def run_example(with_plots=True):
global exp_mod, exp_sim
#Define the rhs
def f(t, y):
ydot = -y[0]
return N.array([ydot])
#Define an Assimulo problem
exp_mod = Explicit_Problem(f, y0=4)
exp_mod.name = 'Simple CVode Example'
#Define an explicit solver
exp_sim = CVode(exp_mod) #Create a CVode solver
#Sets the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = [1e-4] #Default 1e-6
exp_sim.rtol = 1e-4 #Default 1e-6
#Simulate
t1, y1 = exp_sim.simulate(5, 100) #Simulate 5 seconds
t2, y2 = exp_sim.simulate(7) #Simulate 2 seconds more
#Basic test
nose.tools.assert_almost_equal(y2[-1], 0.00347746, 5)
def _solve_cvode(f, t, y0, args, console=False, max_steps=1000, terminate=False):
"""Wrapper for Assimulo's CVODE-Sundials routine
"""
def rhs(t, u):
return f(u, t, *args)
prob = Explicit_Problem(rhs, y0)
sim = CVode(prob)
sim.rtol = 1e-15
sim.atol = 1e-12
sim.discr = 'BDF'
sim.maxord = 5
sim.maxsteps = max_steps
if not console:
sim.verbosity = 50
else:
sim.report_continuously = True
t_end = t[-1]
steps = len(t)
try:
print t_end, steps
t, x = sim.simulate(t_end, steps)
except CVodeError, e:
raise ValueError("Something broke in CVode: %r" % e)
return None, False
def run_example(with_plots=True):
global exp_mod, exp_sim
#Define the rhs
def f(t,y):
ydot = -y[0]
return N.array([ydot])
#Define an Assimulo problem
exp_mod = Explicit_Problem(f, y0=4)
exp_mod.name = 'Simple CVode Example'
#Define an explicit solver
exp_sim = CVode(exp_mod) #Create a CVode solver
#Sets the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = [1e-4] #Default 1e-6
exp_sim.rtol = 1e-4 #Default 1e-6
#Simulate
t1, y1 = exp_sim.simulate(5,100) #Simulate 5 seconds
t2, y2 = exp_sim.simulate(7) #Simulate 2 seconds more
#Basic test
nose.tools.assert_almost_equal(y2[-1], 0.00347746, 5)
def run_example(with_plots=True):
"""
This is the same example from the Sundials package (cvsRoberts_FSA_dns.c)
This simple example problem for CVode, due to Robertson,
is from chemical kinetics, and consists of the following three
equations::
dy1/dt = -p1*y1 + p2*y2*y3
dy2/dt = p1*y1 - p2*y2*y3 - p3*y2**2
dy3/dt = p3*(y2)^2
"""
def f(t, y, p):
yd_0 = -p[0]*y[0]+p[1]*y[1]*y[2]
yd_1 = p[0]*y[0]-p[1]*y[1]*y[2]-p[2]*y[1]**2
yd_2 = p[2]*y[1]**2
return N.array([yd_0,yd_1,yd_2])
#The initial conditions
y0 = [1.0,0.0,0.0] #Initial conditions for y
#Create an Assimulo explicit problem
exp_mod = Explicit_Problem(f,y0)
#Sets the options to the problem
exp_mod.p0 = [0.040, 1.0e4, 3.0e7] #Initial conditions for parameters
exp_mod.pbar = [0.040, 1.0e4, 3.0e7]
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1.e-4
exp_sim.atol = N.array([1.0e-8, 1.0e-14, 1.0e-6])
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
exp_sim.continuous_output = True
#Simulate
t, y = exp_sim.simulate(4,400) #Simulate 4 seconds with 400 communication points
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 9.05518032e-01, 4)
nose.tools.assert_almost_equal(y[-1][1], 2.24046805e-05, 4)
nose.tools.assert_almost_equal(y[-1][2], 9.44595637e-02, 4)
nose.tools.assert_almost_equal(exp_sim.p_sol[0][-1][0], -1.8761, 2) #Values taken from the example in Sundials
nose.tools.assert_almost_equal(exp_sim.p_sol[1][-1][0], 2.9614e-06, 8)
nose.tools.assert_almost_equal(exp_sim.p_sol[2][-1][0], -4.9334e-10, 12)
#Plot
if with_plots:
P.plot(t, y)
P.show()
def _setup_sim(self, **kwargs):
""" Create a simulation interface to Assimulo using CVODE, given
a problem definition
"""
# Create Assimulo interface
sim = CVode(self.prob)
sim.discr = 'BDF'
sim.maxord = 5
# Setup some default arguments for the ODE solver, or override
# if available. This is very hackish, but it's fine for now while
# the number of anticipated tuning knobs is small.
if 'maxh' in kwargs:
sim.maxh = kwargs['maxh']
else:
sim.maxh = np.min([0.1, self.output_dt])
if 'minh' in kwargs:
sim.minh = kwargs['minh']
# else: sim.minh = 0.001
if "iter" in kwargs:
sim.iter = kwargs['iter']
else:
sim.iter = 'Newton'
if "linear_solver" in kwargs:
sim.linear_solver = kwargs['linear_solver']
if "max_steps" in kwargs: # DIFFERENT NAME!!!!
