Source code for elastica.dissipation

__doc__ = """
(added in version 0.3.0)

Built in damper module implementations
"""
__all__ = [
    "DamperBase",
    "AnalyticalLinearDamper",
    "LaplaceDissipationFilter",
]
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, rod: 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 ---------- rod : Union[Type[RodBase], Type[RigidBodyBase]] 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 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: SystemType, 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 self.velocity_filter_term = np.zeros_like(self._system.velocity_collection) self.omega_filter_term = np.zeros_like(self._system.omega_collection)
def dampen_rates(self, rod: RodType, time: float) -> None: nb_filter_rate( rate_collection=rod.velocity_collection, filter_term=self.velocity_filter_term, filter_order=self.filter_order, ) nb_filter_rate( rate_collection=rod.omega_collection, filter_term=self.omega_filter_term, filter_order=self.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