def test_it(self):
actuation_type = 'robin'
bound_cond_type = 'robin'
param = [2., 1.5, -3., -1., -.5]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, alpha, beta = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(param, l, n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, param, dz.bounds) for om in eig_freq])
init_adjoint_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in range(n)]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
rad_pde = ut.get_parabolic_robin_weak_form("eig_funcs", "adjoint_eig_funcs", u, param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(np.real_if_close(eig_val))
B = a2*np.array([adjoint_eig_funcs[i](l) for i in xrange(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B)
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(np.allclose(np.sort(np.linalg.eigvals(ss_weak.A)), np.sort(np.linalg.eigvals(ss_modal.A)),
rtol=1e-05, atol=0.))
self.assertTrue(np.allclose(np.array([i[0] for i in ss_weak.B]), ss_modal.B))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, u, initial_weights, dt)
eval_d = ut.evaluate_approximation("eig_funcs", q, t, dz, spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d[0])
app.exec_()
def test_it(self):
actuation_type = 'dirichlet'
bound_cond_type = 'dirichlet'
param = [1., -2., -1., None, None]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, _, _ = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
omega = np.array([(i+1)*np.pi/l for i in xrange(spatial_disc)])
eig_values = a0 - a2*omega**2 - a1**2/4./a2
norm_fak = np.ones(omega.shape)*np.sqrt(2)
eig_funcs = np.array([ef.SecondOrderDirichletEigenfunction(omega[i], param, dz.bounds, norm_fak[i])
for i in range(spatial_disc)])
register_base("eig_funcs", eig_funcs, overwrite=True)
adjoint_eig_funcs = np.array([ef.SecondOrderDirichletEigenfunction(omega[i],
adjoint_param,
dz.bounds,
norm_fak[i]) for i in range(spatial_disc)])
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
# derive sate-space system
rad_pde = ut.get_parabolic_dirichlet_weak_form("eig_funcs", "adjoint_eig_funcs", u, param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(eig_values)
B = -a2*np.array([adjoint_eig_funcs[i].derive()(l) for i in xrange(spatial_disc)])
ss_modal = sim.StateSpace("eig_funcs", A, B)
# TODO: resolve the big tolerance (rtol=3e-01) between ss_modal.A and ss_weak.A
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(np.allclose(np.sort(np.linalg.eigvals(ss_weak.A)), np.sort(np.linalg.eigvals(ss_modal.A)),
rtol=3e-1, atol=0.))
self.assertTrue(np.allclose(np.array([i[0] for i in ss_weak.B]), ss_modal.B))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, u, initial_weights, dt)
eval_d = ut.evaluate_approximation("eig_funcs", q, t, dz, spat_order=1)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
# original system parameters
a2 = 1.5; a1 = 2.5; a0 = 28; alpha = -2; beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5; a0_t = -25; alpha_t = 3; beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# system/simulation parameters
self.l = 1; self.spatial_domain = (0, self.l); self.spatial_disc = 30
self.n = 10
# create (not normalized) eigenfunctions
self.eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(self.param, self.l, self.n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, self.param, self.spatial_domain) for om in self.eig_freq])
init_adjoint_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.spatial_domain) for om in self.eig_freq])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in range(self.n)]
self.eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenvalues and -frequencies test
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(self.eig_val, eig_val_i)))
calc_eig_freq = np.sqrt((a0_i-eig_val_i)/a2)
self.assertTrue(all(np.isclose(calc_eig_freq, eig_freq_i)))
# intermediate (_i) eigenfunction test
eig_funcs_i = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_i[i],
self.param_i,
self.spatial_domain,
self.eig_funcs[i](0) )
for i in range(self.n)])
self.assertTrue(all(np.isclose([func(0) for func in eig_funcs_i],
[func(0) for func in self.eig_funcs])))
test_vec = np.linspace(0, self.l, 100)
for i in range(self.n):
self.assertTrue(all(np.isclose(self.eig_funcs[i](test_vec),
eig_funcs_i[i](test_vec)*np.exp(-a1/2/a2*test_vec))))
bound_cond_type = 'robin'
l = 1
T = 1
spatial_domain = sim.Domain(bounds=(0, l), num=n_fem)
temporal_domain = sim.Domain(bounds=(0, T), num=1e2)
n = n_modal
show_plots = False
# original system parameters
a2 = .5
a1 = 1
a0 = 2
alpha = -0.5
beta = -1
param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
# target system parameters (controller parameters)
a1_t = -1
a0_t = param_a0_t
alpha_t = 3
beta_t = 2
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = 0.4
k = 5 # = k1 + k2
k1, k2, b = ut.split_domain(k, b_desired, l, mode='coprime')[0:3]
M = np.linalg.inv(ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and target intermediate ("_ti") system parameters
def test_fem(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 30
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 12
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = self.l / 2
k = 51 # = k1 + k2
k1, k2, self.b = ut.split_domain(k, b_desired, self.l, mode='coprime')[0:3]
