# Source code for elastica.interaction

__doc__ = """ Numba implementation module containing interactions between a rod and its environment."""
__all__ = [
"AnisotropicFrictionalPlane",
"InteractionPlane",
"InteractionPlaneRigidBody",
"SlenderBodyTheory",
]

import numpy as np
from elastica.external_forces import NoForces

from numba import njit
from elastica._linalg import (
_batch_matvec,
_batch_cross,
_batch_norm,
_batch_dot,
_batch_product_i_k_to_ik,
_batch_product_i_ik_to_k,
_batch_product_k_ik_to_ik,
_batch_vector_sum,
_batch_matrix_transpose,
_batch_vec_oneD_vec_cross,
)
import warnings

@njit(cache=True)
def find_slipping_elements(velocity_slip, velocity_threshold):
"""
This function takes the velocity of elements and checks if they are larger than the threshold velocity.
If the velocity of elements is larger than threshold velocity, that means those elements are slipping.
In other words, kinetic friction will be acting on those elements, not static friction.
This function outputs an array called slip function, this array has a size of the number of elements.
If the velocity of the element is smaller than the threshold velocity slip function value for that element is 1,
which means static friction is acting on that element. If the velocity of the element is larger than
the threshold velocity slip function value for that element is between 0 and 1, which means kinetic friction is acting
on that element.

Parameters
----------
velocity_slip : numpy.ndarray
2D (dim, blocksize) array containing data with 'float' type.
Rod-like object element velocity.
velocity_threshold : float
Threshold velocity to determine slip.

Returns
-------
slip_function : numpy.ndarray
2D (dim, blocksize) array containing data with 'float' type.
"""
"""
Developer Notes
-----
Benchmark results, for a blocksize of 100 using timeit
python version: 18.9 µs ± 2.98 µs per loop
this version: 1.96 µs ± 58.3 ns per loop
"""
abs_velocity_slip = _batch_norm(velocity_slip)
slip_points = np.where(np.fabs(abs_velocity_slip) > velocity_threshold)
slip_function = np.ones((velocity_slip.shape))
slip_function[slip_points] = np.fabs(
1.0 - np.minimum(1.0, abs_velocity_slip[slip_points] / velocity_threshold - 1.0)
)
return slip_function

@njit(cache=True)
def node_to_element_mass_or_force(input):
"""
This function converts the mass/forces on rod nodes to
elements, where special treatment is necessary at the ends.

Parameters
----------
input: numpy.ndarray
2D (dim, blocksize) array containing nodal mass/forces
with 'float' type.

Returns
-------
output: numpy.ndarray
2D (dim, blocksize) array containing elemental mass/forces
with 'float' type.
"""
"""
Developer Notes
-----
Benchmark results, for a blocksize of 100 using timeit
Python version: 18.1 µs ± 1.03 µs per loop
This version: 1.55 µs ± 13.4 ns per loop
"""
blocksize = input.shape - 1  # nelem
output = np.zeros((3, blocksize))
for i in range(3):
for k in range(0, blocksize):
output[i, k] += 0.5 * (input[i, k] + input[i, k + 1])

# Put extra care for the first and last element
output[..., 0] += 0.5 * input[..., 0]
output[..., -1] += 0.5 * input[..., -1]

return output

def nodes_to_elements(input):
warnings.warn(
# Change the warning to error on v0.3.1
# Remove the function beyond v0.4.0
"This function is now deprecated (issue #80). Please use "
"elastica.interaction.node_to_element_mass_or_force() "
"instead for node-to-element interpolation of mass/forces. "
"The function will be removed in the future (v0.3.1).",
DeprecationWarning,
)
return node_to_element_mass_or_force(input)

@njit(cache=True)
def elements_to_nodes_inplace(vector_in_element_frame, vector_in_node_frame):
"""
Updating nodal forces using the forces computed on elements
Parameters
----------
vector_in_element_frame
vector_in_node_frame

