"""A Cahn-Hilliard equation.
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
from typing import TYPE_CHECKING, Any
from ..fields import ScalarField
from ..grids.boundaries import set_default_bc
from ..tools.docstrings import fill_in_docstring
from .base import PDEBase, expr_prod
if TYPE_CHECKING:
from collections.abc import Callable
from ..backends import BackendBase
from ..grids.boundaries.axes import BoundariesData
from ..tools.typing import TNativeArray
[docs]
class CahnHilliardPDE(PDEBase):
r"""A simple Cahn-Hilliard equation.
The mathematical definition is
.. math::
\partial_t c = \nabla^2 \left(c^3 - c - \gamma \nabla^2 c\right)
where :math:`c` is a scalar field taking values on the interval :math:`[-1, 1]` and
:math:`\gamma` sets the (squared) interfacial width.
"""
explicit_time_dependence = False
default_bc_c = "auto_periodic_neumann"
"""Default boundary condition for order parameter."""
default_bc_mu = "auto_periodic_neumann"
"""Default boundary condition for chemical potential."""
@fill_in_docstring
def __init__(
self,
interface_width: float = 1,
*,
bc_c: BoundariesData | None = None,
bc_mu: BoundariesData | None = None,
):
"""
Args:
interface_width (float):
The width of the interface between the separated phases. This defines
a characteristic length in the simulation. The grid needs to resolve
this length of a stable simulation.
bc_c:
The boundary conditions applied to the field.
{ARG_BOUNDARIES}
bc_mu:
The boundary conditions applied to the chemical potential associated
with the scalar field :math:`c`. Supports the same options as `bc_c`.
"""
super().__init__()
self.interface_width = interface_width
self.bc_c: BoundariesData = set_default_bc(bc_c, self.default_bc_c)
self.bc_mu: BoundariesData = set_default_bc(bc_mu, self.default_bc_mu)
@property
def expression(self) -> str:
"""str: the expression of the right hand side of this PDE"""
return f"∇²(c³ - c - {expr_prod(self.interface_width, '∇²c')})"
[docs]
def evolution_rate( # type: ignore
self,
state: ScalarField,
t: float = 0,
) -> ScalarField:
"""Evaluate the right hand side of the PDE.
Args:
state (:class:`~pde.fields.ScalarField`):
The scalar field describing the concentration distribution
t (float):
The current time point
Returns:
:class:`~pde.fields.ScalarField`:
Scalar field describing the evolution rate of the PDE
"""
assert isinstance(state, ScalarField), "`state` must be ScalarField"
c_laplace = state.laplace(bc=self.bc_c, label="evolution rate", args={"t": t})
result = state**3 - state - self.interface_width * c_laplace
return result.laplace(bc=self.bc_mu, args={"t": t}) # type: ignore
[docs]
def make_evolution_rate(
self, state: ScalarField, backend: BackendBase
) -> Callable[[TNativeArray, float], TNativeArray]:
"""Create a compiled function evaluating the right hand side of the PDE.
Args:
state (:class:`~pde.fields.ScalarField`):
An example for the state defining the grid and data types
backend (str or :class:`~pde.backends.base.BackendBase`):
The backend used for numerical operations
Returns:
A function with signature `(state_data, t)`, which can be called with an
instance of the state data and time to obtain the associated evolution rate.
"""
interface_width = self.interface_width
args: dict[str, Any] = {"backend": backend, "dtype": state.dtype}
laplace_c = state.grid.make_operator(operator="laplace", bc=self.bc_c, **args)
laplace_mu = state.grid.make_operator(operator="laplace", bc=self.bc_mu, **args)
def pde_rhs(state_data, t=0):
"""Evaluate right hand side of PDE."""
mu = (
state_data**3
- state_data
- interface_width * laplace_c(state_data, args={"t": t})
)
return laplace_mu(mu, args={"t": t})
return pde_rhs