def ode_sim(t_final):
y0 = get_initial() # initial conditions from Parameter.py
t0 = 0.0
param = new_param()
ecoli_ode = lambda t, x: rhs(t, x, param)
model = Explicit_Problem(ecoli_ode, y0, t0) # Create an Assimulo problem
model.name = 'E Coli ODE'
sim = CVode(model) # Create the solver CVode
sim.rtol = 1e-8
sim.atol = 1e-8
# pdb.set_trace()
t, y = sim.simulate(t_final) # Use the .simulate method to simulate and provide the final time
# Plot
met_labels = ["ACCOA", "ACEx", "ACO", "ACP", "ADP", "AKG", "AMP", "ASP", "ATP", "BPG", "CAMP", "CIT", "COA", "CYS",
"DAP", "E4P", "ei", "eiia", "eiiaP", "eiicb", "eiicbP", "eiP", "F6P", "FDP", "FUM", "G6P", "GAP",
"GL6P", "GLCx", "GLX", "HCO3", "hpr", "hprP", "icd", "icdP", "ICIT", "KDPG", "MAL", "MG", "MN", "NAD",
"NADH", "NADP", "NADPH", "OAA", "P", "PEP", "PGA2", "PGA3", "PGN", "PYR", "PYRx", "Q", "QH2", "R5P",
"RU5P", "S7P", "SUC", "SUCCOA", "SUCx", "tal", "talC3", "tkt", "tktC2", "X5P", "Px", "Pp", "GLCp",
"ACEp", "ACE", "Hc", "Hp", "FAD", "FADH2", "O2", "FEED"]
for i in y:
if i.any() < 0:
print i
plt.plot([t, t], [y[:, i] for i in [4, 9]], label=["ADP", "ATP"])
plt.title('Metabolite Concentrations Over Time')
plt.xlabel('Time')
plt.ylabel('Concentration')
plt.legend()
plt.savefig('concentration.png')
return model, sim
def simulate(self, Tend, nIntervals, gridWidth):
problem = Explicit_Problem(self.rhs, self.y0)
problem.name = 'CVode'
# 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
simulation = CVode(problem)
# Change multistep method: 'adams' or 'VDF'
if self.discr == 'Adams':
simulation.discr = 'Adams'
simulation.maxord = 12
else:
simulation.discr = 'BDF'
simulation.maxord = 5
# Change iteration algorithm: functional(FixedPoint) or newton
if self.iter == 'FixedPoint':
simulation.iter = 'FixedPoint'
else:
simulation.iter = 'Newton'
# Sets additional parameters
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
# '''Initialize problem '''
# self.t_cur = self.t0
# self.y_cur = self.y0
# 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 = simulation.simulate
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):
ydot = -y[0]
return N.array([ydot])
#Define an Assimulo problem
exp_mod = Explicit_Problem(
f,
t0=5,
y0=0.02695,
name=r'CVode Test Example (reverse time): $\dot y = - y$ ')
#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-8] #Default 1e-6
exp_sim.rtol = 1e-8 #Default 1e-6
exp_sim.backward = True
#Simulate
t, y = exp_sim.simulate(0) #Simulate 5 seconds (t0=5 -> tf=0)
#print 'y(5) = {}, y(0) ={}'.format(y[0][0],y[-1][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.title(exp_mod.name)
P.ylabel('y')
P.xlabel('Time')
P.show()
return exp_mod, exp_sim
def run_example(with_plots=True):
r"""
Demonstration of the use of CVode by solving the
linear test equation :math:`\dot y = - y`
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#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,
name=r'CVode Test Example: $\dot y = - y$')
#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
#Plot
if with_plots:
import pylab as P
P.plot(t1, y1, color="b")
P.plot(t2, y2, color="r")
P.title(exp_mod.name)
P.ylabel('y')
P.xlabel('Time')
P.show()
#Basic test
nose.tools.assert_almost_equal(float(y2[-1]), 0.00347746, 5)
nose.tools.assert_almost_equal(exp_sim.get_last_step(), 0.0222169642893, 3)
return exp_mod, exp_sim
def simulate_prob(t_start, t_stop, x0, p, ncp, with_plots=False):
"""Simulates the problem using Assimulo.
Args:
t_start (double): Simulation start time.
t_stop (double): Simulation stop time.
x0 (list): Initial value.
p (list): Problem specific parameters.
ncp (int): Number of communication points.
with_plots (bool): Plots the solution.
Returns:
tuple: (t,y). Time vector and solution at each time.
"""
# Assimulo
# Define the right-hand side
def f(t, y):
xd_1 = p[0] * y[0]
xd_2 = p[1] * (y[1] - y[0]**2)
return np.array([xd_1, xd_2])
# Define an Assimulo problem
exp_mod = Explicit_Problem(f, y0=x0, name='Planar ODE')
# Define an explicit solver
exp_sim = CVode(exp_mod)
# Sets the solver parameters
exp_sim.atol = 1e-12
exp_sim.rtol = 1e-11
# Simulate
t, y = exp_sim.simulate(tfinal=t_stop, ncp=ncp)
# Plot
if with_plots:
x1 = y[:, 0]
x2 = y[:, 1]
plt.figure()
plt.title('Planar ODE')
plt.plot(t, x1, 'b')
plt.plot(t, x2, 'k')
plt.legend(['x1', 'x2'])
plt.xlim(t_start, t_stop)
plt.xlabel('Time (s)')
plt.ylabel('x')
plt.grid(True)
return t, y.T
def simulate(self, Tend, nIntervals, gridWidth):
problem = Explicit_Problem(self.rhs, self.y0)
problem.name = 'CVode'
# 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
simulation = CVode(problem)
# Change multistep method: 'adams' or 'VDF'
if self.discr == 'Adams':
simulation.discr = 'Adams'
simulation.maxord = 12
else:
simulation.discr = 'BDF'
simulation.maxord = 5
# Change iteration algorithm: functional(FixedPoint) or newton
if self.iter == 'FixedPoint':
simulation.iter = 'FixedPoint'
else:
simulation.iter = 'Newton'
# Sets additional parameters
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
# '''Initialize problem '''
# self.t_cur = self.t0
# self.y_cur = self.y0
# 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 = simulation.simulate
def setUp(self): """ Configures the solver """ #Define an explicit solver simSolver = CVode(self) #Create a CVode solver #Sets the parameters #simSolver.verbosity = LOUD #simSolver.report_continuously = True simSolver.iter = 'Newton' #Default 'FixedPoint' simSolver.discr = 'BDF' #Default 'Adams' #simSolver.discr = 'Adams' simSolver.atol = [1e-6] #Default 1e-6 simSolver.rtol = 1e-6 #Default 1e-6 #simSolver.problem_info['step_events'] = True # activates step events #simSolver.maxh = 1.0 #simSolver.store_event_points = True self.simSolver = simSolver
def setUp(self):
"""
Configures the solver
"""
#Define an explicit solver
simSolver = CVode(self)
#Create a CVode solver
#Sets the parameters
#simSolver.verbosity = LOUD
#simSolver.report_continuously = True
simSolver.iter = 'Newton' #Default 'FixedPoint'
simSolver.discr = 'BDF' #Default 'Adams'
#simSolver.discr = 'Adams'
simSolver.atol = [1e-6] #Default 1e-6
simSolver.rtol = 1e-6 #Default 1e-6
#simSolver.problem_info['step_events'] = True # activates step events
#simSolver.maxh = 1.0
#simSolver.store_event_points = True
self.simSolver = simSolver
def make_explicit_sim(self):
explicit_sim = CVode(self.explicit_problem)
explicit_sim.iter = 'Newton'
explicit_sim.discr = 'BDF'
explicit_sim.rtol = 1e-7
explicit_sim.atol = 1e-7
explicit_sim.sensmethod = 'SIMULTANEOUS'
explicit_sim.suppress_sens = True
explicit_sim.report_continuously = False
explicit_sim.usesens = False
explicit_sim.verbosity = 50
if self.use_jac and self.model_jac is not None:
explicit_sim.usejac = True
else:
explicit_sim.usejac = False
return explicit_sim
def prepareSimulation(self, params=None):
if params == None:
params = AttributeDict({
'absTol': 1e-6,
'relTol': 1e-6,
})
#Define an explicit solver
simSolver = CVode(self)
#Create a CVode solver
#Sets the parameters
#simSolver.verbosity = LOUD
simSolver.report_continuously = True
simSolver.iter = 'Newton' #Default 'FixedPoint'
simSolver.discr = 'BDF' #Default 'Adams'
#simSolver.discr = 'Adams'
simSolver.atol = [params.absTol] #Default 1e-6
simSolver.rtol = params.relTol #Default 1e-6
simSolver.problem_info['step_events'] = True # activates step events
#simSolver.maxh = 1.0
simSolver.store_event_points = True
self.simSolver = simSolver
def prepareSimulation(self, params = None):
if params == None:
params = AttributeDict({
'absTol' : 1e-6,
'relTol' : 1e-6,
})
#Define an explicit solver
simSolver = CVode(self)
#Create a CVode solver
#Sets the parameters
#simSolver.verbosity = LOUD
simSolver.report_continuously = True
simSolver.iter = 'Newton' #Default 'FixedPoint'
simSolver.discr = 'BDF' #Default 'Adams'
#simSolver.discr = 'Adams'
simSolver.atol = [params.absTol] #Default 1e-6
simSolver.rtol = params.relTol #Default 1e-6
simSolver.problem_info['step_events'] = True # activates step events
#simSolver.maxh = 1.0
simSolver.store_event_points = True
self.simSolver = simSolver
def mySolve(xf,boltz_eqs,rtol,atol,verbosity=50):
"""Sets the main options for the ODE solver and solve the equations. Returns the
array of x,y points for all components.
If numerical instabilities are found, re-do the problematic part of the evolution with smaller steps"""
boltz_solver = CVode(boltz_eqs) #Define solver method
boltz_solver.rtol = rtol
boltz_solver.atol = atol
boltz_solver.verbosity = verbosity
boltz_solver.linear_solver = 'SPGMR'
boltz_solver.maxh = xf/300.
xfinal = xf
xres = []
yres = []
sw = boltz_solver.sw[:]
while xfinal <= xf:
try:
boltz_solver.re_init(boltz_eqs.t0,boltz_eqs.y0)
boltz_solver.sw = sw[:]
x,y = boltz_solver.simulate(xfinal)
xres += x
for ypt in y: yres.append(ypt)
if xfinal == xf: break #Evolution has been performed until xf -> exit
except Exception,e:
print e
if not e.t or 'first call' in e.msg[e.value]:
logger.error("Error solving equations:\n "+str(e))
return False
xfinal = max(e.t*random.uniform(0.85,0.95),boltz_eqs.t0+boltz_solver.maxh) #Try again, but now only until the error
logger.warning("Numerical instability found. Restarting evolution from x = "
+str(boltz_eqs.t0)+" to x = "+str(xfinal))
continue
xfinal = xf #In the next step try to evolve from xfinal -> xf
sw = boltz_solver.sw[:]
x0 = float(x[-1])
y0 = [float(yval) for yval in y[-1]]
boltz_eqs.updateValues(x0,y0,sw)
def run_simulation(filename, start_time, save_output, temp, RH, RO2_indices,
H2O, input_dict, simulation_time, batch_step):
from assimulo.solvers import RodasODE, CVode #Choose solver accoring to your need.
from assimulo.problem import Explicit_Problem
# In this function, we import functions that have been pre-compiled for use in the ODE solver
# The function that calculates the RHS of the ODE is also defined within this function, such
# that it can be used by the Assimulo solvers
# The variables passed to this function are defined as follows:
#-------------------------------------------------------------------------------------
# define the ODE function to be called
def dydt_func(t, y):
"""
This function defines the right-hand side [RHS] of the ordinary differential equations [ODEs] to be solved
input:
• t - time variable [internal to solver]
• y - array holding concentrations of all compounds in both gas and particulate [molecules/cc]
output:
dydt - the dy_dt of each compound in both gas and particulate phase [molecules/cc.sec]
"""
#pdb.set_trace()
# Calculate time of day
time_of_day_seconds = start_time + t
# make sure the y array is not a list. Assimulo uses lists
y_asnumpy = numpy.array(y)
#Calculate the concentration of RO2 species, using an index file created during parsing
RO2 = numpy.sum(y[RO2_indices])
#Calculate reaction rate for each equation.
# Note that H2O will change in parcel mode
# The time_of_day_seconds is used for photolysis rates - need to change this if want constant values
rates = evaluate_rates_fortran(RO2, H2O, temp, time_of_day_seconds)
#pdb.set_trace()
# Calculate product of all reactants and stochiometry for each reaction [A^a*B^b etc]
reactants = reactants_fortran(y_asnumpy)
#pdb.set_trace()
#Multiply product of reactants with rate coefficient to get reaction rate
reactants = numpy.multiply(reactants, rates)
#pdb.set_trace()
# Now use reaction rates with the loss_gain matri to calculate the final dydt for each compound
# With the assimulo solvers we need to output numpy arrays
dydt = loss_gain_fortran(reactants)
#pdb.set_trace()
return dydt
#-------------------------------------------------------------------------------------
#-------------------------------------------------------------------------------------
# define jacobian function to be called
def jacobian(t, y):
"""
This function defines Jacobian of the ordinary differential equations [ODEs] to be solved
input:
• t - time variable [internal to solver]
• y - array holding concentrations of all compounds in both gas and particulate [molecules/cc]
output:
dydt_dydt - the N_compounds x N_compounds matrix of Jacobian values
"""
# Different solvers might call jacobian at different stages, so we have to redo some calculations here
# Calculate time of day
time_of_day_seconds = start_time + t
# make sure the y array is not a list. Assimulo uses lists
y_asnumpy = numpy.array(y)
#Calculate the concentration of RO2 species, using an index file created during parsing
RO2 = numpy.sum(y[RO2_indices])
#Calculate reaction rate for each equation.