Returns
-------
Notes
-----
Benchmark results, for a blocksize of 100 using timeit
Python version: 23.1 µs ± 7.57 µs per loop
This version: 696 ns ± 10.2 ns per loop
"""
for i in range(3):
for k in range(vector_in_element_frame.shape):
vector_in_node_frame[i, k] += 0.5 * vector_in_element_frame[i, k]
vector_in_node_frame[i, k + 1] += 0.5 * vector_in_element_frame[i, k]

# base class for interaction
# only applies normal force no friction
[docs]class InteractionPlane:
"""
The interaction plane class computes the plane reaction
force on a rod-like object.  For more details regarding the contact module refer to
Eqn 4.8 of Gazzola et al. RSoS (2018).

Attributes
----------
k: float
Stiffness coefficient between the plane and the rod-like object.
nu: float
Dissipation coefficient between the plane and the rod-like object.
plane_origin: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
Origin of the plane.
plane_normal: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
The normal vector of the plane.
surface_tol: float
Penetration tolerance between the plane and the rod-like object.

"""

[docs]    def __init__(self, k, nu, plane_origin, plane_normal):
"""

Parameters
----------
k: float
Stiffness coefficient between the plane and the rod-like object.
nu: float
Dissipation coefficient between the plane and the rod-like object.
plane_origin: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
Origin of the plane.
plane_normal: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
The normal vector of the plane.
"""
self.k = k
self.nu = nu
self.plane_origin = plane_origin.reshape(3, 1)
self.plane_normal = plane_normal.reshape(3)
self.surface_tol = 1e-4

def apply_normal_force(self, system):
"""
In the case of contact with the plane, this function computes the plane reaction force on the element.

Parameters
----------
system: object
Rod-like object.

Returns
-------
plane_response_force_mag : numpy.ndarray
1D (blocksize) array containing data with 'float' type.
Magnitude of plane response force acting on rod-like object.
no_contact_point_idx : numpy.ndarray
1D (blocksize) array containing data with 'int' type.
Index of rod-like object elements that are not in contact with the plane.
"""
return apply_normal_force_numba(
self.plane_origin,
self.plane_normal,
self.surface_tol,
self.k,
self.nu,
system.mass,
system.position_collection,
system.velocity_collection,
system.internal_forces,
system.external_forces,
)

@njit(cache=True)
def apply_normal_force_numba(
plane_origin,
plane_normal,
surface_tol,
k,
nu,
mass,
position_collection,
velocity_collection,
internal_forces,
external_forces,
):
"""
This function computes the plane force response on the element, in the
case of contact. Contact model given in Eqn 4.8 Gazzola et. al. RSoS 2018 paper
is used.

Parameters
----------
system

Returns
-------
magnitude of the plane response
"""

# Compute plane response force
nodal_total_forces = _batch_vector_sum(internal_forces, external_forces)
element_total_forces = node_to_element_mass_or_force(nodal_total_forces)

force_component_along_normal_direction = _batch_product_i_ik_to_k(
plane_normal, element_total_forces
)
forces_along_normal_direction = _batch_product_i_k_to_ik(
plane_normal, force_component_along_normal_direction
)

# If the total force component along the plane normal direction is greater than zero that means,
# total force is pushing rod away from the plane not towards the plane. Thus, response force
# applied by the surface has to be zero.
forces_along_normal_direction[
..., np.where(force_component_along_normal_direction > 0)
] = 0.0
# Compute response force on the element. Plane response force
# has to be away from the surface and towards the element. Thus
# multiply forces along normal direction with negative sign.
plane_response_force = -forces_along_normal_direction

# Elastic force response due to penetration
element_position = node_to_element_position(position_collection)
distance_from_plane = _batch_product_i_ik_to_k(
plane_normal, (element_position - plane_origin)
)
plane_penetration = np.minimum(distance_from_plane - radius, 0.0)
elastic_force = -k * _batch_product_i_k_to_ik(plane_normal, plane_penetration)

