def new_ballistic_ascent_phase(transcription):
ballistic_ascent = CannonballPhase(transcription=transcription)
# All initial states are free (they will be linked to the final stages of propelled_ascent).
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
ballistic_ascent.set_time_options(fix_initial=False,
initial_bounds=(0, 1),
duration_bounds=(0, 10),
duration_ref=1,
units='s')
ballistic_ascent.set_state_options('r', fix_initial=False, fix_final=False)
ballistic_ascent.set_state_options('h', fix_initial=False, fix_final=False)
ballistic_ascent.set_state_options('gam',
fix_initial=False,
fix_final=True,
upper=89,
units='deg')
ballistic_ascent.set_state_options('v', fix_initial=False, fix_final=False)
ballistic_ascent.add_input_parameter('S', targets=['aero.S'], units='m**2')
ballistic_ascent.add_input_parameter('m_empty',
targets=['eom.m'],
units='kg')
return ballistic_ascent
def new_propelled_ascent_phase(transcription):
propelled_ascent = CannonballPhase(ode_class=WaterPropulsionODE,
transcription=transcription)
# Add states specific for the propelled ascent
propelled_ascent.add_state('p',
units='bar',
rate_source='water_engine.pdot',
targets=['water_engine.p'])
propelled_ascent.add_state('V_w',
units='m**3',
ref=1e-3,
rate_source='water_engine.Vdot',
targets=['water_engine.V_w', 'mass_adder.V_w'])
# All initial states except flight path angle and water volume are fixed
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
# Final water volume is fixed (we will set it to zero so that phase ends when bottle empties)
propelled_ascent.set_time_options(fix_initial=True,
duration_bounds=(0, 0.5),
duration_ref=0.1,
units='s')
propelled_ascent.set_state_options('r', fix_initial=True, fix_final=False)
propelled_ascent.set_state_options('h', fix_initial=True, fix_final=False)
propelled_ascent.set_state_options('gam',
fix_initial=False,
fix_final=False,
lower=0,
upper=89,
units='deg')
propelled_ascent.set_state_options('v', fix_initial=True, fix_final=False)
propelled_ascent.set_state_options('V_w',
fix_initial=False,
fix_final=True)
propelled_ascent.set_state_options('p', fix_initial=True, fix_final=False)
propelled_ascent.add_input_parameter('S', targets=['aero.S'], units='m**2')
propelled_ascent.add_input_parameter('m_empty',
targets=['mass_adder.m_empty'],
units='kg')
propelled_ascent.add_input_parameter('V_b',
targets=['water_engine.V_b'],
units='m**3')
propelled_ascent.add_timeseries_output('water_engine.F', 'T', units='N')
return propelled_ascent
def new_descent_phase(transcription):
descent = CannonballPhase(transcription=transcription)
# All initial states and time are free (they will be linked to the final states of ballistic_ascent).
# Final altitude is fixed (we will set it to zero so that the phase ends at ground impact)
descent.set_time_options(initial_bounds=(.5, 100),
duration_bounds=(.5, 100),
duration_ref=10,
units='s')
descent.add_state('r', )
descent.add_state('h', fix_initial=False, fix_final=True)
descent.add_state('gam', fix_initial=False, fix_final=False, units='deg')
descent.add_state('v', fix_initial=False, fix_final=False)
descent.add_input_parameter('S', targets=['aero.S'], units='m**2')
descent.add_input_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
return descent
