"""
Defines a tensorial field of rank 2 over a grid
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from collections.abc import Sequence
from typing import TYPE_CHECKING, Callable
import numpy as np
from numpy.typing import DTypeLike
from ..grids.base import DimensionError, GridBase
from ..tools.docstrings import fill_in_docstring
from ..tools.misc import get_common_dtype
from ..tools.plotting import PlotReference, plot_on_figure
from ..tools.typing import NumberOrArray
from .datafield_base import DataFieldBase
from .scalar import ScalarField
from .vectorial import VectorField
if TYPE_CHECKING:
from ..grids.boundaries.axes import BoundariesData
[docs]
class Tensor2Field(DataFieldBase):
"""Tensor field of rank 2 discretized on a grid
Warning:
Components of the tensor field are given in the local basis. While the local
basis is identical to the global basis in Cartesian coordinates, the local basis
depends on position in curvilinear coordinate systems. Moreover, the field
always contains all components, even if the underlying grid assumes symmetries.
"""
rank = 2
[docs]
@classmethod
@fill_in_docstring
def from_expression(
cls,
grid: GridBase,
expressions: Sequence[Sequence[str]],
*,
user_funcs: dict[str, Callable] | None = None,
consts: dict[str, NumberOrArray] | None = None,
label: str | None = None,
dtype: DTypeLike | None = None,
) -> Tensor2Field:
"""create a tensor field on a grid from given expressions
Warning:
{WARNING_EXEC}
Args:
grid (:class:`~pde.grids.base.GridBase`):
Grid defining the space on which this field is defined
expressions (list of str):
A 2d list of mathematical expression, one for each component of the
tensor field. The expressions determine the values as a function of the
position on the grid. The expressions may contain standard mathematical
functions and they may depend on the axes labels of the grid.
More information can be found in the
:ref:`expression documentation <documentation-expressions>`.
user_funcs (dict, optional):
A dictionary with user defined functions that can be used in the
expression
consts (dict, optional):
A dictionary with user defined constants that can be used in the
expression. The values of these constants should either be numbers or
:class:`~numpy.ndarray`.
label (str, optional):
Name of the field
dtype (numpy dtype):
The data type of the field. If omitted, it will be determined from
`data` automatically.
"""
from ..tools.expressions import ScalarExpression
if (
isinstance(expressions, str)
or len(expressions) != grid.dim
or any(len(expr) != grid.dim for expr in expressions)
):
axes_names = grid.axes + grid.axes_symmetric
raise DimensionError(
f"Expected a nested list of {grid.dim}x{grid.dim} expressions for the "
f"tensor components of the coordinates {axes_names}."
)
# obtain the coordinates of the grid points
points = [grid.cell_coords[..., i] for i in range(grid.num_axes)]
# evaluate all vector components at all points
data: list[list[np.ndarray]] = [[None] * grid.dim for _ in range(grid.dim)] # type: ignore
for i in range(grid.dim):
for j in range(grid.dim):
expr = ScalarExpression(
expressions[i][j],
signature=grid.axes,
user_funcs=user_funcs,
consts=consts,
repl=grid.c._axes_alt_repl,
)
values = np.broadcast_to(expr(*points), grid.shape)
data[i][j] = values
# create vector field from the data
return cls(grid=grid, data=data, label=label, dtype=dtype)
def _get_axes_index(self, key: tuple[int | str, int | str]) -> tuple[int, int]:
"""turns a general index of two axis into a tuple of two numeric indices"""
try:
if len(key) != 2:
raise IndexError("Index must be given as two integers")
except TypeError:
raise IndexError("Index must be given as two values")
return tuple(self.grid.get_axis_index(k) for k in key) # type: ignore
def __getitem__(self, key: tuple[int | str, int | str]) -> ScalarField:
"""extract a single component of the tensor field as a scalar field"""
return ScalarField(
self.grid,
data=self._data_full[self._get_axes_index(key)],
with_ghost_cells=True,
)
def __setitem__(
self,
key: tuple[int | str, int | str],
value: NumberOrArray | ScalarField,
):
"""set a single component of the tensor field"""
idx = self._get_axes_index(key)
if isinstance(value, ScalarField):
self.grid.assert_grid_compatible(value.grid)
self.data[idx] = value.data
else:
self.data[idx] = value
@DataFieldBase._data_flat.setter # type: ignore
def _data_flat(self, value):
"""set the data from a value from a collection"""
# create a view and reshape it to disallow copying
data_full = value.view()
dim = self.grid.dim
full_grid_shape = tuple(s + 2 for s in self.grid.shape)
data_full.shape = (dim, dim, *full_grid_shape)
# set the result as the full data array
self._data_full = data_full
# ensure that no copying happend
if not np.may_share_memory(self.data, value):
raise RuntimeError("Spurious copy detected!")
