bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D

Bluemira module for the solution of a 2D magnetostatic problem with cylindrical symmetry and toroidal current source using fenics FEM solver

Classes

FixedBoundaryEquilibrium

Simple minimal dataclass for a fixed boundary equilibrium.

FemGradShafranovFixedBoundary

A 2D fem Grad Shafranov solver. The solver is thought as support for the fem fixed

Functions

_plot_array(ax, points, array, title, cmap[, levels])

_add_colorbar(cm, cax, title)

Module Contents

class bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D.FixedBoundaryEquilibrium

Simple minimal dataclass for a fixed boundary equilibrium.

mesh: dolfinx.mesh.Mesh
psi: collections.abc.Callable[[numpy.typing.ArrayLike], float | numpy.typing.NDArray[numpy.float64]]
p_prime: numpy.ndarray
ff_prime: numpy.ndarray
R_0: float
B_0: float
I_p: float
plot(ax=None, *, show: bool = False, show_mesh: bool = False) → matplotlib.pyplot.Axes

Plot the fixed boundary FE equilibrium.

Parameters:

  • ax – The matplotlib axes object to plot on. If None, a new one is created.

  • show_mesh (bool) – Whether or not to show the mesh on the plot.

  • show (bool)

Returns:

The matplotlib axes object

Return type:

ax

class bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D.FemGradShafranovFixedBoundary(p_prime: collections.abc.Callable[[float], float] | None = None, ff_prime: collections.abc.Callable[[float], float] | None = None, mesh: dolfinx.mesh.Mesh | str | None = None, I_p: float | None = None, R_0: float | None = None, B_0: float | None = None, p_order: int = 2, maxiter: int = 10, iter_err_max: float = 1e-05, relaxation: float = 0.0)

Bases: bluemira.magnetostatics.finite_element_2d.FemMagnetostatic2d

Inheritance diagram of bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D.FemGradShafranovFixedBoundary

A 2D fem Grad Shafranov solver. The solver is thought as support for the fem fixed boundary module.

Parameters:

  • p_prime (collections.abc.Callable[[float], float] | None) – p’ flux function. If callable, then used directly (50 points saved in file). If None, these must be specified later on, but before the solve.

  • ff_prime (collections.abc.Callable[[float], float] | None) – FF’ flux function. If callable, then used directly (50 points saved in file). If None, these must be specified later on, but before the solve.

  • mesh (dolfinx.mesh.Mesh | str | None) – Mesh to use when solving the problem. If None, must be specified later on, but before the solve.

  • I_p (float | None) – Plasma current [A]. If None, the plasma current is calculated, otherwise the source term is scaled to match the plasma current.

  • B_0 (float | None) – Toroidal field at R_0 [T]. Used when saving to file.

  • R_0 (float | None) – Major radius [m]. Used when saving to file.

  • p_order (int) – Order of the approximating polynomial basis functions

  • maxiter (int) – Maximum number of iterations

  • iter_err_max (float) – Convergence criterion value

  • relaxation (float) – Relaxation factor for the Picard iteration procedure

_g_func = None
_psi_ax = None
_psi_b = None
_grad_psi = None
_pprime = None
_ffprime = None
_mesh_area = None
_curr_target = None
_R_0 = None
_B_0 = None
iter_err_max = 1e-05
maxiter = 10
relaxation = 0.0
k = 1
property psi_ax: float

Poloidal flux on the magnetic axis

Return type:

float

property psi_b: float

Poloidal flux on the boundary

Return type:

float

grad_psi(point: numpy.ndarray) → numpy.ndarray

Calculate the gradients of psi at a point

Parameters:

point (numpy.ndarray)

Return type:

numpy.ndarray

property psi_norm_2d: collections.abc.Callable[[numpy.ndarray], numpy.ndarray]

Normalised flux function in 2-D

Return type:

collections.abc.Callable[[numpy.ndarray], numpy.ndarray]

set_mesh(mesh: dolfinx.mesh.Mesh | str)

Set the mesh for the solver

Parameters:

mesh (dolfinx.mesh.Mesh | str) – Filename of the xml file with the mesh definition or a dolfin mesh

_create_g_func(pprime: collections.abc.Callable[[numpy.typing.ArrayLike], float | numpy.typing.NDArray[numpy.float64]] | float, ffprime: collections.abc.Callable[[numpy.typing.ArrayLike], float | numpy.typing.NDArray[numpy.float64]] | float, curr_target: float | None = None) → collections.abc.Callable[[numpy.ndarray], float]

Return the density current function given pprime and ffprime.