# Note that H2O will change in parcel mode
rates = evaluate_rates_fortran(RO2, H2O, temp, time_of_day_seconds)
#pdb.set_trace()
# Now use reaction rates with the loss_gain matrix to calculate the final dydt for each compound
# With the assimulo solvers we need to output numpy arrays
dydt_dydt = jacobian_fortran(rates, y_asnumpy)
#pdb.set_trace()
return dydt_dydt
#-------------------------------------------------------------------------------------
#import static compilation of Fortran functions for use in ODE solver
print("Importing pre-compiled Fortran modules")
from rate_coeff_f2py import evaluate_rates as evaluate_rates_fortran
from reactants_conc_f2py import reactants as reactants_fortran
from loss_gain_f2py import loss_gain as loss_gain_fortran
from jacobian_f2py import jacobian as jacobian_fortran
# 'Unpack' variables from input_dict
species_dict = input_dict['species_dict']
species_dict2array = input_dict['species_dict2array']
species_initial_conc = input_dict['species_initial_conc']
equations = input_dict['equations']
#Specify some starting concentrations [ppt]
Cfactor = 2.55e+10 #ppb-to-molecules/cc
# Create variables required to initialise ODE
num_species = len(species_dict.keys())
y0 = [0] * num_species #Initial concentrations, set to 0
t0 = 0.0 #T0
# Define species concentrations in ppb
# You have already set this in the front end script, and now we populate the y array with those concentrations
for specie in species_initial_conc.keys():
y0[species_dict2array[specie]] = species_initial_conc[
specie] * Cfactor #convert from pbb to molcules/cc
#Set the total_time of the simulation to 0 [havent done anything yet]
total_time = 0.0
# Now run through the simulation in batches.
# I do this to enable testing of coupling processes. Some initial investigations with non-ideality in
# the condensed phase indicated that even defining a maximum step was not enough for ODE solvers to
# overshoot a stable region. It also helps with in-simulation debugging. Its up to you if you want to keep this.
# To not run in batches, just define one batch as your total simulation time. This will reduce any overhead with
# initialising the solvers
# Set total simulation time and batch steps in seconds
# Note also that the current module outputs solver information after each batch step. This can be turned off and the
# the batch step change for increased speed
#simulation_time= 3600.0
#batch_step=100.0
t_array = []
time_step = 0
number_steps = int(
simulation_time /
batch_step) # Just cycling through 3 steps to get to a solution
# Define a matrix that stores values as outputs from the end of each batch step. Again, you can remove
# the need to run in batches. You can tell the Assimulo solvers the frequency of outputs.
y_matrix = numpy.zeros((int(number_steps), len(y0)))
print("Starting simulation")
# In the following, we can
while total_time < simulation_time:
if total_time == 0.0:
#Define an Assimulo problem
#Define an explicit solver
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
else:
y0 = y_output[
-1, :] # Take the output from the last batch as the start of this
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
# Define ODE parameters.
# Initial steps might be slower than mid-simulation. It varies.
#exp_mod.jac = dydt_jac
# Define which ODE solver you want to use
exp_sim = CVode(exp_mod)
tol_list = [1.0e-3] * num_species
exp_sim.atol = tol_list #Default 1e-6
exp_sim.rtol = 0.03 #Default 1e-6
exp_sim.inith = 1.0e-6 #Initial step-size
#exp_sim.discr = 'Adams'
exp_sim.maxh = 100.0
# Use of a jacobian makes a big differece in simulation time. This is relatively
# easy to define for a gas phase - not sure for an aerosol phase with composition
# dependent processes.
exp_sim.usejac = True # To be provided as an option in future update.
#exp_sim.fac1 = 0.05
#exp_sim.fac2 = 50.0
exp_sim.report_continuously = True
exp_sim.maxncf = 1000
#Sets the parameters
t_output, y_output = exp_sim.simulate(
batch_step) #Simulate 'batch' seconds
total_time += batch_step
t_array.append(
total_time
) # Save the output from the end step, of the current batch, to a matrix
y_matrix[time_step, :] = y_output[-1, :]
#now save this information into a matrix for later plotting.
time_step += 1
# Do you want to save the generated matrix of outputs?
if save_output:
numpy.save(filename + '_output', y_matrix)
df = pd.DataFrame(y_matrix)
df.to_csv(filename + "_output_matrix.csv")
w = csv.writer(open(filename + "_output_names.csv", "w"))
for specie, number in species_dict2array.items():
w.writerow([specie, number])
with_plots = True
#pdb.set_trace()
#Plot the change in concentration over time for a given specie. For the user to change / remove
#In a future release I will add this as a seperate module
if with_plots:
try:
P.plot(t_array,
numpy.log10(y_matrix[:, species_dict2array['APINENE']]),
marker='o',
label="APINENE")
P.plot(t_array,
numpy.log10(y_matrix[:, species_dict2array['PINONIC']]),
marker='o',
label="PINONIC")
P.title(exp_mod.name)
P.legend(loc='upper left')
P.ylabel("Concetration log10[molecules/cc]")
P.xlabel("Time [seconds] since start of simulation")
P.show()
except:
print(
"There is a problem using Matplotlib in your environment. If using this within a docker container, you will need to transfer the data to the host or configure your container to enable graphical displays. More information can be found at http://wiki.ros.org/docker/Tutorials/GUI "
)
def func_set_system():
##############################################
#% initial conditions
P_0 = depth*9.8*2500. #; % initial chamber pressure (Pa)
T_0 = 1200 #; % initial chamber temperature (K)
eps_g0 = 0.04 #; % initial gas volume fraction
rho_m0 = 2600 #; % initial melt density (kg/m^3)
rho_x0 = 3065 #; % initial crystal density (kg/m^3)
a = 1000 #; % initial radius of the chamber (m)
V_0 = (4.*pi/3.)*a**3. #; % initial volume of the chamber (m^3)
##############################################
##############################################
IC = numpy.array([P_0,T_0,eps_g0,V_0,rho_m0,rho_x0]) # % store initial conditions
## Gas (eps_g = zero), eps_x is zero, too many crystals, 50 % crystallinity,eruption (yes/no)
sw0 = [False,False,False,False,False]
##############################################
#% error tolerances used in ode method
dt = 30e7
N = int(round((end_time-begin_time)/dt))
##############################################
#Define an Assimulo problem
exp_mod = Chamber_Problem(depth=depth,t0=begin_time,y0=IC,sw0=sw0)
exp_mod.param['T_in'] = 1200.
exp_mod.param['eps_g_in'] = 0.0 # Gas fraction of incoming melt - gas phase ..
exp_mod.param['m_eq_in'] = 0.05 # Volatile fraction of incoming melt
exp_mod.param['Mdot_in'] = mdot
exp_mod.param['eta_x_max'] = 0.64 # Locking fraction
exp_mod.param['delta_Pc'] = 20e6
exp_mod.tcurrent = begin_time
exp_mod.radius = a
exp_mod.permeability = perm_val
exp_mod.R_steps = 1500
exp_mod.dt_init = dt
#################
exp_mod.R_outside = numpy.linspace(a,3.*a,exp_mod.R_steps);
exp_mod.T_out_all =numpy.array([exp_mod.R_outside*0.])
exp_mod.P_out_all =numpy.array([exp_mod.R_outside*0.])
exp_mod.sigma_rr_all = numpy.array([exp_mod.R_outside*0.])
exp_mod.sigma_theta_all = numpy.array([exp_mod.R_outside*0.])
exp_mod.sigma_eff_rr_all = numpy.array([exp_mod.R_outside*0.])
exp_mod.sigma_eff_theta_all = numpy.array([exp_mod.R_outside*0.])
exp_mod.max_count = 1 # counting for the append me arrays ..
exp_mod.P_list = append_me()
exp_mod.P_list.update(P_0-exp_mod.plith)
exp_mod.T_list = append_me()
exp_mod.T_list.update(T_0-exp_mod.param['T_S'])
exp_mod.P_flux_list = append_me()
exp_mod.P_flux_list.update(0)
exp_mod.T_flux_list = append_me()
exp_mod.T_flux_list.update(0)
exp_mod.times_list = append_me()
exp_mod.times_list.update(1e-7)
exp_mod.T_out,exp_mod.P_out,exp_mod.sigma_rr,exp_mod.sigma_theta,exp_mod.T_der= Analytical_sol_cavity_T_Use(exp_mod.T_list.data[:exp_mod.max_count],exp_mod.P_list.data[:exp_mod.max_count],exp_mod.radius,exp_mod.times_list.data[:exp_mod.max_count],exp_mod.R_outside,exp_mod.permeability,exp_mod.param['material'])
exp_mod.param['heat_cond'] = 1 # Turn on/off heat conduction
exp_mod.param['visc_relax'] = 1 # Turn on/off viscous relaxation
exp_mod.param['press_relax'] = 1 ## Turn on/off pressure diffusion
exp_mod.param['frac_rad_Temp'] =0.75
exp_mod.param['vol_degass'] = 1.
#exp_mod.state_events = stopChamber #Sets the state events to the problem
#exp_mod.handle_event = handle_event #Sets the event handling to the problem
#Sets the options to the problem
#exp_mod.p0 = [beta_r, beta_m]#, beta_x, alpha_r, alpha_m, alpha_x, L_e, L_m, c_m, c_g, c_x, eruption, heat_cond, visc_relax] #Initial conditions for parameters
#exp_mod.pbar = [beta_r, beta_m]#, beta_x, alpha_r, alpha_m, alpha_x, L_e, L_m, c_m, c_g, c_x, eruption, heat_cond, visc_relax]
#Define an explicit solver
exp_sim = CVode(exp_mod) #Create a CVode solver
#Sets the parameters
#exp_sim.iter = 'Newton'
#exp_sim.discr = 'BDF'
#exp_sim.inith = 1e-7
exp_sim.rtol = 1.e-7
exp_sim.maxh = 3e7
exp_sim.atol = 1e-7
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.usesens = True
#exp_sim.report_continuously = True
return exp_mod,exp_sim,N
def run_example(with_plots=True):
r"""
An example for CVode 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 \\
\dot y_2 &= -9.82
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#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, name='Example using the Jacobian Vector product')
exp_mod.jacv = jacv #Sets the 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
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.xlabel('Time')
P.ylabel('State')
P.title(exp_mod.name)
P.show()
return exp_mod, exp_sim
def simulate(self, Tend, nIntervals, gridWidth):
# define assimulo problem:(has to be done here because of the starting value in Explicit_Problem
solver = Explicit_Problem(self.rhs, self.y0)
''' *******DELETE LATER '''''''''
# problem.handle_event = handle_event
# problem.state_events = state_events
# problem.init_mode = init_mode
solver.handle_result = self.handle_result
solver.name = 'Simple Explicit Example'
simulation = CVode(solver) # Create a RungeKutta34 solver
# simulation.inith = 0.1 #Sets the initial step, default = 0.01
# Change multistep method: 'adams' or 'VDF'
if self.discr == 'Adams':
simulation.discr = 'Adams'
simulation.maxord = 12
else:
simulation.discr = 'BDF'
simulation.maxord = 5
# Change iteration algorithm: functional(FixedPoint) or newton
if self.iter == 'FixedPoint':
simulation.iter = 'FixedPoint'
else:
simulation.iter = 'Newton'
# Sets additional parameters
simulation.atol = self.atol
simulation.rtol = self.rtol
simulation.verbosity = 0
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
# Create Solver and set settings
# noRootFunctions = np.size(self.state_events(self.t0, np.array(self.y0)))
# solver = sundials.CVodeSolver(RHS = self.f, ROOT = self.rootf, SW = [False]*noRootFunctions,
# abstol = self.atol, reltol = self.rtol)
# solver.settings.JAC = None #Add user-dependent jacobian here
'''Initialize problem '''
# solver.init(self.t0, self.y0)
self.handle_result(self.t0, self.y0)
nextTimeEvent = self.time_events(self.t0, self.y0)
self.t_cur = self.t0
self.y_cur = self.y0
state_event = False
#
#
if gridWidth <> None:
nOutputIntervals = int((Tend - self.t0) / gridWidth)
else:
nOutputIntervals = nIntervals
# Define step length depending on if gridWidth or nIntervals has been chosen
if nOutputIntervals > 0:
# Last point on grid (does not have to be Tend:)
if(gridWidth <> None):
dOutput = gridWidth
else:
dOutput = (Tend - self.t0) / nIntervals
else:
dOutput = Tend
outputStepCounter = long(1)
nextOutputPoint = min(self.t0 + dOutput, Tend)
while self.t_cur < Tend:
# Time-Event detection and step time adjustment
if nextTimeEvent is None or nextOutputPoint < nextTimeEvent:
time_event = False
self.t_cur = nextOutputPoint
else:
time_event = True
self.t_cur = nextTimeEvent
try:
# #Integrator step
# self.y_cur = solver.step(self.t_cur)
# self.y_cur = np.array(self.y_cur)
# state_event = False
# Simulate
# take a step to next output point:
t_new, y_new = simulation.simulate(self.t_cur) # 5, 10) #5, 10 self.t_cur self.t_cur 2. argument nsteps Simulate 5 seconds
# t_new, y_new are both vectors of the time and states at t_cur and all intermediate
# points before it! So take last values:
self.t_cur = t_new[-1]
self.y_cur = y_new[-1]
state_event = False
except:
import sys
print "Unexpected error:", sys.exc_info()[0]
# except CVodeRootException, info:
# self.t_cur = info.t
# self.y_cur = info.y
# self.y_cur = np.array(self.y_cur)
# time_event = False
# state_event = True
#
#
# Depending on events have been detected do different tasks
if time_event or state_event:
event_info = [state_event, time_event]
if not self.handle_event(self, event_info):
break
solver.init(self.t_cur, self.y_cur)
nextTimeEvent = self.time_events(self.t_cur, self.y_cur)
# If no timeEvent happens:
if nextTimeEvent <= self.t_cur:
nextTimeEvent = None
if self.t_cur == nextOutputPoint:
# Write output if not happened before:
if not time_event and not state_event:
self.handle_result(nextOutputPoint, self.y_cur)
outputStepCounter += 1
nextOutputPoint = min(self.t0 + outputStepCounter * dOutput, Tend)
self.finalize()
#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 #Plots the result P.plot(t,y) P.show()
def run_simulation(filename, save_output, start_time, temp, RH, RO2_indices,
H2O, PInit, y_cond, input_dict, simulation_time, batch_step,
plot_mass):
from assimulo.solvers import RodasODE, CVode, RungeKutta4, LSODAR #Choose solver accoring to your need.
