"""
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, Literal, TypeVar
import numpy as np
from ..tools.cache import cached_property
from ..tools.plotting import plot_on_axes
from .base import CoordsType, GridBase, _check_shape, discretize_interval
from .cartesian import CartesianGrid
from .coordinates import PolarCoordinates, SphericalCoordinates
if TYPE_CHECKING:
from .boundaries.axes import Boundaries
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 np.pi * radius**2
elif dim == 3:
return 4 / 3 * np.pi * radius**3
else:
raise NotImplementedError(f"Cannot calculate the volume in {dim} dimensions")
[docs]
class SphericalSymGridBase(GridBase, metaclass=ABCMeta):
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
boundary_names = {"inner": (0, False), "outer": (0, True)}
def __init__(self, radius: float | tuple[float, float], shape: int | tuple[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):
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
Returns:
:class:`SphericalGridBase`: represents 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) -> 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: CoordsType = "cartesian",
rng: np.random.Generator | None = 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.c._pos_to_cart(point)
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: Literal["speed", "quality"] = "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:
:dict: 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,
"xs": xs,
"ys": ys,
"extent": (-r_outer, r_outer, -r_outer, r_outer),
"label_x": "x",
"label_y": "y",
}
[docs]
def get_cartesian_grid(
self,
mode: Literal["valid", "inscribed", "full", "circumscribed"] = "valid",
num: int | None = 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.
"""
c = PolarCoordinates()
_axes_symmetric = (1,) # the angular axis is not described
coordinate_constraints = [0, 1] # axes not described explicitly
[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.
"""
c = SphericalCoordinates()
_axes_symmetric = (1, 2) # the angular axes are not described
coordinate_constraints = [0, 1, 2] # axes not described explicitly