Module phi.physics.advect

Container for different advection schemes for grids and particles.

Examples:

  • semi_lagrangian (grid)
  • mac_cormack (grid)
  • runge_kutta_4 (particle)
Expand source code 
"""
Container for different advection schemes for grids and particles.

Examples:

* semi_lagrangian (grid)
* mac_cormack (grid)
* runge_kutta_4 (particle)
"""
from typing import Union

from phiml.math import Solve, channel

from phi import math
from phi.field import SampledField, Field, PointCloud, Grid, sample, reduce_sample, spatial_gradient, unstack, stack, CenteredGrid, StaggeredGrid
from phi.field._field import FieldType
from phi.field._field_math import GridType
from phi.geom import Geometry


def euler(elements: Geometry, velocity: Field, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Euler integrator. """
    if v0 is None:
        v0 = sample(velocity, elements)
    return elements.shifted(v0 * dt)


def rk4(elements: Geometry, velocity: Field, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Runge-Kutta-4 integrator. """
    if v0 is None:
        v0 = sample(velocity, elements)
    vel_half = sample(velocity, elements.shifted(0.5 * dt * v0))
    vel_half2 = sample(velocity, elements.shifted(0.5 * dt * vel_half))
    vel_full = sample(velocity, elements.shifted(dt * vel_half2))
    vel_rk4 = (1 / 6.) * (v0 + 2 * (vel_half + vel_half2) + vel_full)
    return elements.shifted(dt * vel_rk4)


def finite_rk4(elements: Geometry, velocity: Grid, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Runge-Kutta-4 integrator with Euler fallback where velocity values are NaN. """
    v0 = sample(velocity, elements)
    vel_half = sample(velocity, elements.shifted(0.5 * dt * v0))
    vel_half2 = sample(velocity, elements.shifted(0.5 * dt * vel_half))
    vel_full = sample(velocity, elements.shifted(dt * vel_half2))
    vel_rk4 = (1 / 6.) * (v0 + 2 * (vel_half + vel_half2) + vel_full)
    vel_nan = math.where(math.is_finite(vel_rk4), vel_rk4, v0)
    return elements.shifted(dt * vel_nan)



def advect(field: SampledField,
           velocity: Field,
           dt: Union[float, math.Tensor],
           integrator=euler) -> FieldType:
    """
    Advect `field` along the `velocity` vectors using the specified integrator.

    The behavior depends on the type of `field`:

    * `phi.field.PointCloud`: Points are advected forward, see `points`.
    * `phi.field.Grid`: Sample points are traced backward, see `semi_lagrangian`.

    Args:
        field: Field to be advected as `phi.field.SampledField`.
        velocity: Any `phi.field.Field` that can be sampled in the elements of `field`.
        dt: Time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected field of same type as `field`
    """
    if isinstance(field, PointCloud):
        return points(field, velocity, dt=dt, integrator=integrator)
    elif isinstance(field, Grid):
        return semi_lagrangian(field, velocity, dt=dt, integrator=integrator)
    raise NotImplementedError(field)


def finite_difference(grid: Grid,
                      velocity: Field,
                      order=2,
                      implicit: Solve = None) -> Field:

    """
    Finite difference advection using the differentiation Scheme indicated by `scheme` and a simple Euler step

    Args:
        grid: Grid to be advected
        velocity: `Grid` that can be sampled in the elements of `grid`.
        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 `phi.field.spatial_gradient()` and resampling).
            Passing order=4 currently uses 2nd-order resampling. This is work-in-progress.
        implicit: When a `Solve` object is passed, performs an implicit operation with the specified solver and tolerances.
            Otherwise, an explicit stencil is used.

    Returns:
        Advected grid of same type as `grid`
    """
    if isinstance(grid, StaggeredGrid):
        grad_list = [spatial_gradient(field_component, stack_dim=channel('gradient'), order=order, implicit=implicit) for field_component in grid.vector]
        grad_grid = grid.with_values(math.stack([component.values for component in grad_list], channel(velocity)))
        if order == 4:
            amounts = [grad * vel.at(grad, order=2) for grad, vel in zip(grad_grid.gradient, velocity.vector)]  # ToDo resampling does not yet support order=4
        else:
            grad_grid.gradient[0].elements
            velocity.vector[0].at(grad_grid.gradient[0], order=order, implicit=implicit)
            amounts = [grad * vel.at(grad, order=order, implicit=implicit) for grad, vel in zip(grad_grid.gradient, velocity.vector)]
        amount = sum(amounts)
    else:
        assert isinstance(grid, CenteredGrid), f"grid must be CenteredGrid or StaggeredGrid but got {type(grid)}"
        grad = spatial_gradient(grid, stack_dim=channel('gradient'), order=order, implicit=implicit)
        velocity = stack(unstack(velocity, dim='vector'), dim=channel('gradient'))
        amounts = velocity * grad
        amount = sum(amounts.gradient)
    return - amount


def points(field: PointCloud, velocity: Field, dt: float, integrator=euler):
    """
    Advects the sample points of a point cloud using a simple Euler step.
    Each point moves by an amount equal to the local velocity times `dt`.

