Module `phi.physics.diffuse`

Functions to simulate diffusion processes on Field objects.

Functions

def differential(u: phi.field._field.Field, diffusivity: Union[float, phiml.math._tensors.Tensor, phi.field._field.Field], gradient: phi.field._field.Field = None, order: int = 2, implicit: phiml.math._optimize.Solve = None, upwind: phi.field._field.Field = None, correct_skew=True) ‑> phi.field._field.Field

Compute the differential diffusion term, d·∇²u. For grids, uses a finite difference scheme specified by order and implicit(). For FVM, the scheme is specified via order and upwind.

In contrast to explicit() and implicit(), accuracy can be increased by using stencils of higher-order rather than calculating sub-steps.

Args

u: Scalar or vector-valued Field sampled on a CenteredGrid, StaggeredGrid or centered Mesh.
diffusivity: Dynamic viscosity, i.e. diffusion per time. Constant or varying by cell.
gradient: Only used by FVM at the moment. Approximate gradient of u, e.g. ∇u of the previous time step. If None, approximates the gradient as (u_neighbor - u_self) / distance.
order: Spatial order of accuracy. Higher orders entail larger stencils and more computation time but result in more accurate results assuming a large enough resolution. Supported: 2 explicit, 4 explicit, 6 implicit (inherited from laplace()). For FVM, the order is used when interpolating v and prev_v to cell faces if needed.
implicit: When a Solve object is passed, performs an implicit operation with the specified solver and tolerances. Otherwise, an explicit stencil is used.
upwind: For unstructured meshes only. Whether to use upwind interpolation.
correct_skew: If True, adds a correction term for cell skewness. This requires gradient to be passed.

Returns

Differential diffusion as a Field on the same geometry.

def explicit(u: phi.field._field.Field, diffusivity: Union[float, phiml.math._tensors.Tensor, phi.field._field.Field], dt: Union[phiml.math._tensors.Tensor, float], substeps: int = 1, order: int = 2, implicit: phiml.math._optimize.Solve = None, gradient: phi.field._field.Field = None, upwind: phi.field._field.Field = None, correct_skew=True) ‑> phi.field._field.Field

Explicit Euler diffusion with substeps.

Simulate a finite-time diffusion process of the form dF/dt = α · ΔF on a given Field Field with diffusion coefficient α.

Args

u: CenteredGrid, StaggeredGrid or ConstantField
diffusivity: Diffusion per time. diffusion_amount = diffusivity * dt Can be a number, phi.Tensor or Field. If a channel dimension is present, it will be interpreted as non-isotropic diffusion.
dt: Time interval. diffusion_amount = diffusivity * dt
substeps: number of iterations to use (Default value = 1)
order: Spatial order of accuracy. Higher orders entail larger stencils and more computation time but result in more accurate results assuming a large enough resolution. Supported: 2 explicit, 4 explicit, 6 implicit (inherited from laplace()). For FVM, the order is used when interpolating v and prev_v to cell faces if needed.
implicit: When a Solve object is passed, performs a spatially implicit operation with the specified solver and tolerances. Otherwise, an explicit stencil is used.
gradient: Only used by FVM at the moment. Approximate gradient of u, e.g. ∇u of the previous time step. If None, approximates the gradient as (u_neighbor - u_self) / distance.
upwind: For unstructured meshes only. Whether to use upwind interpolation.
correct_skew: If True, adds a correction term for cell skewness. This requires gradient to be passed.

Returns

Diffused field of same type as field.

def finite_difference(u: phi.field._field.Field, diffusivity: Union[float, phiml.math._tensors.Tensor, phi.field._field.Field], gradient: phi.field._field.Field = None, order: int = 2, implicit: phiml.math._optimize.Solve = None, upwind: phi.field._field.Field = None, correct_skew=True) ‑> phi.field._field.Field

Compute the differential diffusion term, d·∇²u. For grids, uses a finite difference scheme specified by order and implicit(). For FVM, the scheme is specified via order and upwind.

In contrast to explicit() and implicit(), accuracy can be increased by using stencils of higher-order rather than calculating sub-steps.

Args

u: Scalar or vector-valued Field sampled on a CenteredGrid, StaggeredGrid or centered Mesh.
diffusivity: Dynamic viscosity, i.e. diffusion per time. Constant or varying by cell.
gradient: Only used by FVM at the moment. Approximate gradient of u, e.g. ∇u of the previous time step. If None, approximates the gradient as (u_neighbor - u_self) / distance.
order: Spatial order of accuracy. Higher orders entail larger stencils and more computation time but result in more accurate results assuming a large enough resolution. Supported: 2 explicit, 4 explicit, 6 implicit (inherited from laplace()). For FVM, the order is used when interpolating v and prev_v to cell faces if needed.
implicit: When a Solve object is passed, performs an implicit operation with the specified solver and tolerances. Otherwise, an explicit stencil is used.
upwind: For unstructured meshes only. Whether to use upwind interpolation.
correct_skew: If True, adds a correction term for cell skewness. This requires gradient to be passed.

Returns

Differential diffusion as a Field on the same geometry.

def fourier(field: phi.field._field.Field, diffusivity: Union[phiml.math._tensors.Tensor, float], dt: Union[phiml.math._tensors.Tensor, float]) ‑> phi.field._field.Field

Exact diffusion of a periodic field in frequency space.

For non-periodic fields or non-constant diffusivity, use another diffusion function such as explicit().

Args

field:
diffusivity: Diffusion per time. diffusion_amount = diffusivity * dt
dt: Time interval. diffusion_amount = diffusivity * dt

Returns

Diffused field of same type as field.

def implicit(field: phi.field._field.Field, diffusivity: Union[float, phiml.math._tensors.Tensor, phi.field._field.Field], dt: Union[phiml.math._tensors.Tensor, float], solve=CG with tolerance None (rel), None (abs), max_iterations=1000, gradient: phi.field._field.Field = None, upwind: phi.field._field.Field = None, correct_skew=True, gradient_for_diffusivity=True) ‑> phi.field._field.Field

Implicit Euler diffusion.

Diffusion by solving a linear system of equations.

Args

field: Field to diffuse.
diffusivity: Diffusion per time. diffusion_amount = diffusivity * dt
dt: Time interval. diffusion_amount = diffusivity * dt
solve: Implicit solve parameters.
gradient: Only used by FVM at the moment. Approximate gradient of u, e.g. ∇u of the previous time step. If None, approximates the gradient as (u_neighbor - u_self) / distance.
upwind: For unstructured meshes only. Whether to use upwind interpolation.
correct_skew: If True, adds a correction term for cell skewness. This requires gradient to be passed.
gradient_for_diffusivity: Whether to compute the gradient w.r.t. the diffusivity parameters.

Returns

Diffused field of same type as field.