M = np.linalg.inv(ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(self.param, self.l, self.n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds) for om in eig_freq])
init_adjoint_eig_funcs = np.array(
[ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.dz.bounds) for om in eig_freq])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in
range(self.n)]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions of the in-domain intermediate (_id) and the intermediate (_i) system
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(eig_val_i, self.eig_val)))
eig_funcs_id = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_i[i],
self.param_i,
self.dz.bounds,
eig_funcs[i](0))
for i in range(self.n)])
eig_funcs_i = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_i[i],
self.param_i,
self.dz.bounds,
eig_funcs[i](0) * eig_funcs_id[i](self.l) /
eig_funcs_id[i](self.b))
for i in range(self.n)])
# eigenfunctions from target system ("_ti")
eig_freq_ti = np.sqrt((a0_ti - self.eig_val) / a2)
eig_funcs_ti = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_ti[i],
self.param_ti,
self.dz.bounds,
eig_funcs_i[i](0))
for i in range(self.n)])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register eigenfunctions
# register_functions("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", self.adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("eig_funcs_i", eig_funcs_i, overwrite=True)
register_base("eig_funcs_ti", eig_funcs_ti, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti, bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable_i = ph.FieldVariable("eig_funcs_i", weight_label="eig_funcs", location=self.l)
d_field_variable_i = ph.SpatialDerivedFieldVariable("eig_funcs_i", 1, weight_label="eig_funcs", location=self.l)
field_variable_ti = ph.FieldVariable("eig_funcs_ti", weight_label="eig_funcs", location=self.l)
d_field_variable_ti = ph.SpatialDerivedFieldVariable("eig_funcs_ti", 1, weight_label="eig_funcs",
location=self.l)
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [ph.ScalarTerm(fem_field_variable)]
self.x_i_at_l = [ph.ScalarTerm(field_variable_i)]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable_i)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_ti)]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_ti)]
# shift transformation
shifted_fem_funcs_i = np.array(
[ef.FiniteTransformFunction(func, M, self.b, self.l, scale_func=lambda z: np.exp(a1 / 2 / a2 * z))
for func in self.fem_funcs])
shifted_eig_funcs_id = np.array([ef.FiniteTransformFunction(func, M, self.b, self.l) for func in eig_funcs_id])
register_base("sh_fem_funcs_i", shifted_fem_funcs_i, overwrite=True)
register_base("sh_eig_funcs_id", shifted_eig_funcs_id, overwrite=True)
sh_fem_field_variable_i = ph.FieldVariable("sh_fem_funcs_i", weight_label="fem_funcs", location=self.l)
sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id", weight_label="eig_funcs", location=self.l)
self.sh_x_fem_i_at_l = [ph.ScalarTerm(sh_fem_field_variable_i),
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (a0_i - a0_ti) / 2 / a2 * z
a2, a1, _, _, _ = self.param
controller = ut.get_parabolic_robin_backstepping_controller(state=self.sh_x_fem_i_at_l,
approx_state=self.x_i_at_l,
d_approx_state=self.xd_i_at_l,
approx_target_state=self.x_ti_at_l,
d_approx_target_state=self.xd_ti_at_l,
integral_kernel_zz=self.int_kernel_zz(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=np.exp(-a1 / 2 / a2 * self.b))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, self.param, self.dz.bounds,
self.b)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, np.zeros((len(self.fem_funcs))), self.dt)
# weights of the intermediate system
mat = cr.calculate_base_transformation_matrix(self.fem_funcs, eig_funcs)
q_i = np.zeros((q.shape[0], len(eig_funcs_i)))
for i in range(q.shape[0]):
q_i[i, :] = np.dot(q[i, :], np.transpose(mat))
eval_i = sim.evaluate_approximation("eig_funcs_i", q_i, t, self.dz)
x_0t = eval_i.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired) ** 2), 1e-2)
# display results
if show_plots:
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
win1 = vis.PgSurfacePlot(eval_i)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 10
# original system parameters
a2 = 1.5
a1_z = cr.Function(lambda z: 1, derivative_handles=[lambda z: 0])
a0_z = lambda z: 3
alpha = -2
beta = -3
self.param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(self.param_t)
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(self.param, d_end=self.l)
self.param_i = a2, 0, a0_i, alpha_i, beta_i
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(self.param_t)
self.param_ti = a2, 0, a0_ti, alpha_ti, beta_ti
# create (not normalized) target (_t) eigenfunctions
eig_freq_t, self.eig_val_t = ef.compute_rad_robin_eigenfrequencies(self.param_t, self.l, self.n)
init_eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(om, self.param_t, self.dz.bounds)
for om in eig_freq_t])
init_adjoint_eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(om, adjoint_param_t, self.dz.bounds)
for om in eig_freq_t])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs_t = [cr.normalize_function(init_eig_funcs_t[i], init_adjoint_eig_funcs_t[i])
for i in range(self.n)]
eig_funcs_t = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs_t])
self.adjoint_eig_funcs_t = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t])
