# Source code for pde.grids.spherical

"""
Spherically-symmetric grids in 2 and 3 dimensions. These are grids that only
discretize the radial direction, assuming symmetry with respect to all angles.
This choice implies that differential operators might not be applicable to all fields.
For instance, the divergence of a vector field on a spherical grid can only be
represented as a scalar field on the same grid if the θ-component of the vector field
vanishes.

.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>

"""

from __future__ import annotations

from abc import ABCMeta
from typing import TYPE_CHECKING, Any, Dict, Generator, Optional, Tuple, TypeVar, Union

import numpy as np

from ..tools.cache import cached_property
from ..tools.plotting import plot_on_axes
from .base import DimensionError, GridBase, _check_shape, discretize_interval
from .cartesian import CartesianGrid

if TYPE_CHECKING:
from .boundaries.axes import Boundaries  # @UnusedImport

π = np.pi
PI_4 = 4 * π
PI_43 = 4 / 3 * π

TNumArr = TypeVar("TNumArr", float, np.ndarray)

[docs]def volume_from_radius(radius: TNumArr, dim: int) -> TNumArr: """Return the volume of a sphere with a given radius Args: radius (float or :class:~numpy.ndarray): Radius of the sphere dim (int): Dimension of the space Returns: float or :class:~numpy.ndarray: Volume of the sphere """ if dim == 1: return 2 * radius elif dim == 2: return π * radius**2 elif dim == 3: return PI_43 * radius**3 else: raise NotImplementedError(f"Cannot calculate the volume in {dim} dimensions")
[docs]class SphericalSymGridBase(GridBase, metaclass=ABCMeta): # lgtm [py/missing-equals] r"""Base class for d-dimensional spherical grids with angular symmetry The angular symmetry implies that states only depend on the radial coordinate :math:r, which is discretized uniformly as .. math:: r_i = R_\mathrm{inner} + \left(i + \frac12\right) \Delta r \quad \text{for} \quad i = 0, \ldots, N - 1 \quad \text{with} \quad \Delta r = \frac{R_\mathrm{outer} - R_\mathrm{inner}}{N} where :math:R_\mathrm{outer} is the outer radius of the grid and :math:R_\mathrm{inner} corresponds to a possible inner radius, which is zero by default. The radial direction is discretized by :math:N support points. """ _periodic = [False] # the radial axis is not periodic num_axes = 1 # the number of independent axes boundary_names = {"inner": (0, False), "outer": (0, True)} def __init__( self, radius: Union[float, Tuple[float, float]], shape: Union[Tuple[int], int] ): 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}). shape (tuple or int): A single number setting the number :math:N of support points along the radial coordinate """ super().__init__() shape_list = _check_shape(shape) if not len(shape_list) == 1: raise ValueError(f"shape must be a single number, not {shape_list}") self._shape: Tuple[int] = (int(shape_list[0]),) 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]) self._axes_coords = (rs,) self._axes_bounds = ((r_inner, r_outer),) self._discretization = np.array((dr,)) @property def state(self) -> Dict[str, Any]: """state: the state of the grid""" return {"radius": self.radius, "shape": self.shape}
[docs] @classmethod def from_state(cls, state: Dict[str, Any]) -> SphericalSymGridBase: # 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"), shape=state_copy.pop("shape")) if state_copy: raise ValueError(f"State items {state_copy.keys()} were not used") return obj
[docs] @classmethod def from_bounds( # type: ignore cls, bounds: Tuple[Tuple[float, float]], shape: Tuple[int], periodic: Tuple[bool], ) -> SphericalSymGridBase: """ Args: bounds (tuple): Give the coordinate range for the radial axis. shape (tuple): The number of support points for the radial axis periodic (bool or list): Not used Returns: SphericalGridBase representing the region chosen by bounds """ if len(bounds) != 1: raise ValueError( f"bounds must be given as ((r_min, r_max),). Got {bounds} instead" ) return cls(bounds[0], shape)
@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 volume(self) -> float: """float: total volume of the grid""" r_inner, r_outer = self.axes_bounds[0] volume = volume_from_radius(r_outer, dim=self.dim) if r_inner > 0: volume -= volume_from_radius(r_inner, dim=self.dim) return volume @cached_property() def cell_volume_data(self) -> Tuple[np.ndarray]: """tuple of :class:~numpy.ndarray: the volumes of all cells""" dr = self.discretization[0] rs = self.axes_coords[0] volumes_h = volume_from_radius(rs + 0.5 * dr, dim=self.dim) volumes_l = volume_from_radius(rs - 0.5 * dr, dim=self.dim) return ((volumes_h - volumes_l).reshape(self.shape[0]),)