    Args:
        field: point cloud to be advected
        velocity: velocity sampled at the same points as the point cloud
        dt: Euler step time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected point cloud
    """
    new_elements = integrator(field.elements, velocity, dt)
    return field.with_elements(new_elements)


def semi_lagrangian(field: GridType,
                    velocity: Field,
                    dt: float,
                    integrator=euler) -> GridType:
    """
    Semi-Lagrangian advection with simple backward lookup.
    
    This method samples the `velocity` at the grid points of `field`
    to determine the lookup location for each grid point by walking backwards along the velocity vectors.
    The new values are then determined by sampling `field` at these lookup locations.

    Args:
        field: quantity to be advected, stored on a grid (CenteredGrid or StaggeredGrid)
        velocity: vector field, need not be compatible with with `field`.
        dt: time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Field with same sample points as `field`

    """
    lookup = integrator(field.elements, velocity, -dt)
    interpolated = reduce_sample(field, lookup)
    return field.with_values(interpolated)


def mac_cormack(field: GridType,
                velocity: Field,
                dt: float,
                correction_strength=1.0,
                integrator=euler) -> GridType:
    """
    MacCormack advection uses a forward and backward lookup to determine the first-order error of semi-Lagrangian advection.
    It then uses that error estimate to correct the field values.
    To avoid overshoots, the resulting value is bounded by the neighbouring grid cells of the backward lookup.

    Args:
        field: Field to be advected, one of `(CenteredGrid, StaggeredGrid)`
        velocity: Vector field, need not be sampled at same locations as `field`.
        dt: Time increment
        correction_strength: The estimated error is multiplied by this factor before being applied.
            The case correction_strength=0 equals semi-lagrangian advection. Set lower than 1.0 to avoid oscillations.
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected field of type `type(field)`

    """
    v0 = sample(velocity, field.elements)
    points_bwd = integrator(field.elements, velocity, -dt, v0=v0)
    points_fwd = integrator(field.elements, velocity, dt, v0=v0)
    # Semi-Lagrangian advection
    field_semi_la = field.with_values(reduce_sample(field, points_bwd))
    # Inverse semi-Lagrangian advection
    field_inv_semi_la = field.with_values(reduce_sample(field_semi_la, points_fwd))
    # correction
    new_field = field_semi_la + correction_strength * 0.5 * (field - field_inv_semi_la)
    # Address overshoots
    limits = field.closest_values(points_bwd)
    lower_limit = math.min(limits, [f'closest_{dim}' for dim in field.shape.spatial.names])
    upper_limit = math.max(limits, [f'closest_{dim}' for dim in field.shape.spatial.names])
    values_clamped = math.clip(new_field.values, lower_limit, upper_limit)
    return new_field.with_values(values_clamped)

Functions

def advect(field: phi.field._field.SampledField, velocity: phi.field._field.Field, dt: Union[phiml.math._tensors.Tensor, float], integrator=<function euler>) ‑> ~FieldType

Advect field along the velocity vectors using the specified integrator.

The behavior depends on the type of field:

Args

field
Field to be advected as SampledField.
velocity
Any Field that can be sampled in the elements of field.
dt
Time increment
integrator
ODE integrator for solving the movement.

Returns

Advected field of same type as field

Expand source code 
def advect(field: SampledField,
           velocity: Field,
           dt: Union[float, math.Tensor],
           integrator=euler) -> FieldType:
    """
    Advect `field` along the `velocity` vectors using the specified integrator.

    The behavior depends on the type of `field`:

    * `phi.field.PointCloud`: Points are advected forward, see `points`.
    * `phi.field.Grid`: Sample points are traced backward, see `semi_lagrangian`.

    Args:
        field: Field to be advected as `phi.field.SampledField`.
        velocity: Any `phi.field.Field` that can be sampled in the elements of `field`.
        dt: Time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected field of same type as `field`
    """
    if isinstance(field, PointCloud):
        return points(field, velocity, dt=dt, integrator=integrator)
    elif isinstance(field, Grid):
        return semi_lagrangian(field, velocity, dt=dt, integrator=integrator)
    raise NotImplementedError(field)
def euler(elements: phi.geom._geom.Geometry, velocity: phi.field._field.Field, dt: float, v0: phiml.math._tensors.Tensor = None) ‑> phi.geom._geom.Geometry

Euler integrator.