# # transformed original eigenfunctions
self.eig_funcs = np.array([ef.TransformedSecondOrderEigenfunction(self.eig_val_t[i],
[eig_funcs_t[i](0), alpha * eig_funcs_t[i](0),
0, 0],
[a2, a1_z, a0_z],
np.linspace(0, self.l, 1e4))
for i in range(self.n)])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register functions
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("adjoint_eig_funcs_t", self.adjoint_eig_funcs_t, overwrite=True)
register_base("eig_funcs", self.eig_funcs, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti, bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable_t = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=self.l)
d_field_variable_t = ph.SpatialDerivedFieldVariable("eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
field_variable = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=self.l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
# x_i = x * transform_i_at_l
self.transform_i_at_l = np.exp(integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x = x_i * inv_transform_i_at_l
self.inv_transform_i_at_l = np.exp(-integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x_ti = x_t * transform_ti_at_l
self.transform_ti_at_l = np.exp(a1_t / 2 / a2 * self.l)
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [ph.ScalarTerm(fem_field_variable, self.transform_i_at_l)]
self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i_at_l)]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable, self.transform_i_at_l),
ph.ScalarTerm(field_variable, self.transform_i_at_l * a1_z(self.l) / 2 / a2)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_t, self.transform_ti_at_l)]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_t, self.transform_ti_at_l),
ph.ScalarTerm(field_variable_t, self.transform_ti_at_l * a1_t / 2 / a2)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = alpha_ti - alpha_i + integrate.quad(lambda z: (a0_i(z) - a0_ti) / 2 / a2, 0, self.l)[0]
controller = ut.get_parabolic_robin_backstepping_controller(state=self.x_fem_i_at_l,
approx_state=self.x_i_at_l,
d_approx_state=self.xd_i_at_l,
approx_target_state=self.x_ti_at_l,
d_approx_target_state=self.xd_ti_at_l,
integral_kernel_zz=self.int_kernel_zz,
original_beta=beta_i,
target_beta=beta_ti,
trajectory=self.traj,
scale=self.inv_transform_i_at_l)
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs", controller, self.param, self.dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, np.zeros((len(self.fem_funcs))), self.dt)
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired) ** 2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def setUp(self):
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and target intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 10
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(self.param, self.l, self.n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds) for om in eig_freq])
init_adjoint_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.dz.bounds)
for om in eig_freq])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(self.n)]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions from target system ("_t")
eig_freq_t = np.sqrt(-a1_t ** 2 / 4 / a2 ** 2 + (a0_t - self.eig_val) / a2)
eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_t[i],
self.param_t, self.dz.bounds).scale(eig_funcs[i](0))
for i in range(self.n)])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", self.adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti, bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=self.l)
field_variable_t = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=self.l)
d_field_variable_t = ph.SpatialDerivedFieldVariable("eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
self.transform_i = lambda z: np.exp(a1 / 2 / a2 * z) # x_i = x * transform_i
self.transform_ti = lambda z: np.exp(a1_t / 2 / a2 * z) # x_ti = x_t * transform_ti
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [ph.ScalarTerm(fem_field_variable, self.transform_i(self.l))]
self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i(self.l))]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable, self.transform_i(self.l)),
ph.ScalarTerm(field_variable, self.transform_i(self.l) * a1 / 2 / a2)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_t, self.transform_ti(self.l))]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_t, self.transform_ti(self.l)),
ph.ScalarTerm(field_variable_t, self.transform_ti(self.l) * a1_t / 2 / a2)]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (a0_i - a0_ti) / 2 / a2 * z
def test_it(self):
actuation_type = 'robin'
bound_cond_type = 'robin'
param = [2., 1.5, -3., -1., -.5]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, alpha, beta = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(param, l, n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, param, dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
rad_pde = ut.get_parabolic_robin_weak_form("eig_funcs",
"adjoint_eig_funcs", u,
param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(np.real_if_close(eig_val))
B = a2 * np.array(
[adjoint_eig_funcs[i](l) for i in range(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=u)
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(
np.allclose(np.sort(np.linalg.eigvals(ss_weak.A[1])),
np.sort(np.linalg.eigvals(ss_modal.A[1])),
rtol=1e-05,
atol=0.))