sim.maxsteps = kwargs['max_steps']
else:
sim.maxsteps = 1000
if "time_limit" in kwargs:
sim.time_limit = kwargs['time_limit']
sim.report_continuously = True
else:
sim.time_limit = 0.0
# Don't save the [t_-, t_+] around events
sim.store_event_points = False
# Setup tolerances
nr = self.args[0]
sim.rtol = state_rtol
sim.atol = state_atol + [1e-12]*nr
if not self.console:
sim.verbosity = 50
else:
sim.verbosity = 40
# sim.report_continuously = False
# Save the Assimulo interface
return sim
def _setup_sim(self, **kwargs):
""" Create a simulation interface to Assimulo using CVODE, given
a problem definition
"""
# Create Assimulo interface
sim = CVode(self.prob)
sim.discr = 'BDF'
sim.maxord = 5
# Setup some default arguments for the ODE solver, or override
# if available. This is very hackish, but it's fine for now while
# the number of anticipated tuning knobs is small.
if 'maxh' in kwargs:
sim.maxh = kwargs['maxh']
else:
sim.maxh = np.min([0.1, self.output_dt])
if 'minh' in kwargs:
sim.minh = kwargs['minh']
# else: sim.minh = 0.001
if "iter" in kwargs:
sim.iter = kwargs['iter']
else:
sim.iter = 'Newton'
if "linear_solver" in kwargs:
sim.linear_solver = kwargs['linear_solver']
if "max_steps" in kwargs: # DIFFERENT NAME!!!!
sim.maxsteps = kwargs['max_steps']
else:
sim.maxsteps = 1000
if "time_limit" in kwargs:
sim.time_limit = kwargs['time_limit']
sim.report_continuously = True
else:
sim.time_limit = 0.0
# Don't save the [t_-, t_+] around events
sim.store_event_points = False
# Setup tolerances
nr = self.args[0]
sim.rtol = state_rtol
sim.atol = state_atol + [1e-12] * nr
if not self.console:
sim.verbosity = 50
else:
sim.verbosity = 40
# sim.report_continuously = False
# Save the Assimulo interface
return sim
def run_example(with_plots=True):
#Defines the rhs
def f(t, y):
yd_0 = y[1]
yd_1 = -9.82
return N.array([yd_0, yd_1])
#Defines the jacobian*vector product
def jacv(t, y, fy, v):
j = N.array([[0, 1.], [0, 0]])
return N.dot(j, v)
y0 = [1.0, 0.0] #Initial conditions
#Defines an Assimulo explicit problem
exp_mod = Explicit_Problem(f, y0)
exp_mod.jacv = jacv #Sets the jacobian
exp_mod.name = 'Example using the Jacobian Vector product'
exp_sim = CVode(exp_mod) #Create a CVode solver
#Set the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = 1e-5 #Default 1e-6
exp_sim.rtol = 1e-5 #Default 1e-6
exp_sim.linear_solver = 'SPGMR' #Change linear solver
#exp_sim.options["usejac"] = False
#Simulate
t, y = exp_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.show()
def run_example(with_plots=True):
global t, y
#Defines the rhs
def f(t, y):
yd_0 = y[1]
yd_1 = -9.82
#print y, yd_0, yd_1
return N.array([yd_0, yd_1])
#Defines the jacobian
def jac(t, y):
j = N.array([[0, 1.], [0, 0]])
return j
#Defines an Assimulo explicit problem
y0 = [1.0, 0.0] #Initial conditions
exp_mod = Explicit_Problem(f, y0)
exp_mod.jac = jac #Sets the jacobian
exp_mod.name = 'Example using Jacobian'
exp_sim = CVode(exp_mod) #Create a CVode solver
#Set the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = 1e-5 #Default 1e-6
exp_sim.rtol = 1e-5 #Default 1e-6
#Simulate
t, y = exp_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, linestyle="dashed", marker="o") #Plot the solution
P.show()
def run_example(with_plots=True):
#Defines the rhs
def f(t,y):
yd_0 = y[1]
yd_1 = -9.82
return N.array([yd_0,yd_1])
#Defines the jacobian*vector product
def jacv(t,y,fy,v):
j = N.array([[0,1.],[0,0]])
return N.dot(j,v)
y0 = [1.0,0.0] #Initial conditions
#Defines an Assimulo explicit problem
exp_mod = Explicit_Problem(f,y0)
exp_mod.jacv = jacv #Sets the jacobian
exp_mod.name = 'Example using the Jacobian Vector product'
exp_sim = CVode(exp_mod) #Create a CVode solver
#Set the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = 1e-5 #Default 1e-6
exp_sim.rtol = 1e-5 #Default 1e-6
exp_sim.linear_solver = 'SPGMR' #Change linear solver
#exp_sim.options["usejac"] = False
#Simulate
t, y = exp_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.show()
def run_example(with_plots=True):
global t,y
#Defines the rhs
def f(t,y):
yd_0 = y[1]
yd_1 = -9.82
#print y, yd_0, yd_1
return N.array([yd_0,yd_1])
#Defines the jacobian
def jac(t,y):
j = N.array([[0,1.],[0,0]])
return j
#Defines an Assimulo explicit problem
y0 = [1.0,0.0] #Initial conditions
exp_mod = Explicit_Problem(f,y0)
exp_mod.jac = jac #Sets the jacobian
exp_mod.name = 'Example using Jacobian'
exp_sim = CVode(exp_mod) #Create a CVode solver
#Set the parameters
exp_sim.iter = 'Newton' #Default 'FixedPoint'
exp_sim.discr = 'BDF' #Default 'Adams'
exp_sim.atol = 1e-5 #Default 1e-6
exp_sim.rtol = 1e-5 #Default 1e-6
#Simulate
t, y = exp_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,linestyle="dashed",marker="o") #Plot the solution
P.show()
def create_model():
def pendulum(t, X, sw):
"""
The ODE to be simulated. The parameter sw should be fixed during
the simulation and only be changed during the event handling.