Expand source code 
def euler(elements: Geometry, velocity: Field, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Euler integrator. """
    if v0 is None:
        v0 = sample(velocity, elements)
    return elements.shifted(v0 * dt)
def finite_difference(grid: phi.field._grid.Grid, velocity: phi.field._field.Field, order=2, implicit: phiml.math._optimize.Solve = None) ‑> phi.field._field.Field

Finite difference advection using the differentiation Scheme indicated by scheme and a simple Euler step

Args

grid
Grid to be advected
velocity
Grid that can be sampled in the elements of grid.
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 spatial_gradient() and resampling). Passing order=4 currently uses 2nd-order resampling. This is work-in-progress.
implicit
When a Solve object is passed, performs an implicit operation with the specified solver and tolerances. Otherwise, an explicit stencil is used.

Returns

Advected grid of same type as grid

Expand source code 
def finite_difference(grid: Grid,
                      velocity: Field,
                      order=2,
                      implicit: Solve = None) -> Field:

    """
    Finite difference advection using the differentiation Scheme indicated by `scheme` and a simple Euler step

    Args:
        grid: Grid to be advected
        velocity: `Grid` that can be sampled in the elements of `grid`.
        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 `phi.field.spatial_gradient()` and resampling).
            Passing order=4 currently uses 2nd-order resampling. This is work-in-progress.
        implicit: When a `Solve` object is passed, performs an implicit operation with the specified solver and tolerances.
            Otherwise, an explicit stencil is used.

    Returns:
        Advected grid of same type as `grid`
    """
    if isinstance(grid, StaggeredGrid):
        grad_list = [spatial_gradient(field_component, stack_dim=channel('gradient'), order=order, implicit=implicit) for field_component in grid.vector]
        grad_grid = grid.with_values(math.stack([component.values for component in grad_list], channel(velocity)))
        if order == 4:
            amounts = [grad * vel.at(grad, order=2) for grad, vel in zip(grad_grid.gradient, velocity.vector)]  # ToDo resampling does not yet support order=4
        else:
            grad_grid.gradient[0].elements
            velocity.vector[0].at(grad_grid.gradient[0], order=order, implicit=implicit)
            amounts = [grad * vel.at(grad, order=order, implicit=implicit) for grad, vel in zip(grad_grid.gradient, velocity.vector)]
        amount = sum(amounts)
    else:
        assert isinstance(grid, CenteredGrid), f"grid must be CenteredGrid or StaggeredGrid but got {type(grid)}"
        grad = spatial_gradient(grid, stack_dim=channel('gradient'), order=order, implicit=implicit)
        velocity = stack(unstack(velocity, dim='vector'), dim=channel('gradient'))
        amounts = velocity * grad
        amount = sum(amounts.gradient)
    return - amount
def finite_rk4(elements: phi.geom._geom.Geometry, velocity: phi.field._grid.Grid, dt: float, v0: phiml.math._tensors.Tensor = None) ‑> phi.geom._geom.Geometry

Runge-Kutta-4 integrator with Euler fallback where velocity values are NaN.

Expand source code 
def finite_rk4(elements: Geometry, velocity: Grid, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Runge-Kutta-4 integrator with Euler fallback where velocity values are NaN. """
    v0 = sample(velocity, elements)
    vel_half = sample(velocity, elements.shifted(0.5 * dt * v0))
    vel_half2 = sample(velocity, elements.shifted(0.5 * dt * vel_half))
    vel_full = sample(velocity, elements.shifted(dt * vel_half2))
    vel_rk4 = (1 / 6.) * (v0 + 2 * (vel_half + vel_half2) + vel_full)
    vel_nan = math.where(math.is_finite(vel_rk4), vel_rk4, v0)
    return elements.shifted(dt * vel_nan)
def mac_cormack(field: ~GridType, velocity: phi.field._field.Field, dt: float, correction_strength=1.0, integrator=<function euler>) ‑> ~GridType

MacCormack advection uses a forward and backward lookup to determine the first-order error of semi-Lagrangian advection. It then uses that error estimate to correct the field values. To avoid overshoots, the resulting value is bounded by the neighbouring grid cells of the backward lookup.

Args

field
Field to be advected, one of (CenteredGrid, StaggeredGrid)
velocity
Vector field, need not be sampled at same locations as field.
dt
Time increment
correction_strength
The estimated error is multiplied by this factor before being applied. The case correction_strength=0 equals semi-lagrangian advection. Set lower than 1.0 to avoid oscillations.
integrator
ODE integrator for solving the movement.

Returns

Advected field of type type(field)

Expand source code 
def mac_cormack(field: GridType,
                velocity: Field,
                dt: float,
                correction_strength=1.0,
                integrator=euler) -> GridType:
    """
    MacCormack advection uses a forward and backward lookup to determine the first-order error of semi-Lagrangian advection.
    It then uses that error estimate to correct the field values.
    To avoid overshoots, the resulting value is bounded by the neighbouring grid cells of the backward lookup.

