def setUp(self):
"""
This sets up the test case.
"""
#Define the residual
def f(t,y,yd):
eps = 1.e-6
my = 1./eps
yd_0 = y[1]
yd_1 = my*((1.-y[0]**2)*y[1]-y[0])
res_0 = yd[0]-yd_0
res_1 = yd[1]-yd_1
return N.array([res_0,res_1])
y0 = [2.0,-0.6] #Initial conditions
yd0 = [-.6,-200000.]
#Define an Assimulo problem
self.mod = Implicit_Problem(f,y0,yd0)
self.mod_t0 = Implicit_Problem(f,y0,yd0,1.0)
#Define an explicit solver
self.sim = _Radau5DAE(self.mod) #Create a Radau5 solve
self.sim_t0 = _Radau5DAE(self.mod_t0)
#Sets the parameters
self.sim.atol = 1e-4 #Default 1e-6
self.sim.rtol = 1e-4 #Default 1e-6
self.sim.inith = 1.e-4 #Initial step-size
def test_problem_name_attribute(self):
res = lambda t,y,yd: y
prob = Implicit_Problem(res, 0.0, 0.0)
assert prob.name == "---"
prob = Implicit_Problem(res, 0.0, 0.0, name="Test")
assert prob.name == "Test"
def simulate(self, Tend, nIntervals, gridWidth):
problem = Implicit_Problem(self.rhs, self.y0, self.yd0)
problem.name = 'IDA'
# solver.rhs = self.right_hand_side
problem.handle_result = self.handle_result
problem.state_events = self.state_events
problem.handle_event = self.handle_event
problem.time_events = self.time_events
problem.finalize = self.finalize
# Create IDA object and set additional parameters
simulation = IDA(problem)
simulation.atol = self.atol
simulation.rtol = self.rtol
simulation.verbosity = self.verbosity
if hasattr(simulation, 'continuous_output'):
simulation.continuous_output = False # default 0, if one step approach should be used
elif hasattr(simulation, 'report_continuously'):
simulation.report_continuously = False # default 0, if one step approach should be used
simulation.tout1 = self.tout1
simulation.lsoff = self.lsoff
# Calculate nOutputIntervals:
if gridWidth <> None:
nOutputIntervals = int((Tend - self.t0) / gridWidth)
else:
nOutputIntervals = nIntervals
# Check for feasible input parameters
if nOutputIntervals == 0:
print 'Error: gridWidth too high or nIntervals set to 0! Continue with nIntervals=1'
nOutputIntervals = 1
# Perform simulation
simulation.simulate(Tend, nOutputIntervals) # to get the values: t_new, y_new, yd_new = simulation.simulate
def run_example():
def residual(t,y,yd):
res_0 = yd[0]-y[2]
res_1 = yd[1]-y[3]
res_2 = yd[2]+y[4]*y[0]
res_3 = yd[3]+y[4]*y[1]+9.82
res_4 = y[2]**2+y[3]**2-y[4]*(y[0]**2+y[1]**2)-y[1]*9.82
return np.array([res_0,res_1,res_2,res_3,res_4])
#The initial conditons
t0 = 0.0 #Initial time
y0 = [1.0, 0.0, 0.0, 0.0, 0.0] #Initial conditions
yd0 = [0.0, 0.0, 0.0, -9.82, 0.0] #Initial conditions
print (residual(t0,y0,yd0))
model = Implicit_Problem(residual, y0, yd0, t0) #Create an Assimulo problem
model.name = 'Pendulum' #Specifies the name of problem (optional)
sim = IDA(model) #Create the IDA solver
tfinal = 10.0 #Specify the final time
ncp = 500 #Number of communcation points (number of return points)
t,y,yd = sim.simulate(tfinal, ncp) #Use the .simulate method to simulate and provide the final time and ncp (optional)
sim.plot()
def test_initstep(self):
"""
This tests the funtionality of the property initstep.
