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 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):
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 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 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 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 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
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
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 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)
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):
"""
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 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()
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_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
N = 60 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:
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
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 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):
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 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()
t0 = 0.0 #Initial time switches0 = [True] #Initial switches #Create an Assimulo Problem mod = Explicit_Problem(pendulum, y0, t0, sw0=switches0) mod.state_events = state_events #Sets the state events to the problem mod.handle_event = handle_event #Sets the event handling to the problem mod.name = 'Pendulum with events' #Sets the name of the problem #Create an Assimulo solver (CVode) sim = CVode(mod) #Specifies options sim.discr = 'Adams' #Sets the discretization method sim.iter = 'FixedPoint' #Sets the iteration method sim.rtol = 1.e-8 #Sets the relative tolerance sim.atol = 1.e-6 #Sets the absolute tolerance #Simulation ncp = 200 #Number of communication points tfinal = 10.0 #Final time t, y = sim.simulate(tfinal, ncp) #Simulate #Plots the result
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"""
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 stiff_ode_solver(matrix,
y_initial,
forward_rate,
rev_rate,
third_body=None,
iteration='Newton',
discr='BDF',
atol=1e-10,
rtol=1e-6,
sim_time=0.001,
num_data_points=500):
"""
Sets up the initial condition for solving the odes
Parameters
----------
matrix : ndarray
stoichiometric matrix
y_initial : list
A list of initial concentrations
forward_rate : list
A list of forward reaction rates
for all the reactions in the mechanism
rev_rate : list
A list of reverse reaction rates
for all the reactions in the mechanism
sim_time : float
total time to simulate in seconds
third_body : ndarray
third body matrix, default = None
iteration : str
determines the iteration method that is be
used by the solver, default='Newton'
discr : determines the discretization method,
default='BDF'
atol : float
absolute tolerance(s) that is to be used
by the solver, default=1e-10
rtol : float
relative tolerance that is to be
used by the solver, default= 1e-7
num_data_points : integer
number of even space data points in output
arrays, default = 500
Returns
----------
t1 : list
A list of time-points at which
the system of ODEs is solved
[t1, t2, t3,...]
y1 : list of lists
A list of concentrations of all the species
at t1 time-points
[[y1(t1), y2(t1),...], [y1(t2), y2(t2),...],...]
"""
# y0[0] = 0
# y0[0] = 0
# dydt = np.zeros((len(species_list)), dtype=float)
# Define the rhs
kf = forward_rate
kr = rev_rate
mat_reac = np.abs(np.asarray(np.where(matrix < 0, matrix, 0)))
mat_prod = np.asarray(np.where(matrix > 0, matrix, 0))
# d = kf * np.prod(y0**np.abs(mat_reac), axis = 1) - kr * np.prod(y0**mat_prod, axis = 1)
def rhs(t, concentration):
# print(t)
y = concentration
if third_body is not None:
third_body_eff = np.dot(third_body, y)
third_body_eff = np.where(third_body_eff > 0, third_body_eff, 1)
# print(third_body_eff)
else:
third_body_eff = np.ones(len(forward_rate))
# print(len(third_body_matrix))
rate_concentration = (kf * np.prod(np.power(y, mat_reac), axis=1) -
kr * np.prod(np.power(y, mat_prod), axis=1))
dydt = np.dot(
third_body_eff,
np.multiply(matrix, rate_concentration.reshape(matrix.shape[0],
1)))
# dydt = [np.sum(third_body_eff * (matrix[:, i] * rate_concentration)) for i in
# range(len(species_list))]
del t
del y
return dydt
t0 = 0
# Define an Assimulo problem
exp_mod = Explicit_Problem(rhs, y_initial, t0)
# Define an explicit solver
exp_sim = CVode(exp_mod) # Create a CVode solver
# Sets the parameters
exp_sim.iter = iteration # Default 'FixedPoint'
exp_sim.discr = discr # Default 'Adams'
exp_sim.atol = [atol] # Default 1e-6
exp_sim.rtol = rtol # Default 1e-6
exp_sim.maxh = 0.1
exp_sim.minh = 1e-18
exp_sim.num_threads = 1
while True:
try:
t1, y1 = exp_sim.simulate(sim_time, num_data_points)
break
except CVodeError:
# reduce absolute error by two orders of magnitude
# and try to solve again.
print("next process started")
atol = atol * 1e-2
exp_sim.atol = atol
# if atol < 1e-15:
# t1 = 0
# y1 = 0
# break
# print(exp_sim.atol)
return t1, y1