[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 in the volume, implying they are not uniformly distributed on the radial axis. 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 if r_max <= r_min: raise RuntimeError("Random points would be too close to boundary") # choose random radius scaled such that points are uniformly distributed r = np.array( [rng.uniform(r_min**self.dim, r_max**self.dim) ** (1 / self.dim)] ) if coords == "cartesian": # choose random angles for the already chosen radius if self.dim == 2: φ = rng.uniform(0, 2 * np.pi) point = np.r_[r, φ] elif self.dim == 3: θ = np.arccos(rng.uniform(-1, 1)) φ = rng.uniform(0, 2 * np.pi) point = np.r_[r, θ, φ] else: raise NotImplementedError(f"{self.dim} dimensions") return self.point_to_cartesian(point, full=True) elif coords == "cell": return self.transform(r, "grid", "cell") elif coords == "grid": return r 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 along the radial axis Args: data (:class:~numpy.ndarray): The values at the grid points extract (str): Determines which cut is done through the grid. This parameter is mainly supplied for a consistent interface and has no effect for polar grids. Returns: A dictionary with information about the line cut, which is convenient for plotting. """ if extract not in {"auto", "r", "radial"}: raise ValueError(f"Unknown extraction method {extract}") return { "data_x": self.axes_coords[0], "data_y": data, "extent_x": self.axes_bounds[0], "label_x": self.axes[0], }
[docs] def get_image_data( self, data: np.ndarray, performance_goal: str = "speed", fill_value: float = 0, masked: bool = True, ) -> Dict[str, Any]: """return a 2d-image of the data Args: data (:class:~numpy.ndarray): The values at the grid points performance_goal (str): Determines the method chosen for interpolation. Possible options are speed and quality. fill_value (float): The value assigned to invalid positions (those inside the hole or outside the region). masked (bool): Whether a :class:numpy.ma.MaskedArray is returned for the data instead of the normal :class:~numpy.ndarray. Returns: A dictionary with information about the image, which is convenient for plotting. """ from scipy import interpolate _, r_outer = self.axes_bounds[0] r_data = self.axes_coords[0] if self.has_hole: num = int(np.ceil(r_outer / self.discretization[0])) x_positive, _ = discretize_interval(0, r_outer, num) else: x_positive = r_data x = np.r_[-x_positive[::-1], x_positive] xs, ys = np.meshgrid(x, x, indexing="ij") r_img = np.hypot(xs, ys) if performance_goal == "speed": # interpolate over the new coordinates using linear interpolation f = interpolate.interp1d( r_data, data, copy=False, bounds_error=False, fill_value=fill_value, assume_sorted=True, ) data_int = f(r_img.flat).reshape(r_img.shape) elif performance_goal == "quality": # interpolate over the new coordinates using radial base function f = interpolate.Rbf(r_data, data, function="cubic") data_int = f(r_img) else: raise ValueError(f"Performance goal {performance_goal} undefined") if masked: mask = (r_img < r_data[0]) | (r_data[-1] < r_img) data_int = np.ma.masked_array(data_int, mask=mask) return { "data": data_int, "x": x, "y": x, "extent": (-r_outer, r_outer, -r_outer, r_outer), "label_x": "x", "label_y": "y", }
[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 """ if with_self: yield np.asanyarray(point, dtype=np.double)
[docs] def point_from_cartesian(self, points: np.ndarray) -> np.ndarray: """convert points given in Cartesian coordinates to this grid 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" return np.linalg.norm(points, axis=-1, keepdims=True) # type: ignore
[docs] def polar_coordinates_real( self, origin=None, *, ret_angle: bool = False, **kwargs ) -> Union[np.ndarray, Tuple[np.ndarray, ...]]: """return spherical coordinates associated with the grid Args: origin: Place holder variable to comply with the interface ret_angle (bool): Determines whether angles are 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. Note that in the case of spherical grids, this angle is zero by convention. """ # check the consistency of the origin argument, which can be set for other grids if origin is not None: origin = np.array(origin, dtype=np.double, ndmin=1) if not np.array_equal(origin, np.zeros(self.dim)): raise RuntimeError(f"Origin must be {str([0]*self.dim)}") # the distance to the origin is exactly the radial coordinate rs = self.axes_coords[0] if ret_angle: return (rs,) + (np.zeros_like(rs),) * (self.dim - 1) else: return rs
[docs] def get_cartesian_grid( self, mode: str = "valid", num: Optional[int] = None ) -> CartesianGrid: """return a Cartesian grid for this spherical one Args: mode (str): Determines how the grid is determined. Setting it to 'valid' (or 'inscribed') only returns points that are fully resolved in the spherical grid, e.g., the Cartesian grid is inscribed in the sphere. Conversely, 'full' (or 'circumscribed') returns all data, so the Cartesian grid is circumscribed. num (int): Number of support points along each axis of the returned grid. Returns: :class:pde.grids.cartesian.CartesianGrid: The requested grid """ # pick how the grid is determined if mode == "valid": if self.has_hole: self._logger.warning("Sphere has holes, so not all points are valid") bounds = self.radius / np.sqrt(self.dim) elif mode == "inscribed": bounds = self.radius / np.sqrt(self.dim) elif mode == "full" or mode == "circumscribed": bounds = self.radius else: raise ValueError(f"Unsupported mode {mode}") # determine the grid points if num is None: num = 2 * round(bounds / self.discretization[0]) grid_bounds = [(-bounds, bounds)] * self.dim return CartesianGrid(grid_bounds, num)
