def plot_traj3d(ax, sol, length, nskip=100):
sol = sol[::nskip]
x, y, z = pos_(sol).T
b = length / 2.
pt0 = pos_(sol)
pt1 = np.array(
[pos_(s) + pct.rot3d(*quat_(s)).dot([b, 0, 0]) for s in sol])
pt2 = np.array(
[pos_(s) + pct.rot3d(*quat_(s)).dot([-b, 0, 0]) for s in sol])
for (x0, y0, z0), (x1, y1, z1), (x2, y2, z2) in zip(pt0, pt1, pt2):
ax.plot([x0, x1], [y0, y1], 'r-', lw=2)
ax.plot([x0, x2], [y0, y2], 'g-', lw=2)
ax.plot(x, y, 'k:', lw=2)
return ax
def torque_dep3d(u, params):
R, (x, y, z) = pct.rot3d(*quat_(u)), pos_(u)
Kcm = np.diag([params.cmf_long, params.cmf_short, params.cmf_short])
E_123 = R.dot(params.elecf(x, y, z))
dip = params.vol * params.eps_f * Kcm.dot(E_123)
return np.cross(dip, E_123)
def torque_drag3d(u, params):
R, (x, y, z) = pct.rot3d(*quat_(u)), pos_(u)
L_xyz = pct.jac3d(params.vf, x, y, z)
L_123 = R.dot(L_xyz).dot(R.T) # velocity gradient
D_123 = (L_123 + L_123.T) / 2 # rate of deformation
W_123 = (L_123 - L_123.T) / 2 # rate of rotation
r, r2 = params.r, params.r2
if r > 1:
a0 = -2 / (r2 - 1) - r * log(
(r - sqrt(r2 - 1)) / (r + sqrt(r2 - 1))) / (r2 - 1)**1.5
b0 = c0 = r2 / (r2 - 1) + r * log(
(r - sqrt(r2 - 1)) / (r + sqrt(r2 - 1))) / (r2 - 1)**1.5 / 2
else:
a0 = b0 = c0 = 2 / 3.
omg1, omg2, omg3 = omg_(u)
k1 = 16 * np.pi * params.mu_f * params.a**3 * r / (b0 + c0) / 3.
k2 = 16 * np.pi * params.mu_f * params.a**3 * r / (c0 + r2 * a0) / 3.
k3 = 16 * np.pi * params.mu_f * params.a**3 * r / (r2 * a0 + b0) / 3.
t1 = 2 * (W_123[2, 1] - omg1)
t2 = (1 - r**2) * D_123[0, 2] + (1 + r**2) * (W_123[0, 2] - omg2)
t3 = (r**2 - 1) * D_123[1, 0] + (r**2 + 1) * (W_123[1, 0] - omg3)
return np.r_[k1 * t1, k2 * t2, k3 * t3]
def force_drag3d(u, params):
r, r2 = params.r, params.r2
if r > 1:
K11 = 8 * (r2 - 1) / (
(2 * r2 - 1) * log(r + sqrt(r2 - 1)) / sqrt(r2 - 1) - r)
K22 = K33 = 16 * (r2 - 1) / (
(2 * r2 - 3) * log(r + sqrt(r2 - 1)) / sqrt(r2 - 1) + r)
else:
K11 = K22 = K33 = 6
K_123 = params.mu_f * np.pi * params.a * np.diag([K11, K22, K33])
R, (x, y, z) = pct.rot3d(*quat_(u)), pos_(u)
return (R.T).dot(K_123).dot(R).dot(params.vf(x, y, z) - vel_(u))
def torque_drag3d_sph(u, params):
R, (x, y, z) = pct.rot3d(*quat_(u)), pos_(u)
L_xyz = pct.jac3d(params.vf, x, y, z)
L_123 = R.dot(L_xyz).dot(R.T) # velocity gradient
W_123 = (L_123 - L_123.T) / 2 # rate of rotation
a0 = b0 = c0 = 2 / 3.
k1 = k2 = k3 = 4 * np.pi * params.mu_f * params.a**3
omg1, omg2, omg3 = omg_(u)
t1 = 2 * (W_123[2, 1] - omg1)
t2 = 2 * (W_123[0, 2] - omg2)
t3 = 2 * (W_123[1, 0] - omg3)
return np.r_[k1 * t1, k2 * t2, k3 * t3]
def force_dep3d(u, params):
R, (x, y, z) = pct.rot3d(*quat_(u)), pos_(u)
Kcm = np.diag([params.cmf_long, params.cmf_short, params.cmf_short])
dip = params.vol * params.eps_f * Kcm.dot(R).dot(params.elecf(x, y, z))
return pct.jac3d(params.elecf, x, y, z).dot(R.T).dot(dip)