"""
def f(t, y, yd):
res_0 = yd[0] - y[1]
res_1 = yd[1] + 9.82 - 0.01 * y[1]**2
return N.array([res_0, res_1])
mod = Implicit_Problem(f, y0=[5.0, 0.0], yd0=[0.0, 9.82])
sim = IDA(mod)
sim.simulate(2.0)
nose.tools.assert_almost_equal(sim.y_sol[-1][0],
-13.4746473811,
places=7)
sim.reset()
sim.inith = 1e-10
sim.simulate(2.0)
nose.tools.assert_almost_equal(sim.y_sol[-1][0],
-13.4746596311,
places=7)
def test_usejac(self):
"""
This tests the functionality of the property usejac.
"""
f = lambda t, x, xd: N.array([xd[0] - x[1], xd[1] - 9.82]
) #Defines the rhs
jac = lambda c, t, x, xd: N.array([[c, -1.], [0., c]]
) #Defines the jacobian
imp_mod = Implicit_Problem(f, [1.0, 0.0], [0., -9.82])
imp_mod.jac = jac
imp_sim = IDA(imp_mod)
imp_sim.simulate(3, 100)
assert imp_sim.statistics["nfevalsLS"] == 0
nose.tools.assert_almost_equal(imp_sim.y_sol[-1][0], 45.1900000, 4)
imp_sim.reset()
imp_sim.usejac = False
imp_sim.simulate(3., 100)
nose.tools.assert_almost_equal(imp_sim.y_sol[-1][0], 45.1900000, 4)
assert imp_sim.statistics["nfevalsLS"] > 0
def test_time_event(self):
f = lambda t, y, yd: y - yd
global tnext
global nevent
tnext = 0.0
nevent = 0
def time_events(t, y, yd, sw):
global tnext, nevent
events = [1.0, 2.0, 2.5, 3.0]
for ev in events:
if t < ev:
tnext = ev
break
else:
tnext = None
nevent += 1
return tnext
def handle_event(solver, event_info):
solver.y += 1.0
global tnext
nose.tools.assert_almost_equal(solver.t, tnext)
assert event_info[0] == []
assert event_info[1] == True
exp_mod = Implicit_Problem(f, 0.0, 0.0)
exp_mod.time_events = time_events
exp_mod.handle_event = handle_event
#CVode
exp_sim = IDA(exp_mod)
exp_sim(5., 100)
assert nevent == 5
def test_completed_step(self):
"""
This tests the functionality of the method completed_step.
"""
global nsteps
nsteps = 0
def f(t, y, yd):
res_1 = y[0] + y[1] + 1.0
res_2 = y[1]
return N.array([res_1, res_2])
def completed_step(solver):
global nsteps
nsteps += 1
y0 = [-1.0, 0.0]
yd0 = [1.0, 0.0]
mod = Implicit_Problem(f, y0, yd0)
mod.step_events = completed_step
sim = IDA(mod)
sim.simulate(2., 100)
assert len(sim.t_sol) == 101
assert nsteps == sim.statistics["nsteps"]
sim = IDA(mod)
nsteps = 0
sim.simulate(2.)
assert len(sim.t_sol) == sim.statistics["nsteps"] + 1
assert nsteps == sim.statistics["nsteps"]
def setUp(self):
"""
This sets up the test case.
