r"""This module implements differential operators on spherical grids.
.. autosummary::
:nosignatures:
CylindricalLaplacian
CylindricalGradient
CylindricalGradientSquared
CylindricalDivergence
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING
import torch
from ....grids import CylindricalSymGrid, GridBase
from ....tools.docstrings import fill_in_docstring
from .. import torch_backend
from .common import TorchDifferentialOperator
if TYPE_CHECKING:
import numpy as np
from torch import Tensor
from ....grids.boundaries import BoundariesList
[docs]
@torch_backend.register_operator(CylindricalSymGrid, "laplace", rank_in=0, rank_out=0)
@fill_in_docstring
class CylindricalLaplacian(TorchDifferentialOperator):
"""Cylindrical Laplace using torch.
{DESCR_CYLINDRICAL_GRID}
"""
rank_in = 0
def __init__(self, grid: GridBase, bcs: BoundariesList | None, *, dtype: np.dtype):
"""Initialize the Cylindrical Laplacian operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
"""
super().__init__(grid, bcs, dtype=dtype)
# calculate preliminary quantities
dr = grid.discretization[0]
self.dr_2, self.dz_2 = 1 / grid.discretization**2
factor_r = 1 / (2 * grid.axes_coords[0] * dr)
self.register_array("factor_r", factor_r[:, None])
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
arr_z_l, arr_z_h = data_full[1:-1, :-2], data_full[1:-1, 2:]
arr_mid = data_full[1:-1, 1:-1]
arr_r_l, arr_r_h = data_full[:-2, 1:-1], data_full[2:, 1:-1]
return ( # type: ignore
(arr_r_h - 2 * arr_mid + arr_r_l) * self.dr_2
+ (arr_r_h - arr_r_l) * self.factor_r # type: ignore
+ (arr_z_l - 2 * arr_mid + arr_z_h) * self.dz_2
)
[docs]
@torch_backend.register_operator(CylindricalSymGrid, "gradient", rank_in=0, rank_out=1)
@fill_in_docstring
class CylindricalGradient(TorchDifferentialOperator):
"""Cylindrical gradient operator using torch.
{DESCR_CYLINDRICAL_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
):
"""Initialize the Cylindrical gradient operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
"""
super().__init__(grid, bcs, dtype=dtype)
# calculate preliminary quantities
self.scale_r, self.scale_z = 1 / (2 * grid.discretization)
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
r = (data_full[2:, 1:-1] - data_full[:-2, 1:-1]) * self.scale_r
z = (data_full[1:-1, 2:] - data_full[1:-1, :-2]) * self.scale_z
phi = torch.zeros_like(r)
return torch.stack((r, z, phi))
[docs]
@torch_backend.register_operator(
CylindricalSymGrid, "gradient_squared", rank_in=0, rank_out=0
)
@fill_in_docstring
class CylindricalGradientSquared(TorchDifferentialOperator):
"""Cylindrical gradient squared operator using torch.
{DESCR_CYLINDRICAL_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
central: bool = True,
dtype: np.dtype,
):
"""Initialize the Cylindrical gradient squared operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
central (bool):
Whether to use central differences. If `False`, forward and backward
differences are used.
dtype:
The data type of the field
"""
super().__init__(grid, bcs, dtype=dtype)
self.central = central
if self.central:
self.scale_r, self.scale_z = 0.25 / grid.discretization**2
else:
self.scale_r, self.scale_z = 0.5 / grid.discretization**2
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
if self.central:
# simple squared sum of central differences
r = (data_full[2:, 1:-1] - data_full[:-2, 1:-1]) ** 2 * self.scale_r
z = (data_full[1:-1, 2:] - data_full[1:-1, :-2]) ** 2 * self.scale_z
return r + z # type: ignore
arr_z_h = data_full[1:-1, 2:]
arr_c = data_full[1:-1, 1:-1]
arr_z_l = data_full[1:-1, :-2]
term_r = (arr[2:, 1:-1] - arr_c) ** 2 + (arr_c - arr[:-2, 1:-1]) ** 2
term_z = (arr_z_h - arr_c) ** 2 + (arr_c - arr_z_l) ** 2
return term_r * self.scale_r + term_z * self.scale_z # type: ignore
[docs]
@torch_backend.register_operator(
CylindricalSymGrid, "divergence", rank_in=1, rank_out=0
)
@fill_in_docstring
class CylindricalDivergence(TorchDifferentialOperator):
"""Cylindrical divergence operator using torch.