# Damping force response due to velocity towards the plane
element_velocity = node_to_element_velocity(
mass=mass, node_velocity_collection=velocity_collection
)
normal_component_of_element_velocity = _batch_product_i_ik_to_k(
plane_normal, element_velocity
)
damping_force = -nu * _batch_product_i_k_to_ik(
plane_normal, normal_component_of_element_velocity
)

# Compute total plane response force
plane_response_force_total = plane_response_force + elastic_force + damping_force

# Check if the rod elements are in contact with plane.
no_contact_point_idx = np.where((distance_from_plane - radius) > surface_tol)
# If rod element does not have any contact with plane, plane cannot apply response
# force on the element. Thus lets set plane response force to 0.0 for the no contact points.
plane_response_force[..., no_contact_point_idx] = 0.0
plane_response_force_total[..., no_contact_point_idx] = 0.0

# Update the external forces
elements_to_nodes_inplace(plane_response_force_total, external_forces)

return (_batch_norm(plane_response_force), no_contact_point_idx)

# class for anisotropic frictional plane
# NOTE: friction coefficients are passed as arrays in the order
# mu_forward : mu_backward : mu_sideways
# head is at x and forward means head to tail
# same convention for kinetic and static
# mu named as to which direction it opposes
[docs]class AnisotropicFrictionalPlane(NoForces, InteractionPlane):
"""
This anisotropic friction plane class is for computing
anisotropic friction forces on rods.
A detailed explanation of the implemented equations
can be found in Gazzola et al. RSoS. (2018).

Attributes
----------
k: float
Stiffness coefficient between the plane and the rod-like object.
nu: float
Dissipation coefficient between the plane and the rod-like object.
plane_origin: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
Origin of the plane.
plane_normal: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
The normal vector of the plane.
slip_velocity_tol: float
Velocity tolerance to determine if the element is slipping or not.
static_mu_array: numpy.ndarray
1D (3,) array containing data with 'float' type.
[forward, backward, sideways] static friction coefficients.
kinetic_mu_array: numpy.ndarray
1D (3,) array containing data with 'float' type.
[forward, backward, sideways] kinetic friction coefficients.
"""

[docs]    def __init__(
self,
k,
nu,
plane_origin,
plane_normal,
slip_velocity_tol,
static_mu_array,
kinetic_mu_array,
):
"""

Parameters
----------
k: float
Stiffness coefficient between the plane and the rod-like object.
nu: float
Dissipation coefficient between the plane and the rod-like object.
plane_origin: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
Origin of the plane.
plane_normal: numpy.ndarray
2D (dim, 1) array containing data with 'float' type.
The normal vector of the plane.
slip_velocity_tol: float
Velocity tolerance to determine if the element is slipping or not.
static_mu_array: numpy.ndarray
1D (3,) array containing data with 'float' type.
[forward, backward, sideways] static friction coefficients.
kinetic_mu_array: numpy.ndarray
1D (3,) array containing data with 'float' type.
[forward, backward, sideways] kinetic friction coefficients.
"""
InteractionPlane.__init__(self, k, nu, plane_origin, plane_normal)
self.slip_velocity_tol = slip_velocity_tol
(
self.static_mu_forward,
self.static_mu_backward,
self.static_mu_sideways,
) = static_mu_array
(
self.kinetic_mu_forward,
self.kinetic_mu_backward,
self.kinetic_mu_sideways,
) = kinetic_mu_array

# kinetic and static friction should separate functions
# for now putting them together to figure out common variables
def apply_forces(self, system, time=0.0):
"""
Call numba implementation to apply friction forces
Parameters
----------
system
time