"""
class Prob_IDA(Implicit_Problem):
def __init__(self):
pass
res = lambda self,t,y,yd,sw: N.array([y[0]-1.0])
state_events = lambda self,t,y,yd,sw: N.array([t-1.0, t])
y0 = [1.0]
yd0 = [1.0]
sw0 = [False, True]
res = Prob_IDA()
class Prob_CVode(Explicit_Problem):
def __init__(self):
pass
rhs = lambda self,t,y,sw: N.array([1.0])
state_events = lambda self,t,y,sw: N.array([t-1.0, t])
y0 = [1.0]
sw0 = [False, True]
f = Prob_CVode()
self.simulators = [IDA(res), CVode(f)]
f = lambda t,y,yd,p: N.array([0.0])
y0 = [1.0]
yd0 = [1.0]
p0 = [1.0]
mod = Implicit_Problem(f, y0,yd0,p0=p0)
self.sim = IDA(mod)
def test_elapsed_step_time(self):
res = lambda t,y,yd: y
prob = Implicit_Problem(res, 0.0, 0.0)
solv = Implicit_ODE(prob)
assert solv.get_elapsed_step_time() == -1.0
def test_implicit_problem(self):
f = lambda t,y,yd: yd + 1
y0 = [1.0, 1.0, 1.0]
yd0 = [-1.0, -1.0, -1.0]
self.problem = Implicit_Problem(f,y0, yd0)
self.simulator = ODASSL(self.problem)
self.simulator.simulate(1)
def solve(self):
# Setup IDA
assert self._initial_time is not None
problem = Implicit_Problem(self._residual_vector_eval,
self.solution.vector(),
self.solution_dot.vector(),
self._initial_time)
problem.jac = self._jacobian_matrix_eval
problem.handle_result = self._monitor
# Define an Assimulo IDA solver
solver = IDA(problem)
# Setup options
assert self._time_step_size is not None
solver.inith = self._time_step_size
if self._absolute_tolerance is not None:
solver.atol = self._absolute_tolerance
if self._max_time_steps is not None:
solver.maxsteps = self._max_time_steps
if self._relative_tolerance is not None:
solver.rtol = self._relative_tolerance
if self._report:
solver.verbosity = 10
solver.display_progress = True
solver.report_continuously = True
else:
solver.display_progress = False
solver.verbosity = 50
# Assert consistency of final time and time step size
assert self._final_time is not None
final_time_consistency = (
self._final_time - self._initial_time) / self._time_step_size
assert isclose(
round(final_time_consistency), final_time_consistency
), ("Final time should be occuring after an integer number of time steps"
)
# Prepare monitor computation if not provided by parameters
if self._monitor_initial_time is None:
self._monitor_initial_time = self._initial_time
assert isclose(
round(self._monitor_initial_time / self._time_step_size),
self._monitor_initial_time / self._time_step_size
), ("Monitor initial time should be a multiple of the time step size"
)
if self._monitor_time_step_size is None:
self._monitor_time_step_size = self._time_step_size
assert isclose(
round(self._monitor_time_step_size / self._time_step_size),
self._monitor_time_step_size / self._time_step_size
), ("Monitor time step size should be a multiple of the time step size"
)
monitor_t = arange(
self._monitor_initial_time,
self._final_time + self._monitor_time_step_size / 2.,
self._monitor_time_step_size)
# Solve
solver.simulate(self._final_time, ncp_list=monitor_t)
def setUp(self):
"""
This function sets up the test case.
"""
f = lambda t, y, yd: y
y0 = [1.0]
yd0 = [1.0]
self.problem = Implicit_Problem(f, y0, yd0)
self.simulator = IDA(self.problem)
def test_time_limit(self):
f = lambda t, y, yd: yd - y
exp_mod = Implicit_Problem(f, 1.0, 1.0)
exp_sim = IDA(exp_mod)
exp_sim.maxh = 1e-8
exp_sim.time_limit = 1 #One second
exp_sim.report_continuously = True
nose.tools.assert_raises(TimeLimitExceeded, exp_sim.simulate, 1)
def solve_squeezer(self):
model = Implicit_Problem(self.squeezer_func, self.y, self.yp, self.t)
sim = IDA(model)
tfinal = 10.0
ncp = 1000
sim.atol = 1.0e-6
sim.suppress_alg = True
for i in range(sim.algvar.size):
if i > 14:
sim.algvar[i] = 0
print(sim.algvar)
t, y, yd = sim.simulate(tfinal, ncp)
sim.plot()
def run_example(with_plots=True):
"""
The same as example :doc:`EXAMPLE_cvode_basic` but now integrated backwards in time.