from assimulo.problem import Explicit_Problem
# In this function, we import functions that have been pre-compiled for use in the ODE solver
# The function that calculates the RHS of the ODE is also defined within this function, such
# that it can be used by the Assimulo solvers
# The variables passed to this function are defined as follows:
#-------------------------------------------------------------------------------------
#-------------------------------------------------------------------------------------
# define the ODE function to be called
def dydt_func(t, y):
"""
This function defines the right-hand side [RHS] of the ordinary differential equations [ODEs] to be solved
input:
• t - time variable [internal to solver]
• y - array holding concentrations of all compounds in both gas and particulate [molecules/cc]
output:
dydt - the dy_dt of each compound in both gas and particulate phase [molecules/cc.sec]
"""
dy_dt = numpy.zeros((total_length_y, 1), )
#pdb.set_trace()
# Calculate time of day
time_of_day_seconds = start_time + t
#pdb.set_trace()
# make sure the y array is not a list. Assimulo uses lists
y_asnumpy = numpy.array(y)
Model_temp = temp
#pdb.set_trace()
#Calculate the concentration of RO2 species, using an index file created during parsing
RO2 = numpy.sum(y[RO2_indices])
#Calculate reaction rate for each equation.
# Note that H2O will change in parcel mode
# The time_of_day_seconds is used for photolysis rates - need to change this if want constant values
rates = evaluate_rates_fortran(RO2, H2O, Model_temp,
time_of_day_seconds)
#pdb.set_trace()
# Calculate product of all reactants and stochiometry for each reaction [A^a*B^b etc]
reactants = reactants_fortran(y_asnumpy[0:num_species - 1])
#pdb.set_trace()
#Multiply product of reactants with rate coefficient to get reaction rate
reactants = numpy.multiply(reactants, rates)
#pdb.set_trace()
# Now use reaction rates with the loss_gain matri to calculate the final dydt for each compound
# With the assimulo solvers we need to output numpy arrays
dydt_gas = loss_gain_fortran(reactants)
#pdb.set_trace()
dy_dt[0:num_species - 1, 0] = dydt_gas
# Change the saturation vapour pressure of water
# Need to re-think the change of organic vapour pressures with temperature.
# At the moment this is kept constant as re-calulation using UManSysProp very slow
sat_vap_water = numpy.exp((-0.58002206E4 / Model_temp) + 0.13914993E1 - \
(0.48640239E-1 * Model_temp) + (0.41764768E-4 * (Model_temp**2.0E0))- \
(0.14452093E-7 * (Model_temp**3.0E0)) + (0.65459673E1 * numpy.log(Model_temp)))
sat_vp[-1] = (numpy.log10(sat_vap_water * 9.86923E-6))
Psat = numpy.power(10.0, sat_vp)
# Convert the concentration of each component in the gas phase into a partial pressure using the ideal gas law
# Units are Pascals
Pressure_gas = (y_asnumpy[0:num_species, ] /
NA) * 8.314E+6 * Model_temp #[using R]
core_mass_array = numpy.multiply(ycore_asnumpy / NA, core_molw_asnumpy)
####### Calculate the thermal conductivity of gases according to the new temperature ########
K_water_vapour = (
5.69 + 0.017 *
(Model_temp - 273.15)) * 1e-3 * 4.187 #[W/mK []has to be in W/m.K]
# Use this value for all organics, for now. If you start using a non-zero enthalpy of
# vapourisation, this needs to change.
therm_cond_air = K_water_vapour
#----------------------------------------------------------------------------
#F2c) Extract the current gas phase concentrations to be used in pressure difference calculations
C_g_i_t = y_asnumpy[0:num_species, ]
#Set the values for oxidants etc to 0 as will force no mass transfer
#C_g_i_t[ignore_index]=0.0
C_g_i_t = C_g_i_t[include_index]
#pdb.set_trace()
total_SOA_mass,aw_array,size_array,dy_dt_calc = dydt_partition_fortran(y_asnumpy,ycore_asnumpy,core_dissociation, \
core_mass_array,y_density_array_asnumpy,core_density_array_asnumpy,ignore_index_fortran,y_mw,Psat, \
DStar_org_asnumpy,alpha_d_org_asnumpy,C_g_i_t,N_perbin,gamma_gas_asnumpy,Latent_heat_asnumpy,GRAV, \
Updraft,sigma,NA,kb,Rv,R_gas,Model_temp,cp,Ra,Lv_water_vapour)
#pdb.set_trace()
# Add the calculated gains/losses to the complete dy_dt array
dy_dt[0:num_species + (num_species_condensed * num_bins),
0] += dy_dt_calc[:]
#pdb.set_trace()
#----------------------------------------------------------------------------
#F4) Now calculate the change in water vapour mixing ratio.
#To do this we need to know what the index key for the very last element is
#pdb.set_trace()
#pdb.set_trace()
#print "elapsed time=", elapsedTime
dydt_func.total_SOA_mass = total_SOA_mass
dydt_func.size_array = size_array
dydt_func.temp = Model_temp
dydt_func.RH = Pressure_gas[-1] / (Psat[-1] * 101325.0)
dydt_func.water_activity = aw_array
#----------------------------------------------------------------------------
return dy_dt
#-------------------------------------------------------------------------------------
#-------------------------------------------------------------------------------------
#import static compilation of Fortran functions for use in ODE solver
print("Importing pre-compiled Fortran modules")
from rate_coeff_f2py import evaluate_rates as evaluate_rates_fortran
from reactants_conc_f2py import reactants as reactants_fortran
from loss_gain_f2py import loss_gain as loss_gain_fortran
from partition_f2py import dydt_partition as dydt_partition_fortran
# 'Unpack' variables from input_dict
species_dict = input_dict['species_dict']
species_dict2array = input_dict['species_dict2array']
species_initial_conc = input_dict['species_initial_conc']
equations = input_dict['equations']
num_species = input_dict['num_species']
num_species_condensed = input_dict['num_species_condensed']
y_density_array_asnumpy = input_dict['y_density_array_asnumpy']
y_mw = input_dict['y_mw']
sat_vp = input_dict['sat_vp']
Delta_H = input_dict['Delta_H']
Latent_heat_asnumpy = input_dict['Latent_heat_asnumpy']
DStar_org_asnumpy = input_dict['DStar_org_asnumpy']
alpha_d_org_asnumpy = input_dict['alpha_d_org_asnumpy']
gamma_gas_asnumpy = input_dict['gamma_gas_asnumpy']
Updraft = input_dict['Updraft']
GRAV = input_dict['GRAV']
Rv = input_dict['Rv']
Ra = input_dict['Ra']
R_gas = input_dict['R_gas']
R_gas_other = input_dict['R_gas_other']
cp = input_dict['cp']
sigma = input_dict['sigma']
NA = input_dict['NA']
kb = input_dict['kb']
Lv_water_vapour = input_dict['Lv_water_vapour']
ignore_index = input_dict['ignore_index']
ignore_index_fortran = input_dict['ignore_index_fortran']
ycore_asnumpy = input_dict['ycore_asnumpy']
core_density_array_asnumpy = input_dict['core_density_array_asnumpy']
y_cond = input_dict['y_cond_initial']
num_bins = input_dict['num_bins']
core_molw_asnumpy = input_dict['core_molw_asnumpy']
core_dissociation = input_dict['core_dissociation']
N_perbin = input_dict['N_perbin']
include_index = input_dict['include_index']
# pdb.set_trace()
#Specify some starting concentrations [ppt]
Cfactor = 2.55e+10 #ppb-to-molecules/cc
# Create variables required to initialise ODE
y0 = [0] * (num_species + num_species_condensed * num_bins
) #Initial concentrations, set to 0
t0 = 0.0 #T0
# Define species concentrations in ppb fr the gas phase
# You have already set this in the front end script, and now we populate the y array with those concentrations
for specie in species_initial_conc.keys():
if specie is not 'H2O':
y0[species_dict2array[specie]] = species_initial_conc[
specie] * Cfactor #convert from pbb to molcules/cc
elif specie is 'H2O':
y0[species_dict2array[specie]] = species_initial_conc[specie]
# Now add the initial condensed phase [including water]
#pdb.set_trace()
y0[num_species:num_species +
((num_bins) * num_species_condensed)] = y_cond[:]
#pdb.set_trace()
#Set the total_time of the simulation to 0 [havent done anything yet]
total_time = 0.0
# Define a 'key' that represents the end of the composition variables to track
total_length_y = len(y0)
key = num_species + ((num_bins) * num_species) - 1
#pdb.set_trace()
# Now run through the simulation in batches.
# I do this to enable testing of coupling processes. Some initial investigations with non-ideality in
# the condensed phase indicated that even defining a maximum step was not enough for ODE solvers to
# overshoot a stable region. It also helps with in-simulation debugging. Its up to you if you want to keep this.
# To not run in batches, just define one batch as your total simulation time. This will reduce any overhead with
# initialising the solvers
# Set total simulation time and batch steps in seconds
# Note also that the current module outputs solver information after each batch step. This can be turned off and the
# the batch step change for increased speed
# simulation_time= 3600.0
# batch_step=300.0
t_array = []
time_step = 0
number_steps = int(
simulation_time /
batch_step) # Just cycling through 3 steps to get to a solution
# Define a matrix that stores values as outputs from the end of each batch step. Again, you can remove
# the need to run in batches. You can tell the Assimulo solvers the frequency of outputs.
y_matrix = numpy.zeros((int(number_steps), len(y0)))
# Also define arrays and matrices that hold information such as total SOA mass
SOA_matrix = numpy.zeros((int(number_steps), 1))
size_matrix = numpy.zeros((int(number_steps), num_bins))
print("Starting simulation")
# In the following, we can
while total_time < simulation_time:
if total_time == 0.0:
#Define an Assimulo problem
#Define an explicit solver
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
else:
y0 = y_output[
-1, :] # Take the output from the last batch as the start of this
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
# Define ODE parameters.
# Initial steps might be slower than mid-simulation. It varies.
#exp_mod.jac = dydt_jac
# Define which ODE solver you want to use
exp_sim = CVode(exp_mod)
tol_list = [1.0e-2] * len(y0)
exp_sim.atol = tol_list #Default 1e-6
exp_sim.rtol = 1.0e-4 #Default 1e-6
exp_sim.inith = 1.0e-6 #Initial step-size
#exp_sim.discr = 'Adams'
exp_sim.maxh = 100.0
# Use of a jacobian makes a big differece in simulation time. This is relatively
# easy to define for a gas phase - not sure for an aerosol phase with composition
# dependent processes.
exp_sim.usejac = False # To be provided as an option in future update.
#exp_sim.fac1 = 0.05
#exp_sim.fac2 = 50.0
exp_sim.report_continuously = True
exp_sim.maxncf = 1000
#Sets the parameters
t_output, y_output = exp_sim.simulate(
batch_step) #Simulate 'batch' seconds
total_time += batch_step
t_array.append(
total_time
) # Save the output from the end step, of the current batch, to a matrix
y_matrix[time_step, :] = y_output[-1, :]
SOA_matrix[time_step, 0] = dydt_func.total_SOA_mass
size_matrix[time_step, :] = dydt_func.size_array
print("SOA [micrograms/m3] = ", dydt_func.total_SOA_mass)
#now save this information into a matrix for later plotting.
time_step += 1
if save_output is True:
print(
"Saving the model output as a pickled object for later retrieval")
# save the dictionary to a file for later retrieval - have to do each seperately.
with open(filename + '_y_output.pickle', 'wb') as handle:
pickle.dump(y_matrix, handle, protocol=pickle.HIGHEST_PROTOCOL)
with open(filename + '_t_output.pickle', 'wb') as handle:
pickle.dump(t_array, handle, protocol=pickle.HIGHEST_PROTOCOL)
with open(filename + '_SOA_output.pickle', 'wb') as handle:
pickle.dump(SOA_matrix, handle, protocol=pickle.HIGHEST_PROTOCOL)
with open(filename + '_size_output.pickle', 'wb') as handle:
pickle.dump(size_matrix, handle, protocol=pickle.HIGHEST_PROTOCOL)
with open(filename + 'include_index.pickle', 'wb') as handle:
pickle.dump(include_index,
handle,
protocol=pickle.HIGHEST_PROTOCOL)
#pdb.set_trace()
#Plot the change in concentration over time for a given specie. For the user to change / remove
#In a future release I will add this as a seperate module
if plot_mass is True:
try:
P.plot(t_array, SOA_matrix[:, 0], marker='o')
P.title(exp_mod.name)
P.ylabel("SOA mass [micrograms/m3]")
P.xlabel("Time [seconds] since start of simulation")
P.show()
except:
print(
"There is a problem using Matplotlib in your environment. If using this within a docker container, you will need to transfer the data to the host or configure your container to enable graphical displays. More information can be found at http://wiki.ros.org/docker/Tutorials/GUI "
)
def run_example(with_plots=True):
r"""
This example shows how to use Assimulo and CVode for simulating sensitivities
for initial conditions.