Returns
-------

"""
anisotropic_friction(
self.plane_origin,
self.plane_normal,
self.surface_tol,
self.slip_velocity_tol,
self.k,
self.nu,
self.kinetic_mu_forward,
self.kinetic_mu_backward,
self.kinetic_mu_sideways,
self.static_mu_forward,
self.static_mu_backward,
self.static_mu_sideways,
system.mass,
system.tangents,
system.position_collection,
system.director_collection,
system.velocity_collection,
system.omega_collection,
system.internal_forces,
system.external_forces,
system.internal_torques,
system.external_torques,
)

@njit(cache=True)
def anisotropic_friction(
plane_origin,
plane_normal,
surface_tol,
slip_velocity_tol,
k,
nu,
kinetic_mu_forward,
kinetic_mu_backward,
kinetic_mu_sideways,
static_mu_forward,
static_mu_backward,
static_mu_sideways,
mass,
tangents,
position_collection,
director_collection,
velocity_collection,
omega_collection,
internal_forces,
external_forces,
internal_torques,
external_torques,
):
plane_response_force_mag, no_contact_point_idx = apply_normal_force_numba(
plane_origin,
plane_normal,
surface_tol,
k,
nu,
mass,
position_collection,
velocity_collection,
internal_forces,
external_forces,
)

# First compute component of rod tangent in plane. Because friction forces acts in plane not out of plane. Thus
# axial direction has to be in plane, it cannot be out of plane. We are projecting rod element tangent vector in
# to the plane. So friction forces can only be in plane forces and not out of plane.
tangent_along_normal_direction = _batch_product_i_ik_to_k(plane_normal, tangents)
tangent_perpendicular_to_normal_direction = tangents - _batch_product_i_k_to_ik(
plane_normal, tangent_along_normal_direction
)
tangent_perpendicular_to_normal_direction_mag = _batch_norm(
tangent_perpendicular_to_normal_direction
)
# Normalize tangent_perpendicular_to_normal_direction. This is axial direction for plane. Here we are adding
# small tolerance (1e-10) for normalization, in order to prevent division by 0.
axial_direction = _batch_product_k_ik_to_ik(
1 / (tangent_perpendicular_to_normal_direction_mag + 1e-14),
tangent_perpendicular_to_normal_direction,
)
element_velocity = node_to_element_velocity(
mass=mass, node_velocity_collection=velocity_collection
)
# first apply axial kinetic friction
velocity_mag_along_axial_direction = _batch_dot(element_velocity, axial_direction)
velocity_along_axial_direction = _batch_product_k_ik_to_ik(
velocity_mag_along_axial_direction, axial_direction
)

# Friction forces depends on the direction of velocity, in other words sign
# of the velocity vector.
velocity_sign_along_axial_direction = np.sign(velocity_mag_along_axial_direction)
# Check top for sign convention
kinetic_mu = 0.5 * (
kinetic_mu_forward * (1 + velocity_sign_along_axial_direction)
+ kinetic_mu_backward * (1 - velocity_sign_along_axial_direction)
)
# Call slip function to check if elements slipping or not
slip_function_along_axial_direction = find_slipping_elements(
velocity_along_axial_direction, slip_velocity_tol
)

