fenicsx.neutronics package

Submodules

fenicsx.neutronics.neutr_diff module

class fenicsx.neutronics.neutr_diff.steady_neutron_diff(domain: dolfinx.mesh.Mesh, ct: dolfinx.cpp.mesh.MeshTags_int32, ft: dolfinx.cpp.mesh.MeshTags_int32, physical_param: dict, regions_markers: list, void_bound_marker: int, albedo: ndarray | None = None, coupling: str | None = None)

Bases: object

” This class implements the solution of the steady multigroup neutron diffusion equation through the \(k\)-eigenvalue method.

Parameters:

• domain (dolfinx.mesh.Mesh) – Mesh imported.

• ct (dolfinx.cpp.mesh.MeshTags_int32) – Cell tags of the regions in the domain.

• ft (dolfinx.cpp.mesh.MeshTags_int32) – Face tags of the boundaries.

• physical_param (dict) – Dictionary with the main physical parameters at reference condition.

• regions_markers (dict) – List of the region markers inside the domain.

• void_bound_marker (int) – Integer describing the index of the void boundary.

• albedo (np.ndarray, optional (Default = None)) – If not None, it, denoted as \(\gamma\), must be an array of \(G\), as the energy groups, to impose the following BC \(-D_g\nabla \phi_g\cdot \mathbf{n} = \gamma \phi_g\).

• coupling (str, optional (Default = None)) – If not None, it indicates the type of coupling with the temperature: log and sqrt are available.

assembleForm(direct: bool = True)

This function assembles the variational forms and the correspondent linear algebra structures.

Parameters:

direct (bool, optional (Default=True)) – If True, a direct linear solver is used (Gauss Elimination), otherwise an iterative one is adopted (preconditioned GMRES with ILU).

solve(temperature: dolfinx.fem.Function | None = None, power: float = 1, nu: float = 2.41, Ef: float = 3.2e-11, tol: float = 1e-10, maxIter: int = 200, LL: int = 50, verbose: bool = False)

This function implements the \(k\)-eigenvalue method: starting from an initial non-trivial guess of eigenvalue-eigenvector pair \(\left(k_{\text{eff}}^{(0)},\,\boldsymbol{\phi}^{(0)}\right)\), at each iteration \(n\geq 1\) the following linear system is solved

\[\mathbb{A}\boldsymbol{\phi}^{(n)} = \frac{1}{k_{\text{eff}}^{(n-1)}} \mathbb{B}\boldsymbol{\phi}^{(n-1)}\]

The convergence check is made on the relative difference for the eigenvalue as

\[\frac{|k_{\text{eff}}^{(n)} - k_{\text{eff}}^{(n-1)}|}{k_{\text{eff}}^{(n)}} < \epsilon\]

Parameters:

• temperature (dolfinx.fem.Function, optional (Default = None)) – Temperature field as a dolfinx.fem.Function.

• power (float, optional (Default = 1)) – Power level \(P\) used to normalise the flux as \(P = \displaystyle\int_V E_f\cdot \sum_{g=1}^G \Sigma_{f,g}\phi_g \, dV\).

• nu (float, optional (Default = 2.41)) – Number of neutrons emitted by the fission event.

• Ef (float, optional (Default = 1.6e-19 * 200e6)) – Energy released by the fission event.

• tol (float, optional (Default = 1e-10)) – Tolerance of the inverse power method, \(\epsilon\).

• maxIter (int, optional (Default = 200)) – Maximum iteration for the inverse power method.

• LL (int, optional (Default = 50)) – Frequency for printing the output.

• verbose (bool, optional (Default = False)) – If True, the output is printed every LL iterations.

Returns:

• normalised_fluxes (list[dolfinx.fem.Function]) – List of dolfinx.fem.Function with the solution of the inverse power method.