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
# Define the rhs
def f(t, y, yd):
res = yd[0] + y[0] * param
return N.array([res])
param = 2
# Define an Assimulo problem
imp_mod = Implicit_Problem(
f,
t0=5,
y0=0.02695,
yd0=-0.02695,
name='IDA Example: $\dot y + y = 0$ (reverse time)')
# Define an explicit solver
imp_sim = IDA(imp_mod) # Create a IDA solver
# Sets the parameters
imp_sim.atol = [1e-8] # Default 1e-6
imp_sim.rtol = 1e-8 # Default 1e-6
imp_sim.backward = True
# Simulate
t, y, yd = imp_sim.simulate(0) # Simulate 5 seconds (t0=5 -> tf=0)
# Basic test
#nose.tools.assert_almost_equal(float(y[-1]), 4.00000000, 3)
# Plot
if with_plots:
P.plot(t, y, color="b")
P.xlabel('Time')
P.ylabel('State')
P.title(imp_mod.name)
P.show()
return imp_mod, imp_sim
def initialize_ode_solver(self, y_0, yd_0, t_0):
model = Implicit_Problem(self.residual, y_0, yd_0, t_0)
model.handle_result = self.handle_result
solver = IDA(model)
solver.rtol = self.solver_rtol
solver.atol = self.solver_atol # * np.array([100, 10, 1e-4, 1e-4])
solver.inith = 0.1 # self.wind.R_g / const.C
solver.maxh = self.dt * self.wind.R_g / const.C
solver.report_continuously = True
solver.display_progress = False
solver.verbosity = 50 # 50 = quiet
solver.num_threads = 3
# solver.display_progress = True
return solver
def test_interpolate(self):
"""
This tests the functionality of the method interpolate.
"""
f = lambda t, y, yd: y**0.25 - yd
prob = Implicit_Problem(f, [1.0], [1.0])
sim = IDA(prob)
sim.simulate(10., 100)
y100 = sim.y_sol
t100 = sim.t_sol
sim.reset()
sim.simulate(10.)
nose.tools.assert_almost_equal(y100[-2], sim.interpolate(9.9, 0), 5)
def test_re_init(self):
res = lambda t,y,yd: y
prob = Implicit_Problem(res, 0.0, 0.0)
solv = Implicit_ODE(prob)
assert solv.t == 0.0
assert solv.y[0] == 0.0
assert solv.yd[0] == 0.0
solv.re_init(1.0, 2.0, 3.0)
assert solv.t == 1.0
assert solv.y[0] == 2.0
assert solv.yd[0] == 3.0
def test_handle_result(self):
"""
This function tests the handle result.
"""
f = lambda t,x,xd: x**0.25-xd
def handle_result(solver, t ,y, yd):
assert solver.t == t
prob = Implicit_Problem(f, [1.0],[1.0])
prob.handle_result = handle_result
sim = IDA(prob)
sim.report_continuously = True
sim.simulate(10.)
def test_switches(self):
"""
This tests that the switches are actually turned when override.
"""
res = lambda t,x,xd,sw: N.array([1.0 - xd])
state_events = lambda t,x,xd,sw: N.array([x[0]-1.])
def handle_event(solver, event_info):
solver.sw = [False] #Override the switches to point to another instance
mod = Implicit_Problem(res,[0.0], [1.0])
mod.sw0 = [True]
mod.state_events = state_events
mod.handle_event = handle_event
sim = Radau5DAE(mod)
assert sim.sw[0] == True
sim.simulate(3)
assert sim.sw[0] == False
def test_time_event(self):
"""
This tests the functionality of the time event function.
"""
f = lambda t, x, xd, sw: xd - x
def time_events(t, y, yd, sw):
if sw[0]:
return 1.0
if sw[1]:
return 3.0
return None
def handle_event(solver, event_info):
if event_info[1]:
solver.y = N.array([1.0])
solver.yd = N.array([1.0])
if not solver.sw[0]:
solver.sw[1] = False
if solver.sw[0]:
solver.sw[0] = False
mod = Implicit_Problem(f, [1.0], [1.0])
mod.time_events = time_events
mod.handle_event = handle_event
mod.switches0 = [True, True]
sim = IDA(mod)
sim.simulate(5.0)
nose.tools.assert_almost_equal(sim.y_sol[38], 1.0000000, 5)
nose.tools.assert_almost_equal(sim.y_sol[87], 1.0000000, 5)
sim = IDA(mod, [1.0], [1.0])
sim.simulate(2.0)
nose.tools.assert_almost_equal(sim.t_sol[-1], 2.0000000, 5)
def run_example(with_plots=True):
#Define the residual
def f(t, y, yd):
eps = 1.e-6
my = 1. / eps
yd_0 = y[1]
yd_1 = my * ((1. - y[0]**2) * y[1] - y[0])
res_0 = yd[0] - yd_0
res_1 = yd[1] - yd_1
return N.array([res_0, res_1])
y0 = [2.0, -0.6] #Initial conditions
yd0 = [-.6, -200000.]