"""
#X = X.copy()
g = 9.81
x = X[0]
vx = X[2]
vy = X[3]
return np.array([vx, vy, -(np.pi**2)*x, -g, 1.0])
def state_events(t, X, sw):
"""
This is our function that keep track of our events, when the sign
of any of the events has changed, we have an event.
"""
x = X[0]
y = X[1]
return [x - y]
#return np.array([x - y])
def handle_event(solver, event_info):
"""
Event handling. This functions is called when Assimulo finds an event as
specified by the event functions.
"""
state_info = event_info[0] #We are only interested in state events info
if state_info[0] != 0: #Check if the first event function have been triggered
if solver.sw[0]: #If the switch is True the pendulum bounces
X = solver.y
X[1] = X[0] + 1e-3
X[3] = 0.9*(X[2] - X[3])
#solver.sw[0] = not solver.sw[0] #Change event function
#Initial values
phi = 1.0241592; Y0 = -13.0666666; A = 2.0003417; w = np.pi
y0 = [A*np.sin(phi), 1+A*np.sin(phi), A*w*np.cos(phi), Y0 - A*w*np.cos(phi)-1, 0] #Initial states
t0 = 0.0 #Initial time
switches0 = [True] #Initial switches
#Create an Assimulo Problem
mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0)
mod.state_events = state_events #Sets the state events to the problem
mod.handle_event = handle_event #Sets the event handling to the problem
mod.name = 'Bouncing Ball on Sonusoidal Platform' #Sets the name of the problem
#Create an Assimulo solver (CVode)
sim = CVode(mod)
#sim = LSODAR(mod)
sim.options['verbosity'] = 40
#Specifies options
sim.discr = 'Adams' #Sets the discretization method
sim.iter = 'FixedPoint' #Sets the iteration method
sim.rtol = 1.e-8 #Sets the relative tolerance
sim.atol = 1.e-6 #Sets the absolute tolerance
return sim
def run_example(with_plots=True):
"""
This example show how to use Assimulo and CVode for simulating sensitivities
for initial conditions.::
dy1/dt = -(k01+k21+k31)*y1 + k12*y2 + k13*y3 + b1
dy2/dt = k21*y1 - (k02+k12)*y2
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
"""
def f(t, y, p):
y1,y2,y3 = y
k01 = 0.0211
k02 = 0.0162
k21 = 0.0111
k12 = 0.0124
k31 = 0.0039
k13 = 0.000035
b1 = 49.3
yd_0 = -(k01+k21+k31)*y1+k12*y2+k13*y3+b1
yd_1 = k21*y1-(k02+k12)*y2
yd_2 = k31*y1-k13*y3
return N.array([yd_0,yd_1,yd_2])
#The initial conditions
y0 = [0.0,0.0,0.0] #Initial conditions for y
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 explicit problem
exp_mod = Explicit_Problem(f, y0, p0=p0)
#Sets the options to the problem
exp_mod.yS0 = yS0
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1e-7
exp_sim.atol = 1e-6
exp_sim.pbar = [1,1,1] #pbar is used to estimate the tolerances for the parameters
exp_sim.continuous_output = True #Need to be able to store the result using the interpolate methods
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
#Simulate
t, y = exp_sim.simulate(400) #Simulate 400 seconds
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 1577.6552477, 5)
nose.tools.assert_almost_equal(y[-1][1], 611.9574565, 5)
nose.tools.assert_almost_equal(y[-1][2], 2215.88563217, 5)
nose.tools.assert_almost_equal(exp_sim.p_sol[0][1][0], 1.0)
#Plot
if with_plots:
P.figure(1)
P.subplot(221)
P.plot(t, N.array(exp_sim.p_sol[0])[:,0],
t, N.array(exp_sim.p_sol[0])[:,1],
t, N.array(exp_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(exp_sim.p_sol[1])[:,0],
t, N.array(exp_sim.p_sol[1])[:,1],
t, N.array(exp_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(exp_sim.p_sol[2])[:,0],
t, N.array(exp_sim.p_sol[2])[:,1],
t, N.array(exp_sim.p_sol[2])[:,2])
P.title("Parameter p3")
P.legend(("p3/dy1","p3/dy2","p3/dy3"))
P.subplot(224)
P.plot(t, y)
P.show()
def run_example():
def pendulum(t,y,sw):
"""
The ODE to be simulated. The parameter sw should be fixed during
the simulation and only be changed during the event handling.
