4.2.5. pde.grids.cylindrical module¶
Cylindrical grids with azimuthal symmetry
- class CylindricalSymGrid(radius: float, bounds_z: Tuple[float, float], shape: Tuple[int, int], periodic_z: bool = False)[source]¶
Bases:
pde.grids.base.GridBase
3-dimensional cylindrical grid assuming polar symmetry
The polar symmetry implies that states only depend on the radial and axial coordinates \(r\) and \(z\), respectively. These are discretized uniformly as
\begin{align*} r_i &= \left(i + \frac12\right) \Delta r &&\quad \text{for} \quad i = 0, \ldots, N_r - 1 &&\quad \text{with} \quad \Delta r = \frac{R}{N_r} \\ z_j &= z_\mathrm{min} + \left(j + \frac12\right) \Delta z &&\quad \text{for} \quad j = 0, \ldots, N_z - 1 &&\quad \text{with} \quad \Delta z = \frac{z_\mathrm{max} - z_\mathrm{min}}{N_z} \end{align*}where \(R\) is the radius of the cylindrical grid, \(z_\mathrm{min}\) and \(z_\mathrm{max}\) denote the respective lower and upper bounds of the axial direction, so that \(z_\mathrm{max} - z_\mathrm{min}\) is the total height. The two axes are discretized by \(N_r\) and \(N_z\) support points, respectively.
Warning
The order of components in the vector and tensor fields defined on this grid is different than in ordinary math. While it is common to use \((r, \phi, z)\), we here use the order \((r, z, \phi)\). It might thus be best to access components by name instead of index, e.g., use
field['z']
instead offield[1]
.- Parameters
- axes_symmetric: List[str] = ['phi']¶
The names of the additional axes that the fields do not depend on, e.g. along which they are constant.
- Type
- classmethod from_state(state: Dict[str, Any]) CylindricalSymGrid [source]¶
create a field from a stored state.
- Parameters
state (dict) – The state from which the grid is reconstructed.
- get_boundary_conditions(bc: BoundariesData = 'auto_periodic_neumann', rank: int = 0) Boundaries [source]¶
constructs boundary conditions from a flexible data format
- Parameters
bc (str or list or tuple or dict) – The boundary conditions applied to the field. Boundary conditions are generally given as a list with one condition for each axis. For periodic axis, only periodic boundary conditions are allowed (indicated by ‘periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using a tuple of two conditions). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘natural’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.
rank (int) – The tensorial rank of the value associated with the boundary conditions.
- Raises
ValueError – If the data given in bc cannot be read
PeriodicityError – If the boundaries are not compatible with the periodic axes of the grid.
- get_cartesian_grid(mode: str = 'valid') CartesianGrid [source]¶
return a Cartesian grid for this Cylindrical one
- Parameters
mode (str) – Determines how the grid is determined. Setting it to ‘valid’ only returns points that are fully resolved in the cylindrical grid, e.g., the cylinder is circumscribed. Conversely, ‘full’ returns all data, so the cylinder is inscribed.
- Returns
The requested grid
- Return type
- get_image_data(data: numpy.ndarray) Dict[str, Any] [source]¶
return a 2d-image of the data
- Parameters
data (
ndarray
) – The values at the grid points- Returns
A dictionary with information about the image, which is convenient for plotting.
- get_line_data(data: numpy.ndarray, extract: str = 'auto') Dict[str, Any] [source]¶
return a line cut for the cylindrical grid
- Parameters
data (
ndarray
) – The values at the grid pointsextract (str) –
Determines which cut is done through the grid. Possible choices are (default is cut_axial):
cut_z or cut_axial: values along the axial coordinate for \(r=0\).
project_z or project_axial: average values for each axial position (radial average).
project_r or project_radial: average values for each radial position (axial average)
- Returns
A dictionary with information about the line cut, which is convenient for plotting.
- get_random_point(*, boundary_distance: float = 0, avoid_center: bool = False, coords: str = 'cartesian', rng: Optional[numpy.random._generator.Generator] = None, cartesian: Optional[bool] = None) numpy.ndarray [source]¶
return a random point within the grid
Note that these points will be uniformly distributed on the radial axis, which implies that they are not uniformly distributed in the volume.
- Parameters
boundary_distance (float) – The minimal distance this point needs to have from all boundaries.
avoid_center (bool) – Determines whether the boundary distance should also be kept from the center, i.e., whether points close to the center are returned.
coords (str) – Determines the coordinate system in which the point is specified. Valid values are cartesian, cell, and grid; see
transform()
.rng (
Generator
) – Random number generator (default:default_rng()
)
- Returns
The coordinates of the point
- Return type
- get_subgrid(indices: Sequence[int]) Union[CartesianGrid, PolarSymGrid] [source]¶
return a subgrid of only the specified axes
- Parameters
indices (list) – Indices indicating the axes that are retained in the subgrid
- Returns
CartesianGrid
orPolarSymGrid
: The subgrid
- iter_mirror_points(point: numpy.ndarray, with_self: bool = False, only_periodic: bool = True) Generator [source]¶
generates all mirror points corresponding to point
- point_from_cartesian(points: numpy.ndarray) numpy.ndarray [source]¶
convert points given in Cartesian coordinates to this grid
This function returns points restricted to the x-z plane, i.e., the y-coordinate will be zero.
- point_to_cartesian(points: numpy.ndarray, *, full: bool = False) numpy.ndarray [source]¶
convert coordinates of a point to Cartesian coordinates
- polar_coordinates_real(origin: numpy.ndarray, *, ret_angle: bool = False) Union[numpy.ndarray, Tuple[numpy.ndarray, numpy.ndarray]] [source]¶
return spherical coordinates associated with the grid
- Parameters
origin (
ndarray
) – Coordinates of the origin at which the polar coordinate system is anchored. Note that this must be of the form [0, 0, z_val], where only z_val can be chosen freely.ret_angle (bool) – Determines whether the azimuthal angle is returned alongside the distance. If False only the distance to the origin is returned for each support point of the grid. If True, the distance and angles are returned.