__doc__ = """
(added in version 0.3.0)
Built in damper module implementations
"""
from abc import ABC, abstractmethod
from elastica.typing import RodType, SystemType
from numba import njit
import numpy as np
[docs]class DamperBase(ABC):
"""Base class for damping module implementations.
Notes
-----
All damper classes must inherit DamperBase class.
Attributes
----------
system : SystemType (RodBase or RigidBodyBase)
"""
_system: SystemType
def __init__(self, *args, **kwargs):
"""Initialize damping module"""
try:
self._system = kwargs["_system"]
except KeyError:
raise KeyError(
"Please use simulator.dampen(...).using(...) syntax to establish "
"damping."
)
@property
def system(self): # -> SystemType: (Return type is not parsed with sphinx book.)
"""
get system (rod or rigid body) reference
Returns
-------
SystemType
"""
return self._system
[docs] @abstractmethod
def dampen_rates(self, system: SystemType, time: float):
# TODO: In the future, we can remove rod and use self.system
"""
Dampen rates (velocity and/or omega) of a rod object.
Parameters
----------
system : SystemType
System (rod or rigid-body) object.
time : float
The time of simulation.
"""
pass
[docs]class AnalyticalLinearDamper(DamperBase):
"""
Analytical linear damper class. This class corresponds to the analytical version of
a linear damper, and uses the following equations to damp translational and
rotational velocities:
.. math::
\\mathbf{v}^{n+1} = \\mathbf{v}^n \\exp \\left( - \\nu~dt \\right)
\\pmb{\\omega}^{n+1} = \\pmb{\\omega}^n \\exp \\left( - \\frac{{\\nu}~m~dt } { \\mathbf{J}} \\right)
Examples
--------
How to set analytical linear damper for rod or rigid body:
>>> simulator.dampen(rod).using(
... AnalyticalLinearDamper,
... damping_constant=0.1,
... time_step = 1E-4, # Simulation time-step
... )
Notes
-----
Since this class analytically treats the damping term, it is unconditionally stable
from a timestep perspective, i.e. the presence of damping does not impose any additional
restriction on the simulation timestep size. This implies that when using
AnalyticalLinearDamper, one can set `damping_constant` as high as possible, without worrying
about the simulation becoming unstable. This now leads to a streamlined procedure
for tuning the `damping_constant`:
1. Set a high value for `damping_constant` to first acheive a stable simulation.
2. If you feel the simulation is overdamped, reduce `damping_constant` until you
feel the simulation is underdamped, and expected dynamics are recovered.
Attributes
----------
translational_damping_coefficient: numpy.ndarray
1D array containing data with 'float' type.
Damping coefficient acting on translational velocity.
rotational_damping_coefficient : numpy.ndarray
1D array containing data with 'float' type.
Damping coefficient acting on rotational velocity.
"""
[docs] def __init__(self, damping_constant, time_step, **kwargs):
"""
Analytical linear damper initializer
Parameters
----------
damping_constant : float
Damping constant for the analytical linear damper.
time_step : float
Time-step of simulation
"""
super().__init__(**kwargs)
# Compute the damping coefficient for translational velocity
nodal_mass = self._system.mass
self.translational_damping_coefficient = np.exp(-damping_constant * time_step)
# Compute the damping coefficient for exponential velocity
if self._system.ring_rod_flag:
element_mass = nodal_mass
else:
element_mass = 0.5 * (nodal_mass[1:] + nodal_mass[:-1])
element_mass[0] += 0.5 * nodal_mass[0]
element_mass[-1] += 0.5 * nodal_mass[-1]
self.rotational_damping_coefficient = np.exp(
-damping_constant
* time_step
* element_mass
* np.diagonal(self._system.inv_mass_second_moment_of_inertia).T
)
def dampen_rates(self, rod: RodType, time: float):
rod.velocity_collection[:] = (
rod.velocity_collection * self.translational_damping_coefficient
)
rod.omega_collection[:] = rod.omega_collection * np.power(
self.rotational_damping_coefficient, rod.dilatation
)
[docs]class LaplaceDissipationFilter(DamperBase):
"""
Laplace Dissipation Filter class. This class corresponds qualitatively to a
low-pass filter generated via the 1D Laplacian operator. It is applied to the
translational and rotational velocities, where it filters out the high
frequency (noise) modes, while having negligible effect on the low frequency
smooth modes.
Examples
--------
How to set Laplace dissipation filter for rod:
>>> simulator.dampen(rod).using(
... LaplaceDissipationFilter,
... filter_order=3, # order of the filter
... )
Notes
-----
The extent of filtering can be controlled by the `filter_order`, which refers
to the number of times the Laplacian operator is applied. Small
integer values (1, 2, etc.) result in aggressive filtering, and can lead to
the "physics" being filtered out. While high values (9, 10, etc.) imply
minimal filtering, and thus negligible effect on the velocities.
Values in the range of 3-7 are usually recommended.
For details regarding the numerics behind the filtering, refer to [1]_, [2]_.
.. [1] Jeanmart, H., & Winckelmans, G. (2007). Investigation of eddy-viscosity
models modified using discrete filters: a simplified “regularized variational
multiscale model” and an “enhanced field model”. Physics of fluids, 19(5), 055110.