{DESCR_CYLINDRICAL_GRID}
"""
rank_in = 1
def __init__(self, grid: GridBase, bcs: BoundariesList | None, *, dtype: np.dtype):
"""Initialize the Cylindrical divergence operator.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid on which the operator acts
bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
The boundary conditions applied to the field. If `None`, no boundary
conditions are enforced.
dtype:
The data type of the field
"""
super().__init__(grid, bcs, dtype=dtype)
self.scale_r, self.scale_z = 1 / (2 * grid.discretization)
rs = grid.axes_coords[0]
self.register_array("rs", rs[:, None])
[docs]
def forward(self, arr: Tensor, args=None) -> Tensor:
"""Fill internal data array, apply operator, and return valid data."""
data_full = self.get_full_data(arr, args=args)
arr_r, arr_z = data_full[0], data_full[1]
return ( # type: ignore
arr_r[1:-1, 1:-1] / self.rs # type: ignore
+ (arr_r[2:, 1:-1] - arr_r[:-2, 1:-1]) * self.scale_r
+ (arr_z[1:-1, 2:] - arr_z[1:-1, :-2]) * self.scale_z
)
# @torch_backend.register_operator(
# CylindricalSymGrid, "vector_gradient", rank_in=1, rank_out=2
# )
# class CylindricalVectorGradient(TorchDifferentialOperator):
# """Cylindrical vector gradient operator using torch."""
# rank_in = 1
# def __init__(
# self,
# grid: GridBase,
# bcs: BoundariesList | None,
# *,
# dtype: np.dtype,
# ):
# """Initialize the Cylindrical divergence operator.
# Args:
# grid (:class:`~pde.grids.base.GridBase`):
# The grid on which the operator acts
# bcs (:class:`~pde.grids.boundaries.axes.BoundariesList` or None):
# The boundary conditions applied to the field. If `None`, no boundary
# conditions are enforced.
# dtype:
# The data type of the field
# """
# super().__init__(grid, bcs, dtype=dtype)
# dr = self.grid.discretization[0]
# self.register_array("rs", self.grid.axes_coords[0])
# self.scale_r = 1 / (2 * dr)
# def forward(self, arr: Tensor, args=None) -> Tensor:
# """Fill internal data array, apply operator, and return valid data."""
# data_full = self.get_full_data(arr, args=args)
# # assign aliases
# arr_r, arr_φ = arr
# out_rr, out_rφ = out[0, 0, :], out[0, 1, :]
# out_φr, out_φφ = out[1, 0, :], out[1, 1, :]
# for i in range(1, dim_r + 1): # iterate radial points
# out_rr[i - 1] = (arr_r[i + 1] - arr_r[i - 1]) * scale_r
# out_rφ[i - 1] = -arr_φ[i] / rs[i - 1]
# out_φr[i - 1] = (arr_φ[i + 1] - arr_φ[i - 1]) * scale_r
# out_φφ[i - 1] = arr_r[i] / rs[i - 1]
# term1 = (data_full[0, 2:] - data_full[0, :-2]) * self.scale_r
# term2 = data_full[0, 1:-1] / self.rs
# return term1 + term2
__all__ = [
"CylindricalDivergence",
"CylindricalGradient",
"CylindricalGradientSquared",
"CylindricalLaplacian",
]