def test_two_phase_cannonball_for_docs(self):
import openmdao.api as om
from openmdao.utils.assert_utils import assert_near_equal
import dymos as dm
from dymos.examples.cannonball.size_comp import CannonballSizeComp
from dymos.examples.cannonball.cannonball_phase import CannonballPhase
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()
external_params = p.model.add_subsystem('external_params',
om.IndepVarComp())
external_params.add_output('radius', val=0.10, units='m')
external_params.add_output('dens', val=7.87, units='g/cm**3')
external_params.add_design_var('radius',
lower=0.01,
upper=0.10,
ref0=0.01,
ref=0.10)
p.model.add_subsystem('size_comp', CannonballSizeComp())
traj = p.model.add_subsystem('traj', dm.Trajectory())
transcription = dm.Radau(num_segments=5, order=3, compressed=True)
ascent = CannonballPhase(transcription=transcription)
ascent = traj.add_phase('ascent', ascent)
# All initial states except flight path angle are fixed
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
ascent.set_time_options(fix_initial=True,
duration_bounds=(1, 100),
duration_ref=100,
units='s')
ascent.set_state_options('r', fix_initial=True, fix_final=False)
ascent.set_state_options('h', fix_initial=True, fix_final=False)
ascent.set_state_options('gam', fix_initial=False, fix_final=True)
ascent.set_state_options('v', fix_initial=False, fix_final=False)
ascent.add_parameter('S', targets=['aero.S'], units='m**2')
ascent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
# Limit the muzzle energy
ascent.add_boundary_constraint('kinetic_energy.ke',
loc='initial',
units='J',
upper=400000,
lower=0,
ref=100000,
shape=(1, ))
# Second Phase (descent)
transcription = dm.GaussLobatto(num_segments=5,
order=3,
compressed=True)
descent = CannonballPhase(transcription=transcription)
traj.add_phase('descent', descent)
# All initial states and time are free (they will be linked to the final states of ascent.
# Final altitude is fixed (we will set it to zero so that the phase ends at ground impact)
descent.set_time_options(initial_bounds=(.5, 100),
duration_bounds=(.5, 100),
duration_ref=100,
units='s')
descent.add_state('r', )
descent.add_state('h', fix_initial=False, fix_final=True)
descent.add_state('gam', fix_initial=False, fix_final=False)
descent.add_state('v', fix_initial=False, fix_final=False)
descent.add_parameter('S', targets=['aero.S'], units='m**2')
descent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
descent.add_objective('r', loc='final', scaler=-1.0)
# Add internally-managed design parameters to the trajectory.
traj.add_parameter('CD',
targets={
'ascent': ['aero.CD'],
'descent': ['aero.CD']
},
val=0.5,
units=None,
opt=False)
traj.add_parameter('CL',
targets={
'ascent': ['aero.CL'],
'descent': ['aero.CL']
},
val=0.0,
units=None,
opt=False)
traj.add_parameter('T',
targets={
'ascent': ['eom.T'],
'descent': ['eom.T']
},
val=0.0,
units='N',
opt=False)
traj.add_parameter('alpha',
targets={
'ascent': ['eom.alpha'],
'descent': ['eom.alpha']
},
val=0.0,
units='deg',
opt=False)
# Add externally-provided design parameters to the trajectory.
# In this case, we connect 'm' to pre-existing input parameters named 'mass' in each phase.
traj.add_parameter('m',
units='kg',
val=1.0,
targets={
'ascent': 'mass',
'descent': 'mass'
})