self.assertTrue(
np.allclose(np.array([i[0] for i in ss_weak.B[1]]), ss_modal.B[1]))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs",
q,
t,
dz,
spat_order=1)
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(
self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# system/simulation parameters
self.l = 1
self.spatial_domain = (0, self.l)
self.spatial_disc = 30
self.n = 10
# create (not normalized) eigenfunctions
self.eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(
self.param, self.l, self.n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param,
self.spatial_domain)
for om in self.eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param,
self.spatial_domain)
for om in self.eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(self.n)
]
self.eig_funcs = np.array(
[f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenvalues and -frequencies test
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(
self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(self.eig_val, eig_val_i)))
calc_eig_freq = np.sqrt((a0_i - eig_val_i) / a2)
self.assertTrue(all(np.isclose(calc_eig_freq, eig_freq_i)))
# intermediate (_i) eigenfunction test
eig_funcs_i = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_i[i], self.param_i,
self.spatial_domain,
self.eig_funcs[i](0))
for i in range(self.n)
])
self.assertTrue(
all(
np.isclose([func(0) for func in eig_funcs_i],
[func(0) for func in self.eig_funcs])))
test_vec = np.linspace(0, self.l, 100)
for i in range(self.n):
self.assertTrue(
all(
np.isclose(
self.eig_funcs[i](test_vec), eig_funcs_i[i](test_vec) *
np.exp(-a1 / 2 / a2 * test_vec))))
def test_it(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 10
# original system parameters
a2 = 1.5
a1_z = cr.Function(lambda z: 1, derivative_handles=[lambda z: 0])
a0_z = lambda z: 3
alpha = -2
beta = -3
self.param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(self.param_t)
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(self.param,
d_end=self.l)
self.param_i = a2, 0, a0_i, alpha_i, beta_i
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, alpha_ti, beta_ti
# create (not normalized) target (_t) eigenfunctions
eig_freq_t, self.eig_val_t = ef.compute_rad_robin_eigenfrequencies(
self.param_t, self.l, self.n)
init_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param_t, self.dz.bounds)
for om in eig_freq_t
])
init_adjoint_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param_t,
self.dz.bounds)
for om in eig_freq_t
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs_t = [
cr.normalize_function(init_eig_funcs_t[i],
init_adjoint_eig_funcs_t[i])
for i in range(self.n)
]
eig_funcs_t = np.array(
[f_tuple[0] for f_tuple in adjoint_and_eig_funcs_t])
self.adjoint_eig_funcs_t = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs_t])
# # transformed original eigenfunctions
self.eig_funcs = np.array([
ef.TransformedSecondOrderEigenfunction(
self.eig_val_t[i],
[eig_funcs_t[i](0), alpha * eig_funcs_t[i](0), 0, 0],
[a2, a1_z, a0_z], np.linspace(0, self.l, 1e4))
for i in range(self.n)
])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register functions
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("adjoint_eig_funcs_t",
self.adjoint_eig_funcs_t,
overwrite=True)
register_base("eig_funcs", self.eig_funcs, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti,
bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable_t = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=self.l)
d_field_variable_t = ph.SpatialDerivedFieldVariable(
"eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
field_variable = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
location=self.l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
# x_i = x * transform_i_at_l
self.transform_i_at_l = np.exp(
integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x = x_i * inv_transform_i_at_l
self.inv_transform_i_at_l = np.exp(
-integrate.quad(lambda z: a1_z(z) / 2 / a2, 0, self.l)[0])
# x_ti = x_t * transform_ti_at_l
self.transform_ti_at_l = np.exp(a1_t / 2 / a2 * self.l)
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [
ph.ScalarTerm(fem_field_variable, self.transform_i_at_l)
]
self.x_i_at_l = [ph.ScalarTerm(field_variable, self.transform_i_at_l)]
self.xd_i_at_l = [
ph.ScalarTerm(d_field_variable, self.transform_i_at_l),
ph.ScalarTerm(field_variable,
self.transform_i_at_l * a1_z(self.l) / 2 / a2)
]
self.x_ti_at_l = [
ph.ScalarTerm(field_variable_t, self.transform_ti_at_l)
]
self.xd_ti_at_l = [
ph.ScalarTerm(d_field_variable_t, self.transform_ti_at_l),
ph.ScalarTerm(field_variable_t,
self.transform_ti_at_l * a1_t / 2 / a2)
]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = alpha_ti - alpha_i + integrate.quad(
lambda z: (a0_i(z) - a0_ti) / 2 / a2, 0, self.l)[0]
controller = ut.get_parabolic_robin_backstepping_controller(
state=self.x_fem_i_at_l,
approx_state=self.x_i_at_l,
d_approx_state=self.xd_i_at_l,
approx_target_state=self.x_ti_at_l,
d_approx_target_state=self.xd_ti_at_l,
integral_kernel_zz=self.int_kernel_zz,
original_beta=beta_i,
target_beta=beta_ti,
trajectory=self.traj,
scale=self.inv_transform_i_at_l)
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs",
controller, self.param,
self.dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, np.zeros(
(len(self.fem_funcs))), self.dt)
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_fem(self):