.. math::
\dot y_1 &= -(k_{01}+k_{21}+k_{31}) y_1 + k_{12} y_2 + k_{13} y_3 + b_1\\
\dot y_2 &= k_{21} y_1 - (k_{02}+k_{12}) y_2 \\
\dot y_3 &= k_{31} y_1 - k_{13} y_3
with the parameter dependent inital conditions
:math:`y_1(0) = 0, y_2(0) = 0, y_3(0) = 0` . The initial values are taken as parameters :math:`p_1,p_2,p_3`
for the computation of the sensitivity matrix,
see http://sundials.2283335.n4.nabble.com/Forward-sensitivities-for-initial-conditions-td3239724.html
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
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,
name='Example: Computing Sensitivities')
#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.report_continuously = 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:
title_text = r"Sensitivity w.r.t. ${}$"
legend_text = r"$\mathrm{{d}}{}/\mathrm{{d}}{}$"
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(title_text.format('p_1'))
P.legend((legend_text.format('y_1',
'p_1'), legend_text.format('y_1', 'p_2'),
legend_text.format('y_1', 'p_3')))
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(title_text.format('p_2'))
P.legend((legend_text.format('y_2',
'p_1'), legend_text.format('y_2', 'p_2'),
legend_text.format('y_2', 'p_3')))
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(title_text.format('p_3'))
P.legend((legend_text.format('y_3',
'p_1'), legend_text.format('y_3', 'p_2'),
legend_text.format('y_3', 'p_3')))
P.subplot(224)
P.title('ODE Solution')
P.plot(t, y)
P.suptitle(exp_mod.name)
P.show()
return exp_mod, exp_sim
def func_set_system():
##############################################
#% initial conditions
P_0 = depth*9.8*2600. #; % initial chamber pressure (Pa)
T_0 = 1200 #; % initial chamber temperature (K)
eps_g0 = 0.04 #; % initial gas volume fraction
rho_m0 = 2600 #; % initial melt density (kg/m^3)
rho_x0 = 3065 #; % initial crystal density (kg/m^3)
a = 2000. #; % initial radius of the chamber (m)
V_0 = (4.*np.pi/3.)*a**3. #; % initial volume of the chamber (m^3)
##############################################
##############################################
IC = np.array([P_0, T_0, eps_g0, V_0, rho_m0, rho_x0]) # % store initial conditions
## Gas (eps_g = zero), eps_x is zero, too many crystals, 50 % crystallinity,eruption (yes/no)
sw0 = [False,False,False,False,False]
##############################################
#% error tolerances used in ode method
dt = 3e7*10.
begin_time = 0 # ; % initialize time
N = int(round((end_time-begin_time)/dt))
##############################################
#Define an Assimulo problem
exp_mod = Chamber_Problem(depth=depth,t0=begin_time,y0=IC,sw0=sw0)
exp_mod.param['T_S'] = 500.#+273.
exp_mod.param['T_in'] = 1200.
exp_mod.param['eps_g_in'] = 0.0 # Gas fraction of incoming melt - gas phase ..
exp_mod.param['m_eq_in'] = 0.03 # Volatile fraction of incoming melt
exp_mod.param['Mdot_in'] = mdot
exp_mod.param['eta_x_max'] = 0.55 # Locking fraction
exp_mod.param['delta_Pc'] = 20e6
exp_mod.allow_diffusion_init = True
exp_mod.radius = a
exp_mod.permeability = perm_val
exp_mod.R_steps = 1500
exp_mod.dt_init = dt
inp_func1 = inp.Input_functions_Degruyer()
exp_mod.set_input_functions(inp_func1)
exp_mod.get_constants()
exp_mod.set_init_crust(material = 'Westerly_Granite')
#################
begin_time = exp_mod.set_init_crust_profile(T_0)
exp_mod.tcurrent = begin_time
P_0 = exp_mod.plith
exp_mod.t0 = begin_time
exp_mod.param['heat_cond'] = 1. # Turn on/off heat conduction
exp_mod.param['visc_relax'] = 1. # Turn on/off viscous relaxation
exp_mod.param['press_relax'] = 1. ## Turn on/off pressure diffusion
exp_mod.param['frac_rad_Temp'] =0.75
exp_mod.param['vol_degass'] = 1.
exp_mod.param['outflow_model'] = None # 'huppert'
IC = np.array([P_0, T_0, eps_g0, V_0, rho_m0, rho_x0]) # % store initial conditions
exp_mod.y0 = IC
#Define an explicit solver
exp_sim = CVode(exp_mod) #Create a CVode solver
# exp_sim = LSODAR(exp_mod) #Create a CVode solver
#Sets the parameters
exp_sim.store_event_points = True
#exp_sim.iter = 'Newton'
#exp_sim.discr = 'BDF'
exp_sim.inith = 1e2
#exp_sim.display_progress = True
exp_sim.rtol = 1.e-7
#exp_sim.maxh = 3e8*5. # 10 years
exp_sim.atol = 1e-7
#exp_sim.sensmethod = 'SIMULTANEOUS' #Defines the sensitvity method used
#exp_sim.suppress_sens = True #Dont suppress the sensitivity variables in the error test.
#exp_sim.usesens = True
#exp_sim.report_continuously = True
return exp_mod,exp_sim,N
def run_sim(Y, time, Y_AER1, YICE):
def dy_dt_func(t, Y):
dy_dt = np.zeros(len(Y))
# add condensed semi-vol mass into bins
if n.SV_flag:
MBIN2[-1 * n.n_sv:, :] = np.reshape(Y[INDSV1:INDSV2],
[n.n_sv, nbins * nmodes])
# calculate saturation vapour pressure over liquid
svp1 = f.svp_liq(Y[ITEMP])
# saturation ratio
SL = svp1 * Y[IRH] / (Y[IPRESS] - svp1)
SL = (SL * Y[IPRESS] / (1 + SL)) / svp1
# water vapour mixing ratio
WV = c.eps * Y[IRH] * svp1 / (Y[IPRESS] - svp1)
WL = np.sum(Y[IND1:IND2] * Y[:IND1]) # LIQUID MIXING RATIO
WI = np.sum(YICE[IND1:IND2] * YICE[:IND1]) # ice mixing ratio
RM = c.RA + WV * c.RV
CPM = c.CP + WV * c.CPV + WL * c.CPW + WI * c.CPI
if simulation_type.lower() == 'chamber':
# CHAMBER MODEL - pressure change
dy_dt[IPRESS] = -100 * PRESS1 * PRESS2 * np.exp(-PRESS2 *
(time + t))
elif simulation_type.lower() == 'parcel':
# adiabatic parcel
dy_dt[IPRESS] = -Y[IPRESS] / RM / Y[
ITEMP] * c.g * w #! HYDROSTATIC EQUATION
else:
print('simulation type unknown')
return
# ----------------------------change in vapour content: -----------------------
# 1. equilibruim size of particles
if n.kappa_flag:
#if n.SV_flag:
# need to recalc kappa taking into acount the condensed semi-vols
Kappa = np.sum(
(MBIN2[:, :] / RHOBIN2[:, :]) * KAPPABIN2[:, :],
axis=0) / np.sum(MBIN2[:, :] / RHOBIN2[:, :], axis=0)
# print(Kappa)
# print(MBIN2/RHOBIN2)
KK01 = f.kk01(Y[0:IND1], Y[ITEMP], MBIN2, RHOBIN2, Kappa)
else:
KK01 = f.K01(Y[0:IND1], Y[ITEMP], MBIN2, n.n_sv, RHOBIN2,
NUBIN2, MOLWBIN2)
# print(KK01[0])
Dw = KK01[2] # wet diameter
RHOAT = KK01[1] # density of particles inc water and aerosol mass
RH_EQ = KK01[0] # equilibrium RH
# print(MBIN2/MOLWBIN2)
# 2. growth rate of particles, Jacobson p455
# rate of change of radius
growth_rate = f.DROPGROWTHRATE(Y[ITEMP], Y[IPRESS], SL, RH_EQ,
RHOAT, Dw)
growth_rate[np.isnan(growth_rate)] = 0 # get rid of nans
growth_rate = np.where(Y[IND1:IND2] < 1e-9, 0.0, growth_rate)
# 3. Mass of water condensing
# change in mass of water per particle
dy_dt[:IND1] = (np.pi * RHOAT * Dw**2) * growth_rate
# 4. Change in vapour content
# change in water vapour mixing ratio
dwv_dt = -1 * np.sum(
Y[IND1:IND2] *
dy_dt[:IND1]) # change to np.sum for speed # mass
# -----------------------------------------------------------------------------
if simulation_type.lower() == 'chamber':
# CHAMBER MODEL - temperature change
dy_dt[ITEMP] = -Temp1 * Temp2 * np.exp(-Temp2 * (time + t))
elif simulation_type.lower() == 'parcel':
# adiabatic parcel
dy_dt[ITEMP] = RM / Y[IPRESS] * dy_dt[IPRESS] * Y[
ITEMP] / CPM # TEMPERATURE CHANGE: EXPANSION
dy_dt[ITEMP] = dy_dt[ITEMP] - c.LV / CPM * dwv_dt
else:
print('simulation type unknown')
return
# --------------------------------RH change------------------------------------
dy_dt[IRH] = svp1 * dwv_dt * (Y[IPRESS] - svp1)
dy_dt[IRH] = dy_dt[IRH] + svp1 * WV * dy_dt[IPRESS]
dy_dt[IRH] = (
dy_dt[IRH] - WV * Y[IPRESS] *
derivative(f.svp_liq, Y[ITEMP], dx=1.0) * dy_dt[ITEMP])
dy_dt[IRH] = dy_dt[IRH] / (c.eps * svp1**2)
# -----------------------------------------------------------------------------
# ------------------------------ SEMI-VOLATILES -------------------------------
if n.SV_flag:
# SV_mass = np.reshape(Y[INDSV1:INDSV2],[n.n_sv,n.nmodes*n.nbins])
# SV_mass = np.where(SV_mass == 0.0,1e-30,SV_mass)
# MBIN2[n.n_sv*-1:,:] = SV_mass
RH_EQ_SV = f.K01SV(Y[:IND1], Y[ITEMP], MBIN2, n.n_sv, RHOBIN2,
NUBIN2, MOLWBIN2)
RH_EQ = RH_EQ_SV[0]
RHOAT = RH_EQ_SV[1]
DW = RH_EQ_SV[2]
SVP_ORG = f.SVP_GASES(n.semi_vols, Y[ITEMP],
n.n_sv) #C-C equation
#RH_ORG = [x*Y[IPRESS]/c.RA/Y[ITEMP] for x in Y[IRH_SV]]
RH_ORG = [x for x in Y[IRH_SV]]
RH_ORG = [(x / c.aerosol_dict[key][0]) * c.R * Y[ITEMP]
for x, key in zip(RH_ORG, n.semi_vols[:n.n_sv])
] # just for n_sv keys in dictionary
RH_ORG = [RH_ORG[x] / SVP_ORG[x] for x in range(n.n_sv)]
dy_dt[INDSV1:INDSV2] = f.SVGROWTHRATE(Y[ITEMP], Y[IPRESS],
SVP_ORG, RH_ORG, RH_EQ,
DW, n.n_sv, n.nbins,
n.nmodes, MOLWBIN2)
dy_dt[IRH_SV] = -np.sum(np.reshape(
dy_dt[INDSV1:INDSV2], [n.n_sv, IND1]) * Y[IND1:IND2],
axis=1) #see line 137
return dy_dt
#--------------------- SET-UP solver ------------------------------------------
y0 = Y
t0 = 0.0
#define assimulo problem
exp_mod = Explicit_Problem(dy_dt_func, y0, t0)
# define an explicit solver
exp_sim = CVode(exp_mod)
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
#set parameters
tol_list = np.zeros_like(Y)
tol_list[0:IND1] = 1e-40 # this is now different to ACPIM (1e-25)
tol_list[IND1:IND2] = 10 # number
tol_list[IND2:IND3] = 1e-30 # capacitance
tol_list[IND3:INDSV2] = 1e-26 #condendensed semi-vol mass
tol_list[IRH_SV] = 1e-26 # RH of each semi-vol compound
tol_list[IPRESS] = 10
tol_list[ITEMP] = 1e-4
tol_list[IRH] = 1e-8 # set tolerance for each dydt function
exp_sim.atol = tol_list
exp_sim.rtol = 1.0e-8
exp_sim.inith = 0 # initial time step-size
exp_sim.usejac = False
exp_sim.maxncf = 100 # max number of convergence failures allowed by solver
exp_sim.verbosity = 40
t_output, y_output = exp_sim.simulate(1)
return y_output[-1, :], t_output[:]
def run_sim_ice(Y, YLIQ):
def dy_dt_func(t, Y):
dy_dt = np.zeros(len(Y))
svp = f.svp_liq(Y[ITEMP])
svp_ice = f.svp_ice(Y[ITEMP])
# vapour mixing ratio
WV = c.eps * Y[IRH_ICE] * svp / (Y[IPRESS_ICE] - svp)
# liquid mixing ratio
WL = sum(YLIQ[IND1:IND2] * YLIQ[0:IND1])
# ice mixing ratio
WI = sum(Y[IND1:IND2] * Y[0:IND1])
Cpm = c.CP + WV * c.CPV + WL * c.CPW + WI * c.CPI
# RH with respect to ice
RH_ICE = WV / (c.eps * svp_ice / (Y[IPRESS_ICE] - svp_ice))
# ------------------------- growth rate of ice --------------------------
RH_EQ = 1e0 # from ACPIM, FPARCELCOLD - MICROPHYSICS.f90
CAP = f.CAPACITANCE01(Y[0:IND1], np.exp(Y[IND2:IND3]))
growth_rate = f.ICEGROWTHRATE(Y[ITEMP_ICE], Y[IPRESS_ICE],
RH_ICE, RH_EQ, Y[0:IND1],
np.exp(Y[IND2:IND3]), CAP)
growth_rate[np.isnan(growth_rate)] = 0 # get rid of nans
growth_rate = np.where(Y[IND1:IND2] < 1e-6, 0.0, growth_rate)
# Mass of water condensing
dy_dt[:IND1] = growth_rate
#---------------------------aspect ratio---------------------------------------
DELTA_RHO = c.eps * svp / (Y[IPRESS_ICE] - svp)
DELTA_RHOI = c.eps * svp_ice / (Y[IPRESS_ICE] - svp_ice)
DELTA_RHO = Y[IRH_ICE] * DELTA_RHO - DELTA_RHOI
DELTA_RHO = DELTA_RHO * Y[IPRESS_ICE] / Y[ITEMP_ICE] / c.RA
RHO_DEP = f.DEP_DENSITY(DELTA_RHO, Y[ITEMP_ICE])
# this is the rate of change of LOG of the aspect ratio
dy_dt[IND2:IND3] = (dy_dt[0:IND1] *
((f.INHERENTGROWTH(Y[ITEMP_ICE]) - 1) /
(f.INHERENTGROWTH(Y[ITEMP_ICE]) + 2)) /
(Y[0:IND1] * c.rhoi * RHO_DEP))
#------------------------------------------------------------------------------
# Change in vapour content
dwv_dt = -1 * sum(Y[IND1:IND2] * dy_dt[0:IND1])
# change in water vapour mixing ratio
DRI = -1 * dwv_dt
dy_dt[ITEMP_ICE] = 0.0 #+c.LS/Cpm*DRI
# if n.Simulation_type.lower() == 'parcel':
# dy_dt[ITEMP_ICE]=dy_dt[ITEMP_ICE] + c.LS/Cpm*DRI
#---------------------------RH change------------------------------------------
dy_dt[IRH_ICE] = (Y[IPRESS_ICE] - svp) * svp * dwv_dt
dy_dt[IRH_ICE] = (
dy_dt[IRH_ICE] - WV * Y[IPRESS_ICE] *
derivative(f.svp_liq, Y[ITEMP_ICE], dx=1.0) * dy_dt[ITEMP_ICE])
dy_dt[IRH_ICE] = dy_dt[IRH_ICE] / (c.eps * svp**2)
#------------------------------------------------------------------------------
return dy_dt
#--------------------- SET-UP solver --------------------------------------
y0 = Y
t0 = 0.0
#define assimulo problem
exp_mod = Explicit_Problem(dy_dt_func, y0, t0)