Parameters:

  • pprime (collections.abc.Callable[[numpy.typing.ArrayLike], float | numpy.typing.NDArray[numpy.float64]] | float) – pprime as function of psi_norm (1-D function)

  • ffprime (collections.abc.Callable[[numpy.typing.ArrayLike], float | numpy.typing.NDArray[numpy.float64]] | float) – ffprime as function of psi_norm (1-D function)

  • curr_target (float | None) – Target current (also used to initialise the solution in case self.psi is still 0 and pprime and ffprime are, then, not defined) [A]

Return type:

Source current callable to solve the magnetostatic problem

define_g()

Return the density current DOLFIN function given pprime and ffprime.

set_profiles(p_prime: collections.abc.Callable[[float], float], ff_prime: collections.abc.Callable[[float], float], I_p: float | None = None, B_0: float | None = None, R_0: float | None = None)

Set the profies for the FEM G-S solver.

Parameters:

  • p_prime (collections.abc.Callable[[float], float]) – pprime as function of psi_norm (1-D function)

  • ff_prime (collections.abc.Callable[[float], float]) – ffprime as function of psi_norm (1-D function)

  • I_p (float | None) – Target current (also used to initialise the solution in case self.psi is still 0 and pprime and ffprime are, then, not defined). If None, plasma current is calculated and not constrained

  • B_0 (float | None) – Toroidal field at R_0 [T]. Used when saving to file.

  • R_0 (float | None) – Major radius [m]. Used when saving to file.

Raises:

TypeError – p_prime and ff_prime must be functions

_calculate_curr_tot() → float

Calculate the total current into the domain

Return type:

float

_update_curr()
_reset_psi_cache()

Reset cached psi-axis and psi-boundary properties.

_check_all_inputs_ready_error()
solve(*, plot: bool = False, debug: bool = False, gif: bool = False, figname: str | None = None, autoclose_plot: bool = True) → FixedBoundaryEquilibrium

Solve the G-S problem.

Parameters:

  • plot (bool) – Whether or not to plot

  • debug (bool) – Whether or not to display debug information

  • gif (bool) – Whether or not to produce a GIF

  • figname (str | None) – The name of the figure. If None, a suitable default is used.

  • autoclose_plot (bool)

Return type:

FixedBoundaryEquilibrium object corresponding to the solve

_equilibrium()
Returns:

Equilibrium data object

static _setup_plot(*, debug: bool) → tuple[matplotlib.figure.Figure, numpy.ndarray, list]
Parameters:

debug (bool)

Return type:

tuple[matplotlib.figure.Figure, numpy.ndarray, list]

_plot_current_iteration(ax, cax, i_iter: int, points: collections.abc.Iterable, prev: numpy.ndarray, diff: numpy.ndarray, *, debug: bool = False)
Parameters:

  • i_iter (int)

  • points (collections.abc.Iterable)

  • prev (numpy.ndarray)

  • diff (numpy.ndarray)

  • debug (bool)

bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D._plot_array(ax, points: numpy.ndarray, array: numpy.ndarray, title: str, cmap: str, levels: numpy.ndarray | None = None)
Parameters:

  • points (numpy.ndarray)

  • array (numpy.ndarray)

  • title (str)

  • cmap (str)

  • levels (numpy.ndarray | None)

bluemira.equilibria.fem_fixed_boundary.fem_magnetostatic_2D._add_colorbar(cm, cax, title)