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(self, inputs, outputs):
P0 = inputs['nodes'][:-1, :]
P1 = inputs['nodes'][1:, :]
norm = compute_norm(P1 - P0)
row0 = (P1 - P0) / norm
cross = compute_cross(row0, self.ref_axis)
norm = compute_norm(cross)
row1 = cross / norm
cross = compute_cross(row0, row1)
row2 = cross
outputs['transform'] = 0.
for k in range(4):
outputs['transform'][:, 3 * k + 0, 3 * k:3 * k + 3] = row0
outputs['transform'][:, 3 * k + 1, 3 * k:3 * k + 3] = row1
outputs['transform'][:, 3 * k + 2, 3 * k:3 * k + 3] = row2
def compute(self, inputs, outputs):
P0 = inputs['nodes'][:-1, :]
P1 = inputs['nodes'][ 1:, :]
norm = compute_norm(P1 - P0)
row0 = (P1 - P0) / norm
cross = compute_cross(row0, self.ref_axis)
norm = compute_norm(cross)
row1 = cross / norm
cross = compute_cross(row0, row1)
row2 = cross
outputs['transform'] = 0.
for k in range(4):
outputs['transform'][:, 3*k + 0, 3*k : 3*k + 3] = row0
outputs['transform'][:, 3*k + 1, 3*k : 3*k + 3] = row1
outputs['transform'][:, 3*k + 2, 3*k : 3*k + 3] = row2
def compute(self, inputs, outputs):
rho = inputs['rho'][0]
horseshoe_circulations = np.outer(inputs['horseshoe_circulations'], np.ones(3))
velocities = inputs['force_pts_velocities']
bound_vecs = inputs['bound_vecs']
# Actually compute the forces by taking the cross of velocities acting
# at the force points with the bound vortex filament vector.
outputs['panel_forces'] = \
rho * horseshoe_circulations * compute_cross(velocities, bound_vecs)
def compute_partials(self, inputs, partials):
rho = inputs['rho'][0]
horseshoe_circulations = np.outer(inputs['horseshoe_circulations'], np.ones(3))
velocities = inputs['force_pts_velocities']
bound_vecs = inputs['bound_vecs']
horseshoe_circulations_ones = np.einsum('i,jk->ijk', inputs['horseshoe_circulations'], np.ones((3, 3)))
deriv_array = np.einsum('i,jk->ijk',
np.ones(self.system_size),
np.eye(3))
partials['panel_forces', 'rho'] = \
(horseshoe_circulations * compute_cross(velocities, bound_vecs)).flatten()
partials['panel_forces', 'horseshoe_circulations'] = \
(rho * compute_cross(velocities, bound_vecs)).flatten()
partials['panel_forces', 'force_pts_velocities'] = \
(rho * horseshoe_circulations_ones * compute_cross_deriv1(deriv_array, bound_vecs)).flatten()
partials['panel_forces', 'bound_vecs'] = \
(rho * horseshoe_circulations_ones * compute_cross_deriv2(velocities, deriv_array)).flatten()
def compute(self, inputs, outputs):
system_size = self.system_size
rho = inputs['rho'][0]
horseshoe_circulations = np.outer(inputs['horseshoe_circulations'], np.ones(3))
velocities = inputs['force_pts_velocities']
bound_vecs = inputs['bound_vecs']