"""
l=1.0
g=9.81
yd_0 = y[1]
yd_1 = -g/l*N.sin(y[0])
return N.array([yd_0, yd_1])
def state_events(t,y,sw):
"""
This is our function that keep track of our events, when the sign
of any of the events has changed, we have an event.
"""
if sw[0]:
e_0 = y[0]+N.pi/4.
else:
e_0 = y[0]
return N.array([e_0])
def handle_event(solver, event_info):
"""
Event handling. This functions is called when Assimulo finds an event as
specified by the event functions.
"""
state_info = event_info[0] #We are only interested in state events info
if state_info[0] != 0: #Check if the first event function have been triggered
if solver.sw[0]: #If the switch is True the pendulum bounces
solver.y[1] = -0.9*solver.y[1] #Change the velocity and lose energy
solver.sw[0] = not solver.sw[0] #Change event function
#Initial values
y0 = [N.pi/2.0, 0.0] #Initial states
t0 = 0.0 #Initial time
switches0 = [True] #Initial switches
#Create an Assimulo Problem
mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0)
mod.state_events = state_events #Sets the state events to the problem
mod.handle_event = handle_event #Sets the event handling to the problem
mod.name = 'Pendulum with events' #Sets the name of the problem
#Create an Assimulo solver (CVode)
sim = CVode(mod)
#Specifies options
sim.discr = 'Adams' #Sets the discretization method
sim.iter = 'FixedPoint' #Sets the iteration method
sim.rtol = 1.e-8 #Sets the relative tolerance
sim.atol = 1.e-6 #Sets the absolute tolerance
#Simulation
ncp = 200 #Number of communication points
tfinal = 10.0 #Final time
t,y = sim.simulate(tfinal, ncp) #Simulate
#Print event information
sim.print_event_data()
#Plot
P.plot(t,y)
P.show()
def integration_assimulo(self, **kwargs):
"""
Perform time integration for ODEs with the assimulo package
"""
assert self.set_time_setting == 1, 'Time discretization must be specified first'
if self.tclose > 0:
close = True
else:
close = False
# Control vector
self.U = interpolate(self.boundary_cntrl_space, self.Vb).vector()[self.bndr_i_b]
if self.discontinous_boundary_values == 1:
self.U[self.Corner_indices] = self.U[self.Corner_indices]/2
# Definition of the sparse solver for the ODE rhs function to
# be defined next
#my_solver = factorized(csc_matrix(self.M))
my_solver = factorized(self.M)
#my_jac_o = csr_matrix(my_solver(self.J @ self.Q))
#my_jac_c = csr_matrix(my_solver((self.J - self.R) @ self.Q))
# Definition of the rhs function required in assimulo
def rhs(t,y):
"""
Definition of the rhs function required in the ODE part of assimulo
"""
if close:
if t < self.tclose:
z = self.my_mult(self.J, self.my_mult(self.Q,y)) + self.my_mult(self.Bext,self.U* self.boundary_cntrl_time(t,self.tclose))
else:
z = self.my_mult((self.J - self.R), self.my_mult(self.Q,y))
else:
z = self.my_mult(self.J, self.my_mult(self.Q,y)) + self.my_mult(self.Bext,self.U* self.boundary_cntrl_time(t,self.tclose))
return my_solver(z)
def jacobian(t,y):
"""
Jacobian related to the ODE formulation
"""
if close:
if t < self.tclose:
my_jac = my_jac_o
else:
my_jac = my_jac_c
else:
my_jac = my_jac_o
return my_jac
def jacv(t,y,fy,v):
"""
Jacobian matrix-vector product related to the ODE formulation
"""
if close:
if t < self.tclose:
z = self.my_mult(self.J, self.my_mult(self.Q,v) )
else:
z = self.my_mult((self.J - self.R), self.my_mult(self.Q,v))
else:
z = self.my_mult(self.J, self.my_mult(self.Q,v))
return my_solver(z)
print('ODE Integration using assimulo built-in functions:')
