"""
Defines a tensorial field of rank 2 over a grid
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING, Callable, List, Optional, Sequence, Tuple, Union
import numba as nb
import numpy as np
from ..grids.base import DimensionError, GridBase
from ..tools.docstrings import fill_in_docstring
from ..tools.misc import get_common_dtype
from ..tools.numba import get_common_numba_dtype, jit
from ..tools.plotting import PlotReference, plot_on_figure
from ..tools.typing import NumberOrArray
from .base import DataFieldBase
from .scalar import ScalarField
from .vectorial import VectorField
if TYPE_CHECKING:
from ..grids.boundaries.axes import BoundariesData # @UnusedImport
[docs]class Tensor2Field(DataFieldBase):
"""Tensor field of rank 2 discretized on a grid"""
rank = 2
[docs] @classmethod
@fill_in_docstring
def from_expression(
cls,
grid: GridBase,
expressions: Sequence[Sequence[str]],
*,
label: str = None,
dtype=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.
label (str, optional):
Name of the field
dtype (numpy dtype):
The data type of the field. All the numpy dtypes are supported. 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 = {name: grid.cell_coords[..., i] for i, name in enumerate(grid.axes)}
# evaluate all vector components at all points
data = [[None] * grid.dim for _ in range(grid.dim)]
for i in range(grid.dim):
for j in range(grid.dim):
expr = ScalarExpression(expressions[i][j], signature=grid.axes)
values = np.broadcast_to(expr(**points), grid.shape)
data[i][j] = values
# create vector field from the data
return cls( # lgtm [py/call-to-non-callable]
grid=grid, data=data, label=label, dtype=dtype
)
def _get_axes_index(
self, key: Tuple[Union[int, str], Union[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[Union[int, str], Union[int, str]]) -> ScalarField:
"""extract a component of the VectorField"""
return ScalarField(self.grid, self.data[self._get_axes_index(key)])
def __setitem__(
self,
key: Tuple[Union[int, str], Union[int, str]],
value: Union[NumberOrArray, ScalarField],
):
"""set a component of the VectorField"""
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
assert np.may_share_memory(self.data, value)
[docs] def dot(
self,
other: Union[VectorField, Tensor2Field],
out: Optional[Union[VectorField, Tensor2Field]] = None,
*,
conjugate: bool = True,
label: str = "dot product",
) -> Union[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:
assert isinstance(out, other.__class__), f"`out` must be {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] def make_dot_operator(
self, backend: str = "numba", *, conjugate: bool = True
) -> Callable[[np.ndarray, np.ndarray, Optional[np.ndarray]], np.ndarray]:
"""return operator calculating the dot product involving vector fields
This supports both products between two vectors as well as products
between a vector and a tensor.
Warning:
This function does not check types or dimensions.
Args:
conjugate (bool):
Whether to use the complex conjugate for the second operand
Returns:
function that takes two instance of :class:`~numpy.ndarray`, which
contain the discretized data of the two operands. An optional third
argument can specify the output array to which the result is
written. Note that the returned function is jitted with numba for
speed.
"""
dim = self.grid.dim
if backend == "numba":
# create the dot product using a numba compiled function
if conjugate:
# create inner function calculating the dot product using conjugate
@jit
def calc(a: np.ndarray, b: np.ndarray, out: np.ndarray) -> np.ndarray:
"""calculate dot product between fields `a` and `b`"""
for i in range(dim):
out[i] = a[i, 0] * b[0].conjugate() # overwrite data in out
for j in range(1, dim):
out[i] += a[i, j] * b[j].conjugate()
return out
else:
# create the inner function calculating the dot product
@jit
def calc(a: np.ndarray, b: np.ndarray, out: np.ndarray) -> np.ndarray:
"""calculate dot product between fields `a` and `b`"""
for i in range(dim):
out[i] = a[i, 0] * b[0] # overwrite potential data in out
for j in range(1, dim):
out[i] += a[i, j] * b[j]
return out
# build the outer function with the correct signature
if nb.config.DISABLE_JIT: # @UndefinedVariable
def dot(
a: np.ndarray, b: np.ndarray, out: np.ndarray = None
) -> np.ndarray:
"""wrapper deciding whether the underlying function is called
with or without `out`."""
if out is None:
out = np.empty(b.shape, dtype=get_common_dtype(a, b))
return calc(a, b, out) # type: ignore
else:
@nb.generated_jit
def dot(
a: np.ndarray, b: np.ndarray, out: np.ndarray = None
) -> np.ndarray:
"""wrapper deciding whether the underlying function is called
with or without `out`."""
if isinstance(a, nb.types.Number):
# simple scalar call -> do not need to allocate anything
raise RuntimeError("Dot needs to be called with fields")
elif isinstance(out, (nb.types.NoneType, nb.types.Omitted)):
# function is called without `out`
dtype = get_common_numba_dtype(a, b)
def f_with_allocated_out(
a: np.ndarray, b: np.ndarray, out: np.ndarray
) -> np.ndarray:
"""helper function allocating output array"""
out = np.empty(b.shape, dtype=dtype)
return calc(a, b, out=out) # type: ignore
return f_with_allocated_out # type: ignore
else:
# function is called with `out` argument
return calc # type: ignore
elif backend == "numpy":
# create the dot product using basic numpy functions
def calc(
a: np.ndarray, b: np.ndarray, out: np.ndarray = None
) -> np.ndarray:
"""calculate dot product between two tensors"""
if a.shape == b.shape:
# dot product between tensor and tensor
if out is None:
# TODO: Remove this construct once we make numpy 1.20 a minimal
# requirement. Earlier version of numpy do not support out=None
# correctly and we thus had to use this work-around
return np.einsum("ij...,jk...->ik...", a, b) # type: ignore
else:
return np.einsum("ij...,jk...->ik...", a, b, out=out)
elif a.shape[1:] == b.shape:
# dot product between tensor and vector
if out is None:
# TODO: Remove this construct once we make numpy 1.20 a minimal
# requirement. Earlier version of numpy do not support out=None
# correctly and we thus had to use this work-around
return np.einsum("ij...,j...->i...", a, b) # type: ignore
else:
return np.einsum("ij...,j...->i...", a, b, out=out)
else:
raise ValueError(f"Unsupported shapes ({a.shape}, {b.shape})")
if conjugate:
# create inner function calculating the dot product using conjugate
def dot(
a: np.ndarray, b: np.ndarray, out: np.ndarray = None
) -> np.ndarray:
"""calculate dot product with conjugated second operand"""
return calc(a, b.conjugate(), out=out) # type: ignore
else:
dot = calc
else:
raise ValueError(f"Undefined backend `{backend}")
return dot
[docs] @fill_in_docstring
def divergence(
self, bc: Optional[BoundariesData], out: Optional[VectorField] = 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
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)
[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: Optional[str] = "scalar `{scalar}`"
) -> ScalarField:
r""" return a scalar field by applying `method`
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: Optional[str] = "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", "create")
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