    Args:
        field: Field to be advected, one of `(CenteredGrid, StaggeredGrid)`
        velocity: Vector field, need not be sampled at same locations as `field`.
        dt: Time increment
        correction_strength: The estimated error is multiplied by this factor before being applied.
            The case correction_strength=0 equals semi-lagrangian advection. Set lower than 1.0 to avoid oscillations.
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected field of type `type(field)`

    """
    v0 = sample(velocity, field.elements)
    points_bwd = integrator(field.elements, velocity, -dt, v0=v0)
    points_fwd = integrator(field.elements, velocity, dt, v0=v0)
    # Semi-Lagrangian advection
    field_semi_la = field.with_values(reduce_sample(field, points_bwd))
    # Inverse semi-Lagrangian advection
    field_inv_semi_la = field.with_values(reduce_sample(field_semi_la, points_fwd))
    # correction
    new_field = field_semi_la + correction_strength * 0.5 * (field - field_inv_semi_la)
    # Address overshoots
    limits = field.closest_values(points_bwd)
    lower_limit = math.min(limits, [f'closest_{dim}' for dim in field.shape.spatial.names])
    upper_limit = math.max(limits, [f'closest_{dim}' for dim in field.shape.spatial.names])
    values_clamped = math.clip(new_field.values, lower_limit, upper_limit)
    return new_field.with_values(values_clamped)
def points(field: phi.field._point_cloud.PointCloud, velocity: phi.field._field.Field, dt: float, integrator=<function euler>)

Advects the sample points of a point cloud using a simple Euler step. Each point moves by an amount equal to the local velocity times dt.

Args

field
point cloud to be advected
velocity
velocity sampled at the same points as the point cloud
dt
Euler step time increment
integrator
ODE integrator for solving the movement.

Returns

Advected point cloud

Expand source code 
def points(field: PointCloud, velocity: Field, dt: float, integrator=euler):
    """
    Advects the sample points of a point cloud using a simple Euler step.
    Each point moves by an amount equal to the local velocity times `dt`.

    Args:
        field: point cloud to be advected
        velocity: velocity sampled at the same points as the point cloud
        dt: Euler step time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Advected point cloud
    """
    new_elements = integrator(field.elements, velocity, dt)
    return field.with_elements(new_elements)
def rk4(elements: phi.geom._geom.Geometry, velocity: phi.field._field.Field, dt: float, v0: phiml.math._tensors.Tensor = None) ‑> phi.geom._geom.Geometry

Runge-Kutta-4 integrator.

Expand source code 
def rk4(elements: Geometry, velocity: Field, dt: float, v0: math.Tensor = None) -> Geometry:
    """ Runge-Kutta-4 integrator. """
    if v0 is None:
        v0 = sample(velocity, elements)
    vel_half = sample(velocity, elements.shifted(0.5 * dt * v0))
    vel_half2 = sample(velocity, elements.shifted(0.5 * dt * vel_half))
    vel_full = sample(velocity, elements.shifted(dt * vel_half2))
    vel_rk4 = (1 / 6.) * (v0 + 2 * (vel_half + vel_half2) + vel_full)
    return elements.shifted(dt * vel_rk4)
def semi_lagrangian(field: ~GridType, velocity: phi.field._field.Field, dt: float, integrator=<function euler>) ‑> ~GridType

Semi-Lagrangian advection with simple backward lookup.

This method samples the velocity at the grid points of field to determine the lookup location for each grid point by walking backwards along the velocity vectors. The new values are then determined by sampling field at these lookup locations.

Args

field
quantity to be advected, stored on a grid (CenteredGrid or StaggeredGrid)
velocity
vector field, need not be compatible with with field.
dt
time increment
integrator
ODE integrator for solving the movement.

Returns

Field with same sample points as field

Expand source code 
def semi_lagrangian(field: GridType,
                    velocity: Field,
                    dt: float,
                    integrator=euler) -> GridType:
    """
    Semi-Lagrangian advection with simple backward lookup.
    
    This method samples the `velocity` at the grid points of `field`
    to determine the lookup location for each grid point by walking backwards along the velocity vectors.
    The new values are then determined by sampling `field` at these lookup locations.

    Args:
        field: quantity to be advected, stored on a grid (CenteredGrid or StaggeredGrid)
        velocity: vector field, need not be compatible with with `field`.
        dt: time increment
        integrator: ODE integrator for solving the movement.

    Returns:
        Field with same sample points as `field`

    """
    lookup = integrator(field.elements, velocity, -dt)
    interpolated = reduce_sample(field, lookup)
    return field.with_values(interpolated)