# Actually compute the forces by taking the cross of velocities acting
# at the force points with the bound vortex filament vector.
outputs['panel_forces'] = \
rho * horseshoe_circulations * compute_cross(velocities, bound_vecs)
def compute_partials(self, inputs, partials):
system_size = self.system_size
rho = inputs['rho'][0]
horseshoe_circulations = np.outer(inputs['horseshoe_circulations'], np.ones(3))
velocities = inputs['force_pts_velocities']
bound_vecs = inputs['bound_vecs']
horseshoe_circulations_ones = np.einsum('i,jk->ijk', inputs['horseshoe_circulations'], np.ones((3, 3)))
deriv_array = np.einsum('i,jk->ijk',
np.ones(self.system_size),
np.eye(3))
partials['panel_forces', 'rho'] = \
(horseshoe_circulations * compute_cross(velocities, bound_vecs)).flatten()
partials['panel_forces', 'horseshoe_circulations'] = \
(rho * compute_cross(velocities, bound_vecs)).flatten()
partials['panel_forces', 'force_pts_velocities'] = \
(rho * horseshoe_circulations_ones * compute_cross_deriv1(deriv_array, bound_vecs)).flatten()
partials['panel_forces', 'bound_vecs'] = \
(rho * horseshoe_circulations_ones * compute_cross_deriv2(velocities, deriv_array)).flatten()
def compute_partials(self, inputs, partials):
surface = self.options['surface']
ny = surface['mesh'].shape[1]
P0 = inputs['nodes'][:-1, :]
P1 = inputs['nodes'][ 1:, :]
P_deriv = np.einsum('i,jk->ijk', np.ones(ny - 1), np.eye(3))
norm = compute_norm(P1 - P0)
norm_deriv = compute_norm_deriv(P1 - P0, P_deriv)
row0 = (P1 - P0) / norm
row0_deriv = P_deriv / add_ones_axis(norm) - add_ones_axis(P1 - P0) / add_ones_axis(norm) ** 2 * norm_deriv
cross = compute_cross(row0, self.ref_axis)
cross_deriv = compute_cross_deriv1(row0_deriv, self.ref_axis)
norm = compute_norm(cross)
norm_deriv = compute_norm_deriv(cross, cross_deriv)
row1 = cross / norm
row1_deriv = cross_deriv / add_ones_axis(norm) - add_ones_axis(cross) / add_ones_axis(norm) ** 2 * norm_deriv
cross = compute_cross(row0, row1)
cross_deriv = (
compute_cross_deriv1(row0_deriv, row1) +
compute_cross_deriv2(row0, row1_deriv)
)
row2 = cross
row2_deriv = cross_deriv
derivs = partials['transform', 'nodes'].reshape((2, ny - 1, 12, 12, 3))
for k in range(4):
derivs[0, :, 3*k + 0, 3*k : 3*k + 3, :] = -row0_deriv
derivs[1, :, 3*k + 0, 3*k : 3*k + 3, :] = row0_deriv
derivs[0, :, 3*k + 1, 3*k : 3*k + 3, :] = -row1_deriv
derivs[1, :, 3*k + 1, 3*k : 3*k + 3, :] = row1_deriv
derivs[0, :, 3*k + 2, 3*k : 3*k + 3, :] = -row2_deriv
derivs[1, :, 3*k + 2, 3*k : 3*k + 3, :] = row2_deriv
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_partials(self, inputs, partials):
surface = self.options['surface']
ny = surface['mesh'].shape[1]
P0 = inputs['nodes'][:-1, :]
P1 = inputs['nodes'][1:, :]
P_deriv = np.einsum('i,jk->ijk', np.ones(ny - 1), np.eye(3))
norm = compute_norm(P1 - P0)
norm_deriv = compute_norm_deriv(P1 - P0, P_deriv)
row0 = (P1 - P0) / norm
row0_deriv = P_deriv / add_ones_axis(norm) - add_ones_axis(
P1 - P0) / add_ones_axis(norm)**2 * norm_deriv
cross = compute_cross(row0, self.ref_axis)
cross_deriv = compute_cross_deriv1(row0_deriv, self.ref_axis)
norm = compute_norm(cross)
norm_deriv = compute_norm_deriv(cross, cross_deriv)
row1 = cross / norm
row1_deriv = cross_deriv / add_ones_axis(norm) - add_ones_axis(
cross) / add_ones_axis(norm)**2 * norm_deriv
cross = compute_cross(row0, row1)
cross_deriv = (compute_cross_deriv1(row0_deriv, row1) +
compute_cross_deriv2(row0, row1_deriv))
row2 = cross
row2_deriv = cross_deriv
derivs = partials['transform', 'nodes'].reshape((2, ny - 1, 12, 12, 3))
for k in range(4):
derivs[0, :, 3 * k + 0, 3 * k:3 * k + 3, :] = -row0_deriv
derivs[1, :, 3 * k + 0, 3 * k:3 * k + 3, :] = row0_deriv
derivs[0, :, 3 * k + 1, 3 * k:3 * k + 3, :] = -row1_deriv
derivs[1, :, 3 * k + 1, 3 * k:3 * k + 3, :] = row1_deriv
derivs[0, :, 3 * k + 2, 3 * k:3 * k + 3, :] = -row2_deriv
derivs[1, :, 3 * k + 2, 3 * k:3 * k + 3, :] = row2_deriv
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