Source code for pde.grids.cylindrical

Cylindrical grids with azimuthal symmetry

.. codeauthor:: David Zwicker <>

from __future__ import annotations

from typing import TYPE_CHECKING, Any, Dict, Generator, Optional, Sequence, Tuple, Union

import numpy as np

from import cached_property
from .base import DimensionError, GridBase, _check_shape, discretize_interval
from .cartesian import CartesianGrid

    from .boundaries.axes import Boundaries, BoundariesData  # @UnusedImport
    from .spherical import PolarSymGrid  # @UnusedImport

[docs]class CylindricalSymGrid(GridBase): # lgtm [py/missing-equals] r""" 3-dimensional cylindrical grid assuming polar symmetry The polar symmetry implies that states only depend on the radial and axial coordinates :math:`r` and :math:`z`, respectively. These are discretized uniformly as .. math:: :nowrap: \begin{align*} r_i &= R_\mathrm{inner} + \left(i + \frac12\right) \Delta r &&\quad \text{for} \quad i = 0, \ldots, N_r - 1 &&\quad \text{with} \quad \Delta r = \frac{R_\mathrm{outer} - R_\mathrm{inner}}{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 :math:`R_\mathrm{outer}` is the outer radius of the grid, :math:`R_\mathrm{inner}` corresponds to a possible inner radius (which is zero by default), and :math:`z_\mathrm{min}` and :math:`z_\mathrm{max}` denote the respective lower and upper bounds of the axial direction, so that :math:`z_\mathrm{max} - z_\mathrm{min}` is the total height. The two axes are discretized by :math:`N_r` and :math:`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 :math:`(r, \phi, z)`, we here use the order :math:`(r, z, \phi)`. It might thus be best to access components by name instead of index, e.g., use :code:`field['z']` instead of :code:`field[1]`. """ dim = 3 # dimension of the described space num_axes = 2 # number of independent axes axes = ["r", "z"] # name of the actual axes axes_symmetric = ["phi"] coordinate_constraints = [0, 1] # constraint Cartesian coordinates boundary_names = { # name all the boundaries "inner": (0, False), "outer": (0, True), "bottom": (1, False), "top": (1, True), } def __init__( self, radius: Union[float, Tuple[float, float]], bounds_z: Tuple[float, float], shape: Union[int, Sequence[int]], periodic_z: bool = False, ): r""" Args: radius (float or tuple of floats): radius :math:`R_\mathrm{outer}` in case a simple float is given. If a tuple is supplied it is interpreted as the inner and outer radius, :math:`(R_\mathrm{inner}, R_\mathrm{outer})`. bounds_z (tuple): The lower and upper bound of the z-axis shape (tuple): The number of support points in r and z direction, respectively. The same number is used for both if a single value is given. periodic_z (bool): Determines whether the z-axis has periodic boundary conditions. """ super().__init__() shape_list = _check_shape(shape) if len(shape_list) == 1: self._shape: Tuple[int, int] = (shape_list[0], shape_list[0]) elif len(shape_list) == 2: self._shape = tuple(shape_list) # type: ignore else: raise DimensionError("`shape` must be two integers") if len(bounds_z) != 2: raise ValueError( "Lower and upper value of the axial coordinate must be specified" ) self._periodic_z: bool = bool(periodic_z) # might cast from np.bool_ self._periodic = [False, self._periodic_z] # radial discretization try: r_inner, r_outer = radius # type: ignore except TypeError: r_inner, r_outer = 0, float(radius) # type: ignore if r_inner < 0: raise ValueError("Inner radius must be positive") if r_inner >= r_outer: raise ValueError("Outer radius must be larger than inner radius") # radial discretization rs, dr = discretize_interval(r_inner, r_outer, self.shape[0]) # axial discretization zs, dz = discretize_interval(*bounds_z, self.shape[1]) assert np.isclose(zs[-1] + dz / 2, bounds_z[1]) self._axes_coords = (rs, zs) self._axes_bounds = ((r_inner, r_outer), tuple(bounds_z)) # type: ignore self._discretization = np.array((dr, dz)) @property def state(self) -> Dict[str, Any]: """state: the state of the grid""" radius = self.axes_bounds[0][1] return { "radius": radius, "bounds_z": self.axes_bounds[1], "shape": self.shape, "periodic_z": self._periodic_z, }
[docs] @classmethod def from_state(cls, state: Dict[str, Any]) -> "CylindricalSymGrid": # type: ignore """create a field from a stored `state`. Args: state (dict): The state from which the grid is reconstructed. """ state_copy = state.copy() obj = cls( radius=state_copy.pop("radius"), bounds_z=state_copy.pop("bounds_z"), shape=state_copy.pop("shape"), periodic_z=state_copy.pop("periodic_z"), ) if state_copy: raise ValueError(f"State items {state_copy.keys()} were not used") return obj
