def kinetic_local(omega_par, disp_vector, passive_el, ind_passive, coord, connect, E, v, rho, const_func):
""" Calculates the local kinetic energy function.
Args:
omega_par (:obj:`float`): 2 * pi * frequency.
disp_vector (:obj:`numpy.array`): Displacement.
passive_el (:obj:`numpy.array`, optional): Passive element nodes.
ind_passive (:obj:`numpy.array`, optional): Index of passive elements.
coord (:obj:`numpy.array`, optional): Coordinates of the element.
connect (:obj:`numpy.array`, optional): Element connectivity.
E (:obj:`float`, optional): Elastic modulus.
v (:obj:`float`, optional): Poisson's ratio.
rho (:obj:`float`, optional): Density.
Returns:
Local kinetic energy on the logarithmic scale and the non-logarithmic local kinetic energy.
"""
ki = 0
for i, ind_el in enumerate(ind_passive):
_, Me = fc.matricesQ4(passive_el[i], coord, connect, E, v, rho)
aux = disp_vector[ind_el].conj().reshape(1, -1)@Me@disp_vector[ind_el]
ki+=aux
fvirg = ((omega_par**2)/4) * ki[0].real
#Log Scale
f = const_func + 10 * np.log10(fvirg)
return f, fvirg
def elastic_potential_local(disp_vector, passive_el, ind_passive, coord, connect, E, v, rho, const_func):
""" Calculates the local elastic potential energy function.
Args:
disp_vector (:obj:`numpy.array`): Displacement.
passive_el (:obj:`numpy.array`, optional): Passive element nodes.
ind_passive (:obj:`numpy.array`, optional): Index of passive elements.
coord (:obj:`numpy.array`, optional): Coordinates of the element.
connect (:obj:`numpy.array`, optional): Element connectivity.
E (:obj:`float`, optional): Elastic modulus.
v (:obj:`float`, optional): Poisson's ratio.
rho (:obj:`float`, optional): Density.
Returns:
Local elastic potential energy on the logarithmic scale and the non-logarithmic local elastic potential energy.
"""
ep2 = 0
for i, ind_el in enumerate(ind_passive):
Ke, _ = fc.matricesQ4(passive_el[i], coord, connect, E, v, rho)
aux = disp_vector[ind_el].reshape(1, -1).conjugate()@Ke@disp_vector[ind_el]
ep2+=aux
fvirg = (1/4) * ep2[0].real
#Log Scale
f = const_func + 10 * np.log10(fvirg)
return f, fvirg
def lambda_local_ep(ngl, ind_passive, passive_el, disp_vector, dyna_stif,
coord, connect, E, v, rho):
""" Calculates the lambda parameter of the local elastic potential energy function.
Args:
ngl (:obj:`int`): Degrees of freedom.
ind_passive (:obj:`numpy.array`): Index of passive elements.
passive_el (:obj:`numpy.array`): Passive element nodes.
disp_vector (:obj:`numpy.array`): Displacement vector.
dyna_stif (:obj:`numpy.array`): Dynamic stiffness matrix.
omega_par (:obj:`float`): 2 * pi * frequency.
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.
Returns:
Lambda parameter solution.
"""
aux1 = np.zeros(ngl, dtype=complex)
fadj = 0
for i, el in enumerate(passive_el):
Ke, _ = fc.matricesQ4(el, coord, connect, E, v, rho)
aux1[ind_passive[i]] = Ke @ disp_vector[ind_passive[i]].conjugate()
fadj += aux1
aux1[:] = 0
fadj *= -1 / 2
lam = spsolve(dyna_stif, fadj)
return lam
def derivative_local_ep(passive_el, lam, ind_dofs, xval, disp_vector, connect,
coord, E, v, rho, x_min_k, x_min_m, omega_par, alpha,
beta, p_par, q_par):
""" Calculates the derivative of the local elastic potential energy function.
Args:
passive_el (:obj:`numpy.array`): Passive element nodes.
lam (:obj:`float`): Lambda parameter.
ind_dofs (:obj:`numpy.array`, optional): TODO
xval (:obj:`numpy.array`): Indicates where there is mass.
disp_vector (:obj:`numpy.array`): Displacement vector.
connect (:obj:`numpy.array`): Element connectivity.
coord (:obj:`numpy.array`): Coordinates of the element.
E (:obj:`float`): Elastic modulus.
v (:obj:`float`): Poisson's ratio.
rho (:obj:`float`): Density.
x_min_m (:obj:`float`): Minimum relative densities to mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
omega_par (:obj:`float`): 2 * pi * frequency.
alpha (:obj:`float`): Damping coefficient proportional to mass.
beta (:obj:`float`): Damping coefficient proportional to stiffness.
p_par (:obj:`int`): Penalization power to stiffness.
q_par (:obj:`int`): Penalization power to mass.
Returns:
Derivative of the local elastic potential energy function.
"""
deriv_f = np.empty((len(connect), 1), dtype=complex)
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_par * 1j * dCe - (omega_par**2) * dMe
if el in passive_el:
deriv_f[el, 0] = (1 / 4) * (
(disp_vector[ind].reshape(1, -1).conjugate() @ dKe
@ disp_vector[ind]) +
(lam[ind].reshape(1, -1) @ dKed @ disp_vector[ind]).real)[0]
else:
deriv_f[el, 0] = ((
lam[ind].reshape(1, -1) @ dKed @ disp_vector[ind]).real)[0]
return deriv_f
def derivative_R(coord, connect, E, v, rho, alpha, beta, omega_par, p_par,
q_par, x_min_m, x_min_k, xval, disp_vector, lam, fvirg,
kinetic_e):
""" Calculates the derivative of the strain-to-kinetic 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_par (:obj:`float`): 2 * pi * frequency
p_par (:obj:`float`): Penalization power to stiffness.
q_par (:obj:`float`): Penalization power to mass.
x_min_m (:obj:`float`): Minimum relative densities to mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
xval (:obj:`numpy.array`): Indicates where there is mass.
disp_vector (:obj:`numpy.array`): Displacement vector.
lam (:obj:`float`): Lambda parameter.
fvirg (:obj:`float`): Strain-to-kinetic function.
kinetic_e (:obj:`float`): Kinetic energy function.
Returns:
Derivative of the strain-to-kinetic function function.
"""
deriv_R = np.empty((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_par * 1j * dCe - (omega_par**2) * dMe
deriv_R[el, 0] = 1/(4*kinetic_e) * (disp_vector[ind].conjugate()@(dKe - (omega_par**2)*fvirg*dMe)@disp_vector[ind]).real + \
(lam[ind]@dKed@disp_vector[ind]).real
return deriv_R
def derivative_input_power(coord, connect, E, v, rho, alpha, beta, omega_par,
p_par, q_par, x_min_m, x_min_k, xval, disp_vector):
""" 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_par (:obj:`float`): 2 * pi * frequency
p_par (:obj:`float`): Penalization power to stiffness.
q_par (:obj:`float`): Penalization power to mass.
x_min_m (:obj:`float`): Minimum relative densities to mass.
x_min_k (:obj:`float`): Minimum relative densities to stiffness.
xval (:obj:`numpy.array`): Indicates where there is mass.
disp_vector (:obj:`numpy.array`): Displacement vector.
Returns:
Derivative of the input power function.
"""
deriv_f = np.empty((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_par * 1j * dCe - (omega_par**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_par * a.real
return deriv_f