#Define an Assimulo problem
imp_mod = Implicit_Problem(f, y0, yd0)
imp_mod.name = 'Van der Pol (implicit)'
#Define an explicit solver
imp_sim = Radau5DAE(imp_mod) #Create a Radau5 solver
#Sets the parameters
imp_sim.atol = 1e-4 #Default 1e-6
imp_sim.rtol = 1e-4 #Default 1e-6
imp_sim.inith = 1.e-4 #Initial step-size
#Simulate
t, y, yd = imp_sim.simulate(2.) #Simulate 2 seconds
#Plot
if with_plots:
P.plot(t, y[:, 0], marker='o')
P.show()
#Basic test
x1 = y[:, 0]
assert N.abs(x1[-1] - 1.706168035) < 1e-3 #For test purpose
def test_make_consistency(self):
"""
This tests the functionality of the method make_consistency.
"""
def f(t, y, yd):
res_1 = y[0] + y[1] + 1.0
res_2 = y[1]
return N.array([res_1, res_2])
y0 = [2.0, 2.0]
yd0 = [1.0, 0.0]
my_Prob = Implicit_Problem(f, y0, yd0)
simulator = IDA(my_Prob)
[flag, y, yd] = simulator.make_consistent('IDA_Y_INIT')
nose.tools.assert_almost_equal(y[1], 0.00000)
nose.tools.assert_almost_equal(y[0], -1.0000)
nose.tools.assert_almost_equal(yd[0], 1.0000)
nose.tools.assert_almost_equal(yd[1], 0.0000)
def test_pbar(self):
"""
Tests the property of pbar.
"""
f = lambda t, y, p: N.array([0.0] * len(y))
y0 = [1.0] * 2
p0 = [1000.0, -100.0]
exp_mod = Explicit_Problem(f, y0, p0=p0)
exp_sim = CVode(exp_mod)
nose.tools.assert_almost_equal(exp_sim.pbar[0], 1000.00000, 4)
nose.tools.assert_almost_equal(exp_sim.pbar[1], 100.000000, 4)
f = lambda t, y, yd, p: N.array([0.0] * len(y))
yd0 = [0.0] * 2
imp_mod = Implicit_Problem(f, y0, yd0, p0=p0)
imp_sim = IDA(imp_mod)
nose.tools.assert_almost_equal(imp_sim.pbar[0], 1000.00000, 4)
nose.tools.assert_almost_equal(imp_sim.pbar[1], 100.000000, 4)
def test_clear_event_log(self):
"""
This tests the functionality of the time event function.
"""
f = lambda t, x, xd, sw: xd - x
def time_events(t, y, yd, sw):
if sw[0]:
return 1.0
if sw[1]:
return 3.0
return None
def handle_event(solver, event_info):
if event_info[1]:
solver.y = N.array([1.0])
solver.yd = N.array([1.0])
if not solver.sw[0]:
solver.sw[1] = False
if solver.sw[0]:
solver.sw[0] = False
mod = Implicit_Problem(f, [1.0], [1.0], sw0=[True, True])
mod.time_events = time_events
mod.handle_event = handle_event
sim = IDA(mod)
sim.verbosity = 10
assert len(sim.event_data) == 0
sim.simulate(5.0)
assert len(sim.event_data) > 0
sim.reset()
assert len(sim.event_data) == 0
sim.simulate(5.0)
assert len(sim.event_data) > 0
def run_example(with_plots=True):
"""
This example show how to use Assimulo and IDA for simulating sensitivities
for initial conditions.::
0 = dy1/dt - -(k01+k21+k31)*y1 - k12*y2 - k13*y3 - b1
0 = dy2/dt - k21*y1 + (k02+k12)*y2
0 = dy3/dt - k31*y1 + k13*y3
y1(0) = p1, y2(0) = p2, y3(0) = p3
p1=p2=p3 = 0
See http://sundials.2283335.n4.nabble.com/Forward-sensitivities-for-initial-conditions-td3239724.html
on return:
- :dfn:`imp_mod` problem instance
- :dfn:`imp_sim` solver instance
"""
def f(t, y, yd, p):
y1, y2, y3 = y
yd1, yd2, yd3 = yd
k01 = 0.0211
k02 = 0.0162
k21 = 0.0111
k12 = 0.0124
k31 = 0.0039
k13 = 0.000035
b1 = 49.3
res_0 = -yd1 - (k01 + k21 + k31) * y1 + k12 * y2 + k13 * y3 + b1
res_1 = -yd2 + k21 * y1 - (k02 + k12) * y2
res_2 = -yd3 + k31 * y1 - k13 * y3
return N.array([res_0, res_1, res_2])
#The initial conditions
y0 = [0.0, 0.0, 0.0] #Initial conditions for y
yd0 = [49.3, 0., 0.]
