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 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 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*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