# Now rolling kinetic friction
rolling_direction = _batch_vec_oneD_vec_cross(axial_direction, plane_normal)
torque_arm = _batch_product_i_k_to_ik(-plane_normal, radius)
velocity_along_rolling_direction = _batch_dot(element_velocity, rolling_direction)
directors_transpose = _batch_matrix_transpose(director_collection)
# w_rot = Q.T @ omega @ Q @ r
rotation_velocity = _batch_matvec(
directors_transpose,
_batch_cross(omega_collection, _batch_matvec(director_collection, torque_arm)),
)
rotation_velocity_along_rolling_direction = _batch_dot(
rotation_velocity, rolling_direction
)
slip_velocity_mag_along_rolling_direction = (
velocity_along_rolling_direction + rotation_velocity_along_rolling_direction
)
slip_velocity_along_rolling_direction = _batch_product_k_ik_to_ik(
slip_velocity_mag_along_rolling_direction, rolling_direction
)
slip_function_along_rolling_direction = find_slipping_elements(
slip_velocity_along_rolling_direction, slip_velocity_tol
)
# Compute unitized total slip velocity vector. We will use this to distribute the weight of the rod in axial
# and rolling directions.
unitized_total_velocity = (
slip_velocity_along_rolling_direction + velocity_along_axial_direction
)
unitized_total_velocity /= _batch_norm(unitized_total_velocity + 1e-14)
# Apply kinetic friction in axial direction.
kinetic_friction_force_along_axial_direction = -(
(1.0 - slip_function_along_axial_direction)
* kinetic_mu
* plane_response_force_mag
* _batch_dot(unitized_total_velocity, axial_direction)
* axial_direction
)
# If rod element does not have any contact with plane, plane cannot apply friction
# force on the element. Thus lets set kinetic friction force to 0.0 for the no contact points.
kinetic_friction_force_along_axial_direction[..., no_contact_point_idx] = 0.0
elements_to_nodes_inplace(
kinetic_friction_force_along_axial_direction, external_forces
)
# Apply kinetic friction in rolling direction.
kinetic_friction_force_along_rolling_direction = -(
(1.0 - slip_function_along_rolling_direction)
* kinetic_mu_sideways
* plane_response_force_mag
* _batch_dot(unitized_total_velocity, rolling_direction)
* rolling_direction
)
# If rod element does not have any contact with plane, plane cannot apply friction
# force on the element. Thus lets set kinetic friction force to 0.0 for the no contact points.
kinetic_friction_force_along_rolling_direction[..., no_contact_point_idx] = 0.0
elements_to_nodes_inplace(
kinetic_friction_force_along_rolling_direction, external_forces
)
# torque = Q @ r @ Fr
external_torques += _batch_matvec(
director_collection,
_batch_cross(torque_arm, kinetic_friction_force_along_rolling_direction),
)

# now axial static friction
nodal_total_forces = _batch_vector_sum(internal_forces, external_forces)
element_total_forces = node_to_element_mass_or_force(nodal_total_forces)
force_component_along_axial_direction = _batch_dot(
element_total_forces, axial_direction
)
force_component_sign_along_axial_direction = np.sign(
force_component_along_axial_direction
)
# check top for sign convention
static_mu = 0.5 * (
static_mu_forward * (1 + force_component_sign_along_axial_direction)
+ static_mu_backward * (1 - force_component_sign_along_axial_direction)
)
max_friction_force = (
slip_function_along_axial_direction * static_mu * plane_response_force_mag
)
# friction = min(mu N, pushing force)
static_friction_force_along_axial_direction = -(
np.minimum(np.fabs(force_component_along_axial_direction), max_friction_force)
* force_component_sign_along_axial_direction
* axial_direction
)
# If rod element does not have any contact with plane, plane cannot apply friction
# force on the element. Thus lets set static friction force to 0.0 for the no contact points.
static_friction_force_along_axial_direction[..., no_contact_point_idx] = 0.0
elements_to_nodes_inplace(
static_friction_force_along_axial_direction, external_forces
)

# now rolling static friction
# there is some normal, tangent and rolling directions inconsitency from Elastica
total_torques = _batch_matvec(
directors_transpose, (internal_torques + external_torques)
)
# Elastica has opposite defs of tangents in interaction.h and rod.cpp
total_torques_along_axial_direction = _batch_dot(total_torques, axial_direction)
force_component_along_rolling_direction = _batch_dot(
element_total_forces, rolling_direction
)
noslip_force = -(
(
- 2.0 * total_torques_along_axial_direction
)
/ 3.0
)
max_friction_force = (
slip_function_along_rolling_direction
* static_mu_sideways
* plane_response_force_mag
)
noslip_force_sign = np.sign(noslip_force)
static_friction_force_along_rolling_direction = (
np.minimum(np.fabs(noslip_force), max_friction_force)
* noslip_force_sign
* rolling_direction
)
# If rod element does not have any contact with plane, plane cannot apply friction
# force on the element. Thus lets set plane static friction force to 0.0 for the no contact points.
static_friction_force_along_rolling_direction[..., no_contact_point_idx] = 0.0
elements_to_nodes_inplace(
static_friction_force_along_rolling_direction, external_forces
)
external_torques += _batch_matvec(
director_collection,
_batch_cross(torque_arm, static_friction_force_along_rolling_direction),
)

