r"""This module implements differential operators on polar grids.
.. autosummary::
:nosignatures:
PolarLaplacian
PolarGradient
PolarGradientSquared
PolarDivergence
PolarVectorGradient
PolarTensorDivergence
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING, Literal
import torch
from ....grids import GridBase, PolarSymGrid
from ....tools.docstrings import fill_in_docstring
from ..backend import TorchBackend
from .common import TorchDifferentialOperator
if TYPE_CHECKING:
import numpy as np
from torch import Tensor
from ....grids.boundaries import BoundariesList
[docs]
@TorchBackend.register_operator(PolarSymGrid, "laplace", rank_in=0, rank_out=0)
@fill_in_docstring
class PolarLaplacian(TorchDifferentialOperator):
"""Polar Laplace using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
):
"""Initialize the Polar 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]
factor_r = 1 / (2 * self.grid.axes_coords[0] * dr)
self.register_array("factor_r", factor_r)
self.dr_2 = 1 / dr**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)
term1 = (data_full[2:] - 2 * data_full[1:-1] + data_full[:-2]) * self.dr_2
term2 = (data_full[2:] - data_full[:-2]) * self.factor_r # type: ignore
return term1 + term2 # type: ignore
[docs]
@TorchBackend.register_operator(PolarSymGrid, "gradient", rank_in=0, rank_out=1)
@fill_in_docstring
class PolarGradient(TorchDifferentialOperator):
"""Polar gradient operator using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
method: Literal["central", "forward", "backward"] = "central",
):
"""Initialize the Polar 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
method (str):
The method for calculating the derivative. Possible values are
'central', 'forward', and 'backward'.
"""
super().__init__(grid, bcs, dtype=dtype)
# calculate preliminary quantities
self.method = method
if method == "central":
self.scale_r = 0.5 / grid.discretization[0]
elif method in {"forward", "backward"}:
self.scale_r = 1 / grid.discretization[0]
else:
msg = f"Unknown derivative type `{method}`"
raise ValueError(msg)
[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.method == "central":
r = (data_full[2:] - data_full[:-2]) * self.scale_r
elif self.method == "forward":
r = (data_full[2:] - data_full[1:-1]) * self.scale_r
elif self.method == "backward":
r = (data_full[1:-1] - data_full[:-2]) * self.scale_r
# no angular dependence by definition
return torch.stack((r, torch.zeros_like(r)))
[docs]
@TorchBackend.register_operator(PolarSymGrid, "gradient_squared", rank_in=0, rank_out=0)
@fill_in_docstring
class PolarGradientSquared(TorchDifferentialOperator):
"""Polar gradient squared operator using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 0
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
central: bool = True,
dtype: np.dtype,
):
"""Initialize the Polar 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
dr = grid.discretization[0]
if self.central:
self.scale = 0.25 / dr**2
else:
self.scale = 0.5 / dr**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
return (data_full[2:] - data_full[:-2]) ** 2 * self.scale # type: ignore
term1 = (data_full[2:] - data_full[1:-1]) ** 2
term2 = (data_full[1:-1] - data_full[:-2]) ** 2
return (term1 + term2) * self.scale # type: ignore
[docs]
@TorchBackend.register_operator(PolarSymGrid, "divergence", rank_in=1, rank_out=0)
@fill_in_docstring
class PolarDivergence(TorchDifferentialOperator):
"""Polar divergence operator using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 1
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
):
"""Initialize the Polar 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)
[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)
term1 = (data_full[0, 2:] - data_full[0, :-2]) * self.scale_r
term2 = data_full[0, 1:-1] / self.rs # type: ignore
return term1 + term2 # type: ignore
[docs]
@TorchBackend.register_operator(PolarSymGrid, "vector_gradient", rank_in=1, rank_out=2)
@fill_in_docstring
class PolarVectorGradient(TorchDifferentialOperator):
"""Polar vector gradient operator using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 1
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
):
"""Initialize the Polar vector 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)
dr = self.grid.discretization[0]
self.result_shape = (2, 2, *grid.shape)
self.register_array("rs", self.grid.axes_coords[0])
self.scale_r = 1 / (2 * dr)
[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)
result = torch.empty(self.result_shape, dtype=arr.dtype, device=arr.device)
arr_r, arr_φ = data_full[0], data_full[1]
result[0, 0] = (arr_r[2:] - arr_r[:-2]) * self.scale_r # rr component
result[0, 1] = -arr_φ[1:-1] / self.rs # type: ignore # rφ component
result[1, 0] = (arr_φ[2:] - arr_φ[:-2]) * self.scale_r # φr component
result[1, 1] = arr_r[1:-1] / self.rs # type: ignore # φφ component
return result
[docs]
@TorchBackend.register_operator(
PolarSymGrid, "tensor_divergence", rank_in=2, rank_out=1
)
@fill_in_docstring
class PolarTensorDivergence(TorchDifferentialOperator):
"""Polar tensor divergence operator using torch.
{DESCR_POLAR_GRID}
"""
rank_in = 2
def __init__(
self,
grid: GridBase,
bcs: BoundariesList | None,
*,
dtype: np.dtype,
):
"""Initialize the Polar tensor 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)
[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_rr, arr_rφ = data_full[0, 0], data_full[0, 1]
arr_φr, arr_φφ = data_full[1, 0], data_full[1, 1]
out_r = (arr_rr[2:] - arr_rr[:-2]) * self.scale_r
out_r += (arr_rr[1:-1] - arr_φφ[1:-1]) / self.rs # type: ignore
out_φ = (arr_φr[2:] - arr_φr[:-2]) * self.scale_r
out_φ += (arr_rφ[1:-1] + arr_φr[1:-1]) / self.rs # type: ignore
return torch.stack((out_r, out_φ))
__all__ = [
"PolarDivergence",
"PolarGradient",
"PolarGradientSquared",
"PolarLaplacian",
"PolarTensorDivergence",
"PolarVectorGradient",
]