#
# https://jmodelica.org/assimulo/_modules/assimulo/examples/cvode_with_preconditioning.html#run_example
#
model = Explicit_Problem(rhs,self.A0,self.tinit)
#model.jac = jacobian
model.jacv = jacv
sim = CVode(model,**kwargs)
sim.atol = 1e-3
sim.rtol = 1e-3
sim.linear_solver = 'SPGMR'
sim.maxord = 3
#sim.usejac = True
#sim = RungeKutta34(model,**kwargs)
time_span, ODE_solution = sim.simulate(self.tfinal)
A_ode = ODE_solution.transpose()
# Hamiltonian
self.Nt = A_ode.shape[1]
self.tspan = np.array(time_span)
Ham_ode = np.zeros(self.Nt)
for k in range(self.Nt):
#Ham_ode[k] = 1/2 * A_ode[:,k] @ self.M @ self.Q @ A_ode[:,k]
Ham_ode[k] = 1/2 * self.my_mult(A_ode[:,k].T, \
self.my_mult(self.M, self.my_mult(self.Q, A_ode[:,k])))
# Get q variables
Aq_ode = A_ode[:self.Nq,:]
# Get p variables
Ap_ode = A_ode[self.Nq:,:]
# Get Deformation
Rho = np.zeros(self.Np)
for i in range(self.Np):
Rho[i] = self.rho(self.coord_p[i])
W_ode = np.zeros((self.Np,self.Nt))
theta = .5
for k in range(self.Nt-1):
W_ode[:,k+1] = W_ode[:,k] + self.dt * 1/Rho[:] * ( theta * Ap_ode[:,k+1] + (1-theta) * Ap_ode[:,k] )
self.Ham_ode = Ham_ode
return Aq_ode, Ap_ode, Ham_ode, W_ode, np.array(time_span)
def run_example(with_plots=True):
def curl(v):
return array([[0, v[2], -v[1]], [-v[2], 0, v[0]], [v[1], -v[0], 0]])
#Defines the rhs
def f(t, u):
"""
Simulations for the Gyro (Heavy Top) example in Celledoni/Safstrom:
Journal of Physics A, Vol 39, 5463-5478, 2006
"""
I1 = 1000.
I2 = 5000.
I3 = 6000.
u0 = [0, 0, 1.]
pi = u[0:3]
Q = (u[3:12]).reshape((3, 3))
Qu0 = dot(Q, u0)
f = array([Qu0[1], -Qu0[0], 0.])
f = 0
omega = array([pi[0] / I1, pi[1] / I2, pi[2] / I3])
pid = dot(curl(omega), pi) + f
Qd = dot(curl(omega), Q)
return hstack([pid, Qd.reshape((9, ))])
def energi(state):
energi = []
for st in state:
Q = (st[3:12]).reshape((3, 3))
pi = st[0:3]
u0 = [0, 0, 1.]
Qu0 = dot(Q, u0)
V = Qu0[2] # potential energy
T = 0.5 * (pi[0]**2 / 1000. + pi[1]**2 / 5000. + pi[2]**2 / 6000.)
energi.append([T])
return energi
#Initial conditions
y0 = hstack([[1000. * 10, 5000. * 10, 6000 * 10], eye(3).reshape((9, ))])
#Create an Assimulo explicit problem
gyro_mod = Explicit_Problem(f, y0)
#Create an Assimulo explicit solver (CVode)
gyro_sim = CVode(gyro_mod)
#Sets the parameters
gyro_sim.discr = 'BDF'
gyro_sim.iter = 'Newton'
gyro_sim.maxord = 2 #Sets the maxorder
gyro_sim.atol = 1.e-10
gyro_sim.rtol = 1.e-10
#Simulate
t, y = gyro_sim.simulate(0.1)
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], 692.800241862)
nose.tools.assert_almost_equal(y[-1][8], 7.08468221e-1)
#Plot
if with_plots:
P.plot(t, y)
P.show()
def run_example(with_plots=True):
"""
This example show how to use Assimulo and CVode for simulating sensitivities
for initial conditions.::
dy1/dt = -(k01+k21+k31)*y1 + k12*y2 + k13*y3 + b1
dy2/dt = k21*y1 - (k02+k12)*y2
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
"""
def f(t, y, p):
y1, y2, y3 = y
k01 = 0.0211
k02 = 0.0162
k21 = 0.0111
k12 = 0.0124
k31 = 0.0039
k13 = 0.000035
b1 = 49.3
yd_0 = -(k01 + k21 + k31) * y1 + k12 * y2 + k13 * y3 + b1
yd_1 = k21 * y1 - (k02 + k12) * y2
yd_2 = k31 * y1 - k13 * y3
return N.array([yd_0, yd_1, yd_2])
#The initial conditions
y0 = [0.0, 0.0, 0.0] #Initial conditions for y
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 explicit problem
exp_mod = Explicit_Problem(f, y0, p0=p0)
#Sets the options to the problem
exp_mod.yS0 = yS0
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1e-7
