10.1.2. pyopencl_scripts.rk4_functions¶
This module provides functions for simulating networks or particles using python.
10.1.2.1. Overview¶
10.1.2.2. Classes and Functions¶
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ilpm.pyopencl_scripts.rk4_functions.rk4_damp(xy, v, NL, KL, BM, Mm, beta, h)[source]¶ Performs Runge-Kutta 4th order time step increment for spring system with damping.
Parameters: x : array NP x nd
initial position (extension of spring)
v : array NP x nd
initial velocity
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
BM : array NP x nn
unstretched (reference) bond length matrix
beta : float
damping coefficient
h : float
time step
Returns
———-
xout,vout : arrays NP x nd
output positions and velocities
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ilpm.pyopencl_scripts.rk4_functions.rk4_damp_constvBC(xy, v, NL, KL, BM, Mm, beta, h, BND)[source]¶ Performs Runge-Kutta 4th order time step increment for spring system with damping, while moving boundary particles (BND index particles) at a prescribed velocity (prescribed in v, by v[BND])
Parameters: x : array NP x nd
initial position (extension of spring)
v : array NP x nd
initial velocity
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
BM : array NP x nn
unstretched (reference) bond length matrix
beta : float
damping coefficient
h : float
time step
Returns
———-
xout, vout : arrays NP x nd
output positions and velocities
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ilpm.pyopencl_scripts.rk4_functions.fdspring(xy, v, NL, KL, BM, Mm, beta)[source]¶ Computes dv/dt for massive particles connected by damped springs (in 2D or 3D).
Parameters: xy : matrix of dimensions Nx2 (N = number of masses/gyros)
Current positions of the masses/gyroscopes
v : array NP x spatial dimensions
velocity
NL : matrix of dimension N x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
KL : matrix of dimension N x (max number of neighbors)
Spring constant – like NL matrix, with spring const values (k) corresponding to a true connection while 0 signifies that there is not a connection
BM : NP x NN array or float
Rest bond lengths, arranged like KL
Mm : matrix of dimension Nx1
Masses of each particle
beta : float
damping coefficient
Returns: ftx : martix of dimension N x 2
dv/dt for all particles
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ilpm.pyopencl_scripts.rk4_functions.fspring(xy, NL, KL, BM, Mm)[source]¶ Computes dv/dt for massive particles connected by undamped springs (in 2D or 3D).
Parameters: t : float
simulation time
xy : matrix of dimensions Nx2 (N = number of masses/gyros)
Current positions of the masses/gyroscopes
f : function
function which calculates the right hand side of the differential equation. Equation of motion
NL : matrix of dimension N x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
KL : matrix of dimension N x (max number of neighbors)
Spring constant – like NL matrix, with spring const values (k) corresponding to a true connection while 0 signifies that there is not a connection
BM : NP x NN array or float
Rest bond lengths, arranged like KL
Mm : matrix of dimension Nx1
Masses of each particle
Returns: ftx : martix of dimension N x 2
dv/dt for all particles
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ilpm.pyopencl_scripts.rk4_functions.rk4_1stO(Xn, R, dt, f, Ni, Nk, spring, pin, b_l=1)[source]¶ Calculates the change in position for a lattice of gyroscopes according to 1st order eqn (‘Nash evolution’). calc_forces_and_cross(X,R, NL, KL, k=1., pin=1., bond_length = 1. )
Parameters: Xn : nx3 array (where n = number of gyros)
Current positions of the gyroscopes
R : n x 2 array
Rest positions of the gyroscopes
dt : float
simulation time step
f : function
function which calculates the right hand side of the differential equation. Equation of motion
Ni : matrix of dimension n x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
Nk : matrix of dimension n x (max number of neighbors)
Correponds to Ni matrix. 1 corresponds to a true connection while 0 signifies that there is not a connection
spring : float
spring constant
pin : float
gravitational spring constant
R : matrix of dimension nx3
Equilibrium positions of all the gyroscopes