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 30
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 12
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = self.l / 2
k = 51 # = k1 + k2
k1, k2, self.b = ut.split_domain(k, b_desired, self.l,
mode='coprime')[0:3]
M = np.linalg.inv(
ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(
self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(
self.param, self.l, self.n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(self.n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions of the in-domain intermediate (_id) and the intermediate (_i) system
eig_freq_i, eig_val_i = ef.compute_rad_robin_eigenfrequencies(
self.param_i, self.l, self.n)
self.assertTrue(all(np.isclose(eig_val_i, self.eig_val)))
eig_funcs_id = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_i[i], self.param_i,
self.dz.bounds, eig_funcs[i](0))
for i in range(self.n)
])
eig_funcs_i = np.array([
ef.SecondOrderRobinEigenfunction(
eig_freq_i[i], self.param_i, self.dz.bounds, eig_funcs[i](0) *
eig_funcs_id[i](self.l) / eig_funcs_id[i](self.b))
for i in range(self.n)
])
# eigenfunctions from target system ("_ti")
eig_freq_ti = np.sqrt((a0_ti - self.eig_val) / a2)
eig_funcs_ti = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_ti[i], self.param_ti,
self.dz.bounds, eig_funcs_i[i](0))
for i in range(self.n)
])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register eigenfunctions
# register_functions("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs",
self.adjoint_eig_funcs,
overwrite=True)
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("eig_funcs_i", eig_funcs_i, overwrite=True)
register_base("eig_funcs_ti", eig_funcs_ti, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti,
bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable_i = ph.FieldVariable("eig_funcs_i",
weight_label="eig_funcs",
location=self.l)
d_field_variable_i = ph.SpatialDerivedFieldVariable(
"eig_funcs_i", 1, weight_label="eig_funcs", location=self.l)
field_variable_ti = ph.FieldVariable("eig_funcs_ti",
weight_label="eig_funcs",
location=self.l)
d_field_variable_ti = ph.SpatialDerivedFieldVariable(
"eig_funcs_ti", 1, weight_label="eig_funcs", location=self.l)
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [ph.ScalarTerm(fem_field_variable)]
self.x_i_at_l = [ph.ScalarTerm(field_variable_i)]
self.xd_i_at_l = [ph.ScalarTerm(d_field_variable_i)]
self.x_ti_at_l = [ph.ScalarTerm(field_variable_ti)]
self.xd_ti_at_l = [ph.ScalarTerm(d_field_variable_ti)]
# shift transformation
shifted_fem_funcs_i = np.array([
ef.FiniteTransformFunction(
func,
M,
self.b,
self.l,
scale_func=lambda z: np.exp(a1 / 2 / a2 * z))
for func in self.fem_funcs
])
shifted_eig_funcs_id = np.array([
ef.FiniteTransformFunction(func, M, self.b, self.l)
for func in eig_funcs_id
])
register_base("sh_fem_funcs_i", shifted_fem_funcs_i, overwrite=True)
register_base("sh_eig_funcs_id", shifted_eig_funcs_id, overwrite=True)
sh_fem_field_variable_i = ph.FieldVariable("sh_fem_funcs_i",
weight_label="fem_funcs",
location=self.l)
sh_field_variable_id = ph.FieldVariable("sh_eig_funcs_id",
weight_label="eig_funcs",
location=self.l)
self.sh_x_fem_i_at_l = [
ph.ScalarTerm(sh_fem_field_variable_i),
ph.ScalarTerm(field_variable_i),
ph.ScalarTerm(sh_field_variable_id, -1)
]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (
a0_i - a0_ti) / 2 / a2 * z
a2, a1, _, _, _ = self.param
controller = ut.get_parabolic_robin_backstepping_controller(
state=self.sh_x_fem_i_at_l,
approx_state=self.x_i_at_l,
d_approx_state=self.xd_i_at_l,
approx_target_state=self.x_ti_at_l,
d_approx_target_state=self.xd_ti_at_l,
integral_kernel_zz=self.int_kernel_zz(self.l),
original_beta=self.beta_i,
target_beta=self.beta_ti,
trajectory=self.traj,
scale=np.exp(-a1 / 2 / a2 * self.b))
# determine (A,B) with modal-transfomation
rad_pde = ut.get_parabolic_robin_weak_form("fem_funcs", "fem_funcs",
controller, self.param,
self.dz.bounds, self.b)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# simulate
t, q = sim.simulate_state_space(ss_weak, np.zeros(
(len(self.fem_funcs))), self.dt)
# weights of the intermediate system
mat = cr.calculate_base_transformation_matrix(self.fem_funcs,
eig_funcs)
q_i = np.zeros((q.shape[0], len(eig_funcs_i)))
for i in range(q.shape[0]):
q_i[i, :] = np.dot(q[i, :], np.transpose(mat))
eval_i = sim.evaluate_approximation("eig_funcs_i", q_i, t, self.dz)
x_0t = eval_i.output_data[:, 0]
yc, tc = tr.gevrey_tanh(self.T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-2)
# display results
if show_plots:
eval_d = sim.evaluate_approximation("fem_funcs", q, t, self.dz)
win1 = vis.PgSurfacePlot(eval_i)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def setUp(self):
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
self.param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(self.param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
self.param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and target intermediate ("_ti") system parameters
_, _, a0_i, self.alpha_i, self.beta_i = ef.transform2intermediate(
self.param)
self.param_i = a2, 0, a0_i, self.alpha_i, self.beta_i
_, _, a0_ti, self.alpha_ti, self.beta_ti = ef.transform2intermediate(
self.param_t)
self.param_ti = a2, 0, a0_ti, self.alpha_ti, self.beta_ti
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
self.dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
self.T = 1.
