4.3.3 pde.pdes.cahn_hilliard module

A Cahn-Hilliard equation.

class CahnHilliardPDE(interface_width=1, *, bc_c=None, bc_mu=None)[source]

Bases: PDEBase

A simple Cahn-Hilliard equation.

The mathematical definition is

\[\partial_t c = \nabla^2 \left(c^3 - c - \gamma \nabla^2 c\right)\]

where \(c\) is a scalar field taking values on the interval \([-1, 1]\) and \(\gamma\) sets the (squared) interfacial width.

Parameters:

  • 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 (BoundariesData | None) – The boundary conditions applied to the field. Boundary conditions are generally given as a dictionary with one condition for each axis side. For periodic axes, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using specific identifiers, like x- and y+). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘auto_periodic_neumann’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

  • bc_mu (BoundariesData | None) – The boundary conditions applied to the chemical potential associated with the scalar field \(c\). Supports the same options as bc_c.

default_bc_c = 'auto_periodic_neumann'

Default boundary condition for order parameter.

default_bc_mu = 'auto_periodic_neumann'

Default boundary condition for chemical potential.

evolution_rate(state, t=0)[source]

Evaluate the right hand side of the PDE.

Parameters:

  • state (ScalarField) – The scalar field describing the concentration distribution

  • t (float) – The current time point

Returns:

Scalar field describing the evolution rate of the PDE

Return type:

ScalarField

explicit_time_dependence: bool | None = False

Flag indicating whether the right hand side of the PDE has an explicit time dependence.

Type:

bool

property expression: str

the expression of the right hand side of this PDE

Type:

str