• k_eff_ (float) – Converged eigenvalue.

update_xs()

This function updates the cross sections according to the coupling scheme chosen.

class fenicsx.neutronics.neutr_diff.transient_neutron_diff(domain: dolfinx.mesh.Mesh, ct: dolfinx.cpp.mesh.MeshTags_int32, ft: dolfinx.cpp.mesh.MeshTags_int32, physical_param: dict, regions_markers: list, void_bound_marker: int, albedo: ndarray | None = None, coupling: str | None = None)

Bases: object

” This class implements the solution of the transient multigroup neutron diffusion equation coupled with the precursors equations with a semi-implicit first-order advancement scheme.

Parameters:

• domain (dolfinx.mesh.Mesh) – Mesh imported.

• ct (dolfinx.cpp.mesh.MeshTags_int32) – Cell tags of the regions in the domain.

• ft (dolfinx.cpp.mesh.MeshTags_int32) – Face tags of the boundaries.

• physical_param (dict) – Dictionary with the main physical parameters at reference condition.

• regions_markers (dict) – List of the region markers inside the domain.

• void_bound_marker (int) – Integer describing the index of the void boundary.

• albedo (np.ndarray, optional (Default = None)) – If not None, it, denoted as \(\gamma\), must be an array of \(G\), as the energy groups, to impose the following BC \(-D_g\nabla \phi_g\cdot \mathbf{n} = \gamma \phi_g\).

• coupling (str, optional (Default = None)) – If not None, it indicates the type of coupling with the temperature: log and sqrt are available.

advance(t: float, sigma_ag_transient_value: list, temperature: dolfinx.fem.Function | None = None, return_prec: bool = False)

This function implements a single step forward in time.

Parameters:

• t (float) – Time to advance to, say \(t_{n+1}\).

• sigma_ag_transient_value (list) – List of callable function dependent on time used to impose variation of the reactivity.

• temperature (dolfinx.fem.Function, optional (Default = None)) – Temperature field as a dolfinx.fem.Function.

• return_prec (bool, optional (Default = False)) – If True, the precursors are returned in the solution.

Returns:

• power (float) – Power at time \(t_{n+1}\), defined as \(P(t) = \displaystyle\int_V E_f\cdot \sum_{g=1}^G \Sigma_{f,g}\phi_g \, dV\).

• solution (list[dolfinx.fem.Function]) – List of dolfinx.fem.Function with the output of this time step.

assembleForm(phi_ss: list[dolfinx.fem.Function], dt: float, nu: float = 2.41, Ef: float = 3.2e-11, direct=True)

This function assembles the variational forms and the correspondent linear algebra structures.

Parameters:

• phi_ss (list[dolfinx.fem.Function) – List of the steady state group fluxes.

• dt (float) – Time step size.

• nu (float, optional (Default = 2.41)) – Number of neutrons emitted by the fission event.

• Ef (float, optional (Default = 1.6e-19 * 200e6)) – Energy released by the fission event.

• direct (bool, optional (Default=True)) – If True, a direct linear solver is used (Gauss Elimination), otherwise an iterative one is adopted (preconditioned GMRES with ILU).

update_xs()

This function updates the cross sections according to the coupling scheme chosen.

fenicsx.neutronics.thermal module

class fenicsx.neutronics.thermal.steady_thermal_diffusion(domain: dolfinx.mesh.Mesh, ct: dolfinx.cpp.mesh.MeshTags_int32, ft: dolfinx.cpp.mesh.MeshTags_int32, physical_param: dict, regions_markers: list, void_bound_marker: int, h: float | None = None, TD: float = 300.0, coupling=None)

Bases: object

This class implements the solution of the steady thermal diffusion equation with internal power generation made by neutrons.

Parameters:

• domain (dolfinx.mesh.Mesh) – Mesh imported.

• ct (dolfinx.cpp.mesh.MeshTags_int32) – Cell tags of the regions in the domain.

• ft (dolfinx.cpp.mesh.MeshTags_int32) – Face tags of the boundaries.

• physical_param (dict) – Dictionary with the main physical parameters at reference condition.

• regions_markers (dict) – List of the region markers inside the domain.