temporal_disc = 1e2
self.dt = sim.Domain(bounds=(0, self.T), num=temporal_disc)
self.n = 10
# create (not normalized) eigenfunctions
eig_freq, self.eig_val = ef.compute_rad_robin_eigenfrequencies(
self.param, self.l, self.n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, self.param, self.dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, self.dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(self.n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
self.adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions from target system ("_t")
eig_freq_t = np.sqrt(-a1_t**2 / 4 / a2**2 + (a0_t - self.eig_val) / a2)
eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_t[i], self.param_t,
self.dz.bounds).scale(
eig_funcs[i](0))
for i in range(self.n)
])
# create testfunctions
nodes, self.fem_funcs = sf.cure_interval(sf.LagrangeFirstOrder,
self.dz.bounds,
node_count=self.n)
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs",
self.adjoint_eig_funcs,
overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
register_base("fem_funcs", self.fem_funcs, overwrite=True)
# init trajectory
self.traj = tr.RadTrajectory(self.l, self.T, self.param_ti,
bound_cond_type, actuation_type)
# original () and target (_t) field variable
fem_field_variable = ph.FieldVariable("fem_funcs", location=self.l)
field_variable = ph.FieldVariable("eig_funcs", location=self.l)
d_field_variable = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
location=self.l)
field_variable_t = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=self.l)
d_field_variable_t = ph.SpatialDerivedFieldVariable(
"eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
# intermediate (_i) and target intermediate (_ti) transformations by z=l
self.transform_i = lambda z: np.exp(a1 / 2 / a2 * z
) # x_i = x * transform_i
self.transform_ti = lambda z: np.exp(a1_t / 2 / a2 * z
) # x_ti = x_t * transform_ti
# intermediate (_i) and target intermediate (_ti) field variable (list of scalar terms = sum of scalar terms)
self.x_fem_i_at_l = [
ph.ScalarTerm(fem_field_variable, self.transform_i(self.l))
]
self.x_i_at_l = [
ph.ScalarTerm(field_variable, self.transform_i(self.l))
]
self.xd_i_at_l = [
ph.ScalarTerm(d_field_variable, self.transform_i(self.l)),
ph.ScalarTerm(field_variable,
self.transform_i(self.l) * a1 / 2 / a2)
]
self.x_ti_at_l = [
ph.ScalarTerm(field_variable_t, self.transform_ti(self.l))
]
self.xd_ti_at_l = [
ph.ScalarTerm(d_field_variable_t, self.transform_ti(self.l)),
ph.ScalarTerm(field_variable_t,
self.transform_ti(self.l) * a1_t / 2 / a2)
]
# discontinuous operator (Kx)(t) = int_kernel_zz(l)*x(l,t)
self.int_kernel_zz = lambda z: self.alpha_ti - self.alpha_i + (
a0_i - a0_ti) / 2 / a2 * z
def test_it(self):
# original system parameters
a2 = 1.5
a1 = 2.5
a0 = 28
alpha = -2
beta = -3
param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
# target system parameters (controller parameters)
a1_t = -5
a0_t = -25
alpha_t = 3
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param)
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t)
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
# create (not normalized) eigenfunctions
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(
param, self.l, n)
init_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, param, dz.bounds)
for om in eig_freq
])
init_adjoint_eig_funcs = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq
])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [
cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i])
for i in range(n)
]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array(
[f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions from target system ("_t")
eig_freq_t = np.sqrt(-a1_t**2 / 4 / a2**2 + (a0_t - eig_val) / a2)
eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(eig_freq_t[i], param_t,
dz.bounds).scale(eig_funcs[i](0))
for i in range(n)
])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, self.l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# controller initialization
x_at_l = ph.FieldVariable("eig_funcs", location=self.l)
xd_at_l = ph.SpatialDerivedFieldVariable("eig_funcs",
1,
location=self.l)
x_t_at_l = ph.FieldVariable("eig_funcs_t",
weight_label="eig_funcs",
location=self.l)
xd_t_at_l = ph.SpatialDerivedFieldVariable("eig_funcs_t",
1,
weight_label="eig_funcs",
location=self.l)
combined_transform = lambda z: np.exp((a1_t - a1) / 2 / a2 * z)
int_kernel_zz = lambda z: alpha_ti - alpha_i + (a0_i - a0_ti
) / 2 / a2 * z
controller = ct.Controller(
ct.ControlLaw([
ph.ScalarTerm(x_at_l,
(beta_i - beta_ti - int_kernel_zz(self.l))),
ph.ScalarTerm(x_t_at_l, -beta_ti * combined_transform(self.l)),
ph.ScalarTerm(x_at_l, beta_ti),
ph.ScalarTerm(xd_t_at_l, -combined_transform(self.l)),
ph.ScalarTerm(x_t_at_l,
-a1_t / 2 / a2 * combined_transform(self.l)),
ph.ScalarTerm(xd_at_l, 1),
ph.ScalarTerm(x_at_l, a1 / 2 / a2 + int_kernel_zz(self.l))
]))
# init trajectory
traj = tr.RadTrajectory(self.l, T, param_t, bound_cond_type,
actuation_type)
traj.scale = combined_transform(self.l)
# input with feedback
control_law = sim.SimulationInputSum([traj, controller])
# control_law = sim.simInputSum([traj])
# determine (A,B) with modal-transformation
A = np.diag(np.real(eig_val))
B = a2 * np.array(
[adjoint_eig_funcs[i](self.l) for i in range(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=control_law)
# simulate
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired)**2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
def test_it(self):
actuation_type = 'dirichlet'
bound_cond_type = 'dirichlet'
param = [1., -2., -1., None, None]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
a2, a1, a0, _, _ = param
l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
omega = np.array([(i + 1) * np.pi / l for i in range(spatial_disc)])
eig_values = a0 - a2 * omega**2 - a1**2 / 4. / a2
norm_fak = np.ones(omega.shape) * np.sqrt(2)
eig_funcs = np.array([
ef.SecondOrderDirichletEigenfunction(omega[i], param, dz.bounds,
norm_fak[i])
for i in range(spatial_disc)
])
register_base("eig_funcs", eig_funcs, overwrite=True)
adjoint_eig_funcs = np.array([
ef.SecondOrderDirichletEigenfunction(omega[i], adjoint_param,
dz.bounds, norm_fak[i])
for i in range(spatial_disc)
])
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# init trajectory
u = tr.RadTrajectory(l, T, param, bound_cond_type, actuation_type)
# determine (A,B) with weak-formulation (pyinduct)
# derive sate-space system
rad_pde = ut.get_parabolic_dirichlet_weak_form("eig_funcs",
"adjoint_eig_funcs", u,
param, dz.bounds)
cf = sim.parse_weak_formulation(rad_pde)
ss_weak = cf.convert_to_state_space()
# determine (A,B) with modal-transfomation
A = np.diag(eig_values)
B = -a2 * np.array(
[adjoint_eig_funcs[i].derive()(l) for i in range(spatial_disc)])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=u)
# TODO: resolve the big tolerance (rtol=3e-01) between ss_modal.A and ss_weak.A
# check if ss_modal.(A,B) is close to ss_weak.(A,B)
self.assertTrue(
np.allclose(np.sort(np.linalg.eigvals(ss_weak.A[1])),
np.sort(np.linalg.eigvals(ss_modal.A[1])),
rtol=3e-1,
atol=0.))