# Slender body module
@njit(cache=True)
def sum_over_elements(input):
"""
This function sums all elements of the input array.
Using a Numba njit decorator shows better performance
compared to python sum(), .sum() and np.sum()

Parameters
----------
input: numpy.ndarray
1D (blocksize) array containing data with 'float' type.

Returns
-------
float

"""
"""
Developer Note
-----
Faster than sum(), .sum() and np.sum()

For blocksize = 200

sum(): 36.9 µs ± 3.99 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

.sum(): 3.17 µs ± 90.1 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

np.sum(): 5.17 µs ± 364 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

This version: 513 ns ± 24.6 ns per loop (mean ± std. dev. of 7 runs, 1000000 loops each)
"""

output = 0.0
for i in range(input.shape):
output += input[i]

return output

@njit(cache=True)
def node_to_element_position(node_position_collection):
"""
This function computes the position of the elements
from the nodal values.
Here we define a separate function because benchmark results
showed that using Numba, we get more than 3 times faster calculation.

Parameters
----------
node_position_collection: numpy.ndarray
2D (dim, blocksize) array containing nodal positions with
'float' type.

Returns
-------
element_position_collection: numpy.ndarray
2D (dim, blocksize) array containing elemental positions with
'float' type.
"""
"""
Developer Notes
-----
Benchmark results, for a blocksize of 100,

Python version: 3.5 µs ± 149 ns per loop

This version: 729 ns ± 14.3 ns per loop

"""
n_elem = node_position_collection.shape - 1
element_position_collection = np.empty((3, n_elem))
for k in range(n_elem):
element_position_collection[0, k] = 0.5 * (
node_position_collection[0, k + 1] + node_position_collection[0, k]
)
element_position_collection[1, k] = 0.5 * (
node_position_collection[1, k + 1] + node_position_collection[1, k]
)
element_position_collection[2, k] = 0.5 * (
node_position_collection[2, k + 1] + node_position_collection[2, k]
)

return element_position_collection

@njit(cache=True)
def node_to_element_velocity(mass, node_velocity_collection):
"""
This function computes the velocity of the elements
from the nodal values. Uses the velocity of center of mass
in order to conserve momentum during computation.

Parameters
----------
mass: numpy.ndarray
2D (dim, blocksize) array containing nodal masses with
'float' type.
node_velocity_collection: numpy.ndarray
2D (dim, blocksize) array containing nodal velocities with
'float' type.

Returns
-------
element_velocity_collection: numpy.ndarray
2D (dim, blocksize) array containing elemental velocities with
'float' type.
"""
n_elem = node_velocity_collection.shape - 1
element_velocity_collection = np.empty((3, n_elem))
for k in range(n_elem):
element_velocity_collection[0, k] = (
mass[k + 1] * node_velocity_collection[0, k + 1]
+ mass[k] * node_velocity_collection[0, k]
)
element_velocity_collection[1, k] = (
mass[k + 1] * node_velocity_collection[1, k + 1]
+ mass[k] * node_velocity_collection[1, k]
)
element_velocity_collection[2, k] = (
mass[k + 1] * node_velocity_collection[2, k + 1]
+ mass[k] * node_velocity_collection[2, k]
)
element_velocity_collection[:, k] /= mass[k + 1] + mass[k]

return element_velocity_collection

def node_to_element_pos_or_vel(vector_in_node_frame):
# Remove the function beyond v0.4.0
raise NotImplementedError(
"This function is removed in v0.3.0. For node-to-element interpolation please use: \n"
"elastica.interaction.node_to_element_position() for rod position \n"
"elastica.interaction.node_to_element_velocity() for rod velocity. \n"
"For detail, refer to issue #80."
)