p0 = [0.0, 0.0, 0.0] #Initial conditions for parameters
yS0 = N.array([[1, 0, 0], [0, 1, 0], [0, 0, 1.]])
#Create an Assimulo implicit problem
imp_mod = Implicit_Problem(f,
y0,
yd0,
p0=p0,
name='Example: Computing Sensitivities')
#Sets the options to the problem
imp_mod.yS0 = yS0
#Create an Assimulo explicit solver (IDA)
imp_sim = IDA(imp_mod)
#Sets the paramters
imp_sim.rtol = 1e-7
imp_sim.atol = 1e-6
imp_sim.pbar = [
1, 1, 1
] #pbar is used to estimate the tolerances for the parameters
imp_sim.report_continuously = True #Need to be able to store the result using the interpolate methods
imp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
imp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
#Simulate
t, y, yd = imp_sim.simulate(400) #Simulate 400 seconds
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 1577.6552477, 3)
nose.tools.assert_almost_equal(y[-1][1], 611.9574565, 3)
nose.tools.assert_almost_equal(y[-1][2], 2215.88563217, 3)
nose.tools.assert_almost_equal(imp_sim.p_sol[0][1][0], 1.0)
#Plot
if with_plots:
P.figure(1)
P.subplot(221)
P.plot(t,
N.array(imp_sim.p_sol[0])[:, 0], t,
N.array(imp_sim.p_sol[0])[:, 1], t,
N.array(imp_sim.p_sol[0])[:, 2])
P.title("Parameter p1")
P.legend(("p1/dy1", "p1/dy2", "p1/dy3"))
P.subplot(222)
P.plot(t,
N.array(imp_sim.p_sol[1])[:, 0], t,
N.array(imp_sim.p_sol[1])[:, 1], t,
N.array(imp_sim.p_sol[1])[:, 2])
P.title("Parameter p2")
P.legend(("p2/dy1", "p2/dy2", "p2/dy3"))
P.subplot(223)
P.plot(t,
N.array(imp_sim.p_sol[2])[:, 0], t,
N.array(imp_sim.p_sol[2])[:, 1], t,
N.array(imp_sim.p_sol[2])[:, 2])
P.title("Parameter p3")
P.legend(("p3/dy1", "p3/dy2", "p3/dy3"))
P.subplot(224)
P.title('ODE Solution')
P.plot(t, y)
P.suptitle(imp_mod.name)
P.show()
return imp_mod, imp_sim
+ ct.omega ** 2 / ct.vA(x,ct.z) ** 2 * u_perp
res = np.zeros(6 * ct.nz)
res[0:ct.nz] = np.real(dux + 1j * ct.omega * b_par + nabla_perp_u_perp)
res[ct.nz:2 * ct.nz] = np.imag(dux + 1j * ct.omega * b_par +
nabla_perp_u_perp)
res[2 * ct.nz:3 * ct.nz] = np.real(db_par + 1j / ct.omega * L_ux)
res[3 * ct.nz:4 * ct.nz] = np.imag(db_par + 1j / ct.omega * L_ux)
res[4 * ct.nz:5 * ct.nz] = np.real(L_u_perp -
1j * ct.omega * nabla_perp_b_par)
res[5 * ct.nz:] = np.imag(L_u_perp - 1j * ct.omega * nabla_perp_b_par)
return res
model = Implicit_Problem(residual, ct.U_x_min, ct.dU_x_min, ct.x_min)
model.name = 'Resonant abosprotion'
sim = IDA(model)
x, U, dU = sim.simulate(ct.x_max, ct.nx)
# sim.plot()
# fig = plt.figure()
# ax = fig.add_subplot(111)
# ax.plot(ct.z, ct.dU_perp_x_min[0:ct.nz])
# ax.plot(ct.z, np.real(ct.du_perp_ana(ct.x_min,ct.z)))
# fig = plt.figure()
# ax = fig.add_subplot(111)
def run_example(with_plots=True):
r"""
An example for IDA with scaled preconditioned GMRES method
as a special linear solver.
