evoxels.problem_definition

Classes

AllenCahnEquation(vg[, eps, gab, M, force, ...])
CoupledReactionDiffusion(vg[, D_A, D_B, ...])
ODE()
PeriodicCahnHilliard(vg[, eps, D, mu_hom, A])
ReactionDiffusion(vg, D, BC_type[, bcs, f, A])
ReactionDiffusionSBM(vg, D, BC_type[, bcs, ...])
SemiLinearODE()
SmoothedBoundaryODE()

class evoxels.problem_definition.AllenCahnEquation(vg: evoxels.voxelgrid.VoxelGrid, eps: float = 2.0, gab: float = 1.0, M: float = 1.0, force: float = 0.0, curvature: float = 0.01, potential: Callable | None = None)

M: float = 1.0

__init__(vg: VoxelGrid, eps: float = 2.0, gab: float = 1.0, M: float = 1.0, force: float = 0.0, curvature: float = 0.01, potential: Callable | None = None) → None

property bc_type

E.g. ‘periodic’, ‘dirichlet’, or ‘neumann’.

curvature: float = 0.01

eps: float = 2.0

force: float = 0.0

property fourier_symbol

Symbol of the highest order spatial operator

The symbol of an operator is its representation in the Fourier (spectral) domain. For instance the: - Laplacian operator $

abla^2$ has a symbol $-k^2$,

• diffusion operator $D

abla^2$ corresponds to $-k^2D$

The symbol is required for pseudo-spectral timesteppers.

gab: float = 1.0

property order

Spatial order of convergence for numerical right-hand side.

pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

potential: Callable | None = None

rhs(t, phi)

Two-phase Allen-Cahn equation

Microstructural evolution of the order parameter \phi which can be interpreted as a phase fraction. \(M\) denotes the mobility, \(\epsilon\) controls the diffuse interface width, \(\gamma\) denotes the interfacial energy. The laplacian leads to a phase evolution driven by curvature minimization which can be controlled by setting curvature= in range \([0,1]\).

Parameters:

• phi (array-like) – order parameter.

• t (float) – Current time.

Returns:

Backend array of the same shape as \phi containing d\phi/dt.

rhs_analytic(t, phi)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

vg: VoxelGrid

class evoxels.problem_definition.CoupledReactionDiffusion(vg: evoxels.voxelgrid.VoxelGrid, D_A: float = 1.0, D_B: float = 0.5, feed: float = 0.055, kill: float = 0.117, interaction: Callable | None = None)

D_A: float = 1.0

D_B: float = 0.5

__init__(vg: VoxelGrid, D_A: float = 1.0, D_B: float = 0.5, feed: float = 0.055, kill: float = 0.117, interaction: Callable | None = None) → None

property bc_type

E.g. ‘periodic’, ‘dirichlet’, or ‘neumann’.

feed: float = 0.055

property fourier_symbol

Symbol of the highest order spatial operator

The symbol of an operator is its representation in the Fourier (spectral) domain. For instance the: - Laplacian operator $

abla^2$ has a symbol $-k^2$,

• diffusion operator $D

abla^2$ corresponds to $-k^2D$

The symbol is required for pseudo-spectral timesteppers.

interaction: Callable | None = None

kill: float = 0.117

property order

Spatial order of convergence for numerical right-hand side.

pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

rhs(t, u)

Two-component reaction-diffusion system

Use batch channels for multiple species: - Species A with concentration c_A = u[0] - Species B with concentration c_B = u[1]

Parameters:

• u (array-like) – species

• t (float) – Current time.

Returns:

Backend array of the same shape as u containing du/dt.

rhs_analytic(t, u)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

vg: VoxelGrid

class evoxels.problem_definition.ODE

abstract property bc_type: str

E.g. ‘periodic’, ‘dirichlet’, or ‘neumann’.

abstract property order: int

Spatial order of convergence for numerical right-hand side.

abstractmethod pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

abstractmethod rhs(t, u)

Numerical right-hand side of the ODE system.

Parameters:

• t (float) – Current time.

• u (array) – Current state.

