def solver(alpha, ic, T, dt=0.05):
def f(u, t):
x_A, vx_A, y_A, vy_A, x_B, vx_B, y_B, vy_B = u
distance3 = np.sqrt((x_B - x_A)**2 + (y_B - y_A)**2)**3
system = [
vx_A,
1 / (1.0 + alpha) * (x_B - x_A) / distance3,
vy_A,
1 / (1.0 + alpha) * (y_B - y_A) / distance3,
vx_B,
-1 / (1.0 + alpha**(-1)) * (x_B - x_A) / distance3,
vy_B,
-1 / (1.0 + alpha**(-1)) * (y_B - y_A) / distance3,
]
return system
Nt = int(round(T / dt))
t_mesh = np.linspace(0, Nt * dt, Nt + 1)
solver = odespy.RK4(f)
solver.set_initial_condition(ic)
u, t = solver.solve(t_mesh)
x_A = u[:, 0]
x_B = u[:, 2]
y_A = u[:, 4]
y_B = u[:, 6]
return x_A, x_B, y_A, y_B, t
def varying_gravity(epsilon):
def ode_a(u, t):
h, v = u
return [v, -epsilon**(-2) / (1 + h)**2]
def ode_b(u, t):
h, v = u
return [v, -1.0 / (1 + h)**2]
def ode_c(u, t):
h, v = u
return [v, -1.0 / (1 + epsilon**2 * h)**2]
problems = [ode_a, ode_b, ode_c] # right-hand sides
ics = [[0, 1], [0, epsilon], [0, 1]] # initial conditions
for problem, ic, legend in zip(problems, ics, ['a', 'b', 'c']):
solver = odespy.RK4(problem)
solver.set_initial_condition(ic)
t = np.linspace(0, 5, 5001)
# Solve ODE until h < 0 (h is in u[:,0])
u, t = solver.solve(t, terminate=lambda u, t, n: u[n, 0] < 0)
h = u[:, 0]
plt.figure()
plt.plot(t, h)
plt.legend(legend, loc='upper left')
plt.title(r'$\epsilon^2=%g$' % epsilon**2)
plt.xlabel(r'$\bar t$')
plt.ylabel(r'$\bar h(\bar t)$')
plt.savefig('tmp_%s.png' % legend)
plt.savefig('tmp_%s.pdf' % legend)
def solver(R0, alpha, gamma, delta, T, dt=0.1):
def f(u, t):
S, I, R, V = u
return [
-R0 * S * I - delta(t) * S + gamma * R, R0 * S * I - I,
I - gamma * R,
delta(t) * S
]
Nt = int(round(T / dt))
t_mesh = np.linspace(0, Nt * dt, Nt + 1)
solver = odespy.RK4(f)
solver.set_initial_condition([1 - alpha, alpha, 0, 0])
u, t = solver.solve(t_mesh)
S = u[:, 0]
I = u[:, 1]
R = u[:, 2]
V = u[:, 3]
# Consistency check
N = 1
tol = 1E-13
for i in range(len(S)):
if abs(S[i] + I[i] + R[i] + V[i] - N) > tol:
print 'Consistency error: S+I+R+V=%g != %g' % \
(S[i] + I[i] + R[i] + V[i], N)
return S, I, R, V, t
def run_solvers_and_plot(solvers, timesteps_per_period=20, num_periods=1, b=0):
w = 2 * np.pi # frequency of undamped free oscillations
P = 2 * np.pi / w # duration of one period
dt = P / timesteps_per_period
Nt = num_periods * timesteps_per_period
T = Nt * dt
t_mesh = np.linspace(0, T, Nt + 1)
t_fine = np.linspace(0, T, 8 * Nt + 1) # used for very accurate solution
legends = []
solver_exact = odespy.RK4(f)
for solver in solvers:
solver.set_initial_condition([solver.users_f.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
# Make plots (plot last 10 periods????)
if num_periods <= 80:
plt.figure(1)
if len(t_mesh) <= 50:
plt.plot(t, u[:, 0]) # markers by default
else:
plt.plot(t, u[:, 0], '-2') # no markers
plt.hold('on')
legends.append(solver_name)
# Compare with exact solution plotted on a very fine mesh
#t_fine = np.linspace(0, T, 10001)
#u_e = solver.users_f.exact(t_fine)
# Compare with RK4 on a much finer mesh
solver_exact.set_initial_condition([solver.users_f.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_e, u_e[:, 0], '-') # avoid markers by spec. line type
legends.append('exact (RK4)')
plt.legend(legends, loc='upper left')
plt.xlabel('t')
plt.ylabel('u')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_u.eps' % (timesteps_per_period, num_periods))
def get_solver(self, func):
"""
Returns the solver method from odespy package.
Args:
func: function
function with ODE system.
