def derivative_R(coord, connect, E, v, rho, alpha, beta, omega, p_par, q_par,
x_min_k, x_min_m, xval, disp_vector, lam, fvirg, elastic_p,
kinetic_e):
""" calculates the derivative of the kinetic energy function.
Args:
coord (:obj:`numpy.array`): Coordinates of the element.
connect (:obj:`numpy.array`): Element connectivity.
E (:obj:`float`): Elastic modulus.
v (:obj:`float`): Poisson's ratio.
rho (:obj:`float`): Density.
alpha (:obj:`float`): Damping coefficient proportional to mass.
beta (:obj:`float`): Damping coefficient proportional to stiffness.
omega (:obj:`float`): 2 * pi * frequency
p_par (:obj:`float`): Penalization power to stiffness.
q_par (:obj:`float`): Penalization power to mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
x_min_m (:obj:`float`): Minimum relative densities to mass.
xval (:obj:`numpy.array`): Indicates where there is mass.
disp_vector (:obj:`numpy.array`): Displacement.
lam (:obj:`float`): Lambda parameter.
fvirg (:obj:`float`): R Ratio function value.
elastic_p (:obj:`float`): Elastic potential energy function value.
kinetic_e (:obj:`float`): Kinetic energy function value.
Returns:
Derivative values.
"""
deriv_R = np.zeros((len(connect), 1), dtype=complex)
dofs = 2
ind_dofs = (np.array([
dofs * connect[:, 1] - 1, dofs * connect[:, 1], dofs * connect[:, 2] -
1, dofs * connect[:, 2], dofs * connect[:, 3] - 1,
dofs * connect[:, 3], dofs * connect[:, 4] - 1, dofs * connect[:, 4]
],
dtype=int) - 1).T
for el in range(len(connect)):
Ke, Me = fc.matricesQ4(el, coord, connect, E, v, rho)
ind = ind_dofs[el, :]
dKe = p_par * (xval[el]**(p_par - 1)) * (1 - x_min_k) * Ke
dCe = alpha * Me + beta * dKe
if xval[el] > 0.1:
dMe = q_par * (xval[el]**(q_par - 1)) * (1 - x_min_m) * Me
else:
dMe = ((9 * 3.512e7 * xval[el]**8 - 10 * 2.081e8 * xval[el]**9) *
(1 - x_min_m)) * Me
dKed = dKe + omega * 1j * dCe - (omega**2) * dMe
#
deriv_R[el,
0] = 1 / (4 * kinetic_e) * (disp_vector[ind].conjugate() @ (
dKe -
(omega**2) * fvirg * dMe) @ disp_vector[ind]).real + (
lam[ind] @ dKed @ disp_vector[ind]).real
#Log Scale
deriv_R[el, 0] = 10.0 * deriv_R[el, 0] * np.log10(np.exp(1)) / fvirg
return deriv_R.real
def solution2D(coord, connect, ind_rows, ind_cols, nelx, nely, ngl, E, v, rho,
alpha, beta, eta, omega, xval, x_min_k, x_min_m, p_par, q_par):
""" Assembly matrices.
Args:
coord (:obj:`numpy.array`): Coordinates of the element.
connect (:obj:`numpy.array`): Element connectivity.
ind_rows (:obj:`numpy.array`): Node indexes to make the assembly.
ind_cols (:obj:`numpy.array`): Node indexes to make the assembly.
nelx (:obj:`int`): Number of elements on the X-axis.
nely (:obj:`int`): Number of elements on the Y-axis.
ngl (:obj:`int`): Degrees of freedom.
E (:obj:`float`): Elastic modulus.
v (:obj:`float`): Poisson's ratio.
rho (:obj:`float`): Density.
alpha (:obj:`float`): Damping coefficient proportional to mass.
beta (:obj:`float`): Damping coefficient proportional to stiffness.
eta (:obj:`float`): Damping coefficient.
omega (:obj:`float`): 2 pi frequency
xval (:obj:`numpy.array`): Indicates where there is mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
x_min_m (:obj:`float`): Minimum relative densities to mass.
p_par (:obj:`int`): Penalization power to stiffness.
q_par (:obj:`int`): Penalization power to mass.