self.assertTrue(
np.allclose(np.array([i[0] for i in ss_weak.B[1]]), ss_modal.B[1]))
# display results
if show_plots:
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs",
q,
t,
dz,
spat_order=1)
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
bound_cond_type = 'robin'
l = 1
T = 1
spatial_domain = sim.Domain(bounds=(0, l), num=n_fem)
temporal_domain = sim.Domain(bounds=(0, T), num=1e2)
n = n_modal
show_plots = False
# original system parameters
a2 = .5
a1 = 1
a0 = 2
alpha = -0.5
beta = -1
param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
# target system parameters (controller parameters)
a1_t = -1
a0_t = param_a0_t
alpha_t = 3
beta_t = 2
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# actuation_type by b which is close to b_desired on a k times subdivided spatial domain
b_desired = 0.4
k = 5 # = k1 + k2
k1, k2, b = ut.split_domain(k, b_desired, l, mode='coprime')[0:3]
M = np.linalg.inv(ut.get_inn_domain_transformation_matrix(k1, k2, mode="2n"))
# original intermediate ("_i") and target intermediate ("_ti") system parameters
def test_it(self):
# original system parameters
a2 = 1.5;
a1 = 2.5;
a0 = 28;
alpha = -2;
beta = -3
param = [a2, a1, a0, alpha, beta]
adjoint_param = ef.get_adjoint_rad_evp_param(param)
# target system parameters (controller parameters)
a1_t = -5;
a0_t = -25;
alpha_t = 3;
beta_t = 2
# a1_t = a1; a0_t = a0; alpha_t = alpha; beta_t = beta
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
# original intermediate ("_i") and traget intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param)
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t)
# system/simulation parameters
actuation_type = 'robin'
bound_cond_type = 'robin'
self.l = 1.
spatial_disc = 10
dz = sim.Domain(bounds=(0, self.l), num=spatial_disc)
T = 1.
temporal_disc = 1e2
dt = sim.Domain(bounds=(0, T), num=temporal_disc)
n = 10
# create (not normalized) eigenfunctions
eig_freq, eig_val = ef.compute_rad_robin_eigenfrequencies(param, self.l, n)
init_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, param, dz.bounds) for om in eig_freq])
init_adjoint_eig_funcs = np.array([ef.SecondOrderRobinEigenfunction(om, adjoint_param, dz.bounds)
for om in eig_freq])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs = [cr.normalize_function(init_eig_funcs[i], init_adjoint_eig_funcs[i]) for i in range(n)]
eig_funcs = np.array([f_tuple[0] for f_tuple in adjoint_and_eig_funcs])
adjoint_eig_funcs = np.array([f_tuple[1] for f_tuple in adjoint_and_eig_funcs])
# eigenfunctions from target system ("_t")
eig_freq_t = np.sqrt(-a1_t ** 2 / 4 / a2 ** 2 + (a0_t - eig_val) / a2)
eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(eig_freq_t[i],
param_t, dz.bounds).scale(eig_funcs[i](0))
for i in range(n)])
# register eigenfunctions
register_base("eig_funcs", eig_funcs, overwrite=True)
register_base("adjoint_eig_funcs", adjoint_eig_funcs, overwrite=True)
register_base("eig_funcs_t", eig_funcs_t, overwrite=True)
# derive initial field variable x(z,0) and weights
start_state = cr.Function(lambda z: 0., domain=(0, self.l))
initial_weights = cr.project_on_base(start_state, adjoint_eig_funcs)
# controller initialization
x_at_l = ph.FieldVariable("eig_funcs", location=self.l)
xd_at_l = ph.SpatialDerivedFieldVariable("eig_funcs", 1, location=self.l)
x_t_at_l = ph.FieldVariable("eig_funcs_t", weight_label="eig_funcs", location=self.l)
xd_t_at_l = ph.SpatialDerivedFieldVariable("eig_funcs_t", 1, weight_label="eig_funcs", location=self.l)
combined_transform = lambda z: np.exp((a1_t - a1) / 2 / a2 * z)
int_kernel_zz = lambda z: alpha_ti - alpha_i + (a0_i - a0_ti) / 2 / a2 * z