# define an explicit solver
exp_sim = CVode(exp_mod)
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
# set tolerance for each dydt function
tol_list = np.zeros_like(Y)
tol_list[0:IND1] = 1e-30 # mass
tol_list[IND1:IND2] = 10 # number
tol_list[IND2:IND3] = 1e-30 # aspect ratio
# tol_list[IND3:IRH_SV_ICE] = 1e-26
#tol_list[IRH_SV_ICE] = 1e-26
tol_list[IPRESS_ICE] = 10
tol_list[ITEMP_ICE] = 1e-4
tol_list[IRH_ICE] = 1e-8
exp_sim.atol = tol_list
exp_sim.rtol = 1.0e-8
exp_sim.inith = 1.0e-2 # initial time step-size
exp_sim.usejac = False
exp_sim.maxncf = 100 # max number of convergence failures allowed by solver
exp_sim.verbosity = 40
t_output, y_output = exp_sim.simulate(1)
return y_output[-1, :]
def run_example(with_plots=True):
r"""
Example to demonstrate the use of a preconditioner
.. math::
\dot y_1 & = 2 t \sin y_1 + t \sin y_2 \\
\dot y_2 & = 3 t \sin y_1 + 2 t \sin y_2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#Define the rhs
def rhs(t, y):
A = np.array([[2.0, 1.0], [3.0, 2.0]])
yd = np.dot(A * t, np.sin(y))
return yd
#Define the preconditioner setup function
def prec_setup(t, y, fy, jok, gamma, data):
A = np.array([[2.0, 1.0], [3.0, 2.0]])
#If jok is false the jacobian data needs to be recomputed
if jok == False:
#Extract the diagonal of the jacobian to form a Jacobi preconditioner
a0 = A[0, 0] * t * np.cos(y[0])
a1 = A[1, 1] * t * np.cos(y[1])
a = np.array([(1. - gamma * a0), (1. - gamma * a1)])
#Return true (jacobian data was recomputed) and the new data
return [True, a]
#If jok is true the existing jacobian data can be reused
if jok == True:
#Return false (jacobian data was reused) and the old data
return [False, data]
#Define the preconditioner solve function
def prec_solve(t, y, fy, r, gamma, delta, data):
#Solve the system Pz = r
z0 = r[0] / data[0]
z1 = r[1] / data[1]
z = np.array([z0, z1])
return z
#Initial conditions
y0 = [1.0, 2.0]
#Define an Assimulo problem
exp_mod = Explicit_Problem(
rhs, y0, name="Example of using a preconditioner in SUNDIALS")
#Set the preconditioner setup and solve function for the problem
exp_mod.prec_setup = prec_setup
exp_mod.prec_solve = prec_solve
#Create a CVode solver
exp_sim = CVode(exp_mod)
#Set the parameters for the solver
exp_sim.iter = 'Newton'
exp_sim.discr = 'BDF'
exp_sim.atol = 1e-5
exp_sim.rtol = 1e-5
exp_sim.linear_solver = 'SPGMR'
exp_sim.precond = "PREC_RIGHT" #Set the desired type of preconditioning
#Simulate
t, y = exp_sim.simulate(5)
if with_plots:
exp_sim.plot()
#Basic verification
nose.tools.assert_almost_equal(y[-1, 0], 3.11178295, 4)
nose.tools.assert_almost_equal(y[-1, 1], 3.19318992, 4)
return exp_mod, exp_sim
def run_example(with_plots=True):
"""
Simulations for the Gyro (Heavy Top) example in Celledoni/Safstrom:
Journal of Physics A, Vol 39, 5463-5478, 2006
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
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
exp_mod = Explicit_Problem(f, y0, name="Gyroscope Example")
#Create an Assimulo explicit solver (CVode)
exp_sim = CVode(exp_mod)
#Sets the parameters
exp_sim.discr = 'BDF'
exp_sim.iter = 'Newton'
exp_sim.maxord = 2 #Sets the maxorder
exp_sim.atol = 1.e-10
exp_sim.rtol = 1.e-10
#Simulate
t, y = exp_sim.simulate(0.1)
#Plot
if with_plots:
import pylab as P
P.plot(t, y / 10000.)
P.xlabel('Time')
P.ylabel('States, scaled by $10^4$')
P.title(exp_mod.name)
P.show()
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], 692.800241862)
nose.tools.assert_almost_equal(y[-1][8], 7.08468221e-1)
return exp_mod, exp_sim
def run_simulation(filename, save_output, start_time, temp, RH, RO2_indices,
H2O, input_dict, simulation_time, batch_step):
from assimulo.solvers import RodasODE, CVode #Choose solver accoring to your need.
from assimulo.problem import Explicit_Problem
# In this function, we import functions that have been pre-compiled for use in the ODE solver
# The function that calculates the RHS of the ODE is also defined within this function, such
# that it can be used by the Assimulo solvers
# In the standard Python version [not using Numba] I use Sparse matrix operations in calculating loss/gain of each compound.
# This function loads the matrix created at the beginning of the module.
def load_sparse_csr(filename):
loader = numpy.load('loss_gain_' + filename + '.npz')
return csr_matrix(
(loader['data'], loader['indices'], loader['indptr']),
shape=loader['shape'])
def load_sparse_csr_reactants(filename):
loader = numpy.load('reactants_indices_sparse_' + filename + '.npz')
return csr_matrix(
(loader['data'], loader['indices'], loader['indptr']),
shape=loader['shape'])
#-------------------------------------------------------------------------------------
# define the ODE function to be called
def dydt_func(t, y):
"""
This function defines the right-hand side [RHS] of the ordinary differential equations [ODEs] to be solved
input:
• t - time variable [internal to solver]
• y - array holding concentrations of all compounds in both gas and particulate [molecules/cc]
output:
dydt - the dy_dt of each compound in both gas and particulate phase [molecules/cc.sec]
"""
#pdb.set_trace()
#Here we use the pre-created Numba based functions to arrive at our value for dydt
# Calculate time of day
time_of_day_seconds = start_time + t
# make sure the y array is not a list. Assimulo uses lists
y_asnumpy = numpy.array(y)
#pdb.set_trace()
# reactants=numpy.zeros((equations),)
#pdb.set_trace()
#Calculate the concentration of RO2 species, using an index file created during parsing
RO2 = numpy.sum(y[RO2_indices])
#Calculate reaction rate for each equation.
# Note that H2O will change in parcel mode [to be changed in the full aerosol mode]
# The time_of_day_seconds is used for photolysis rates - need to change this if want constant values
#pdb.set_trace()
rates = evaluate_rates(time_of_day_seconds, RO2, H2O, temp,
numpy.zeros((equations)), numpy.zeros((63)))
# Calculate product of all reactants and stochiometry for each reaction [A^a*B^b etc]
reactants = reactant_product(y_asnumpy, equations,
numpy.zeros((equations)))
#Multiply product of reactants with rate coefficient to get reaction rate
#pdb.set_trace()
reactants = numpy.multiply(reactants, rates)
# Now use reaction rates with the loss_gain information in a pre-created Numba file to calculate the final dydt for each compound
dydt = dydt_eval(numpy.zeros((len(y_asnumpy))), reactants)
#pdb.set_trace()
############ Development place-holder ##############
# ----------------------------------------------------------------------------------
# The following demonstrates the same procedure but using only Numpy and pure python
# For the full MCM this is too slow, but is useful for demonstrations and testing
#Calculate reaction rate for each equation.
## rates=test(time_of_day_seconds,RO2,H2O,temp)
# Calculate product of all reactants and stochiometry for each reaction [A^a*B^b etc]
# Take the approach of using sparse matrix operations from a python perspective
# This approach uses the rule of logarithms and sparse matrix multiplication
##temp_array=reactants_indices_sparse @ numpy.log(y_asnumpy)
##indices=numpy.where(temp_array > 0.0)
##reactants[indices]=numpy.exp(temp_array[indices])
#Multiply product of reactants with rate coefficient to get reaction rate
## reactants = numpy.multiply(reactants,rates)
# Now use reaction rates with the loss_gain matri to calculate the final dydt for each compound
# With the assimulo solvers we need to output numpy arrays
##dydt=numpy.array(loss_gain @ reactants)
# ----------------------------------------------------------------------------------
return dydt
#-------------------------------------------------------------------------------------
print(
"Importing Numba modules [compiling if first import or clean build...please be patient]"
)
#import Numba functions for use in ODE solver
from Rate_coefficients_numba import evaluate_rates
from Reactants_conc_numba import reactants as reactant_product
from Loss_Gain_numba import dydt as dydt_eval
# 'Unpack' variables from input_dict
species_dict = input_dict['species_dict']
species_dict2array = input_dict['species_dict2array']
species_initial_conc = input_dict['species_initial_conc']
equations = input_dict['equations']
# Set dive by zero to ignore for use of any sparse matrix multiplication
numpy.errstate(divide='ignore')
# --- For Numpy and pure Python runs ----
# Load the sparse matrix used in calculating the reactant products and dydt function
## reactants_indices_sparse = load_sparse_csr_reactants(filename)
## loss_gain = load_sparse_csr(filename)
#Specify some starting concentrations [ppt]
Cfactor = 2.55e+10 #ppb-to-molecules/cc
# Create variables required to initialise ODE
num_species = len(species_dict.keys())
y0 = [0] * num_species #Initial concentrations, set to 0
t0 = 0.0 #T0
# Define species concentrations in ppb
# You have already set this in the front end script, and now we populate the y array with those concentrations
for specie in species_initial_conc.keys():
y0[species_dict2array[specie]] = species_initial_conc[
specie] * Cfactor #convert from pbb to molcules/cc
#Set the total_time of the simulation to 0 [havent done anything yet]
total_time = 0.0
# Now run through the simulation in batches.
# I do this to enable testing of coupling processes. Some initial investigations with non-ideality in
# the condensed phase indicated that even defining a maximum step was not enough for ODE solvers to
# overshoot a stable region. It also helps with in-simulation debugging. Its up to you if you want to keep this.
# To not run in batches, just define one batch as your total simulation time. This will reduce any overhead with
# initialising the solvers
# Set total simulation time and batch steps in seconds
# Note also that the current module outputs solver information after each batch step. This can be turned off and the
# the batch step change for increased speed
# simulation_time= 3600.0 # seconds
# batch_step=100.0 # seconds
t_array = []
time_step = 0
number_steps = int(
simulation_time /
batch_step) # Just cycling through 3 steps to get to a solution
# Define a matrix that stores values as outputs from the end of each batch step. Again, you can remove
# the need to run in batches. You can tell the Assimulo solvers the frequency of outputs.
y_matrix = numpy.zeros((int(number_steps), len(y0)))
print("Starting simulation")
#pdb.set_trace()
while total_time < simulation_time:
if total_time == 0.0:
#Define an Assimulo problem
#Define an explicit solver
#pdb.set_trace()
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
else:
y0 = y_output[
-1, :] # Take the output from the last batch as the start of this
exp_mod = Explicit_Problem(dydt_func, y0, t0, name=filename)
# Define ODE parameters.
# Initial steps might be slower than mid-simulation. It varies.
#exp_mod.jac = dydt_jac
# Define which ODE solver you want to use
exp_sim = CVode(exp_mod)
tol_list = [1.0e-3] * num_species
exp_sim.atol = tol_list #Default 1e-6
exp_sim.rtol = 1e-6 #Default 1e-6
exp_sim.inith = 1.0e-6 #Initial step-size
#exp_sim.discr = 'Adams'
exp_sim.maxh = 100.0
# Use of a jacobian makes a big differece in simulation time. This is relatively
# easy to define for a gas phase - not sure for an aerosol phase with composition
# dependent processes.
exp_sim.usejac = False # To be provided as an option in future update. See Fortran variant for use of Jacobian
#exp_sim.fac1 = 0.05
#exp_sim.fac2 = 50.0
exp_sim.report_continuously = True
exp_sim.maxncf = 1000
#Sets the parameters
t_output, y_output = exp_sim.simulate(
batch_step) #Simulate 'batch' seconds
total_time += batch_step
t_array.append(
total_time
) # Save the output from the end step, of the current batch, to a matrix
y_matrix[time_step, :] = y_output[-1, :]
#pdb.set_trace()
#now save this information into a matrix for later plotting.