Returns: dx1,dx2,dx3,dx4,xout : arrays of dimension nx3
displacements in this time step
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ilpm.pyopencl_scripts.rk4_functions.fglidingHST_exact(xy, v, NL, KL, BM, Mm, params)[source]¶ Increment velocity for lattice of gliding Heavy Symmetric Tops (HSTs)
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ilpm.pyopencl_scripts.rk4_functions.rk4_glidingHST_exact(xy, v, NL, KL, BM, Mm, params)[source]¶ Performs Runge-Kutta 4th order time step increment for gliding heavy symmetric top (Rutstam-like evolution)
Parameters: xy : array NP x 5
current position of COM of gyroscopes: x, y, theta, phi, psi
v : array NP x 5
current velocity of the pivot points: vx, vy, dtheta/dt, dphi/dt, dpsi/dt
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
BM : array NP x nn
unstretched (reference) bond length matrix
h : float
time step
Returns: dx1,dv1,dx2,dv2,dx3,dv3,dx4,dv4,xout,vout : arrays NP x nd
all parameters for integration
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ilpm.pyopencl_scripts.rk4_functions.fglidingHST_PL(xy, v, NL, KL, BM, Mm, params)[source]¶ Increment velocity for lattice of gliding Heavy Symmetric Tops (HSTs) in Planar Limit
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ilpm.pyopencl_scripts.rk4_functions.rk4_glidingHST_PL(xy, v, NL, KL, BM, Mm, params)[source]¶ Performs Runge-Kutta 4th order time step increment for gliding heavy symmetric top in Planar Limit (hanging gyros only) Note that b=0 here (hanging gyros)!
Parameters: xy : array NP x 4
current position of COM of gyroscopes: X, Y, deltaX, deltaY
v : array NP x 4
current velocity of the pivot points: vX, vY, vdeltaX, vdeltaY
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
BM : array NP x nn
unstretched (reference) bond length matrix
h : float
time step
Returns
———-
dx1,dv1,dx2,dv2,dx3,dv3,dx4,dv4,xout,vout : arrays NP x nd
all parameters for integration
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ilpm.pyopencl_scripts.rk4_functions.calc_forces_and_cross(X, R, NL, KL, k=1.0, pin=1.0, bond_length=1.0)[source]¶ Calculates the forces on gyroscopes from springs and pinning and does a cross product for EOM . Currently CANNOT handle inhomogeneous bond_lengths or spring constants. FIX THIS. CANNOT handle correctly the case when bond_length !=1 either. FIX THIS TOO.
Parameters: X : matrix of dimensions nx3 (n = number of gyros)
Current positions of the gyroscopes
R : matrix of dimension nx3
Equilibrium positions of all the gyroscopes
NL : matrix of dimension n x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
KL : matrix of dimension n x (max number of neighbors)
Correponds to Ni matrix. 1 corresponds to a true connection while 0 signifies that there is not a connection
k : float or NP x 1 array
spring constant array. Default value = 1
pin : float or NP array
gravitational spring constant = lmg/Iw. Default value = 1
bond_length : float
Length of unstretched springs. Default value = 1
Returns: crossed : matrix of dimension nx3
cross product of forces and zhat
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ilpm.pyopencl_scripts.rk4_functions.calc_forces_and_cross_magnetic(X, Ni, Nk, spring, pin, R, bond_length=1, out_index=1)[source]¶ Calculates the forces on gyroscopes from magnetic interactions and pinning and does a cross product for EOM .
Parameters: X : matrix of dimensions 2nx3 (n = number of gyros)
Current positions of the gyroscopes
Ni : matrix of dimension n x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
Nk : matrix of dimension n x (max number of neighbors)
Correponds to Ni matrix. 1 corresponds to a true connection while 0 signifies that there is not a connection
spring : float
spring constant
pin : float
gravitational spring constant
R : matrix of dimension 2nx3
Equilibrium positions of all the gyroscopes
bond_length : float
Length of unstretched springs. Default value = 1
Returns: crossed : matrix of dimension nx3
cross product of forces and zhat
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ilpm.pyopencl_scripts.rk4_functions.fspring_perp(xy, NL, KL, KGdivKL, Mm, NP, nn)[source]¶ Computes dv/dt for all particles in 2D at time t, with a perpendicular force between particles.