[docs]
def dot(
self,
other: VectorField | Tensor2Field,
out: VectorField | Tensor2Field | None = None,
*,
conjugate: bool = True,
label: str = "dot product",
) -> VectorField | Tensor2Field:
"""calculate the dot product involving a tensor field
This supports the dot product between two tensor fields as well as the
product between a tensor and a vector. The resulting fields will be a
tensor or vector, respectively.
Args:
other (VectorField or Tensor2Field):
the second field
out (VectorField or Tensor2Field, optional):
Optional field to which the result is written.
conjugate (bool):
Whether to use the complex conjugate for the second operand
label (str, optional):
Name of the returned field
Returns:
:class:`~pde.fields.vectorial.VectorField` or
:class:`~pde.fields.tensorial.Tensor2Field`: result of applying the dot operator
"""
# check input
self.grid.assert_grid_compatible(other.grid)
if not isinstance(other, (VectorField, Tensor2Field)):
raise TypeError("Second term must be a vector or tensor field")
# create and check the output instance
if out is None:
out = other.__class__(self.grid, dtype=get_common_dtype(self, other))
else:
if not isinstance(out, other.__class__):
raise TypeError(f"`out` must be of type `{other.__class__}`")
self.grid.assert_grid_compatible(out.grid)
# calculate the result
other_data = other.data.conjugate() if conjugate else other.data
np.einsum("ij...,j...->i...", self.data, other_data, out=out.data)
if label is not None:
out.label = label
return out
__matmul__ = dot # support python @-syntax for matrix multiplication
[docs]
@fill_in_docstring
def divergence(
self, bc: BoundariesData | None, out: VectorField | None = None, **kwargs
) -> VectorField:
r"""apply tensor divergence and return result as a field
The tensor divergence is a vector field :math:`v_\alpha` resulting from a
contracting of the derivative of the tensor field :math:`t_{\alpha\beta}`:
.. math::
v_\alpha = \sum_\beta \frac{\partial t_{\alpha\beta}}{\partial x_\beta}
Args:
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES_OPTIONAL}
out (VectorField, optional):
Optional scalar field to which the result is written.
label (str, optional):
Name of the returned field
**kwargs:
Additional arguments affecting how the operator behaves.
Returns:
:class:`~pde.fields.vectorial.VectorField`: result of applying the operator
"""
return self.apply_operator("tensor_divergence", bc=bc, out=out, **kwargs) # type: ignore
@property
def integral(self) -> np.ndarray:
""":class:`~numpy.ndarray`: integral of each component over space"""
return self.grid.integrate(self.data) # type: ignore
[docs]
def transpose(self, label: str = "transpose") -> Tensor2Field:
"""return the transpose of the tensor field
Args:
label (str, optional): Name of the returned field
Returns:
:class:`~pde.fields.tensorial.Tensor2Field`: transpose of the tensor field
"""
axes = (1, 0) + tuple(range(2, 2 + self.grid.num_axes))
return Tensor2Field(self.grid, self.data.transpose(axes), label=label)
[docs]
def symmetrize(
self, make_traceless: bool = False, inplace: bool = False
) -> Tensor2Field:
"""symmetrize the tensor field in place
Args:
make_traceless (bool):
Determines whether the result is also traceless
inplace (bool):
Flag determining whether to symmetrize the current field or
return a new one
Returns:
:class:`~pde.fields.tensorial.Tensor2Field`: result of the operation
"""
if inplace:
out = self
else:
out = self.copy()
out += self.transpose()
out *= 0.5
if make_traceless:
dim = self.grid.dim
value = self.trace() / dim
for i in range(dim):
out.data[i, i] -= value.data
return out
[docs]
def to_scalar(
self, scalar: str = "auto", *, label: str | None = "scalar `{scalar}`"
) -> ScalarField:
r"""return scalar variant of the field
The invariants of the tensor field :math:`\boldsymbol{A}` are
.. math::
I_1 &= \mathrm{tr}(\boldsymbol{A}) \\
I_2 &= \frac12 \left[
(\mathrm{tr}(\boldsymbol{A})^2 -
\mathrm{tr}(\boldsymbol{A}^2)
\right] \\
I_3 &= \det(A)
where `tr` denotes the trace and `det` denotes the determinant. Note that the
three invariants can only be distinct and non-zero in three dimensions. In two
dimensional spaces, we have the identity :math:`2 I_2 = I_3` and in
one-dimensional spaces, we have :math:`I_1 = I_3` as well as
:math:`I_2 = 0`.