• void_bound_marker (int) – Integer describing the index of the void boundary onto which BC is imposed (Dirichlet or Robin).

• h (float, optional (Default = None)) – If not None, it, denoted as \(h\), represents the heat transfer coefficients in Newton’s cooling law.

• TD (float, optional (Default = None)) – If not None, it, denoted as \(T_D\), represents either the bulk temperature in Newton’s cooling law or the Dirichlet condition to impose.

• coupling (str, optional (Default = None)) – If not None, it indicates the type of coupling with the temperature: log and sqrt are available.

assembleForm(direct=True)

This function assembles the variational forms and the correspondent linear algebra structures.

Parameters:

direct (bool, optional (Default=True)) – If True, a direct linear solver is used (Gauss Elimination), otherwise an iterative one is adopted (preconditioned GMRES with ILU).

solve(phi: list[dolfinx.fem.Function], k_eff: float, temperature: dolfinx.fem.Function | None = None)

This function solves the steady equation.

Parameters:

• phi (list[dolfinx.fem.Function]) – List of dolfinx.fem.Function for the internal power generation, \(q''' = E_f\cdot \sum_{g=1}^G \Sigma_{f,g}\phi_g\).

• k_eff (float) – Effective multiplication factor.

• temperature (dolfinx.fem.Function, optional (Default = None)) – Temperature field as a dolfinx.fem.Function used to calculate the cross sections.

Returns:

solution – Output of the linear system.

Return type:

dolfinx.fem.Function

update_xs()

This function updates the cross sections according to the coupling scheme chosen.

class fenicsx.neutronics.thermal.transient_thermal_diffusion(domain: dolfinx.mesh.Mesh, ct: dolfinx.cpp.mesh.MeshTags_int32, ft: dolfinx.cpp.mesh.MeshTags_int32, physical_param: dict, regions_markers: list, void_bound_marker: int, h: float | None = None, TD: float = 300.0, coupling=None)

Bases: object

This class implements the solution of the transient thermal diffusion equation with internal power generation made by neutrons.

Parameters:

• domain (dolfinx.mesh.Mesh) – Mesh imported.

• ct (dolfinx.cpp.mesh.MeshTags_int32) – Cell tags of the regions in the domain.

• ft (dolfinx.cpp.mesh.MeshTags_int32) – Face tags of the boundaries.

• physical_param (dict) – Dictionary with the main physical parameters at reference condition.

• regions_markers (dict) – List of the region markers inside the domain.

• void_bound_marker (int) – Integer describing the index of the void boundary onto which BC is imposed (Dirichlet or Robin).

• h (float, optional (Default = None)) – If not None, it, denoted as \(h\), represents the heat transfer coefficients in Newton’s cooling law.

• TD (float, optional (Default = None)) – If not None, it, denoted as \(T_D\), represents either the bulk temperature in Newton’s cooling law or the Dirichlet condition to impose.

• coupling (str, optional (Default = None)) – If not None, it indicates the type of coupling with the temperature: log and sqrt are available.

advance(phi: list)

This function advances in time with a single step.

Parameters:

phi (list[dolfinx.fem.Function]) – List of dolfinx.fem.Function for the internal power generation, \(q''' = E_f\cdot \sum_{g=1}^G \Sigma_{f,g}\phi_g``\)

Returns:

solution – Solution at the new time step..

Return type:

dolfinx.fem.Function

assembleForm(T_ss: dolfinx.fem.Function, dt: float, direct=True)

This function assembles the variational forms and the correspondent linear algebra structures.

Parameters:

• T_ss (dolfinx.fem.Function) – Steady solution, i.e. the initial condition.

• dt (float) – Time step size.

• direct (bool, optional (Default=True)) – If True, a direct linear solver is used (Gauss Elimination), otherwise an iterative one is adopted (preconditioned GMRES with ILU).

update_xs()

This function updates the cross sections according to the coupling scheme chosen.

Module contents

models/fenicsx/neutronics.

OFELIA: FEniCSx models for MultiPhysics.