Returns:

Same type as u containing the time derivative.

abstractmethod rhs_analytic(t, u)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

class evoxels.problem_definition.PeriodicCahnHilliard(vg: evoxels.voxelgrid.VoxelGrid, eps: float = 3.0, D: float = 1.0, mu_hom: Callable | None = None, A: float = 0.25)

A: float = 0.25

D: float = 1.0

__init__(vg: VoxelGrid, eps: float = 3.0, D: float = 1.0, mu_hom: Callable | None = None, A: float = 0.25) → None

property bc_type

E.g. ‘periodic’, ‘dirichlet’, or ‘neumann’.

eps: float = 3.0

property fourier_symbol

Symbol of the highest order spatial operator

The symbol of an operator is its representation in the Fourier (spectral) domain. For instance the: - Laplacian operator $

abla^2$ has a symbol $-k^2$,

• diffusion operator $D

abla^2$ corresponds to $-k^2D$

The symbol is required for pseudo-spectral timesteppers.

mu_hom: Callable | None = None

property order

Spatial order of convergence for numerical right-hand side.

pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

rhs(t, c)

Evaluate \(\partial c / \partial t\) for the CH equation.

Numerical computation of

\[\frac{\partial c}{\partial t} = \nabla \cdot \bigl( M \, \nabla \mu \bigr), \quad \mu = \frac{\delta F}{\delta c} = f'(c) - \kappa \, \nabla^2 c\]

where \(M\) is the (possibly concentration-dependent) mobility, \(\mu\) the chemical potential, and \(\kappa\) the gradient energy coefficient.

Parameters:

• c (array-like) – Concentration field.

• t (float) – Current time.

Returns:

Backend array of the same shape as c containing dc/dt.

rhs_analytic(t, c)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

vg: VoxelGrid

class evoxels.problem_definition.ReactionDiffusion(vg: evoxels.voxelgrid.VoxelGrid, D: float, BC_type: str, bcs: tuple = (0, 0), f: Callable | None = None, A: float = 0.25)

A: float = 0.25

BC_type: str

D: float

__init__(vg: VoxelGrid, D: float, BC_type: str, bcs: tuple = (0, 0), f: Callable | None = None, A: float = 0.25) → None

property bc_type

E.g. ‘periodic’, ‘dirichlet’, or ‘neumann’.

bcs: tuple = (0, 0)

f: Callable | None = None

property fourier_symbol

Symbol of the highest order spatial operator

The symbol of an operator is its representation in the Fourier (spectral) domain. For instance the: - Laplacian operator $

abla^2$ has a symbol $-k^2$,

• diffusion operator $D

abla^2$ corresponds to $-k^2D$

The symbol is required for pseudo-spectral timesteppers.

property order

Spatial order of convergence for numerical right-hand side.

pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

rhs(t, u)

Numerical right-hand side of the ODE system.

Parameters:

• t (float) – Current time.

• u (array) – Current state.

Returns:

Same type as u containing the time derivative.

rhs_analytic(t, u)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

vg: VoxelGrid

class evoxels.problem_definition.ReactionDiffusionSBM(vg: evoxels.voxelgrid.VoxelGrid, D: float, BC_type: str, bcs: tuple = (0, 0), f: Callable | None = None, A: float = 0.25, mask: Any | None = None, bc_flux: Callable | float = 0.0)

__init__(vg: VoxelGrid, D: float, BC_type: str, bcs: tuple = (0, 0), f: Callable | None = None, A: float = 0.25, mask: Any | None = None, bc_flux: Callable | float = 0.0) → None

bc_flux: Callable | float = 0.0

mask: Any | None = None

pad_bc(u)

Function to pad and impose boundary conditions.

Enables applying boundary conditions on u within and outside of the right-hand-side function.

Parameters:

u – field

Returns:

Field padded with boundary values.

rhs(t, u)

Numerical right-hand side of the ODE system.

Parameters:

• t (float) – Current time.

• u (array) – Current state.

Returns:

Same type as u containing the time derivative.

rhs_analytic(t, u, mask)

Sympy expression of the problem right-hand side.

Parameters:

• t (float) – Current time.

• u – Sympy function of current state.

Returns:

Sympy function of problem right-hand side.

class evoxels.problem_definition.SemiLinearODE

abstract property fourier_symbol

Symbol of the highest order spatial operator

The symbol of an operator is its representation in the Fourier (spectral) domain. For instance the: - Laplacian operator $

abla^2$ has a symbol $-k^2$,

• diffusion operator $D

abla^2$ corresponds to $-k^2D$

The symbol is required for pseudo-spectral timesteppers.

class evoxels.problem_definition.SmoothedBoundaryODE

abstract property mask: Any | float

A field (same shape as the state) that remains fixed.