time_step += 1
# Do you want to save the generated matrix of outputs?
if save_output:
numpy.save(filename + '_output', y_matrix)
df = pd.DataFrame(y_matrix)
df.to_csv(filename + "_output_matrix.csv")
w = csv.writer(open(filename + "_output_names.csv", "w"))
for specie, number in species_dict2array.items():
w.writerow([specie, number])
with_plots = True
#pdb.set_trace()
#Plot the change in concentration over time for a given specie. For the user to change / remove
#In a future release I will add this as a seperate module
if with_plots:
try:
P.plot(t_array,
numpy.log10(y_matrix[:, species_dict2array['APINENE']]),
marker='o',
label="APINENE")
P.plot(t_array,
numpy.log10(y_matrix[:, species_dict2array['PINONIC']]),
marker='o',
label="PINONIC")
P.title(exp_mod.name)
P.legend(loc='upper left')
P.ylabel("Concetration log10[molecules/cc]")
P.xlabel("Time [seconds] since start of simulation")
P.show()
except:
print(
"There is a problem using Matplotlib in your environment. If using this within a docker container, you will need to transfer the data to the host or configure your container to enable graphical displays. More information can be found at http://wiki.ros.org/docker/Tutorials/GUI "
)
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:
.. math::
\dot y_1 &= -p_1 y_1 + p_2 y_2 y_3 \\
\dot y_2 &= p_1 y_1 - p_2 y_2 y_3 - p_3 y_2^2 \\
\dot y_3 &= p_3 y_2^2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
def f(t, y, p):
p3 = 3.0e7
yd_0 = -p[0] * y[0] + p[1] * y[1] * y[2]
yd_1 = p[0] * y[0] - p[1] * y[1] * y[2] - p3 * y[1]**2
yd_2 = p3 * 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,
name='Sundials test example: Chemical kinetics')
#Sets the options to the problem
exp_mod.p0 = [0.040, 1.0e4] #Initial conditions for parameters
exp_mod.pbar = [0.040, 1.0e4]
#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.report_continuously = True
#Simulate
t, y = exp_sim.simulate(
4, 400) #Simulate 4 seconds with 400 communication points
#Plot
if with_plots:
import pylab as P
P.plot(t, y)
P.xlabel('Time')
P.ylabel('State')
P.title(exp_mod.name)
P.show()
#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)
return exp_mod, exp_sim
def run_example(with_plots=True):
r"""
This is the same example from the Sundials package (cvsRoberts_FSA_dns.c)
Its purpose is to demonstrate the use of parameters in the differential equation.
This simple example problem for CVode, due to Robertson
see http://www.dm.uniba.it/~testset/problems/rober.php,
is from chemical kinetics, and consists of the system:
.. math::
\dot y_1 &= -p_1 y_1 + p_2 y_2 y_3 \\
\dot y_2 &= p_1 y_1 - p_2 y_2 y_3 - p_3 y_2^2 \\
\dot y_3 &= p_3 y_ 2^2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
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])
def jac(t,y, p):
J = N.array([[-p[0], p[1]*y[2], p[1]*y[1]],
[p[0], -p[1]*y[2]-2*p[2]*y[1], -p[1]*y[1]],
[0.0, 2*p[2]*y[1],0.0]])
return J
def fsens(t, y, s, p):
J = N.array([[-p[0], p[1]*y[2], p[1]*y[1]],
[p[0], -p[1]*y[2]-2*p[2]*y[1], -p[1]*y[1]],
[0.0, 2*p[2]*y[1],0.0]])
P = N.array([[-y[0],y[1]*y[2],0],
[y[0], -y[1]*y[2], -y[1]**2],
[0,0,y[1]**2]])
return J.dot(s)+P
#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, name='Robertson Chemical Kinetics Example')
exp_mod.rhs_sens = fsens
exp_mod.jac = jac
#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 solver 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.report_continuously = 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.title(exp_mod.name)
P.xlabel('Time')
P.ylabel('State')
P.show()
return exp_mod, exp_sim
def ode_gen(t, y, num_speci, num_eqn, rindx, pindx, rstoi, pstoi, H2Oi, TEMP,
RO2_indices, num_sb, Psat, mfp, accom_coeff, surfT, y_dens,
N_perbin, DStar_org, y_mw, x, core_diss, Varr, Vbou, RH, rad0,
Vol0, end_sim_time, pconc, save_step, rbou, therm_sp, Cw,
light_time, light_stat, nreac, nprod, prodn, reacn, new_partr, MV,
nucv1, nucv2, nucv3, inflectDp, pwl_xpre, pwl_xpro, inflectk,
nuc_comp, ChamR, Rader, PInit, testf, kwgt):
# ----------------------------------------------------------
# inputs
# t - suggested time step length (s)
# num_speci - number of components
# num_eqn - number of equations
# Psat - saturation vapour pressures (molecules/cm3 (air))
# y_dens - components' densities (g/cc)
# y_mw - components' molecular weights (g/mol)
# x - radii of particle size bins (um) (excluding walls)
# therm_sp - thermal speed of components (m/s)
# DStar_org - gas-phase diffusion coefficient of components (m2/s)
# Cw - concentration of wall (molecules/cm3 (air))
# light_time - times (s) of when lights on and lights off (corresponding to light
# status in light_stat)
# light_stat - order of lights on (1) and lights off (0)
# chamA - chamber area (m2)
# nreac - number of reactants per equation
# nprod - number of products per equation
# pindx - indices of equation products (cols) in each equation (rows)
# prodn - pindx no. of columns
# reacn - rindx no. of columns
# rindx - index of reactants per equation
# rstoi - stoichometry of reactants per equation
# pstoi - stoichometry of products
# pconc - concentration of seed particles (#/cc (air)) (1)
# new_partr - radius of two ELVOC molecules together in a newly nucleating
# particle (cm)
# MV - molar volume (cc/mol) (1D array)
# nuc_comp - index of the nucleating component
# ChamR - spherical equivalent radius of chamber (below eq. 2 Charan (2018)) (m)
# Rader - flag of whether or not to use Rader and McMurry approach
# PInit - pressure inside chamber (Pa)
# testf - flag to say whether in normal mode (0) or testing mode (1)
# kgwt - mass transfer coefficient for vapour-wall partitioning (cm3/molecule.s)
# ----------------------------------------------------------
if testf == 1:
return (0, 0, 0, 0) # return dummies
R_gas = si.R # ideal gas constant (kg.m2.s-2.K-1.mol-1)
NA = si.Avogadro # Avogadro's number (molecules/mol)
step = 0 # ode time interval step number
t0 = t # remember original suggested time step (s)
# final +1 for ELVOC in newly nucleating particles
y0 = np.zeros((num_speci + num_sb * num_speci))
y0[:] = y[:] # initial concentrations (molecules/cc (air))
y00 = np.zeros((num_speci + num_sb * num_speci))
y00[:] = y[:] # initial concentrations (molecules/cc (air))
# initial volumes of particles in size bins at start of time steps
if num_sb > 1:
Vstart = np.zeros((num_sb - 1))
Vstart[:] = Vol0[:] * N_perbin
else:
Vstart = 0.0
sumt = 0.0 # total time integrated over (s)
# record initial values
if num_sb > 0:
# particle-phase concentrations (molecules/cc (air))
yp = np.transpose(y[num_speci:-(num_speci)].reshape(
num_sb - 1, num_speci))
else:
yp = 0.0
[t_out, y_mat, Nresult,
x2] = recording(y, N_perbin, x, step, sumt, 0, 0, 0, 0,
int(end_sim_time / save_step), num_speci, num_sb,
y_mw[:, 0], y_dens[:, 0] * 1.0e-3, yp, Vbou)
tnew = 0.46875 # relatively small time step (s) for first bit
# number concentration of nucleated particles formed (# particles/cc (air))
new_part_sum1 = 0.0
save_count = int(1) # count on number of times saving code called
# count in number of time steps since time interval was last reduced
tinc_count = 10
from rate_valu_calc import rate_valu_calc # function to update rate coefficients
print('starting ode solver')
while sumt < end_sim_time: # step through time intervals to do ode
# increase time step if time step not been decreased for 10 steps
if tinc_count == 0 and tnew < t0:
tnew = tnew * 2.0
# check whether lights on or off
timediff = sumt - np.array(light_time)
timedish = (timediff == np.min(timediff[timediff >= 0])
) # reference time index
lightm = (
np.array(light_stat))[timedish] # whether lights on or off now
# update reaction rate coefficients
reac_coef = rate_valu_calc(RO2_indices, y[H2Oi], TEMP, lightm, y)
y0[:] = y[:] # update initial concentrations (molecules/cc (air))
# update particle volumes at start of time step (um3)
Vstart = Varr * N_perbin
redt = 1 # reset time reduction flag
t = tnew # reset integration time (s)
if num_sb > 1:
# update partitioning coefficients
[kimt, kelv_fac
] = kimt_calc(y, mfp, num_sb, num_speci, accom_coeff, y_mw, surfT,
R_gas, TEMP, NA, y_dens, N_perbin, DStar_org,
x.reshape(1, -1) * 1.0e-6, Psat, therm_sp, H2Oi)
# ensure no confusion that components are present due to low value fillers for
# concentrations (molecules/cc (air))
y0[y0 == 1.0e-40] = 0.0
print()
# enter a while loop that continues to decrease the time step until particle
# size bins don't change by more than one size bin (given by moving centre)
while redt == 1:
print('cumulative time', sumt)
# numba compiler to convert to machine code
@jit(f8[:](f8, f8[:]), nopython=True)
# ode solver -------------------------------------------------------------
def dydt(t, y):
# empty array to hold rate of change (molecules/cc(air).s)
dydt = np.zeros((len(y)))
# gas-phase rate of change ------------------------------------
for i in range(num_eqn): # equation loop
# gas-phase rate of change (molecules/cc (air).s)
gprate = ((y[rindx[i, 0:nreac[i]]]**
rstoi[i, 0:nreac[i]]).prod()) * reac_coef[i]
dydt[rindx[i, 0:nreac[i]]] -= gprate # loss of reactants
dydt[pindx[i, 0:nprod[i]]] += gprate # gain of products
if num_sb > 1:
# -----------------------------------------------------------
for ibin in range(num_sb - 1): # size bin loop
Csit = y[num_speci * (ibin + 1):num_speci * (ibin + 2)]
# sum of molecular concentrations per bin (molecules/cc (air))
conc_sum = np.zeros((1))
if pconc > 0.0: # if seed particles present
conc_sum[0] = ((Csit[0:-1].sum()) +
Csit[-1] * core_diss)
else:
conc_sum[0] = Csit.sum()
# prevent numerical error due to division by zero
ish = conc_sum == 0.0
conc_sum[ish] = 1.0e-40
# particle surface gas-phase concentration (molecules/cc (air))
Csit = (Csit / conc_sum) * Psat[:, 0] * kelv_fac[ibin]
# partitioning rate (molecules/cc.s)
dydt_all = kimt[:, ibin] * (y[0:num_speci] - Csit)