controller = ct.Controller(
ct.ControlLaw([ph.ScalarTerm(x_at_l, (beta_i - beta_ti - int_kernel_zz(self.l))),
ph.ScalarTerm(x_t_at_l, -beta_ti * combined_transform(self.l)),
ph.ScalarTerm(x_at_l, beta_ti),
ph.ScalarTerm(xd_t_at_l, -combined_transform(self.l)),
ph.ScalarTerm(x_t_at_l, -a1_t / 2 / a2 * combined_transform(self.l)),
ph.ScalarTerm(xd_at_l, 1),
ph.ScalarTerm(x_at_l, a1 / 2 / a2 + int_kernel_zz(self.l))
]))
# init trajectory
traj = tr.RadTrajectory(self.l, T, param_t, bound_cond_type, actuation_type)
traj.scale = combined_transform(self.l)
# input with feedback
control_law = sim.SimulationInputSum([traj, controller])
# control_law = sim.simInputSum([traj])
# determine (A,B) with modal-transformation
A = np.diag(np.real(eig_val))
B = a2 * np.array([adjoint_eig_funcs[i](self.l) for i in range(len(eig_freq))])
ss_modal = sim.StateSpace("eig_funcs", A, B, input_handle=control_law)
# simulate
t, q = sim.simulate_state_space(ss_modal, initial_weights, dt)
eval_d = sim.evaluate_approximation("eig_funcs", q, t, dz)
x_0t = eval_d.output_data[:, 0]
yc, tc = tr.gevrey_tanh(T, 1)
x_0t_desired = np.interp(t, tc, yc[0, :])
self.assertLess(np.average((x_0t - x_0t_desired) ** 2), 1e-4)
# display results
if show_plots:
win1 = vis.PgAnimatedPlot([eval_d], title="Test")
win2 = vis.PgSurfacePlot(eval_d)
app.exec_()
# original system parameters
a2 = .5
a1_z = cr.Function(lambda z: 0.1 * np.exp(4 * z),
derivative_handles=[lambda z: 0.4 * np.exp(4 * z)])
a0_z = lambda z: 1 + 10 * z + 2 * np.sin(4 * np.pi / l * z)
alpha = -1
beta = -1
param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -0
a0_t = -6
alpha_t = 3
beta_t = 3
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(param_t)
# original intermediate ("_i") and target intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param, d_end=l)
param_i = a2, 0, a0_i, alpha_i, beta_i
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t)
param_ti = a2, 0, a0_ti, alpha_ti, beta_ti
# create (not normalized) target (_t) eigenfunctions
eig_freq_t, eig_val_t = ef.compute_rad_robin_eigenfrequencies(param_t, l, n)
init_eig_funcs_t = np.array([ef.SecondOrderRobinEigenfunction(om, param_t, spatial_domain.bounds) for om in eig_freq_t])
init_adjoint_eig_funcs_t = np.array(
[ef.SecondOrderRobinEigenfunction(om, adjoint_param_t, spatial_domain.bounds) for om in eig_freq_t])
# normalize eigenfunctions and adjoint eigenfunctions
adjoint_and_eig_funcs_t = [cr.normalize_function(init_eig_funcs_t[i], init_adjoint_eig_funcs_t[i]) for i in range(n)]
# original system parameters
a2 = .5
a1_z = cr.Function(lambda z: 0.1 * np.exp(4 * z),
derivative_handles=[lambda z: 0.4 * np.exp(4 * z)])
a0_z = lambda z: 1 + 10 * z + 2 * np.sin(4 * np.pi / l * z)
alpha = -1
beta = -1
param = [a2, a1_z, a0_z, alpha, beta]
# target system parameters (controller parameters)
a1_t = -0
a0_t = -6
alpha_t = 3
beta_t = 3
param_t = [a2, a1_t, a0_t, alpha_t, beta_t]
adjoint_param_t = ef.get_adjoint_rad_evp_param(param_t)
# original intermediate ("_i") and target intermediate ("_ti") system parameters
_, _, a0_i, alpha_i, beta_i = ef.transform2intermediate(param, d_end=l)
param_i = a2, 0, a0_i, alpha_i, beta_i
_, _, a0_ti, alpha_ti, beta_ti = ef.transform2intermediate(param_t)
param_ti = a2, 0, a0_ti, alpha_ti, beta_ti
# create (not normalized) target (_t) eigenfunctions
eig_freq_t, eig_val_t = ef.compute_rad_robin_eigenfrequencies(param_t, l, n)
init_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, param_t, spatial_domain.bounds)
for om in eig_freq_t
])
init_adjoint_eig_funcs_t = np.array([
ef.SecondOrderRobinEigenfunction(om, adjoint_param_t,