Note, how the operation Jacobian times vector is provided.
ODE:
.. math::
\dot y_1 - y_2 &= 0\\
\dot y_2 -9.82 &= 0
on return:
- :dfn:`imp_mod` problem instance
- :dfn:`imp_sim` solver instance
"""
#Defines the residual
def res(t, y, yd):
res_0 = yd[0] - y[1]
res_1 = yd[1] + 9.82
return N.array([res_0, res_1])
#Defines the Jacobian*vector product
def jacv(t, y, yd, res, v, c):
jy = N.array([[0, -1.], [0, 0]])
jyd = N.array([[1, 0.], [0, 1]])
j = jy + c * jyd
return N.dot(j, v)
#Initial conditions
y0 = [1.0, 0.0]
yd0 = [0.0, -9.82]
#Defines an Assimulo implicit problem
imp_mod = Implicit_Problem(
res, y0, yd0, name='Example using the Jacobian Vector product')
imp_mod.jacv = jacv #Sets the jacobian
imp_sim = IDA(imp_mod) #Create an IDA solver instance
#Set the parameters
imp_sim.atol = 1e-5 #Default 1e-6
imp_sim.rtol = 1e-5 #Default 1e-6
imp_sim.linear_solver = 'SPGMR' #Change linear solver
#imp_sim.options["usejac"] = False
#Simulate
t, y, yd = imp_sim.simulate(
5, 1000) #Simulate 5 seconds with 1000 communication points
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], -121.75000000, 4)
nose.tools.assert_almost_equal(y[-1][1], -49.100000000)
#Plot
if with_plots:
P.plot(t, y)
P.xlabel('Time')
P.ylabel('State')
P.title(imp_mod.name)
P.show()
return imp_mod, imp_sim
def dae_solver(residual,
y0,
yd0,
t0,
p0=None,
jac=None,
name='DAE',
solver='IDA',
algvar=None,
atol=1e-6,
backward=False,
display_progress=True,
pbar=None,
report_continuously=False,
rtol=1e-6,
sensmethod='STAGGERED',
suppress_alg=False,
suppress_sens=False,
usejac=False,
usesens=False,
verbosity=30,
tfinal=10.,
ncp=500):
'''
DAE solver.
Parameters
----------
residual: function
Implicit DAE model.
y0: List[float]
Initial model state.
yd0: List[float]
Initial model state derivatives.
t0: float
Initial simulation time.
p0: List[float]
Parameters for which sensitivites are to be calculated.
jac: function
Model jacobian.
name: string
Model name.
solver: string
DAE solver.
algvar: List[bool]
A list for defining which variables are differential and which are algebraic.
The value True(1.0) indicates a differential variable and the value False(0.0) indicates an algebraic variable.
atol: float
Absolute tolerance.
backward: bool
Specifies if the simulation is done in reverse time.
display_progress: bool
Actives output during the integration in terms of that the current integration is periodically printed to the stdout.
Report_continuously needs to be activated.
pbar: List[float]
An array of positive floats equal to the number of parameters. Default absolute values of the parameters.
Specifies the order of magnitude for the parameters. Useful if IDAS is to estimate tolerances for the sensitivity solution vectors.
report_continuously: bool
Specifies if the solver should report the solution continuously after steps.
rtol: float
Relative tolerance.
sensmethod: string
Specifies the sensitivity solution method.
Can be either ‘SIMULTANEOUS’ or ‘STAGGERED’. Default is 'STAGGERED'.
suppress_alg: bool
Indicates that the error-tests are suppressed on algebraic variables.
suppress_sens: bool
Indicates that the error-tests are suppressed on the sensitivity variables.
usejac: bool
Sets the option to use the user defined jacobian.
usesens: bool
Aactivates or deactivates the sensitivity calculations.
verbosity: int
Determines the level of the output.