exp_sim.atol = 1e-6
exp_sim.pbar = [
1, 1, 1
] #pbar is used to estimate the tolerances for the parameters
exp_sim.continuous_output = True #Need to be able to store the result using the interpolate methods
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
#Simulate
t, y = exp_sim.simulate(400) #Simulate 400 seconds
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 1577.6552477, 5)
nose.tools.assert_almost_equal(y[-1][1], 611.9574565, 5)
nose.tools.assert_almost_equal(y[-1][2], 2215.88563217, 5)
nose.tools.assert_almost_equal(exp_sim.p_sol[0][1][0], 1.0)
#Plot
if with_plots:
P.figure(1)
P.subplot(221)
P.plot(t,
N.array(exp_sim.p_sol[0])[:, 0], t,
N.array(exp_sim.p_sol[0])[:, 1], t,
N.array(exp_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(exp_sim.p_sol[1])[:, 0], t,
N.array(exp_sim.p_sol[1])[:, 1], t,
N.array(exp_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(exp_sim.p_sol[2])[:, 0], t,
N.array(exp_sim.p_sol[2])[:, 1], t,
N.array(exp_sim.p_sol[2])[:, 2])
P.title("Parameter p3")
P.legend(("p3/dy1", "p3/dy2", "p3/dy3"))
P.subplot(224)
P.plot(t, y)
P.show()
def run_example(with_plots=True):
"""
This is the same example from the Sundials package (cvsRoberts_FSA_dns.c)
This simple example problem for CVode, due to Robertson,
is from chemical kinetics, and consists of the following three
equations::
dy1/dt = -p1*y1 + p2*y2*y3
dy2/dt = p1*y1 - p2*y2*y3 - p3*y2**2
dy3/dt = p3*(y2)^2
"""
def f(t, y, p):
yd_0 = -p[0] * y[0] + p[1] * y[1] * y[2]
yd_1 = p[0] * y[0] - p[1] * y[1] * y[2] - p[2] * y[1]**2
yd_2 = p[2] * y[1]**2
return N.array([yd_0, yd_1, yd_2])
#The initial conditions
y0 = [1.0, 0.0, 0.0] #Initial conditions for y
#Create an Assimulo explicit problem
exp_mod = Explicit_Problem(f, y0)
#Sets the options to the problem
exp_mod.p0 = [0.040, 1.0e4, 3.0e7] #Initial conditions for parameters
exp_mod.pbar = [0.040, 1.0e4, 3.0e7]
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the paramters
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.rtol = 1.e-4
exp_sim.atol = N.array([1.0e-8, 1.0e-14, 1.0e-6])
exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
exp_sim.suppress_sens = False #Dont suppress the sensitivity variables in the error test.
exp_sim.continuous_output = True
#Simulate
t, y = exp_sim.simulate(
4, 400) #Simulate 4 seconds with 400 communication points
#Basic test
nose.tools.assert_almost_equal(y[-1][0], 9.05518032e-01, 4)
nose.tools.assert_almost_equal(y[-1][1], 2.24046805e-05, 4)
nose.tools.assert_almost_equal(y[-1][2], 9.44595637e-02, 4)
nose.tools.assert_almost_equal(
exp_sim.p_sol[0][-1][0], -1.8761,
2) #Values taken from the example in Sundials
nose.tools.assert_almost_equal(exp_sim.p_sol[1][-1][0], 2.9614e-06, 8)
nose.tools.assert_almost_equal(exp_sim.p_sol[2][-1][0], -4.9334e-10, 12)
#Plot
if with_plots:
P.plot(t, y)
P.show()
def run_example(with_plots=True):
def curl(v):
return array([[0,v[2],-v[1]],[-v[2],0,v[0]],[v[1],-v[0],0]])
#Defines the rhs
def f(t,u):
"""
Simulations for the Gyro (Heavy Top) example in Celledoni/Safstrom:
Journal of Physics A, Vol 39, 5463-5478, 2006
"""
I1=1000.
I2=5000.
I3=6000.
u0=[0,0,1.]
pi=u[0:3]
Q=(u[3:12]).reshape((3,3))
Qu0=dot(Q,u0)
f=array([Qu0[1],-Qu0[0],0.])
f=0
omega=array([pi[0]/I1,pi[1]/I2,pi[2]/I3])
pid=dot(curl(omega),pi)+f
Qd=dot(curl(omega),Q)
return hstack([pid,Qd.reshape((9,))])
def energi(state):
energi=[]
for st in state:
Q=(st[3:12]).reshape((3,3))
pi=st[0:3]
u0=[0,0,1.]