[docs] @plot_on_axes() def plot(self, ax, **kwargs): r"""visualize the spherically symmetric grid in two dimensions Args: {PLOT_ARGS} \**kwargs: Extra arguments are passed on the to the matplotlib plotting routines, e.g., to set the color of the lines """ from matplotlib import collections, patches kwargs.setdefault("edgecolor", kwargs.get("color", "k")) kwargs.setdefault("facecolor", "none") (rb,) = self.axes_bounds rmax = rb[1] # draw circular parts circles = [] for r in np.linspace(*rb, self.shape[0] + 1): if r == 0: c = patches.Circle((0, 0), 0.01 * rmax) else: c = patches.Circle((0, 0), r) circles.append(c) ax.add_collection(collections.PatchCollection(circles, **kwargs)) ax.set_xlim(-rmax, rmax) ax.set_xlabel("x") ax.set_ylim(-rmax, rmax) ax.set_ylabel("y") ax.set_aspect(1)
[docs]class PolarSymGrid(SphericalSymGridBase): r"""2-dimensional polar grid assuming angular symmetry The angular symmetry implies that states only depend on the radial coordinate :math:r, which is discretized uniformly as .. math:: r_i = R_\mathrm{inner} + \left(i + \frac12\right) \Delta r \quad \text{for} \quad i = 0, \ldots, N - 1 \quad \text{with} \quad \Delta r = \frac{R_\mathrm{outer} - R_\mathrm{inner}}{N} where :math:R_\mathrm{outer} is the outer radius of the grid and :math:R_\mathrm{inner} corresponds to a possible inner radius, which is zero by default. The radial direction is discretized by :math:N support points. """ dim = 2 # dimension of the described space axes = ["r"] axes_symmetric = ["phi"] coordinate_constraints = [0, 1] # axes not described explicitly
[docs] def point_to_cartesian( self, points: np.ndarray, *, full: bool = False ) -> np.ndarray: """convert coordinates of a point to Cartesian coordinates This function returns points along the y-coordinate, i.e, the x coordinates will be zero. 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) if full: # angles are supplied if points.shape[-1] != self.dim: raise DimensionError(f"Shape {points.shape} cannot denote points") x = points[..., 0] * np.cos(points[..., 1]) y = points[..., 0] * np.sin(points[..., 1]) else: # angles are not supplied if points.shape[-1] != self.num_axes: raise DimensionError(f"Shape {points.shape} cannot denote points") y = points[..., 0] x = np.zeros_like(y) return np.stack((x, y), axis=-1)
[docs]class SphericalSymGrid(SphericalSymGridBase): r"""3-dimensional spherical grid assuming spherical symmetry The symmetry implies that states only depend on the radial coordinate :math:r, which is discretized as follows: .. math:: r_i = R_\mathrm{inner} + \left(i + \frac12\right) \Delta r \quad \text{for} \quad i = 0, \ldots, N - 1 \quad \text{with} \quad \Delta r = \frac{R_\mathrm{outer} - R_\mathrm{inner}}{N} where :math:R_\mathrm{outer} is the outer radius of the grid and :math:R_\mathrm{inner} corresponds to a possible inner radius, which is zero by default. The radial direction is discretized by :math:N support points. Warning: Not all results of differential operators on vectorial and tensorial fields can be expressed in terms of fields that only depend on the radial coordinate :math:r. In particular, the gradient of a vector field can only be calculated if the azimuthal component of the vector field vanishes. Similarly, the divergence of a tensor field can only be taken in special situations. """ dim = 3 # dimension of the described space axes = ["r"] axes_symmetric = ["theta", "phi"] coordinate_constraints = [0, 1, 2] # axes not described explicitly
[docs] def point_to_cartesian( self, points: np.ndarray, *, full: bool = False ) -> np.ndarray: """convert coordinates of a point to Cartesian coordinates This function returns points along the z-coordinate, i.e, the x and y coordinates will be zero. 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) if full: # angles are supplied if points.shape[-1] != self.dim: raise DimensionError(f"Shape {points.shape} cannot denote points") rsinθ = points[..., 0] * np.sin(points[..., 1]) x = rsinθ * np.cos(points[..., 2]) y = rsinθ * np.sin(points[..., 2]) z = points[..., 0] * np.cos(points[..., 1]) else: # angles are not supplied if points.shape[-1] != self.num_axes: raise DimensionError(f"Shape {points.shape} cannot denote points") z = points[..., 0] x = y = np.zeros_like(z) return np.stack((x, y, z), axis=-1)