QUIET = 50 WHISPER = 40 NORMAL = 30 LOUD = 20 SCREAM = 10
tfinal: float
Simulation final time.
ncp: int
Number of communication points (number of return points).
Returns
-------
sol: solution [time, model states], List[float]
'''
if usesens is True: # parameter sensitivity
model = Implicit_Problem(residual, y0, yd0, t0, p0=p0)
else:
model = Implicit_Problem(residual, y0, yd0, t0)
model.name = name
if usejac is True: # jacobian
model.jac = jac
if algvar is not None: # differential or algebraic variables
model.algvar = algvar
if solver == 'IDA': # solver
from assimulo.solvers import IDA
sim = IDA(model)
sim.atol = atol
sim.rtol = rtol
sim.backward = backward # backward in time
sim.report_continuously = report_continuously
sim.display_progress = display_progress
sim.suppress_alg = suppress_alg
sim.verbosity = verbosity
if usesens is True: # sensitivity
sim.sensmethod = sensmethod
sim.pbar = np.abs(p0)
sim.suppress_sens = suppress_sens
# Simulation
# t, y, yd = sim.simulate(tfinal, ncp=(ncp - 1))
ncp_list = np.linspace(t0, tfinal, num=ncp, endpoint=True)
t, y, yd = sim.simulate(tfinal, ncp=0, ncp_list=ncp_list)
# Plot
# sim.plot()
# plt.figure()
# plt.subplot(221)
# plt.plot(t, y[:, 0], 'b.-')
# plt.legend([r'$\lambda$'])
# plt.subplot(222)
# plt.plot(t, y[:, 1], 'r.-')
# plt.legend([r'$\dot{\lambda}$'])
# plt.subplot(223)
# plt.plot(t, y[:, 2], 'k.-')
# plt.legend([r'$\eta$'])
# plt.subplot(224)
# plt.plot(t, y[:, 3], 'm.-')
# plt.legend([r'$\dot{\eta}$'])
# plt.figure()
# plt.subplot(221)
# plt.plot(t, yd[:, 0], 'b.-')
# plt.legend([r'$\dot{\lambda}$'])
# plt.subplot(222)
# plt.plot(t, yd[:, 1], 'r.-')
# plt.legend([r'$\ddot{\lambda}$'])
# plt.subplot(223)
# plt.plot(t, yd[:, 2], 'k.-')
# plt.legend([r'$\dot{\eta}$'])
# plt.subplot(224)
# plt.plot(t, yd[:, 3], 'm.-')
# plt.legend([r'$\ddot{\eta}$'])
# plt.figure()
# plt.subplot(121)
# plt.plot(y[:, 0], y[:, 1])
# plt.xlabel(r'$\lambda$')
# plt.ylabel(r'$\dot{\lambda}$')
# plt.subplot(122)
# plt.plot(y[:, 2], y[:, 3])
# plt.xlabel(r'$\eta$')
# plt.ylabel(r'$\dot{\eta}$')
# plt.figure()
# plt.subplot(121)
# plt.plot(yd[:, 0], yd[:, 1])
# plt.xlabel(r'$\dot{\lambda}$')
# plt.ylabel(r'$\ddot{\lambda}$')
# plt.subplot(122)
# plt.plot(yd[:, 2], yd[:, 3])
# plt.xlabel(r'$\dot{\eta}$')
# plt.ylabel(r'$\ddot{\eta}$')
# plt.figure()
# plt.subplot(121)
# plt.plot(y[:, 0], y[:, 2])
# plt.xlabel(r'$\lambda$')
# plt.ylabel(r'$\eta$')
# plt.subplot(122)
# plt.plot(y[:, 1], y[:, 3])
# plt.xlabel(r'$\dot{\lambda}$')
# plt.ylabel(r'$\dot{\eta}$')
# plt.figure()
# plt.subplot(121)
# plt.plot(yd[:, 0], yd[:, 2])
# plt.xlabel(r'$\dot{\lambda}$')
# plt.ylabel(r'$\dot{\eta}$')
# plt.subplot(122)
# plt.plot(yd[:, 1], yd[:, 3])
# plt.xlabel(r'$\ddot{\lambda}$')
# plt.ylabel(r'$\ddot{\eta}$')
# plt.show()
sol = [t, y, yd]
return sol