[docs] @classmethod def from_bounds( cls, bounds: Sequence[Tuple[float, float]], shape: Sequence[int], periodic: Sequence[bool], ) -> CylindricalSymGrid: """ Args: bounds (tuple): Give the coordinate range for each axis. This should be a tuple of two number (lower and upper bound) for each axis. The length of `bounds` must be 2. shape (tuple): The number of support points for each axis. The length of `shape` needs to be 2. periodic (bool or list): Specifies which axes possess periodic boundary conditions. The first entry is ignored. Returns: CylindricalGrid representing the region chosen by bounds """ radii, bounds_z = bounds if radii[0] != 0: raise NotImplementedError("Cylinders with hollow core are not implemented.") return cls(radii[1], bounds_z, shape, periodic_z=periodic[1])
@property def has_hole(self) -> bool: """returns whether the inner radius is larger than zero""" return self.axes_bounds[0][0] > 0 @property def radius(self) -> Union[float, Tuple[float, float]]: """float: radius of the sphere""" r_inner, r_outer = self.axes_bounds[0] if r_inner == 0: return r_outer else: return r_inner, r_outer @property def length(self) -> float: """float: length of the cylinder""" return self.axes_bounds[1][1] - self.axes_bounds[1][0] @property def volume(self) -> float: """float: total volume of the grid""" r_inner, r_outer = self.axes_bounds[0] return float(np.pi * self.length * (r_outer**2 - r_inner**2))
[docs] def get_random_point( self, *, boundary_distance: float = 0, avoid_center: bool = False, coords: str = "cartesian", rng: Optional[np.random.Generator] = None, ) -> np.ndarray: """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. Args: 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 :meth:`~pde.grids.base.GridBase.transform`. rng (:class:`~numpy.random.Generator`): Random number generator (default: :func:`~numpy.random.default_rng()`) Returns: :class:`~numpy.ndarray`: The coordinates of the point """ if rng is None: rng = np.random.default_rng() # handle the boundary distance r_inner, r_outer = self.axes_bounds[0] r_min = r_inner + boundary_distance if avoid_center else r_inner r_max = r_outer - boundary_distance z_min = self.axes_bounds[1][0] + boundary_distance z_max = self.axes_bounds[1][1] - boundary_distance if r_max <= r_min or z_max <= z_min: raise RuntimeError("Random points would be too close to boundary") # create random point r = np.sqrt(rng.uniform(r_min**2, r_max**2)) z = rng.uniform(z_min, z_max) if coords == "cartesian": φ = rng.uniform(0, 2 * np.pi) # additional random angle return self.point_to_cartesian(np.array([r, z, φ]), full=True) elif coords == "cell": return self.transform(np.array([r, z]), "grid", "cell") elif coords == "grid": return np.array([r, z]) else: raise ValueError(f"Unknown coordinate system `{coords}`")
[docs] def get_line_data(self, data: np.ndarray, extract: str = "auto") -> Dict[str, Any]: """return a line cut for the cylindrical grid Args: data (:class:`~numpy.ndarray`): The values at the grid points extract (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 :math:`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. """ if extract == "auto": extract = "cut_axial" if extract == "cut_z" or extract == "cut_axial": # do a cut along the z axis for r=0 axis = 1 data_y: Union[np.ndarray, Tuple[np.ndarray]] = data[..., 0, :] label_y = "Cut along z" elif extract == "project_z" or extract == "project_axial": # project on the axial coordinate (average radially) axis = 1 data_y = (data.mean(axis=-2),) label_y = "Projection onto z" elif extract == "project_r" or extract == "project_radial": # project on the radial coordinate (average axially) axis = 0 data_y = (data.mean(axis=-1),) label_y = "Projection onto r" else: raise ValueError(f"Unknown extraction method {extract}") return { "data_x": self.axes_coords[axis], "data_y": data_y, "extent_x": self.axes_bounds[axis], "label_x": self.axes[axis], "label_y": label_y, }
[docs] def get_image_data(self, data: np.ndarray) -> Dict[str, Any]: """return a 2d-image of the data Args: data (:class:`~numpy.ndarray`): The values at the grid points Returns: A dictionary with information about the image, which is convenient for plotting. """ bounds_r, bounds_z = self.axes_bounds if self.has_hole: image_data = data extent = (bounds_r[0], bounds_r[1], bounds_z[0], bounds_z[1]) else: image_data = np.vstack((data[::-1, :], data)) extent = (-bounds_r[1], bounds_r[1], bounds_z[0], bounds_z[1]) return { "data": image_data, "x": self.axes_coords[0], "y": self.axes_coords[1], "extent": extent, "label_x": self.axes[0], "label_y": self.axes[1], }