# In this case, by omitting targets, we're connecting these parameters to parameters
# with the same name in each phase.
traj.add_parameter('S', units='m**2', val=0.005)
# Link Phases (link time and all state variables)
traj.link_phases(phases=['ascent', 'descent'], vars=['*'])
# Issue Connections
p.model.connect('external_params.radius', 'size_comp.radius')
p.model.connect('external_params.dens', 'size_comp.dens')
p.model.connect('size_comp.mass', 'traj.parameters:m')
p.model.connect('size_comp.S', 'traj.parameters:S')
# Finish Problem Setup
p.model.linear_solver = om.DirectSolver()
p.driver.add_recorder(om.SqliteRecorder('ex_two_phase_cannonball.db'))
p.setup()
# Set Initial Guesses
p.set_val('external_params.radius', 0.05, units='m')
p.set_val('external_params.dens', 7.87, units='g/cm**3')
p.set_val('traj.parameters:CD', 0.5)
p.set_val('traj.parameters:CL', 0.0)
p.set_val('traj.parameters:T', 0.0)
p.set_val('traj.ascent.t_initial', 0.0)
p.set_val('traj.ascent.t_duration', 10.0)
p.set_val('traj.ascent.states:r',
ascent.interpolate(ys=[0, 100], nodes='state_input'))
p.set_val('traj.ascent.states:h',
ascent.interpolate(ys=[0, 100], nodes='state_input'))
p.set_val('traj.ascent.states:v',
ascent.interpolate(ys=[200, 150], nodes='state_input'))
p.set_val('traj.ascent.states:gam',
ascent.interpolate(ys=[25, 0], nodes='state_input'),
units='deg')
p.set_val('traj.descent.t_initial', 10.0)
p.set_val('traj.descent.t_duration', 10.0)
p.set_val('traj.descent.states:r',
descent.interpolate(ys=[100, 200], nodes='state_input'))
p.set_val('traj.descent.states:h',
descent.interpolate(ys=[100, 0], nodes='state_input'))
p.set_val('traj.descent.states:v',
descent.interpolate(ys=[150, 200], nodes='state_input'))
p.set_val('traj.descent.states:gam',
descent.interpolate(ys=[0, -45], nodes='state_input'),
units='deg')
dm.run_problem(p)
assert_near_equal(p.get_val('traj.descent.states:r')[-1],
3183.25,
tolerance=1.0E-2)
exp_out = traj.simulate()
print('optimal radius: {0:6.4f} m '.format(
p.get_val('external_params.radius', units='m')[0]))
print('cannonball mass: {0:6.4f} kg '.format(
p.get_val('size_comp.mass', units='kg')[0]))
print('launch angle: {0:6.4f} '
'deg '.format(
p.get_val('traj.ascent.timeseries.states:gam',
units='deg')[0, 0]))
print('maximum range: {0:6.4f} '
'm '.format(
p.get_val('traj.descent.timeseries.states:r')[-1, 0]))
fig, axes = plt.subplots(nrows=1, ncols=1, figsize=(10, 6))
time_imp = {
'ascent': p.get_val('traj.ascent.timeseries.time'),
'descent': p.get_val('traj.descent.timeseries.time')
}
time_exp = {
'ascent': exp_out.get_val('traj.ascent.timeseries.time'),
'descent': exp_out.get_val('traj.descent.timeseries.time')
}
r_imp = {
'ascent': p.get_val('traj.ascent.timeseries.states:r'),
'descent': p.get_val('traj.descent.timeseries.states:r')
}
r_exp = {
'ascent': exp_out.get_val('traj.ascent.timeseries.states:r'),
'descent': exp_out.get_val('traj.descent.timeseries.states:r')
}
h_imp = {
'ascent': p.get_val('traj.ascent.timeseries.states:h'),
'descent': p.get_val('traj.descent.timeseries.states:h')
}
h_exp = {
'ascent': exp_out.get_val('traj.ascent.timeseries.states:h'),
'descent': exp_out.get_val('traj.descent.timeseries.states:h')
}
axes.plot(r_imp['ascent'], h_imp['ascent'], 'bo')
axes.plot(r_imp['descent'], h_imp['descent'], 'ro')
axes.plot(r_exp['ascent'], h_exp['ascent'], 'b--')
axes.plot(r_exp['descent'], h_exp['descent'], 'r--')
axes.set_xlabel('range (m)')
axes.set_ylabel('altitude (m)')
fig, axes = plt.subplots(nrows=4, ncols=1, figsize=(10, 6))
states = ['r', 'h', 'v', 'gam']
for i, state in enumerate(states):
x_imp = {
'ascent':
p.get_val('traj.ascent.timeseries.states:{0}'.format(state)),
'descent':
p.get_val('traj.descent.timeseries.states:{0}'.format(state))
}
x_exp = {
'ascent':
exp_out.get_val(
'traj.ascent.timeseries.states:{0}'.format(state)),
'descent':
exp_out.get_val(
'traj.descent.timeseries.states:{0}'.format(state))
}
axes[i].set_ylabel(state)
axes[i].plot(time_imp['ascent'], x_imp['ascent'], 'bo')
axes[i].plot(time_imp['descent'], x_imp['descent'], 'ro')
axes[i].plot(time_exp['ascent'], x_exp['ascent'], 'b--')
axes[i].plot(time_exp['descent'], x_exp['descent'], 'r--')
params = ['CL', 'CD', 'T', 'alpha', 'mass', 'S']
fig, axes = plt.subplots(nrows=6, ncols=1, figsize=(12, 6))
for i, param in enumerate(params):
p_imp = {
'ascent':
p.get_val(
'traj.ascent.timeseries.parameters:{0}'.format(param)),
'descent':
p.get_val(
'traj.descent.timeseries.parameters:{0}'.format(param))
}
p_exp = {
'ascent':
exp_out.get_val('traj.ascent.timeseries.'