# gas-phase change
dydt[0:num_speci] -= dydt_all
# particle-phase change
dydt[num_speci * (ibin + 1):num_speci *
(ibin + 2)] += dydt_all
# -----------------------------------------------------------
# gas-wall partitioning eq. 14 of Zhang et al.
# (2015) (https://www.atmos-chem-phys.net/15/4197/2015/
# acp-15-4197-2015.pdf) (molecules/cc.s (air))
# concentration at wall (molecules/cc (air))
Csit = y[num_speci * num_sb:num_speci * (num_sb + 1)]
Csit = (Psat[:, 0] * (Csit / Cw))
# eq. 14 of Zhang et al. (2015)
# (https://www.atmos-chem-phys.net/15/4197/2015/
# acp-15-4197-2015.pdf) (molecules/cc.s (air))
dydt_all = (kwgt * Cw) * (y[0:num_speci] - Csit)
# gas-phase change
dydt[0:num_speci] -= dydt_all
# wall concentration change
dydt[num_speci * num_sb:num_speci * (num_sb + 1)] += dydt_all
return (dydt)
mod = Explicit_Problem(dydt, y0)
mod_sim = CVode(mod) # define a solver instance
mod_sim.atol = 1.0e-3 # absolute tolerance
mod_sim.rtol = 1.0e-3 # relative tolerance
mod_sim.discr = 'BDF' # the integration approach, default is 'Adams'
t_array, res = mod_sim.simulate(t)
y = res[-1, :]
# low value filler for concentrations (molecules/cc (air)) to prevent
# numerical errors
y0[y0 == 0.0] = 1.0e-40
y[y == 0.0] = 1.0e-40
if num_sb > 1 and (N_perbin > 1.0e-10).sum() > 0:
# call on the moving centre method for rebinning particles
(N_perbin, Varr, y[num_speci::], x, redt, t, tnew) = movcen(
N_perbin, Vbou,
np.transpose(y[num_speci::].reshape(num_sb, num_speci)),
(np.squeeze(y_dens * 1.0e-3)), num_sb, num_speci, y_mw, x,
Vol0, t, t0, tinc_count, y0[num_speci::], MV)
else:
redt = 0
# if time step needs reducing then reset gas-phase concentrations to their
# values preceding the ode, this will already have been done inside moving
# centre module for particle-phase
if redt == 1:
y[0:num_speci] = y0[0:num_speci]
# start counter to determine when to next try increasing time interval
if redt == 1:
tinc_count = 10
if redt == 0 and tinc_count > -1:
tinc_count -= 1
if tinc_count == -1:
tinc_count = 10
sumt += t # total time covered (s)
step += 1 # ode time interval step number
if num_sb > 1:
if (N_perbin > 1.0e-10).sum() > 0:
# coagulation
# y indices due to final element in y being number of ELVOC molecules
# contributing to newly nucleated particles
[N_perbin, y[num_speci:-(num_speci)], x, Gi, eta_ai,
Varr] = coag(
RH, TEMP, x * 1.0e-6, (Varr * 1.0e-18).reshape(1, -1),
y_mw.reshape(-1, 1), x * 1.0e-6,
np.transpose(y[num_speci::].reshape(num_sb, num_speci)),
(N_perbin).reshape(1, -1), t,
(Vbou * 1.0e-18).reshape(1, -1), num_speci, 0,
(np.squeeze(y_dens * 1.0e-3)), rad0, PInit)
# particle loss to walls
[N_perbin, y[num_speci:-(num_speci)]
] = wallloss(N_perbin.reshape(-1,
1), y[num_speci:-(num_speci)],
Gi, eta_ai, x * 2.0e-6, y_mw, Varr * 1.0e-18,
num_sb, num_speci, TEMP, t, inflectDp, pwl_xpre,
pwl_xpro, inflectk, ChamR, Rader)
# particle nucleation
if sumt < 3600.0 and pconc == 0.0:
[N_perbin, y, x[0], Varr[0],
new_part_sum1] = nuc(sumt, new_part_sum1, N_perbin, y,
y_mw.reshape(-1, 1),
np.squeeze(y_dens * 1.0e-3), num_speci,
x[0], new_partr, t, MV, nucv1, nucv2,
nucv3, nuc_comp)
if sumt >= save_step * save_count: # save at every time step given by save_step (s)
if num_sb > 0:
# particle-phase concentrations (molecules/cc (air))
yp = np.transpose(y[num_speci:-(num_speci)].reshape(
num_sb - 1, num_speci))
else:
yp = 0.0
# record new values
[t_out, y_mat, Nresult,
x2] = recording(y, N_perbin, x, save_count, sumt,
y_mat, Nresult, x2, t_out,
int(end_sim_time / save_step), num_speci, num_sb,
y_mw[:, 0], y_dens[:, 0] * 1.0e-3, yp, Vbou)
save_count += int(1)
return (t_out, y_mat, Nresult, x2)
X = 165e-6 # [m] ### Initial conditions c_init = 1000.0 # [mol/m^3] c_centered = c_init*numpy.ones( N, dtype='d' ) exp_mod = MyProblem( N, X, c_centered, 'ce only model, explicit CVode' ) #exp_mod = MyProblem( N, X, numpy.linspace(c_init-c_init/5.,c_init+c_init/5.,N), 'ce only model, explicit CVode' ) # Set the ODE solver 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 exp_mod.set_j_vec( 1.e-4 ) t1, y1 = exp_sim.simulate(100, 100) exp_mod.set_j_vec( 0.0 ) t2, y2 = exp_sim.simulate(200, 100) #exp_mod.set_j_vec( 0.0 ) #t3, y3 = exp_sim.simulate(10000, 200) #Plot #if with_plots: f, ax = plt.subplots(1,2)
def ode_solv(y, integ_step, rindx, pindx, rstoi, pstoi, nreac, nprod, rrc,
jac_stoi, njac, jac_den_indx, jac_indx, Cinfl_now, y_arr, y_rind,
uni_y_rind, y_pind, uni_y_pind, reac_col, prod_col, rstoi_flat,
pstoi_flat, rr_arr, rr_arr_p, rowvals, colptrs, num_comp, num_sb,
wall_on, Psat, Cw, act_coeff, kw, jac_wall_indx, seedi, core_diss,
kelv_fac, kimt, num_asb, jac_part_indx, rindx_aq, pindx_aq,
rstoi_aq, pstoi_aq, nreac_aq, nprod_aq, jac_stoi_aq, njac_aq,
jac_den_indx_aq, jac_indx_aq, y_arr_aq, y_rind_aq, uni_y_rind_aq,
y_pind_aq, uni_y_pind_aq, reac_col_aq, prod_col_aq, rstoi_flat_aq,
pstoi_flat_aq, rr_arr_aq, rr_arr_p_aq, eqn_num):
# inputs: -------------------------------------
# y - initial concentrations (moleucles/cm3)
# integ_step - the maximum integration time step (s)
# rindx - index of reactants per equation
# pindx - index of products per equation
# rstoi - stoichiometry of reactants
# pstoi - stoichiometry of products
# nreac - number of reactants per equation
# nprod - number of products per equation
# rrc - reaction rate coefficient
# jac_stoi - stoichiometries relevant to Jacobian
# njac - number of Jacobian elements affected per equation
# jac_den_indx - index of component denominators for Jacobian
# jac_indx - index of Jacobian to place elements per equation (rows)
# Cinfl_now - influx of components with constant influx
# (molecules/cc/s)
# y_arr - index for matrix used to arrange concentrations,
# enabling calculation of reaction rate coefficients
# y_rind - index of y relating to reactants for reaction rate
# coefficient equation
# uni_y_rind - unique index of reactants
# y_pind - index of y relating to products
# uni_y_pind - unique index of products
# reac_col - column indices for sparse matrix of reaction losses
# prod_col - column indices for sparse matrix of production gains
# rstoi_flat - 1D array of reactant stoichiometries per equation
# pstoi_flat - 1D array of product stoichiometries per equation
# rr_arr - index for reaction rates to allow reactant loss
# calculation
# rr_arr_p - index for reaction rates to allow reactant loss
# calculation
# rowvals - row indices of Jacobian elements
# colptrs - indices of rowvals corresponding to each column of the
# Jacobian
# num_comp - number of components
# num_sb - number of size bins
# wall_on - flag saying whether to include wall partitioning
# Psat - pure component saturation vapour pressures (molecules/cc)
# Cw - effective absorbing mass concentration of wall (molecules/cc)
# act_coeff - activity coefficient of components
# kw - mass transfer coefficient to wall (/s)
# jac_wall_indx - index of inputs to Jacobian by wall partitioning
# seedi - index of seed material
# core_diss - dissociation constant of seed material
# kelv_fac - kelvin factor for particles
# kimt - mass transfer coefficient for gas-particle partitioning (s)
# num_asb - number of actual size bins (excluding wall)
# jac_part_indx - index for sparse Jacobian for particle influence
# eqn_num - number of gas- and aqueous-phase reactions
# ---------------------------------------------
def dydt(t, y): # define the ODE(s)
# empty array to hold rate of change per component
dd = np.zeros((len(y)))
# gas-phase reactions -------------------------
# empty array to hold relevant concentrations for
# reaction rate coefficient calculation
rrc_y = np.ones((rindx.shape[0] * rindx.shape[1]))
rrc_y[y_arr] = y[y_rind]
rrc_y = rrc_y.reshape(rindx.shape[0], rindx.shape[1], order='C')
# reaction rate (molecules/cc/s)
rr = rrc[0:rindx.shape[0]] * ((rrc_y**rstoi).prod(axis=1))
# loss of reactants
data = rr[rr_arr] * rstoi_flat # prepare loss values
# convert to sparse matrix
loss = SP.csc_matrix((data, y_rind, reac_col))
# register loss of reactants
dd[uni_y_rind] -= np.array((loss.sum(axis=1))[uni_y_rind])[:, 0]
# gain of products
data = rr[rr_arr_p] * pstoi_flat # prepare loss values
# convert to sparse matrix
loss = SP.csc_matrix((data, y_pind, prod_col))
# register gain of products
dd[uni_y_pind] += np.array((loss.sum(axis=1))[uni_y_pind])[:, 0]
# particle-phase reactions -------------------------
if (eqn_num[1] > 0):
# empty array to hold relevant concentrations for
# reaction rate coefficient calculation
# tile aqueous-phase reaction rate coefficients
rr_aq = np.tile(rrc[rindx.shape[0]::], num_asb)
# prepare aqueous-phase concentrations
rrc_y = np.ones((rindx_aq.shape[0] * rindx_aq.shape[1]))
rrc_y[y_arr_aq] = y[y_rind_aq]
rrc_y = rrc_y.reshape(rindx_aq.shape[0],
rindx_aq.shape[1],
order='C')
# reaction rate (molecules/cc/s)
rr = rr_aq * ((rrc_y**rstoi_aq).prod(axis=1))
# loss of reactants
data = rr[rr_arr_aq] * rstoi_flat_aq # prepare loss values
# convert to sparse matrix
loss = SP.csc_matrix((data[0, :], y_rind_aq, reac_col_aq))
# register loss of reactants
dd[uni_y_rind_aq] -= np.array((loss.sum(axis=1))[uni_y_rind_aq])[:,
0]
# gain of products
data = rr[rr_arr_p_aq] * pstoi_flat_aq # prepare loss values
# convert to sparse matrix
loss = SP.csc_matrix((data[0, :], y_pind_aq, prod_col_aq))
# register gain of products
dd[uni_y_pind_aq] += np.array((loss.sum(axis=1))[uni_y_pind_aq])[:,
0]
# gas-particle partitioning-----------------
# transform particle phase concentrations into
# size bins in rows, components in columns
ymat = (y[num_comp:num_comp * (num_asb + 1)]).reshape(
num_asb, num_comp)
# total particle-phase concentration per size bin (molecules/cc (air))
csum = ((ymat.sum(axis=1) - ymat[:, seedi].sum(axis=1)) + (
(ymat[:, seedi] * core_diss).sum(axis=1)).reshape(-1)).reshape(
-1, 1)
# size bins with contents
isb = (csum[:, 0] > 0.)
# container for gas-phase concentrations at particle surface
Csit = np.zeros((num_asb, num_comp))
# mole fraction of components at particle surface
Csit[isb, :] = (ymat[isb, :] / csum[isb, :])
# gas-phase concentration of components at
# particle surface (molecules/cc (air))
Csit[isb, :] = Csit[isb, :] * Psat[isb, :] * kelv_fac[isb] * act_coeff[
isb, :]
# partitioning rate (molecules/cc/s)
dd_all = kimt * (y[0:num_comp].reshape(1, -1) - Csit)
# gas-phase change
dd[0:num_comp] -= dd_all.sum(axis=0)
# particle change
dd[num_comp:num_comp * (num_asb + 1)] += (dd_all.flatten())
# gas-wall partitioning ----------------
# concentration on wall (molecules/cc (air))
Csit = y[num_comp * num_sb:num_comp * (num_sb + 1)]
# saturation vapour pressure on wall (molecules/cc (air))
# note, just using the top rows of Psat and act_coeff
# as do not need the repetitions over size bins
if (Cw > 0.):
Csit = Psat[0, :] * (Csit / Cw) * act_coeff[0, :]
# rate of transfer (molecules/cc/s)
dd_all = kw * (y[0:num_comp] - Csit)
dd[0:num_comp] -= dd_all # gas-phase change
dd[num_comp * num_sb:num_comp *
(num_sb + 1)] += dd_all # wall change
return (dd)
def jac(t, y): # define the Jacobian
# elements of sparse Jacobian matrix
data = np.zeros((94))
for i in range(rindx.shape[0]): # gas-phase reaction loop
# reaction rate (molecules/cc/s)
rr = rrc[i] * (y[rindx[i, 0:nreac[i]]].prod())
# prepare Jacobian inputs
jac_coeff = np.zeros((njac[i, 0]))
# only fill Jacobian if reaction rate sufficient
if (rr != 0.):
jac_coeff = (rr * (jac_stoi[i, 0:njac[i, 0]]) /
(y[jac_den_indx[i, 0:njac[i, 0]]]))
data[jac_indx[i, 0:njac[i, 0]]] += jac_coeff
n_aqr = nreac_aq.shape[0] # number of aqueous-phase reactions
aqi = 0 # aqueous-phase reaction counter
for i in range(rindx.shape[0],
rrc.shape[0]): # aqueous-phase reaction loop
# reaction rate (molecules/cc/s)
rr = rrc[i] * (y[rindx_aq[aqi::n_aqr,
0:nreac_aq[aqi]]].prod(axis=1))
# spread along affected components
rr = rr.reshape(-1, 1)
rr = (np.tile(rr, int(njac_aq[aqi, 0] /
(num_sb - wall_on)))).flatten(order='C')
# prepare Jacobian inputs
jac_coeff = np.zeros((njac_aq[aqi, 0]))
nzi = (rr != 0)
jac_coeff[nzi] = (
rr[nzi] * ((jac_stoi_aq[aqi, 0:njac_aq[aqi, 0]])[nzi]) /
((y[jac_den_indx_aq[aqi, 0:njac_aq[aqi, 0]]])[nzi]))
# stack size bins
jac_coeff = jac_coeff.reshape(int(num_sb - wall_on),
int(njac_aq[aqi, 0] /
(num_sb - wall_on)),
order='C')
data[jac_indx_aq[aqi::n_aqr,
0:(int(njac_aq[aqi, 0] /
(num_sb - wall_on)))]] += jac_coeff
aqi += 1
# gas-particle partitioning
part_eff = np.zeros((56))
part_eff[0:24:3] = -kimt.sum(axis=0) # effect of gas on gas
# transform particle phase concentrations into
# size bins in rows, components in columns
ymat = (y[num_comp:num_comp * (num_asb + 1)]).reshape(
num_asb, num_comp)
# total particle-phase concentration per size bin (molecules/cc (air))
csum = ymat.sum(axis=1) - ymat[:, seedi].sum(
axis=1) + (ymat[:, seedi] * core_diss).sum(axis=1)
# effect of particle on gas
for isb in range(int(num_asb)): # size bin loop
if csum[isb] > 0: # if particles present
# effect of gas on particle
part_eff[1 + isb:num_comp * (num_asb + 1):num_asb +
1] = +kimt[isb, :]
# start and finish index
sti = int((num_asb + 1) * num_comp + isb * (num_comp * 2))
fii = int(sti + (num_comp * 2.))