[docs] def iter_mirror_points( self, point: np.ndarray, with_self: bool = False, only_periodic: bool = True ) -> Generator: """generates all mirror points corresponding to `point` Args: point (:class:`~numpy.ndarray`): the point within the grid with_self (bool): whether to include the point itself only_periodic (bool): whether to only mirror along periodic axes Returns: A generator yielding the coordinates that correspond to mirrors """ point = np.asanyarray(point, dtype=np.double) if with_self: yield point if not only_periodic or self._periodic_z: yield point - np.array([self.length, 0, 0]) yield point + np.array([self.length, 0, 0])
@cached_property() def cell_volume_data(self) -> Tuple[np.ndarray, float]: """:class:`~numpy.ndarray`: the volumes of all cells""" dr, dz = self.discretization rs = self.axes_coords[0] r_vols = 2 * np.pi * dr * rs # same as r_vols = np.pi * ((rs + dr / 2) ** 2 - (rs - dr / 2)**2) return (r_vols, dz)
[docs] def point_to_cartesian( self, points: np.ndarray, *, full: bool = False ) -> np.ndarray: """convert coordinates of a point to Cartesian coordinates Args: points (:class:`~numpy.ndarray`): The grid coordinates of the points full (bool): Flag indicating whether angular coordinates are specified Returns: :class:`~numpy.ndarray`: The Cartesian coordinates of the point """ points = np.atleast_1d(points) z = points[..., 1] if full: if points.shape[-1] != self.dim: raise DimensionError(f"Shape {points.shape} cannot denote full points") x = points[..., 0] * np.cos(points[..., 2]) y = points[..., 0] * np.sin(points[..., 2]) else: if points.shape[-1] != self.num_axes: raise DimensionError(f"Shape {points.shape} cannot denote grid points") x = points[..., 0] y = np.zeros_like(x) return np.stack((x, y, z), axis=-1)
[docs] def point_from_cartesian(self, points: np.ndarray) -> np.ndarray: """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. Args: points (:class:`~numpy.ndarray`): Points given in Cartesian coordinates. Returns: :class:`~numpy.ndarray`: Points given in the coordinates of the grid """ points = np.atleast_1d(points) assert points.shape[-1] == self.dim, f"Point must have {self.dim} coordinates" rs = np.hypot(points[..., 0], points[..., 1]) zs = points[..., 2] return np.stack((rs, zs), axis=-1)
[docs] def polar_coordinates_real( self, origin: np.ndarray, *, ret_angle: bool = False ) -> Union[np.ndarray, Tuple[np.ndarray, np.ndarray]]: """return spherical coordinates associated with the grid Args: origin (:class:`~numpy.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. """ origin = np.array(origin, dtype=np.double, ndmin=1) if len(origin) != self.dim: raise DimensionError("Dimensions are not compatible") if origin[0] != 0 or origin[1] != 0: raise RuntimeError("Origin must lie on symmetry axis for cylindrical grid") # calculate the difference vector between all cells and the origin diff = self.difference_vector_real(np.array([0, origin[2]]), self.cell_coords) dist: np.ndarray = np.linalg.norm(diff, axis=-1) # get distance if ret_angle: return dist, np.arctan2(diff[:, :, 0], diff[:, :, 1]) else: return dist
[docs] def get_cartesian_grid(self, mode: str = "valid") -> CartesianGrid: """return a Cartesian grid for this Cylindrical one Args: 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: :class:`pde.grids.cartesian.CartesianGrid`: The requested grid """ # Pick the grid instance radius_outer = self.axes_bounds[0][1] if mode == "valid": bounds = radius_outer / np.sqrt(self.dim) elif mode == "full": bounds = radius_outer else: raise ValueError(f"Unsupported mode `{mode}`") # determine the Cartesian grid num = round(bounds / self.discretization[0]) grid_bounds = [(-bounds, bounds), (-bounds, bounds), self.axes_bounds[1]] grid_shape = 2 * num, 2 * num, self.shape[1] return CartesianGrid(grid_bounds, grid_shape)
[docs] def slice(self, indices: Sequence[int]) -> Union["CartesianGrid", "PolarSymGrid"]: """return a subgrid of only the specified axes Args: indices (list): Indices indicating the axes that are retained in the subgrid Returns: :class:`~pde.grids.cartesian.CartesianGrid` or :class:`~pde.grids.spherical.PolarSymGrid`: The subgrid """ if len(indices) != 1: raise ValueError(f"Can only get sub-grid for one axis.") if indices[0] == 0: # return a radial grid from .spherical import PolarSymGrid # @Reimport return PolarSymGrid(self.radius, self.shape[0]) elif indices[0] == 1: # return a Cartesian grid along the z-axis subgrid = CartesianGrid( bounds=[self.axes_bounds[1]], shape=self.shape[1], periodic=self.periodic[1], ) subgrid.axes = [self.axes[1]] return subgrid else: raise ValueError(f"Cannot get sub-grid for index {indices[0]}")