'parameters:{0}'.format(param)),
'descent':
exp_out.get_val('traj.descent.timeseries.'
'parameters:{0}'.format(param))
}
axes[i].set_ylabel(param)
axes[i].plot(time_imp['ascent'], p_imp['ascent'], 'bo')
axes[i].plot(time_imp['descent'], p_imp['descent'], 'ro')
axes[i].plot(time_exp['ascent'], p_exp['ascent'], 'b--')
axes[i].plot(time_exp['descent'], p_exp['descent'], 'r--')
plt.show()
def test_connect_control_to_parameter(self):
""" Test that the final value of a control in one phase can be connected as the value
of a parameter in a subsequent phase. """
import openmdao.api as om
from openmdao.utils.assert_utils import assert_near_equal
import dymos as dm
from dymos.examples.cannonball.size_comp import CannonballSizeComp
from dymos.examples.cannonball.cannonball_phase import CannonballPhase
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()
external_params = p.model.add_subsystem('external_params',
om.IndepVarComp())
external_params.add_output('radius', val=0.10, units='m')
external_params.add_output('dens', val=7.87, units='g/cm**3')
external_params.add_design_var('radius',
lower=0.01,
upper=0.10,
ref0=0.01,
ref=0.10)
p.model.add_subsystem('size_comp', CannonballSizeComp())
traj = p.model.add_subsystem('traj', dm.Trajectory())
transcription = dm.Radau(num_segments=5, order=3, compressed=True)
ascent = CannonballPhase(transcription=transcription)
ascent = traj.add_phase('ascent', ascent)
# All initial states except flight path angle are fixed
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
ascent.set_time_options(fix_initial=True,
duration_bounds=(1, 100),
duration_ref=100,
units='s')
ascent.set_state_options('r', fix_initial=True, fix_final=False)
ascent.set_state_options('h', fix_initial=True, fix_final=False)
ascent.set_state_options('gam', fix_initial=False, fix_final=True)
ascent.set_state_options('v', fix_initial=False, fix_final=False)
ascent.add_parameter('S', targets=['aero.S'], units='m**2')
ascent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
ascent.add_control('CD', targets=['aero.CD'], opt=False, val=0.05)
# Limit the muzzle energy
ascent.add_boundary_constraint('kinetic_energy.ke',
loc='initial',
units='J',
upper=400000,
lower=0,
ref=100000,
shape=(1, ))
# Second Phase (descent)
transcription = dm.GaussLobatto(num_segments=5,
order=3,
compressed=True)
descent = CannonballPhase(transcription=transcription)
traj.add_phase('descent', descent)
# All initial states and time are free (they will be linked to the final states of ascent.
# Final altitude is fixed (we will set it to zero so that the phase ends at ground impact)
descent.set_time_options(initial_bounds=(.5, 100),
duration_bounds=(.5, 100),
duration_ref=100,
units='s')
descent.add_state('r', )
descent.add_state('h', fix_initial=False, fix_final=True)
descent.add_state('gam', fix_initial=False, fix_final=False)
descent.add_state('v', fix_initial=False, fix_final=False)
descent.add_parameter('S', targets=['aero.S'], units='m**2')
descent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
descent.add_parameter('CD', targets=['aero.CD'], val=0.01)
descent.add_objective('r', loc='final', scaler=-1.0)
# Add internally-managed design parameters to the trajectory.
traj.add_parameter('CL',
targets={
'ascent': ['aero.CL'],
'descent': ['aero.CL']
},
val=0.0,
units=None,
opt=False)
traj.add_parameter('T',
targets={
'ascent': ['eom.T'],
'descent': ['eom.T']
},
val=0.0,
units='N',
opt=False)
traj.add_parameter('alpha',
targets={
'ascent': ['eom.alpha'],
'descent': ['eom.alpha']
},
val=0.0,
units='deg',
opt=False)