# diagonal index
diag_indxg = sti + np.arange(0, num_comp * 2, 2).astype('int')
diag_indxp = sti + np.arange(1, num_comp * 2, 2).astype('int')
# prepare for diagonal (component effect on itself)
diag = kimt[isb, :] * Psat[0, :] * act_coeff[0, :] * kelv_fac[
isb, 0] * (csum[isb] - ymat[isb, :]) / (csum[isb]**2.)
# implement to part_eff
part_eff[diag_indxg] = +diag
part_eff[diag_indxp] = -diag
data[jac_part_indx] += part_eff
if (Cw > 0.):
wall_eff = np.zeros((32))
wall_eff[0:16:2] = -kw # effect of gas on gas
wall_eff[1:16:2] = +kw # effect of gas on wall
# effect of wall on gas
wall_eff[16:32:2] = +kw * (Psat[0, :] * act_coeff[0, :] / Cw)
# effect of wall on wall
wall_eff[16 + 1:32:2] = -kw * (Psat[0, :] * act_coeff[0, :] / Cw)
data[jac_wall_indx] += wall_eff
# create Jacobian
j = SP.csc_matrix((data, rowvals, colptrs))
return (j)
mod = Explicit_Problem(dydt, y) # instantiate solver
mod.jac = jac # set the Jacobian
mod_sim = CVode(mod) # define a solver instance
# there is a general trend of an inverse relationship
# between tolerances and computation time
mod_sim.atol = 0.001
mod_sim.rtol = 0.0001
# integration approach (backward differentiation formula)
mod_sim.discr = 'BDF'
# call solver
res_t, res = mod_sim.simulate(integ_step)
# return concentrations following integration
return (res, res_t)
def run_example(with_plots=True):
r"""
Example for demonstrating the use of a user supplied Jacobian
ODE:
.. math::
\dot y_1 &= y_2 \\
\dot y_2 &= -9.82
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#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
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, name='Example using analytic Jacobian')
exp_mod.jac = jac #Sets the 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
#Plot
if with_plots:
import pylab as P
P.plot(t, y, linestyle="dashed", marker="o") #Plot the solution
P.xlabel('Time')
P.ylabel('State')
P.title(exp_mod.name)
P.show()
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], -121.75000000, 4)
nose.tools.assert_almost_equal(y[-1][1], -49.100000000)
return exp_mod, exp_sim
gamma = p[3]
flux = np.array(
[alpha * y[0], beta * y[0] * y[1], delta * y[0] * y[1], gamma * y[1]])
rhs = np.array([flux[0] - flux[1], flux[2] - flux[3]])
return rhs
if __name__ == '__main__':
p = np.array([.5, .02, .4, .004])
ode_function = lambda t, x: rhs_fun(t, x, p)
# define explicit assimulo problem
prob = Explicit_Problem(ode_function, y0=np.array([10, .0001]))
# create solver instance
solver = CVode(prob)
solver.iter = 'Newton'
solver.discr = 'Adams'
solver.atol = 1e-10
solver.rtol = 1e-10
solver.display_progress = True
solver.verbosity = 10
# simulate system
time_course, y_result = solver.simulate(10, 200)
print time_course
print y_result
return ydot
#Create Assimulo problem model
mod_pen = Explicit_Problem(rhs, y0, t0)
mod_pen.name = 'Elastic Pendulum with CVode'
"""
Run simulation
"""
#Create solver object
sim_pen = CVode(mod_pen)
#Simulation parameters
#atol = 1.e-6*ones(shape(y0))
atol = rtol * N.array([1, 1, 1, 1]) #default 1e-06
sim_pen.atol = atol
sim_pen.rtol = rtol
sim_pen.maxh = hmax
sim_pen.maxord = maxord
sim_pen.inith = h0
#Simulate
t, y = sim_pen.simulate(tf)
"""
Plot results
"""
#Plot
#P.plot(t,y)
#P.legend(['$x$','$y$','$\dot{x}$','$\dot{y}$'])
#P.title('Elastic pendulum with k = ' + str(k) + ', stretch = ' + str(stretch))
#P.xlabel('time')
from assimulo.solvers import CVode
import matplotlib.pyplot as plt
def rhs(t,y):
k = 1000
L = k*(math.sqrt(y[0]**2+y[1]**2)-1)/math.sqrt(y[0]**2+y[1]**2)
result = numpy.array([y[2],y[3],-y[0]*L,-y[1]*L-1])
return result
t0 = 0
y0 = numpy.array([0.8,-0.8,0,0])
model = Explicit_Problem(rhs,y0,t0)
model.name = 'task1'
sim = CVode(model)
sim.atol=numpy.array([1,1,1,1])*1e-5
sim.rtol=1e-6
sim.maxord=3
#sim.discr='BDF'
#sim.iter='Newton'
tfinal = 70
(t,y) = sim.simulate(tfinal)
#sim.plot()
plt.plot(y[:,0],y[:,1])
plt.axis('equal')
plt.show()
def simulate(self, Tend, nIntervals, gridWidth):
# define assimulo problem:(has to be done here because of the starting value in Explicit_Problem
solver = Explicit_Problem(self.rhs, self.y0)
''' *******DELETE LATER '''''''''
# problem.handle_event = handle_event
# problem.state_events = state_events
# problem.init_mode = init_mode
solver.handle_result = self.handle_result
solver.name = 'Simple Explicit Example'
simulation = CVode(solver) # Create a RungeKutta34 solver
# simulation.inith = 0.1 #Sets the initial step, default = 0.01
# Change multistep method: 'adams' or 'VDF'
if self.discr == 'Adams':
simulation.discr = 'Adams'
simulation.maxord = 12
else:
simulation.discr = 'BDF'
simulation.maxord = 5
# Change iteration algorithm: functional(FixedPoint) or newton
if self.iter == 'FixedPoint':
simulation.iter = 'FixedPoint'
else:
simulation.iter = 'Newton'
# Sets additional parameters
simulation.atol = self.atol
simulation.rtol = self.rtol
simulation.verbosity = 0
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
# Create Solver and set settings
# noRootFunctions = np.size(self.state_events(self.t0, np.array(self.y0)))
# solver = sundials.CVodeSolver(RHS = self.f, ROOT = self.rootf, SW = [False]*noRootFunctions,
# abstol = self.atol, reltol = self.rtol)
# solver.settings.JAC = None #Add user-dependent jacobian here
'''Initialize problem '''
# solver.init(self.t0, self.y0)
self.handle_result(self.t0, self.y0)
nextTimeEvent = self.time_events(self.t0, self.y0)
self.t_cur = self.t0
self.y_cur = self.y0
state_event = False
#
#
if gridWidth <> None:
nOutputIntervals = int((Tend - self.t0) / gridWidth)
else:
nOutputIntervals = nIntervals
# Define step length depending on if gridWidth or nIntervals has been chosen
if nOutputIntervals > 0:
# Last point on grid (does not have to be Tend:)
if(gridWidth <> None):
dOutput = gridWidth
else:
dOutput = (Tend - self.t0) / nIntervals
else:
dOutput = Tend
outputStepCounter = long(1)
nextOutputPoint = min(self.t0 + dOutput, Tend)
while self.t_cur < Tend:
# Time-Event detection and step time adjustment
if nextTimeEvent is None or nextOutputPoint < nextTimeEvent:
time_event = False
self.t_cur = nextOutputPoint
else:
time_event = True
self.t_cur = nextTimeEvent
try:
# #Integrator step
# self.y_cur = solver.step(self.t_cur)
# self.y_cur = np.array(self.y_cur)
# state_event = False
# Simulate
# take a step to next output point:
t_new, y_new = simulation.simulate(self.t_cur) # 5, 10) #5, 10 self.t_cur self.t_cur 2. argument nsteps Simulate 5 seconds
# t_new, y_new are both vectors of the time and states at t_cur and all intermediate
# points before it! So take last values:
self.t_cur = t_new[-1]
self.y_cur = y_new[-1]
state_event = False
except:
import sys
print "Unexpected error:", sys.exc_info()[0]
# except CVodeRootException, info:
# self.t_cur = info.t
# self.y_cur = info.y
# self.y_cur = np.array(self.y_cur)
# time_event = False
# state_event = True
#
#
# Depending on events have been detected do different tasks
if time_event or state_event:
event_info = [state_event, time_event]
if not self.handle_event(self, event_info):
break
solver.init(self.t_cur, self.y_cur)
nextTimeEvent = self.time_events(self.t_cur, self.y_cur)
# If no timeEvent happens:
if nextTimeEvent <= self.t_cur:
nextTimeEvent = None
if self.t_cur == nextOutputPoint:
# Write output if not happened before:
if not time_event and not state_event:
self.handle_result(nextOutputPoint, self.y_cur)
outputStepCounter += 1
nextOutputPoint = min(self.t0 + outputStepCounter * dOutput, Tend)
self.finalize()
# Define the rhs
def rhs(t, y):
k = 10
A = np.array([[0, 0, 1, 0],
[0, 0, 0, 1],
[-L(y, k), 0, 0, 0],
[0, -L(y, k), 0, 0]])
b = np.array([0, 0, 0, -1])
yd = np.dot(A, y) + b
return yd
def L(y, k):
norm = ln.norm(y[0:2])
return k * (norm - 1) / norm
pend_mod = Explicit_Problem(rhs, y0 = np.array([1.0, 1.0, 1.0, 1.0]))
pend_mod.name = 'Nonlinear Pendulum'
# Define an explicit solver
exp_sim = BDF_3(pend_mod) # Create a BDF solver
t, y = exp_sim.simulate(4)
exp_sim.plot()
mpl.show()
toltmp=0.1
cvod= CVode(pend_mod)
cvod.atol=toltmp
cvod.rtol=toltmp
cvod.maxord=19
cvod.simulate(5)
cvod.plot()
def run_example(with_plots=True):
r"""
Example for demonstrating the use of a user supplied Jacobian (sparse).
Note that this will only work if Assimulo has been configured with
Sundials + SuperLU. Based on the SUNDIALS example cvRoberts_sps.c
ODE:
.. math::
\dot y_1 &= -0.04y_1 + 1e4 y_2 y_3 \\
\dot y_2 &= - \dot y_1 - \dot y_3 \\
\dot y_3 &= 3e7 y_2^2
on return:
- :dfn:`exp_mod` problem instance
- :dfn:`exp_sim` solver instance
"""
#Defines the rhs
def f(t, y):
yd_0 = -0.04 * y[0] + 1e4 * y[1] * y[2]
yd_2 = 3e7 * y[1] * y[1]
yd_1 = -yd_0 - yd_2
return N.array([yd_0, yd_1, yd_2])
#Defines the Jacobian
def jac(t, y):
colptrs = [0, 3, 6, 9]
rowvals = [0, 1, 2, 0, 1, 2, 0, 1, 2]
data = [
-0.04, 0.04, 0.0, 1e4 * y[2], -1e4 * y[2] - 6e7 * y[1], 6e7 * y[1],
1e4 * y[1], -1e4 * y[1], 0.0
]
J = SP.csc_matrix((data, rowvals, colptrs))
return J
#Defines an Assimulo explicit problem
y0 = [1.0, 0.0, 0.0] #Initial conditions
exp_mod = Explicit_Problem(f,
y0,
name='Example using analytic (sparse) Jacobian')
exp_mod.jac = jac #Sets the Jacobian
exp_mod.jac_nnz = 9
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-8, 1e-14, 1e-6] #Default 1e-6
exp_sim.rtol = 1e-4 #Default 1e-6
exp_sim.linear_solver = "sparse"
#Simulate
t, y = exp_sim.simulate(0.4) #Simulate 0.4 seconds
#Basic tests
nose.tools.assert_almost_equal(y[-1][0], 0.9851, 3)
#Plot
if with_plots:
P.plot(t, y[:, 1], linestyle="dashed", marker="o") #Plot the solution
P.xlabel('Time')
P.ylabel('State')
P.title(exp_mod.name)
P.show()
return exp_mod, exp_sim