Qu0=dot(Q,u0)
V=Qu0[2] # potential energy
T=0.5*(pi[0]**2/1000.+pi[1]**2/5000.+pi[2]**2/6000.)
energi.append([T])
return energi
#Initial conditions
y0=hstack([[1000.*10,5000.*10,6000*10],eye(3).reshape((9,))])
#Create an Assimulo explicit problem
gyro_mod = Explicit_Problem(f,y0)
#Create an Assimulo explicit solver (CVode)
gyro_sim=CVode(gyro_mod)
#Sets the parameters
gyro_sim.discr='BDF'
gyro_sim.iter='Newton'
gyro_sim.maxord=2 #Sets the maxorder
gyro_sim.atol=1.e-10
gyro_sim.rtol=1.e-10
#Simulate
t, y = gyro_sim.simulate(0.1)
#Basic tests
nose.tools.assert_almost_equal(y[-1][0],692.800241862)
nose.tools.assert_almost_equal(y[-1][8],7.08468221e-1)
#Plot
if with_plots:
P.plot(t,y)
P.show()
def integration_assimulo(self, **kwargs):
"""
Perform time integration for ODEs with the assimulo package
"""
assert self.set_time_setting == 1, 'Time discretization must be specified first'
if self.tclose > 0:
close = True
else:
close = False
# Control vector
self.U = interpolate(self.boundary_cntrl_space, self.Vb).vector()[self.bndr_i_b]
if self.discontinous_boundary_values == 1:
self.U[self.Corner_indices] = self.U[self.Corner_indices]/2
# Definition of the sparse solver for the ODE rhs function to
# be defined next
my_solver = factorized(csc_matrix(self.Mp_rho_Cv))
C = self.A
if self.sparse == 1:
my_jac_o = csr_matrix(my_solver(C.toarray()))
else:
my_jac_o = my_solver(C)
# Definition of the rhs function required in assimulo
def rhs(t,y):
"""
Definition of the rhs function required in the ODE part of assimulo
"""
z = self.my_mult(self.A, y) + self.my_mult(self.Bext,self.U* self.boundary_cntrl_time(t,self.tclose))
return my_solver(z)
#z = np.zeros(shape=y.shape[:])
#z[0:self.Np] = self.my_mult(self.Mp, yd[0:self.Np]) - self.my_mult(D, y[self.Np+self.Nq:self.Np+2*self.Nq])
#z[self.Np:self.Np+self.Nq] = self.my_mult(self.Mq, y[self.Np:self.Np+self.Nq]) + self.my_mult(D.T, y[0:self.Np]) - self.my_mult(self.Bp, self.U* self.boundary_cntrl_time(t,self.tclose))
#z[self.Np+self.Nq:self.Np+2*self.Nq] = self.my_mult(self.Mq, y[self.Np+self.Nq:self.Np+2*self.Nq]) - self.my_mult(self.L,y[self.Np:self.Np+self.Nq])
#return z
def jacobian(t,y):
"""
Jacobian related to the ODE formulation
"""
my_jac = my_jac_o
return my_jac
def jacv(t,y,fy,v):
"""
Jacobian matrix-vector product related to the ODE formulation
"""
return None
print('ODE Integration using assimulo built-in functions:')
#
# https://jmodelica.org/assimulo/_modules/assimulo/examples/cvode_with_preconditioning.html#run_example
#
model = Explicit_Problem(rhs,self.Tp0,self.tinit)
sim = CVode(model,**kwargs)
sim.atol = 1e-3
sim.rtol = 1e-3
sim.linear_solver = 'SPGMR'
sim.maxord = 3
#sim.usejac = True
#sim = RungeKutta34(model,**kwargs)
time_span, ODE_solution = sim.simulate(self.tfinal)
A_ode = ODE_solution.transpose()
# Hamiltonian
self.Nt = A_ode.shape[1]
self.tspan = np.array(time_span)
Ham_ode = np.zeros(self.Nt)
for k in range(self.Nt):
Ham_ode[k] = 1/2 * self.my_mult(A_ode[:,k].T, \
self.my_mult(self.Mp_rho_Cv, A_ode[:,k]))
self.Ham_ode = Ham_ode
return Ham_ode, np.array(time_span)
def create_model():
def pendulum(t, X, sw):
"""
The ODE to be simulated. The parameter sw should be fixed during
the simulation and only be changed during the event handling.
"""
#X = X.copy()
g = 9.81
x = X[0]
vx = X[2]
vy = X[3]
return np.array([vx, vy, -(np.pi**2) * x, -g, 1.0])
def state_events(t, X, sw):
"""
This is our function that keep track of our events, when the sign
of any of the events has changed, we have an event.
"""
x = X[0]
y = X[1]
return [x - y]
#return np.array([x - y])
def handle_event(solver, event_info):
"""
Event handling. This functions is called when Assimulo finds an event as
specified by the event functions.