@njit(cache=True)
def slender_body_forces(
tangents, velocity_collection, dynamic_viscosity, lengths, radius, mass
):
r"""
This function computes hydrodynamic forces on a body using slender body theory.
The below implementation is from Eq. 4.13 in Gazzola et al. RSoS. (2018).

.. math::
F_{h}=\frac{-4\pi\mu}{\ln{(L/r)}}\left(\mathbf{I}-\frac{1}{2}\mathbf{t}^{\textrm{T}}\mathbf{t}\right)\mathbf{v}

Parameters
----------
tangents: numpy.ndarray
2D (dim, blocksize) array containing data with 'float' type.
Rod-like element tangent directions.
velocity_collection: numpy.ndarray
2D (dim, blocksize) array containing data with 'float' type.
Rod-like object velocity collection.
dynamic_viscosity: float
Dynamic viscosity of the fluid.
length: numpy.ndarray
1D (blocksize) array containing data with 'float' type.
Rod-like object element lengths.
1D (blocksize) array containing data with 'float' type.
Rod-like object element radius.
mass: numpy.ndarray
1D (blocksize) array containing data with 'float' type.
Rod-like object node mass.

Returns
-------
stokes_force: numpy.ndarray
2D (dim, blocksize) array containing data with 'float' type.
"""

"""
Developer Note
----
Faster than numpy einsum implementation for blocksize 100

numpy: 39.5 µs ± 6.78 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

this version: 3.91 µs ± 310 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
"""

f = np.empty((tangents.shape, tangents.shape))
total_length = sum_over_elements(lengths)
element_velocity = node_to_element_velocity(
mass=mass, node_velocity_collection=velocity_collection
)

for k in range(tangents.shape):
# compute the entries of tt. a[#][#] are the the
# entries of tt matrix
a11 = tangents[0, k] * tangents[0, k]
a12 = tangents[0, k] * tangents[1, k]
a13 = tangents[0, k] * tangents[2, k]

a21 = tangents[1, k] * tangents[0, k]
a22 = tangents[1, k] * tangents[1, k]
a23 = tangents[1, k] * tangents[2, k]

a31 = tangents[2, k] * tangents[0, k]
a32 = tangents[2, k] * tangents[1, k]
a33 = tangents[2, k] * tangents[2, k]

# factor = - 4*pi*mu/ln(L/r)
factor = (
-4.0
* np.pi
* dynamic_viscosity
/ np.log(total_length / radius[k])
* lengths[k]
)

# Fh = factor * ((I - 0.5 * a) * v)
f[0, k] = factor * (
(1.0 - 0.5 * a11) * element_velocity[0, k]
+ (0.0 - 0.5 * a12) * element_velocity[1, k]
+ (0.0 - 0.5 * a13) * element_velocity[2, k]
)
f[1, k] = factor * (
(0.0 - 0.5 * a21) * element_velocity[0, k]
+ (1.0 - 0.5 * a22) * element_velocity[1, k]
+ (0.0 - 0.5 * a23) * element_velocity[2, k]
)
f[2, k] = factor * (
(0.0 - 0.5 * a31) * element_velocity[0, k]
+ (0.0 - 0.5 * a32) * element_velocity[1, k]
+ (1.0 - 0.5 * a33) * element_velocity[2, k]
)

return f

# slender body theory
[docs]class SlenderBodyTheory(NoForces):
"""
This slender body theory class is for flow-structure
interaction problems. This class applies hydrodynamic
forces on the body using the slender body theory given in
Eq. 4.13 of Gazzola et al. RSoS (2018).