# Add externally-provided design parameters to the trajectory.
# In this case, we connect 'm' to pre-existing input parameters named 'mass' in each phase.
traj.add_parameter('m',
units='kg',
val=1.0,
targets={
'ascent': 'mass',
'descent': 'mass'
})
# In this case, by omitting targets, we're connecting these parameters to parameters
# with the same name in each phase.
traj.add_parameter('S', units='m**2', val=0.005)
# Link Phases (link time and all state variables)
traj.link_phases(phases=['ascent', 'descent'], vars=['*'])
# Issue Connections
p.model.connect('external_params.radius', 'size_comp.radius')
p.model.connect('external_params.dens', 'size_comp.dens')
p.model.connect('size_comp.mass', 'traj.parameters:m')
p.model.connect('size_comp.S', 'traj.parameters:S')
traj.connect('ascent.timeseries.controls:CD',
'descent.parameters:CD',
src_indices=[-1])
# A linear solver at the top level can improve performance.
p.model.linear_solver = om.DirectSolver()
# Finish Problem Setup
p.setup()
# Set Initial Guesses
p.set_val('external_params.radius', 0.05, units='m')
p.set_val('external_params.dens', 7.87, units='g/cm**3')
p.set_val('traj.ascent.controls:CD', 0.5)
p.set_val('traj.parameters:CL', 0.0)
p.set_val('traj.parameters:T', 0.0)
p.set_val('traj.ascent.t_initial', 0.0)
p.set_val('traj.ascent.t_duration', 10.0)
p.set_val('traj.ascent.states:r',
ascent.interpolate(ys=[0, 100], nodes='state_input'))
p.set_val('traj.ascent.states:h',
ascent.interpolate(ys=[0, 100], nodes='state_input'))
p.set_val('traj.ascent.states:v',
ascent.interpolate(ys=[200, 150], nodes='state_input'))
p.set_val('traj.ascent.states:gam',
ascent.interpolate(ys=[25, 0], nodes='state_input'),
units='deg')
p.set_val('traj.descent.t_initial', 10.0)
p.set_val('traj.descent.t_duration', 10.0)
p.set_val('traj.descent.states:r',
descent.interpolate(ys=[100, 200], nodes='state_input'))
p.set_val('traj.descent.states:h',
descent.interpolate(ys=[100, 0], nodes='state_input'))
p.set_val('traj.descent.states:v',
descent.interpolate(ys=[150, 200], nodes='state_input'))
p.set_val('traj.descent.states:gam',
descent.interpolate(ys=[0, -45], nodes='state_input'),
units='deg')
dm.run_problem(p, simulate=True)
assert_near_equal(p.get_val('traj.descent.states:r')[-1],
3183.25,
tolerance=1.0E-2)
assert_near_equal(
p.get_val('traj.ascent.timeseries.controls:CD')[-1],
p.get_val('traj.descent.timeseries.parameters:CD')[0])
def test_link_bounded_times_final_to_initial(self):
""" Test that linking phases with times that are fixed at the linkage point raises an exception. """
import openmdao.api as om
import dymos as dm
from dymos.examples.cannonball.size_comp import CannonballSizeComp
from dymos.examples.cannonball.cannonball_phase import CannonballPhase
p = om.Problem(model=om.Group())
p.driver = om.pyOptSparseDriver()
p.driver.options['optimizer'] = 'SLSQP'
p.driver.declare_coloring()
external_params = p.model.add_subsystem('external_params',
om.IndepVarComp())
external_params.add_output('radius', val=0.10, units='m')
external_params.add_output('dens', val=7.87, units='g/cm**3')
external_params.add_design_var('radius',
lower=0.01,
upper=0.10,
ref0=0.01,
ref=0.10)
p.model.add_subsystem('size_comp', CannonballSizeComp())
traj = p.model.add_subsystem('traj', dm.Trajectory())
transcription = dm.Radau(num_segments=5, order=3, compressed=True)
ascent = CannonballPhase(transcription=transcription)
ascent = traj.add_phase('ascent', ascent)