"""
state_info = event_info[
0] #We are only interested in state events info
if state_info[
0] != 0: #Check if the first event function have been triggered
if solver.sw[0]: #If the switch is True the pendulum bounces
X = solver.y
X[1] = X[0] + 1e-3
X[3] = 0.9 * (X[2] - X[3])
#solver.sw[0] = not solver.sw[0] #Change event function
#Initial values
phi = 1.0241592
Y0 = -13.0666666
A = 2.0003417
w = np.pi
y0 = [
A * np.sin(phi), 1 + A * np.sin(phi), A * w * np.cos(phi),
Y0 - A * w * np.cos(phi) - 1, 0
] #Initial states
t0 = 0.0 #Initial time
switches0 = [True] #Initial switches
#Create an Assimulo Problem
mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0)
mod.state_events = state_events #Sets the state events to the problem
mod.handle_event = handle_event #Sets the event handling to the problem
mod.name = 'Bouncing Ball on Sonusoidal Platform' #Sets the name of the problem
#Create an Assimulo solver (CVode)
sim = CVode(mod)
#sim = LSODAR(mod)
sim.options['verbosity'] = 40
#Specifies options
sim.discr = 'Adams' #Sets the discretization method
sim.iter = 'FixedPoint' #Sets the iteration method
sim.rtol = 1.e-8 #Sets the relative tolerance
sim.atol = 1.e-6 #Sets the absolute tolerance
return sim
def create_model():
def pendulum(t, X, sw):
"""
The ODE to be simulated. The parameter sw should be fixed during
the simulation and only be changed during the event handling.
"""
X = X.copy()
X[0] = X[1]
X[1] = -1 + 0.04 * (X[1]**2) * np.sin(X[1])
X[2] = 1
return X
def state_events(t, X, sw):
"""
This is our function that keep track of our events, when the sign
of any of the events has changed, we have an event.
"""
return [X[0] - np.sin(X[2])]
def handle_event(solver, event_info):
"""
Event handling. This functions is called when Assimulo finds an event as
specified by the event functions.
"""
state_info = event_info[0] #We are only interested in state events info
if state_info[0] != 0: #Check if the first event function have been triggered
if solver.sw[0]: #If the switch is True the pendulum bounces
X = solver.y
if X[1] - np.cos(X[2]) < 0:
X[1] = -0.9*X[1] + 1.9*np.cos(X[2])
#solver.sw[0] = not solver.sw[0] #Change event function
#Initial values
y0 = [0, 0, 0] #Initial states
t0 = 0.0 #Initial time
switches0 = [True] #Initial switches
#Create an Assimulo Problem
mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0)
mod.state_events = state_events #Sets the state events to the problem
mod.handle_event = handle_event #Sets the event handling to the problem
mod.name = 'Bouncing Ball on Sonusoidal Platform' #Sets the name of the problem
#Create an Assimulo solver (CVode)
sim = CVode(mod)
#sim = LSODAR(mod)
#sim.options['verbosity'] = 20 #LOUD
sim.options['verbosity'] = 40 #WHISPER
#sim.options['minh'] = 1e-4
#sim.options['rtol'] = 1e-3
#Specifies options
sim.discr = 'Adams' #Sets the discretization method
sim.iter = 'FixedPoint' #Sets the iteration method
sim.rtol = 1.e-8 #Sets the relative tolerance
sim.atol = 1.e-6 #Sets the absolute tolerance
return sim
def create_model():
def pendulum(t, X, sw):
"""
The ODE to be simulated. The parameter sw should be fixed during
the simulation and only be changed during the event handling.
"""
X = X.copy()
X[0] = X[1]
X[1] = -1 + 0.04 * (X[1]**2) * np.sin(X[1])
X[2] = 1
return X
def state_events(t, X, sw):
"""
This is our function that keep track of our events, when the sign
of any of the events has changed, we have an event.
"""
return [X[0] - np.sin(X[2])]
def handle_event(solver, event_info):
"""
Event handling. This functions is called when Assimulo finds an event as
specified by the event functions.
"""
state_info = event_info[
0] #We are only interested in state events info
if state_info[
0] != 0: #Check if the first event function have been triggered
if solver.sw[0]: #If the switch is True the pendulum bounces
X = solver.y
if X[1] - np.cos(X[2]) < 0:
X[1] = -0.9 * X[1] + 1.9 * np.cos(X[2])
#solver.sw[0] = not solver.sw[0] #Change event function
#Initial values
y0 = [0, 0, 0] #Initial states
t0 = 0.0 #Initial time
switches0 = [True] #Initial switches
#Create an Assimulo Problem
mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0)
mod.state_events = state_events #Sets the state events to the problem
mod.handle_event = handle_event #Sets the event handling to the problem
mod.name = 'Bouncing Ball on Sonusoidal Platform' #Sets the name of the problem
#Create an Assimulo solver (CVode)
sim = CVode(mod)
#sim = LSODAR(mod)
#sim.options['verbosity'] = 20 #LOUD
sim.options['verbosity'] = 40 #WHISPER
#sim.options['minh'] = 1e-4
#sim.options['rtol'] = 1e-3
#Specifies options
sim.discr = 'Adams' #Sets the discretization method
sim.iter = 'FixedPoint' #Sets the iteration method
sim.rtol = 1.e-8 #Sets the relative tolerance
sim.atol = 1.e-6 #Sets the absolute tolerance
return sim