pde_opt.numerics.equations.cahn_hilliard

This module contains various Cahn-Hilliard equation classes.

Classes

CahnHilliard2DPeriodic(domain, kappa, mu, D)	Cahn–Hilliard equation in 2D with periodic boundary conditions.
CahnHilliard2DSmoothedBoundary(domain, ...)	Cahn–Hilliard equation with smoothed boundary method for arbitrary geometries.
CahnHilliard3DPeriodic(domain, kappa, mu, D)	Cahn–Hilliard equation in 3D with periodic boundary conditions.

class pde_opt.numerics.equations.cahn_hilliard.CahnHilliard2DPeriodic(domain: Domain, kappa: float, mu: Callable | equinox.Module, D: Callable | equinox.Module, derivs: str = 'fd')

Cahn–Hilliard equation in 2D with periodic boundary conditions.

The Cahn-Hilliard equation describes phase separation and coarsening dynamics. The equation is:

\[\frac{\partial u}{\partial t} = \nabla \cdot (D(u) \nabla \mu)\]

where u is the concentration, D(u) is the mobility, and μ is the chemical potential. The chemical potential is given by:

\[\mu = \mu_h(u) - \kappa \nabla^2 u\]

domain: Domain

Domain of the equation

kappa: float

Gradient energy coefficient

mu: Callable | equinox.Module

Function for the chemical potential

D: Callable | equinox.Module

Function for the mobility

derivs: str = 'fd'

Type of derivative computation

fft = None

ifft = None

fourier_symbol = None

rhs(state, t)

Right hand side of the equation.

rhs_fourier(state, t)

rhs_fd(state, t)

__init__(domain: Domain, kappa: float, mu: Callable | equinox.Module, D: Callable | equinox.Module, derivs: str = 'fd') → None

class pde_opt.numerics.equations.cahn_hilliard.CahnHilliard3DPeriodic(domain: Domain, kappa: float, mu: Callable | equinox.Module, D: Callable | equinox.Module, derivs: str = 'fd')

Cahn–Hilliard equation in 3D with periodic boundary conditions.

The Cahn-Hilliard equation describes phase separation and coarsening dynamics. The equation is:

\[\frac{\partial u}{\partial t} = \nabla \cdot (D(u) \nabla \mu)\]

where u is the concentration, D(u) is the mobility, and μ is the chemical potential. The chemical potential is given by:

\[\mu = \mu_h(u) - \kappa \nabla^2 u\]

domain: Domain

Domain of the equation

kappa: float

Gradient energy coefficient

mu: Callable | equinox.Module

Function for the chemical potential

D: Callable | equinox.Module

Function for the mobility

derivs: str = 'fd'

Type of derivative computation

fft = None

ifft = None

fourier_symbol = None

rhs(state, t)

Right hand side of the equation.

rhs_fourier(state, t)

rhs_fd(state, t)

__init__(domain: Domain, kappa: float, mu: Callable | equinox.Module, D: Callable | equinox.Module, derivs: str = 'fd') → None

class pde_opt.numerics.equations.cahn_hilliard.CahnHilliard2DSmoothedBoundary(domain: Domain, kappa: float, f: Callable | equinox.Module, mu: Callable | equinox.Module, D: Callable | equinox.Module, theta: Callable | equinox.Module, flux: Callable | equinox.Module, derivs: str = 'fd')

Cahn–Hilliard equation with smoothed boundary method for arbitrary geometries.

This class implements the Cahn-Hilliard equation using the smoothed boundary method, which allows for complex domain geometries through a smooth level-set function ψ.

The equation is:

\[\frac{\partial u}{\partial t} = \frac{1}{\psi} \nabla \cdot (\psi D(u) \nabla \mu) + \frac{|\nabla \psi|}{\psi} J_n\]

where the chemical potential includes boundary effects:

\[\mu = \mu_h(u) - \frac{\kappa}{\psi} \nabla \cdot (\psi \nabla u) - \sqrt{\kappa} \frac{|\nabla \psi|}{\psi} \sqrt{2f} \cos(\theta)\]

__init__(domain: Domain, kappa: float, f: Callable | equinox.Module, mu: Callable | equinox.Module, D: Callable | equinox.Module, theta: Callable | equinox.Module, flux: Callable | equinox.Module, derivs: str = 'fd') → None

domain: Domain

Domain of the equation

kappa: float

Gradient energy coefficient

f: Callable | equinox.Module

Function for the free energy density

mu: Callable | equinox.Module

Function for the chemical potential

D: Callable | equinox.Module

Function for the mobility

theta: Callable | equinox.Module

Function for the contact angle

flux: Callable | equinox.Module

Function for the normal flux

derivs: str = 'fd'

Type of derivative computation

rhs(state, t)

Right hand side of the equation.

rhs_fd(state, t)