.. [2] Lorieul, G. (2018). Development and validation of a 2D Vortex Particle-Mesh
method for incompressible multiphase flows (Doctoral dissertation,
Université Catholique de Louvain).
Attributes
----------
filter_order : int
Filter order, which corresponds to the number of times the Laplacian
operator is applied. Increasing `filter_order` implies higher-order/weaker
filtering.
velocity_filter_term: numpy.ndarray
2D array containing data with 'float' type.
Filter term that modifies rod translational velocity.
omega_filter_term: numpy.ndarray
2D array containing data with 'float' type.
Filter term that modifies rod rotational velocity.
"""
[docs] def __init__(self, filter_order: int, **kwargs):
"""
Filter damper initializer
Parameters
----------
filter_order : int
Filter order, which corresponds to the number of times the Laplacian
operator is applied. Increasing `filter_order` implies higher-order/weaker
filtering.
"""
super().__init__(**kwargs)
if not (filter_order > 0 and isinstance(filter_order, int)):
raise ValueError(
"Invalid filter order! Filter order must be a positive integer."
)
self.filter_order = filter_order
if self._system.ring_rod_flag:
# There are two periodic boundaries
blocksize = self._system.n_elems + 2
self.velocity_filter_term = np.zeros((3, blocksize))
self.omega_filter_term = np.zeros((3, blocksize))
self.filter_function = _filter_function_periodic_condition_ring_rod
else:
self.velocity_filter_term = np.zeros_like(self._system.velocity_collection)
self.omega_filter_term = np.zeros_like(self._system.omega_collection)
self.filter_function = _filter_function_periodic_condition
def dampen_rates(self, rod: RodType, time: float) -> None:
self.filter_function(
rod.velocity_collection,
self.velocity_filter_term,
rod.omega_collection,
self.omega_filter_term,
self.filter_order,
)
@njit(cache=True)
def _filter_function_periodic_condition_ring_rod(
velocity_collection,
velocity_filter_term,
omega_collection,
omega_filter_term,
filter_order,
):
blocksize = velocity_filter_term.shape[1]
# Transfer velocity to an array which has periodic boundaries and synchornize boundaries
velocity_collection_with_periodic_bc = np.empty((3, blocksize))
velocity_collection_with_periodic_bc[:, 1:-1] = velocity_collection[:]
velocity_collection_with_periodic_bc[:, 0] = velocity_collection[:, -1]
velocity_collection_with_periodic_bc[:, -1] = velocity_collection[:, 0]
# Transfer omega to an array which has periodic boundaries and synchornize boundaries
omega_collection_with_periodic_bc = np.empty((3, blocksize))
omega_collection_with_periodic_bc[:, 1:-1] = omega_collection[:]
omega_collection_with_periodic_bc[:, 0] = omega_collection[:, -1]
omega_collection_with_periodic_bc[:, -1] = omega_collection[:, 0]
nb_filter_rate(
rate_collection=velocity_collection_with_periodic_bc,
filter_term=velocity_filter_term,
filter_order=filter_order,
)
nb_filter_rate(
rate_collection=omega_collection_with_periodic_bc,
filter_term=omega_filter_term,
filter_order=filter_order,
)
# Transfer filtered velocity back
velocity_collection[:] = velocity_collection_with_periodic_bc[:, 1:-1]
omega_collection[:] = omega_collection_with_periodic_bc[:, 1:-1]
@njit(cache=True)
def _filter_function_periodic_condition(
velocity_collection,
velocity_filter_term,
omega_collection,
omega_filter_term,
filter_order,
):
nb_filter_rate(
rate_collection=velocity_collection,
filter_term=velocity_filter_term,
filter_order=filter_order,
)
nb_filter_rate(
rate_collection=omega_collection,
filter_term=omega_filter_term,
filter_order=filter_order,
)
@njit(cache=True)
def nb_filter_rate(
rate_collection: np.ndarray, filter_term: np.ndarray, filter_order: int
) -> None:
"""
Filters the rod rates (velocities) in numba njit decorator
Parameters
----------
rate_collection : numpy.ndarray
2D array containing data with 'float' type.
Array containing rod rates (velocities).
filter_term: numpy.ndarray
2D array containing data with 'float' type.
Filter term that modifies rod rates (velocities).
filter_order : int
Filter order, which corresponds to the number of times the Laplacian
operator is applied. Increasing `filter_order` implies higher order/weaker
filtering.
Notes
-----
For details regarding the numerics behind the filtering, refer to:
.. [1] Jeanmart, H., & Winckelmans, G. (2007). Investigation of eddy-viscosity
models modified using discrete filters: a simplified “regularized variational
multiscale model” and an “enhanced field model”. Physics of fluids, 19(5), 055110.
.. [2] Lorieul, G. (2018). Development and validation of a 2D Vortex Particle-Mesh
method for incompressible multiphase flows (Doctoral dissertation,
Université Catholique de Louvain).
"""
filter_term[...] = rate_collection
for i in range(filter_order):
filter_term[..., 1:-1] = (
-filter_term[..., 2:] - filter_term[..., :-2] + 2.0 * filter_term[..., 1:-1]
) / 4.0
# dont touch boundary values
filter_term[..., 0] = 0.0
filter_term[..., -1] = 0.0
rate_collection[...] = rate_collection - filter_term