Parameters: t : float
simulation time
xy : matrix of dimensions Nx2 (N = number of masses/gyros)
Current positions of the masses/gyroscopes
f : function
function which calculates the right hand side of the differential equation. Equation of motion
NL : matrix of dimension N x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
KL : matrix of dimension N x (max number of neighbors)
Spring constant – like NL matrix, with spring const values (k) corresponding to a true connection while 0 signifies that there is not a connection
Mm : matrix of dimension Nx1
Masses of each particle
Returns: ftx : martix of dimension N x 2
dv/dt for all particles, in 2D
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ilpm.pyopencl_scripts.rk4_functions.rk4_perp(xy, v, NL, KL, KGdivKL, Mm, NP, nn, h)[source]¶ Performs Runge-Kutta 4th order time step increment for spring system with perpendicular force btwn particles.
Parameters: x : array NP x nd
initial position (extension of spring)
v : array NP x nd
initial velocity
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
h : float
time step
Returns
———-
dx1,dv1,dx2,dv2,dx3,dv3,dx4,dv4,xout,vout : arrays NP x nd
all parameters for integration
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ilpm.pyopencl_scripts.rk4_functions.fgyro_Nash(xy, xy0, NL, theta, OmK, Omg, Mm, NP, nn)[source]¶ Computes dx/dt for gyros in 2D with linearized approximation (Schrodinger-like).
Parameters: xy : matrix of dimensions Nx2 (N = number of masses/gyros)
Current positions of the masses/gyroscopes
xy0 : array NP x nd
initial position (extension of spring)
NL : matrix of dimension N x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
theta : array NP x nn
matrix of angles between particles
OmK : matrix of dimension N x (max number of neighbors)
Spring constant – like NL matrix, with spring const values (k l^2 /I omega) corresponding to a true connection while 0 signifies that there is not a connection
Mm : matrix of dimension Nx1
Masses of each particle
Returns: ftx : martix of dimension N x 2
dv/dt for all particles, in 2D
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ilpm.pyopencl_scripts.rk4_functions.fBrunGyro(xy, xy0, NL, KL, KG, Mm, NP, nn)[source]¶ Computes dv/dt for all gyros in 2D at time t in steady state frequency determined by KG=I omega^2/l^2, according to the model of Brun et al 2012.
Parameters: t : float
simulation time
xy : matrix of dimensions Nx2 (N = number of masses/gyros)
Current positions of the masses/gyroscopes
f : function
function which calculates the right hand side of the differential equation. Equation of motion
NL : matrix of dimension N x (max number of neighbors)
Each row corresponds to a gyroscope. The entries tell the numbers of the neighboring gyroscopes
KL : matrix of dimension N x (max number of neighbors)
Spring constant – like NL matrix, with spring const values (k) corresponding to a true connection while 0 signifies that there is not a connection
KG : array of dimension N x 1
Gyro constant ‘rotational force’
Mm : matrix of dimension Nx1
Masses of each particle
Returns: ftx : martix of dimension N x 2
dv/dt for all particles, in 2D
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ilpm.pyopencl_scripts.rk4_functions.rk4_BrunGyro(xy, xy0, v, NL, KL, KG, Mm, NP, nn, h)[source]¶ Performs Runge-Kutta 4th order time step increment for spring-gyro system in steady state frequency determined by KG=I omega^2/l^2, according to the model of Brun et al 2012.
Parameters: x : array NP x nd
initial position (extension of spring)
v : array NP x nd
initial velocity
NL : array NP x nn
ith row contains indices of neighbors for ith particle
KL : array NP x nn
bond strength for each bond (same format as NL)
h : float
time step
Returns
———-
dx1,dv1,dx2,dv2,dx3,dv3,dx4,dv4,xout,vout : arrays NP x nd
all parameters for integration