# All initial states except flight path angle are fixed
# Final flight path angle is fixed (we will set it to zero so that the phase ends at apogee)
ascent.set_time_options(initial_bounds=(0, 0),
duration_bounds=(10, 10),
duration_ref=100,
units='s')
ascent.set_state_options('r', fix_initial=True, fix_final=False)
ascent.set_state_options('h', fix_initial=True, fix_final=False)
ascent.set_state_options('gam', fix_initial=False, fix_final=True)
ascent.set_state_options('v', fix_initial=False, fix_final=False)
ascent.add_parameter('S', targets=['aero.S'], units='m**2')
ascent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
# Limit the muzzle energy
ascent.add_boundary_constraint('kinetic_energy.ke',
loc='initial',
units='J',
upper=400000,
lower=0,
ref=100000,
shape=(1, ))
# Second Phase (descent)
transcription = dm.GaussLobatto(num_segments=5,
order=3,
compressed=True)
descent = CannonballPhase(transcription=transcription)
traj.add_phase('descent', descent)
# All initial states and time are free (they will be linked to the final states of ascent.
# Final altitude is fixed (we will set it to zero so that the phase ends at ground impact)
descent.set_time_options(initial_bounds=(10, 10),
duration_bounds=(10, 10),
duration_ref=100,
units='s')
descent.add_state('r', )
descent.add_state('h', fix_initial=False, fix_final=True)
descent.add_state('gam', fix_initial=False, fix_final=False)
descent.add_state('v', fix_initial=False, fix_final=False)
descent.add_parameter('S', targets=['aero.S'], units='m**2')
descent.add_parameter('mass',
targets=['eom.m', 'kinetic_energy.m'],
units='kg')
descent.add_objective('r', loc='final', scaler=-1.0)
# Add internally-managed design parameters to the trajectory.
traj.add_parameter('CD',
targets={
'ascent': ['aero.CD'],
'descent': ['aero.CD']
},
val=0.5,
units=None,
opt=False)
traj.add_parameter('CL',
targets={
'ascent': ['aero.CL'],
'descent': ['aero.CL']
},
val=0.0,
units=None,
opt=False)
traj.add_parameter('T',
targets={
'ascent': ['eom.T'],
'descent': ['eom.T']
},
val=0.0,
units='N',
opt=False)
traj.add_parameter('alpha',
targets={
'ascent': ['eom.alpha'],
'descent': ['eom.alpha']
},
val=0.0,
units='deg',
opt=False)
# Add externally-provided design parameters to the trajectory.
# In this case, we connect 'm' to pre-existing input parameters named 'mass' in each phase.
traj.add_parameter('m',
units='kg',
val=1.0,
targets={
'ascent': 'mass',
'descent': 'mass'
})
# In this case, by omitting targets, we're connecting these parameters to parameters
# with the same name in each phase.
traj.add_parameter('S', units='m**2', val=0.005)
# Link Phases (link time and all state variables)
# Note velocity is not included here. Doing so is equivalent to linking kinetic energy,
# and causes a duplicate row in the constraint jacobian.
traj.link_phases(phases=['ascent', 'descent'],
vars=['time', 'r', 'h', 'gam'],
connected=False)
traj.add_linkage_constraint('ascent',
'descent',
'kinetic_energy.ke',
'kinetic_energy.ke',
ref=100000,
connected=False)
# Issue Connections
p.model.connect('external_params.radius', 'size_comp.radius')
p.model.connect('external_params.dens', 'size_comp.dens')
p.model.connect('size_comp.mass', 'traj.parameters:m')
p.model.connect('size_comp.S', 'traj.parameters:S')
with self.assertRaises(ValueError) as e:
p.setup()
self.assertEqual(
str(e.exception),
'Invalid linkage in Trajectory traj: Cannot link final '
'value of "time" in ascent to initial value of "time" in '
'descent. Values on both sides of the linkage are fixed.')