Returns:
A tuple with stiffnes, mass and dynamic matrices and time to assembly the matrices.
"""
#
t01 = time()
data_k = np.zeros((nelx * nely, 64), dtype=complex)
data_m = np.zeros((nelx * nely, 64), dtype=complex)
#
for el in range(nelx * nely):
Ke, Me = fc.matricesQ4(el, coord, connect, E, v, rho)
data_k[el, :] = (x_min_k + (xval[el]**p_par) *
(1 - x_min_k)) * Ke.flatten()
if xval[el] > 0.1:
data_m[el, :] = (x_min_m + (xval[el]**q_par) *
(1 - x_min_m)) * Me.flatten()
else:
data_m[el, :] = (x_min_m +
(3.512e7 * xval[el]**9 - 2.081e8 * xval[el]**10) *
(1 - x_min_m)) * Me.flatten()
#
data_k = data_k.flatten()
data_m = data_m.flatten()
stif_matrix = csc_matrix((data_k, (ind_rows, ind_cols)), shape=(ngl, ngl))
mass_matrix = csc_matrix((data_m, (ind_rows, ind_cols)), shape=(ngl, ngl))
damp_matrix = alpha * mass_matrix + (beta) * stif_matrix
dyna_stif = -(omega**
2) * mass_matrix + 1j * omega * damp_matrix + stif_matrix
tf1 = time()
t_assembly = str(round((tf1 - t01), 6))
return stif_matrix, mass_matrix, dyna_stif, t_assembly
def derivative_input_power(coord, connect, E, v, rho, alpha, beta, omega,
p_par, q_par, x_min_k, x_min_m, xval, disp_vector,
fvirg):
""" calculates the derivative of the input power function.
Args:
coord (:obj:`numpy.array`): Coordinates of the element.
connect (:obj:`numpy.array`): Element connectivity.
E (:obj:`float`): Elastic modulus.
v (:obj:`float`): Poisson's ratio.
rho (:obj:`float`): Density.
alpha (:obj:`float`): Damping coefficient proportional to mass.
beta (:obj:`float`): Damping coefficient proportional to stiffness.
omega (:obj:`float`): 2 * pi * frequency
p_par (:obj:`float`): Penalization power to stiffness.
q_par (:obj:`float`): Penalization power to mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
x_min_m (:obj:`float`): Minimum relative densities to mass.
xval (:obj:`numpy.array`): Indicates where there is mass.
disp_vector (:obj:`numpy.array`): Displacement.
fvirg (:obj:`float`): Input power function value.
Returns:
Derivative values.
"""
deriv_f = np.zeros((len(connect), 1))
dofs = 2
ind_dofs = (np.array([
dofs * connect[:, 1] - 1, dofs * connect[:, 1], dofs * connect[:, 2] -
1, dofs * connect[:, 2], dofs * connect[:, 3] - 1,
dofs * connect[:, 3], dofs * connect[:, 4] - 1, dofs * connect[:, 4]
],
dtype=int) - 1).T
for el in range(len(connect)):
Ke, Me = fc.matricesQ4(el, coord, connect, E, v, rho)
ind = ind_dofs[el, :]
dKe = p_par * (xval[el]**(p_par - 1)) * (1 - x_min_k) * Ke
dCe = alpha * Me + beta * dKe
if xval[el] > 0.1:
dMe = q_par * (xval[el]**(q_par - 1)) * (1 - x_min_m) * Me
else:
dMe = ((9 * 3.512e7 * xval[el]**8 - 10 * 2.081e8 * xval[el]**9) *
(1 - x_min_m)) * Me
dKed = dKe + omega * 1j * dCe - (omega**2) * dMe
a = 1j * (disp_vector[ind].reshape(1, 8) @ dKed
@ disp_vector[ind].reshape(8, 1))[0, 0]
deriv_f[el, 0] = -0.5 * omega * a.real
#Log Scale
deriv_f[el, 0] = 10.0 * deriv_f[el, 0] * np.log10(np.exp(1)) / fvirg
return deriv_f