k = 0.01 # stiffness
M = sps.eye(n, format='csc')
E = d * sps.eye(n, format='csc')
K = sps.diags([n * [2 * k * n ** 2],
(n - 1) * [-k * n ** 2],
(n - 1) * [-k * n ** 2]],
[0, -1, 1],
format='csc')
B = np.zeros((n, 1))
B[n2 - 1, 0] = n
Cp = np.zeros((1, n))
Cp[0, n2 - 1] = 1
# Second-order system
so_sys = SecondOrderModel.from_matrices(M, E, K, B, Cp)
print(f'order of the model = {so_sys.order}')
print(f'number of inputs = {so_sys.input_dim}')
print(f'number of outputs = {so_sys.output_dim}')
poles = so_sys.poles()
fig, ax = plt.subplots()
ax.plot(poles.real, poles.imag, '.')
ax.set_title('System poles')
plt.show()
w = np.logspace(-4, 2, 200)
fig, ax = plt.subplots()
so_sys.mag_plot(w, ax=ax)
ax.set_title('Bode plot of the full model')
d = 10 # damping
k = 0.01 # stiffness
M = sps.eye(n, format='csc')
E = d * sps.eye(n, format='csc')
K = sps.diags(
[n * [2 * k * n**2], (n - 1) * [-k * n**2],
(n - 1) * [-k * n**2]], [0, -1, 1],
format='csc')
B = np.zeros((n, 1))
B[n2 - 1, 0] = n
Cp = np.zeros((1, n))
Cp[0, n2 - 1] = 1
# Second-order system
so_sys = SecondOrderModel.from_matrices(M, E, K, B, Cp)
print(f'order of the model = {so_sys.order}')
print(f'number of inputs = {so_sys.input_dim}')
print(f'number of outputs = {so_sys.output_dim}')
poles = so_sys.poles()
fig, ax = plt.subplots()
ax.plot(poles.real, poles.imag, '.')
ax.set_title('System poles')
plt.show()
w = np.logspace(-4, 2, 200)
fig, ax = plt.subplots()
so_sys.mag_plot(w, ax=ax)
ax.set_title('Magnitude plot of the full model')
def main(
n: int = Argument(
101, help='Order of the full second-order model (odd number).'),
r: int = Argument(5, help='Order of the ROMs.'),
):
"""String equation example."""
set_log_levels({'pymor.algorithms.gram_schmidt.gram_schmidt': 'ERROR'})
# Assemble matrices
assert n % 2 == 1, 'The order has to be an odd integer.'
n2 = (n + 1) // 2
d = 10 # damping
k = 0.01 # stiffness
M = sps.eye(n, format='csc')
E = d * sps.eye(n, format='csc')
K = sps.diags(
[n * [2 * k * n**2], (n - 1) * [-k * n**2],
(n - 1) * [-k * n**2]], [0, -1, 1],
format='csc')
B = np.zeros((n, 1))
B[n2 - 1, 0] = n
Cp = np.zeros((1, n))
Cp[0, n2 - 1] = 1
# Second-order system
so_sys = SecondOrderModel.from_matrices(M, E, K, B, Cp)
print(f'order of the model = {so_sys.order}')
print(f'number of inputs = {so_sys.dim_input}')
print(f'number of outputs = {so_sys.dim_output}')
poles = so_sys.poles()
fig, ax = plt.subplots()
ax.plot(poles.real, poles.imag, '.')
ax.set_title('System poles')
plt.show()
w = np.logspace(-4, 2, 200)
fig, ax = plt.subplots()
so_sys.mag_plot(w, ax=ax)
ax.set_title('Magnitude plot of the full model')
plt.show()
psv = so_sys.psv()
vsv = so_sys.vsv()
pvsv = so_sys.pvsv()
vpsv = so_sys.vpsv()
fig, ax = plt.subplots(2, 2, figsize=(12, 8), sharey=True)
ax[0, 0].semilogy(range(1, len(psv) + 1), psv, '.-')
ax[0, 0].set_title('Position singular values')
ax[0, 1].semilogy(range(1, len(vsv) + 1), vsv, '.-')
ax[0, 1].set_title('Velocity singular values')
ax[1, 0].semilogy(range(1, len(pvsv) + 1), pvsv, '.-')
ax[1, 0].set_title('Position-velocity singular values')
ax[1, 1].semilogy(range(1, len(vpsv) + 1), vpsv, '.-')
ax[1, 1].set_title('Velocity-position singular values')
plt.show()
print(f'FOM H_2-norm: {so_sys.h2_norm():e}')
if config.HAVE_SLYCOT:
print(f'FOM H_inf-norm: {so_sys.hinf_norm():e}')
else:
print('H_inf-norm calculation is skipped due to missing slycot.')
print(f'FOM Hankel-norm: {so_sys.hankel_norm():e}')
# Model order reduction
run_mor_method(so_sys, w, SOBTpReductor(so_sys), 'SOBTp', r)
run_mor_method(so_sys, w, SOBTvReductor(so_sys), 'SOBTv', r)
run_mor_method(so_sys, w, SOBTpvReductor(so_sys), 'SOBTpv', r)
run_mor_method(so_sys, w, SOBTvpReductor(so_sys), 'SOBTvp', r)
run_mor_method(so_sys, w, SOBTfvReductor(so_sys), 'SOBTfv', r)
run_mor_method(so_sys, w, SOBTReductor(so_sys), 'SOBT', r)
run_mor_method(so_sys,
w,
SORIRKAReductor(so_sys),
'SOR-IRKA',
r,
irka_options={'maxit': 10})
run_mor_method(so_sys, w, BTReductor(so_sys.to_lti()), 'BT', r)
run_mor_method(so_sys, w, IRKAReductor(so_sys.to_lti()), 'IRKA', r)