Args:
scalar (str):
The method to calculate the scalar. Possible choices include `norm` (the
default chosen when the value is `auto`), `min`, `max`, `squared_sum`,
`norm_squared`, `trace` (or `invariant1`), `invariant2`, and
`determinant` (or `invariant3`)
label (str, optional):
Name of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: the scalar field after
applying the operation
"""
if scalar == "auto":
scalar = "norm"
if scalar == "norm":
data = np.linalg.norm(self.data, axis=(0, 1))
elif scalar == "min":
data = np.min(self.data, axis=(0, 1))
elif scalar == "max":
data = np.max(self.data, axis=(0, 1))
elif scalar == "squared_sum":
data = np.sum(self.data**2, axis=(0, 1))
elif scalar == "norm_squared":
data = np.sum(self.data * self.data.conjugate(), axis=(0, 1))
elif scalar == "trace" or scalar == "invariant1":
data = self.data.trace(axis1=0, axis2=1)
elif scalar == "invariant2":
data = np.zeros(self.grid.shape)
for i in range(self.grid.dim):
for j in range(i):
data += (
self.data[i, i] * self.data[j, j]
- self.data[i, j] * self.data[j, i]
)
data *= 0.5
elif scalar in {"det", "determinant", "invariant3"}:
if self.grid.dim == 1:
data = self.data[0, 0]
else:
data = np.zeros(self.grid.shape)
# this iterates over all of space and might thus be slow, but
# the interface of np.linalg.det is not very flexible. We could
# in principle use the definition of np.linalg.det without the
# multiple checks to gain some speed
for idx in np.ndindex(*self.grid.shape):
data[idx] = np.linalg.det(self.data[(...,) + idx])
else:
raise ValueError(
f"Unknown method `{scalar}` for `to_scalar`. Valid methods are `norm`, "
"`min`, `max`, squared_sum`, `norm_squared`, `trace`, `determinant`, "
"and `invariant#`, where # is 1, 2, or 3"
)
# determine label of the result
if self.label is None:
if label is not None:
label = label.format(scalar=scalar)
else:
label = f"{scalar} of {self.label}"
return ScalarField(self.grid, data, label=label)
[docs]
def trace(self, label: str | None = "trace") -> ScalarField:
"""return the trace of the tensor field as a scalar field
Args:
label (str, optional): Name of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: scalar field of traces
"""
return self.to_scalar(scalar="trace", label=label)
def _update_plot_components(self, reference: list[list[PlotReference]]) -> None:
"""update a plot collection with the current field values
Args:
reference (list of :class:`PlotReference`):
All references of the plot to update
"""
for i in range(self.grid.dim):
for j in range(self.grid.dim):
self[i, j]._update_plot(reference[i][j])
[docs]
@plot_on_figure(update_method="_update_plot_components")
def plot_components(
self,
kind: str = "auto",
fig=None,
**kwargs,
) -> list[list[PlotReference]]:
r"""visualize all the components of this tensor field
Args:
kind (str or list of str):
Determines the kind of the visualizations. Supported values are `image`
or `line`. Alternatively, `auto` determines the best visualization based
on the grid.
{PLOT_ARGS}
\**kwargs:
All additional keyword arguments are forwarded to the actual plotting
function of all subplots.
Returns:
2d list of :class:`PlotReference`: Instances that contain information
to update all the plots with new data later.
"""
# create all the subpanels
dim = self.grid.dim
axs = fig.subplots(nrows=dim, ncols=dim, squeeze=False)
# plot all the elements onto the respective axes
kwargs.setdefault("action", "none")
kwargs["kind"] = kind
comps = self.grid.axes + self.grid.axes_symmetric
references = [
[
self[i, j].plot(
ax=axs[i][j],
title=f"{comps[i]}{comps[j]} Component",
**kwargs,
)
for j in range(dim)
]
for i in range(dim)
]
# return the references for all subplots
return references