +It is also necessary to install `vtk` and `openmc_data_downloader`, which can be done with: + +``` +conda install -c conda-forge vtk +pip install openmc_data_downloader +``` +```` + ++++ + +Let's take a look at the OpenMC tritium source output: + +```{code-cell} ipython3 +:tags: [hide-input] + +import pyvista as pv + +pv.set_jupyter_backend("html") + +grid = pv.read("openmc_data/tritium_production_mesh.vtk") +scalar_name = "tritium_production" +grid.set_active_scalars(scalar_name) + +plotter = pv.Plotter() +plotter.add_mesh(grid, scalars=scalar_name, cmap="hot", show_edges=False) +plotter.view_xy() +plotter.show() +``` + +### Read data from OpenMC output + +Now that we have our tritium source data, we can use `openmc2dolfinx` to create a structured grid reader that outputs a `dolfinx.fem.Function` to be used in FESTIM: + +```{code-cell} ipython3 +from openmc2dolfinx import StructuredGridReader + +reader = StructuredGridReader("openmc_data/tritium_production_mesh.vtk") +reader.create_dolfinx_mesh() +``` + +### Setting up our FESTIM model + +Let us set up our FESTIM model now. We will create a refined box mesh and use a dummy material. We will consider insulated boundary conditions ($c=0$ on all boundaries): + +```{code-cell} ipython3 +import festim as F +import dolfinx +from mpi4py import MPI + +dim = 60 +my_model = F.HydrogenTransportProblem() + +tritium = F.Species("T") + +refined_mesh = dolfinx.mesh.create_box( + MPI.COMM_WORLD, + points=[(-dim, -dim, -dim), (dim, dim, dim)], + n=(30, 30, 70), + cell_type=dolfinx.mesh.CellType.tetrahedron, +) + +my_model.mesh = F.Mesh(refined_mesh) +boundary = F.SurfaceSubdomain(id=1) + +my_mat = F.Material(D_0=1, E_D=0) +volume = F.VolumeSubdomain(id=1, material=my_mat) +my_model.subdomains = [boundary, volume] + +my_model.species = [tritium] +my_model.boundary_conditions = [F.FixedConcentrationBC(subdomain=boundary, value=0.0, species=tritium)] +my_model.temperature = 300 +my_model.settings = F.Settings(atol=1e-10, rtol=1e-10, transient=False) +``` + +````{note} +Here we chose to create more refined mesh from the hydrogen transport model than was used in OpenMC. We could, however, use the same mesh from OpenMC by running: +``` +my_model.mesh = F.Mesh(reader.dolfinx_mesh) +``` +```` + ++++ + +Since we chose not to use the OpenMC mesh, we need to interpolate the results onto our new FESTIM mesh: + +```{code-cell} ipython3 +original_source_term = reader.create_dolfinx_function("tritium_production") +V = dolfinx.fem.functionspace(refined_mesh, ("DG", 0)) +source_term = dolfinx.fem.Function(V) +F.helpers.nmm_interpolate(source_term, original_source_term) +``` + +Now we can add our *interpolated* tritium source into our problem: + +```{code-cell} ipython3 +tritium_source = F.ParticleSource( + value=source_term, species=tritium, volume=volume +) + +my_model.sources = [tritium_source] +``` + +Finally, let's solve and take a look at the results: + +```{code-cell} ipython3 +my_model.initialise() +my_model.run() +``` + +```{code-cell} ipython3 +:tags: [hide-input] + +from dolfinx import plot +import pyvista + +c = tritium.post_processing_solution + +topology, cell_types, geometry = plot.vtk_mesh(c.function_space) +u_grid = pyvista.UnstructuredGrid(topology, cell_types, geometry) +u_grid.point_data["c"] = c.x.array.real +u_grid.set_active_scalars("c") +u_plotter = pyvista.Plotter() + +u_plotter.add_mesh(u_grid, cmap="viridis", show_edges=False) +u_plotter.view_xy() + +if not pyvista.OFF_SCREEN: + u_plotter.show() +else: + figure = u_plotter.screenshot("concentration.png") +``` + +As we'd expect, there is no diffusion on the outer boundaries of our cube (recall our insulated boundary conditions). Let's take a look inside the cube to how the tritium source diffuses inside the material: + +```{code-cell} ipython3 +:tags: [hide-input] + +bounds = [0, dim, 0, dim, 0, dim] + +clipped = u_grid.clip_box(bounds, invert=True) + +pl = pv.Plotter() +pl.add_mesh(clipped, cmap="viridis", show_edges=False) +pl.view_xy() +pl.show() +``` + +We see a high concentration where the tritium is generated and it diffuses throughout the material. 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81c0b96..9ec1204 100644 --- a/environment.yml +++ b/environment.yml @@ -35,4 +35,6 @@ dependencies: - trame-vuetify - nest-asyncio2 - pip: - - h-transport-materials==0.16 \ No newline at end of file + - h-transport-materials==0.16 + - foam2dolfinx + - openmc2dolfinx