Attributes
----------
dynamic_viscosity: float
Dynamic viscosity of the fluid.

"""

[docs]    def __init__(self, dynamic_viscosity):
"""

Parameters
----------
dynamic_viscosity : float
Dynamic viscosity of the fluid.
"""
super(SlenderBodyTheory, self).__init__()
self.dynamic_viscosity = dynamic_viscosity

def apply_forces(self, system, time=0.0):
"""
This function applies hydrodynamic forces on body
using the slender body theory given in
Eq. 4.13 Gazzola et. al. RSoS 2018 paper

Parameters
----------
system

Returns
-------

"""

stokes_force = slender_body_forces(
system.tangents,
system.velocity_collection,
self.dynamic_viscosity,
system.lengths,
system.mass,
)
elements_to_nodes_inplace(stokes_force, system.external_forces)

# base class for interaction
# only applies normal force no friction
class InteractionPlaneRigidBody:
def __init__(self, k, nu, plane_origin, plane_normal):
self.k = k
self.nu = nu
self.plane_origin = plane_origin.reshape(3, 1)
self.plane_normal = plane_normal.reshape(3)
self.surface_tol = 1e-4

def apply_normal_force(self, system):
"""
This function computes the plane force response on the rigid body, in the
case of contact. Contact model given in Eqn 4.8 Gazzola et. al. RSoS 2018 paper
is used.
Parameters
----------
system

Returns
-------
magnitude of the plane response
"""
return apply_normal_force_numba_rigid_body(
self.plane_origin,
self.plane_normal,
self.surface_tol,
self.k,
self.nu,
system.length,
system.position_collection,
system.velocity_collection,
system.external_forces,
)

@njit(cache=True)
def apply_normal_force_numba_rigid_body(
plane_origin,
plane_normal,
surface_tol,
k,
nu,
length,
position_collection,
velocity_collection,
external_forces,
):

# Compute plane response force
# total_forces = system.internal_forces + system.external_forces
total_forces = external_forces
force_component_along_normal_direction = _batch_product_i_ik_to_k(
plane_normal, total_forces
)
forces_along_normal_direction = _batch_product_i_k_to_ik(
plane_normal, force_component_along_normal_direction
)
# If the total force component along the plane normal direction is greater than zero that means,
# total force is pushing rod away from the plane not towards the plane. Thus, response force
# applied by the surface has to be zero.
forces_along_normal_direction[
..., np.where(force_component_along_normal_direction > 0)
] = 0.0
# Compute response force on the element. Plane response force
# has to be away from the surface and towards the element. Thus
# multiply forces along normal direction with negative sign.
plane_response_force = -forces_along_normal_direction

# Elastic force response due to penetration
element_position = position_collection
distance_from_plane = _batch_product_i_ik_to_k(
plane_normal, (element_position - plane_origin)
)
plane_penetration = np.minimum(distance_from_plane - length / 2, 0.0)
elastic_force = -k * _batch_product_i_k_to_ik(plane_normal, plane_penetration)

# Damping force response due to velocity towards the plane
element_velocity = velocity_collection
normal_component_of_element_velocity = _batch_product_i_ik_to_k(
plane_normal, element_velocity
)
damping_force = -nu * _batch_product_i_k_to_ik(
plane_normal, normal_component_of_element_velocity
)

# Compute total plane response force
plane_response_force_total = plane_response_force + elastic_force + damping_force

# Check if the rigid body is in contact with plane.
no_contact_point_idx = np.where((distance_from_plane - length / 2) > surface_tol)
# If rod element does not have any contact with plane, plane cannot apply response
# force on the element. Thus lets set plane response force to 0.0 for the no contact points.
plane_response_force[..., no_contact_point_idx] = 0.0
plane_response_force_total[..., no_contact_point_idx] = 0.0

# Update the external forces
external_forces += plane_response_force_total

return (_batch_norm(plane_response_force), no_contact_point_idx)