def _compute_finite_vortex_deriv2(r1, r2, r2_deriv):
r1_norm = add_ones_axis(compute_norm(r1))
r2_norm = add_ones_axis(compute_norm(r2))
r2_norm_deriv = compute_norm_deriv(r2, r2_deriv)
r1_x_r2 = add_ones_axis(compute_cross(r1, r2))
r1_d_r2 = add_ones_axis(compute_dot(r1, r2))
r1_x_r2_deriv = compute_cross_deriv2(r1, r2_deriv)
r1_d_r2_deriv = compute_dot_deriv(r1, r2_deriv)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
num_deriv = (-r2_norm_deriv / r2_norm ** 2) * r1_x_r2 \
+ (1. / r1_norm + 1. / r2_norm) * r1_x_r2_deriv
den = r1_norm * r2_norm + r1_d_r2
den_deriv = r1_norm * r2_norm_deriv + r1_d_r2_deriv
result = np.divide(
num_deriv * den - num * den_deriv,
den ** 2 * 4 * np.pi,
out=np.zeros_like(num),
where=np.abs(den)>tol
)
return result
def _compute_semi_infinite_vortex(u, r):
r_norm = compute_norm(r)
u_x_r = compute_cross(u, r)
u_d_r = compute_dot(u, r)
num = u_x_r
den = r_norm * (r_norm - u_d_r)
return num / den / 4 / np.pi
def _compute_semi_infinite_vortex(u, r):
r_norm = compute_norm(r)
u_x_r = compute_cross(u, r)
u_d_r = compute_dot(u, r)
num = u_x_r
den = r_norm * (r_norm - u_d_r)
return num / den / 4 / np.pi
def _compute_finite_vortex(r1, r2):
r1_norm = compute_norm(r1)
r2_norm = compute_norm(r2)
r1_x_r2 = compute_cross(r1, r2)
r1_d_r2 = compute_dot(r1, r2)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
den = r1_norm * r2_norm + r1_d_r2
result = num / den / 4 / np.pi
result[np.abs(den) < tol] = 0.
return result
def _compute_finite_vortex(r1, r2):
r1_norm = compute_norm(r1)
r2_norm = compute_norm(r2)
r1_x_r2 = compute_cross(r1, r2)
r1_d_r2 = compute_dot(r1, r2)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
den = r1_norm * r2_norm + r1_d_r2
result = np.divide(num, den * 4 * np.pi, out=np.zeros_like(num), where=np.abs(den)>tol)
return result
def _compute_finite_vortex(r1, r2):
r1_norm = compute_norm(r1)
r2_norm = compute_norm(r2)
r1_x_r2 = compute_cross(r1, r2)
r1_d_r2 = compute_dot(r1, r2)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
den = r1_norm * r2_norm + r1_d_r2
result = num / den / 4 / np.pi
result[np.abs(den) < tol] = 0.
return result
def _compute_semi_infinite_vortex_deriv(u, r, r_deriv):
r_norm = add_ones_axis(compute_norm(r))
r_norm_deriv = compute_norm_deriv(r, r_deriv)
u_x_r = add_ones_axis(compute_cross(u, r))
u_x_r_deriv = compute_cross_deriv2(u, r_deriv)
u_d_r = add_ones_axis(compute_dot(u, r))
u_d_r_deriv = compute_dot_deriv(u, r_deriv)
num = u_x_r
num_deriv = u_x_r_deriv
den = r_norm * (r_norm - u_d_r)
den_deriv = r_norm_deriv * (r_norm - u_d_r) + r_norm * (r_norm_deriv - u_d_r_deriv)
return (num_deriv * den - num * den_deriv) / den ** 2 / 4 / np.pi
def _compute_semi_infinite_vortex_deriv(u, r, r_deriv):
r_norm = add_ones_axis(compute_norm(r))
r_norm_deriv = compute_norm_deriv(r, r_deriv)
u_x_r = add_ones_axis(compute_cross(u, r))
u_x_r_deriv = compute_cross_deriv2(u, r_deriv)
u_d_r = add_ones_axis(compute_dot(u, r))
u_d_r_deriv = compute_dot_deriv(u, r_deriv)
num = u_x_r
num_deriv = u_x_r_deriv
den = r_norm * (r_norm - u_d_r)
den_deriv = r_norm_deriv * (r_norm - u_d_r) + r_norm * (r_norm_deriv - u_d_r_deriv)
return (num_deriv * den - num * den_deriv) / den ** 2 / 4 / np.pi
def _compute_finite_vortex_deriv2(r1, r2, r2_deriv):
r1_norm = add_ones_axis(compute_norm(r1))
r2_norm = add_ones_axis(compute_norm(r2))
r2_norm_deriv = compute_norm_deriv(r2, r2_deriv)
r1_x_r2 = add_ones_axis(compute_cross(r1, r2))
r1_d_r2 = add_ones_axis(compute_dot(r1, r2))
r1_x_r2_deriv = compute_cross_deriv2(r1, r2_deriv)
r1_d_r2_deriv = compute_dot_deriv(r1, r2_deriv)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
num_deriv = (-r2_norm_deriv / r2_norm ** 2) * r1_x_r2 \
+ (1. / r1_norm + 1. / r2_norm) * r1_x_r2_deriv
den = r1_norm * r2_norm + r1_d_r2
den_deriv = r1_norm * r2_norm_deriv + r1_d_r2_deriv
result = (num_deriv * den - num * den_deriv) / den**2 / 4 / np.pi
result[np.abs(den) < tol] = 0.
return result
def _compute_finite_vortex_deriv2(r1, r2, r2_deriv):
r1_norm = add_ones_axis(compute_norm(r1))
r2_norm = add_ones_axis(compute_norm(r2))
r2_norm_deriv = compute_norm_deriv(r2, r2_deriv)
r1_x_r2 = add_ones_axis(compute_cross(r1, r2))
r1_d_r2 = add_ones_axis(compute_dot(r1, r2))
r1_x_r2_deriv = compute_cross_deriv2(r1, r2_deriv)
r1_d_r2_deriv = compute_dot_deriv(r1, r2_deriv)
num = (1. / r1_norm + 1. / r2_norm) * r1_x_r2
num_deriv = (-r2_norm_deriv / r2_norm ** 2) * r1_x_r2 \
+ (1. / r1_norm + 1. / r2_norm) * r1_x_r2_deriv
den = r1_norm * r2_norm + r1_d_r2
den_deriv = r1_norm * r2_norm_deriv + r1_d_r2_deriv
result = (num_deriv * den - num * den_deriv) / den ** 2 / 4 / np.pi
result[np.abs(den) < tol] = 0.
return result