Returns: an instance of odeSolver
"""
if self.solverMethod is solverMethod.LSODA:
solver = odespy.Lsoda(func)
elif self.solverMethod is solverMethod.LSODAR:
solver = odespy.Lsodar(func)
elif self.solverMethod is solverMethod.LSODE:
solver = odespy.Lsode(func)
elif self.solverMethod is solverMethod.HEUN:
solver = odespy.Heun(func)
elif self.solverMethod is solverMethod.EULER:
solver = odespy.Euler(func)
elif self.solverMethod is solverMethod.RK4:
solver = odespy.RK4(func)
elif self.solverMethod is solverMethod.DORMAN_PRINCE:
solver = odespy.DormandPrince(func)
elif self.solverMethod is solverMethod.RKFehlberg:
solver = odespy.RKFehlberg(func)
elif self.solverMethod is solverMethod.Dopri5:
solver = odespy.Dopri5(func)
elif self.solverMethod is solverMethod.Dop853:
solver = odespy.Dop853(func)
elif self.solverMethod is solverMethod.Vode:
solver = odespy.Vode(func)
elif self.solverMethod is solverMethod.AdamsBashforth2:
solver = odespy.AdamsBashforth2(func, method='bdf')
elif self.solverMethod is solverMethod.Radau5:
solver = odespy.Radau5(func)
elif self.solverMethod is solverMethod.AdamsBashMoulton2:
solver = odespy.AdamsBashMoulton2(func)
# update default parameters
solver.nsteps = SolverConfigurations.N_STEPS
solver.atol = SolverConfigurations.ABSOLUTE_TOL
solver.rtol = SolverConfigurations.RELATIVE_TOL
return solver
def simulate_pendulum(alpha, theta0, dt, T):
import odespy
def f(u, t, alpha):
omega, theta = u
return [-alpha * omega * abs(omega) - sin(theta), omega]
import numpy as np
Nt = int(round(T / float(dt)))
t = np.linspace(0, Nt * dt, Nt + 1)
solver = odespy.RK4(f, f_args=[alpha])
solver.set_initial_condition([0, theta0])
u, t = solver.solve(t, terminate=lambda u, t, n: abs(u[n, 1]) < 1E-3)
omega = u[:, 0]
theta = u[:, 1]
S = omega**2 + np.cos(theta)
drag = -alpha * np.abs(omega) * omega
return t, theta, omega, S, drag
def simulate(alpha, beta=0, num_periods=8, time_steps_per_period=60):
# Use oscillations without friction to set dt and T
P = 2 * np.pi
dt = P / time_steps_per_period
T = num_periods * P
t = np.linspace(0, T, time_steps_per_period * num_periods + 1)
import odespy
def f(u, t, alpha):
# Note the sequence of unknowns: v, u (v=du/dt)
v, u = u
return [-alpha * np.sign(v) - u, v]
solver = odespy.RK4(f, f_args=[alpha])
solver.set_initial_condition([beta, 1]) # sequence must match f
uv, t = solver.solve(t)
u = uv[:, 1] # recall sequence in f: v, u
v = uv[:, 0]
return u, t
def simulate(Theta, num_periods=8, time_steps_per_period=60, scaling=1):
# Use oscillations for small Theta to set dt and T
P = 2 * np.pi
dt = P / time_steps_per_period
T = num_periods * P
t = np.linspace(0, T, time_steps_per_period * num_periods + 1)
import odespy
def f(u, t, Theta):
# Note the sequence of unknowns: omega, theta
# omega = d(theta)/dt, angular velocity
omega, theta = u
return [-Theta**(-1) * np.sin(Theta * theta), omega]
solver = odespy.RK4(f, f_args=[Theta])
solver.set_initial_condition([0, 1]) # sequence must match f
u, t = solver.solve(t)
theta = u[:, 1] # recall sequence in f: omega, theta
return theta, t
def solver(alpha, beta, epsilon, T, dt=0.1):
def f(u, t):
Q, P, S, E = u
return [
alpha*(E*S - Q),
beta*Q,
-E*S + (1-beta/alpha)*Q,
(-E*S + Q)/epsilon,
]
Nt = int(round(T/dt))
t_mesh = np.linspace(0, Nt*dt, Nt+1)
solver = odespy.RK4(f)
solver.set_initial_condition([0, 0, 1, 1])
u, t = solver.solve(t_mesh)
Q = u[:,0]
P = u[:,1]
S = u[:,2]
E = u[:,3]
return Q, P, S, E
def simulate2(mu, nu, omega, T=20):
def f(u, t, mu):
H, L = u
return [H * (1 - L), mu * L * (H - 1)]
t = np.linspace(0, T, 1501)
solver = odespy.RK4(f, f_args=(mu, ))
solver.set_initial_condition([nu, omega])
u, t = solver.solve(t)
H = u[:, 0]
L = u[:, 1]
plt.figure()
global fig_counter
fig_counter += 1
plt.plot(t, H, t, L)
plt.legend(['H', 'L'])
plt.title(r'$\mu=%g$, $\nu=%g$, $\omega=%g$' % (mu, nu, omega))
plt.savefig('tmp%d.png' % fig_counter)
plt.savefig('tmp%d.pdf' % fig_counter)
return H, L, t
def solver(alpha, ic, T, dt=0.05):
def f(u, t):
x, vx, y, vy = u
v = np.sqrt(vx**2 + vy**2) # magnitude of velocity
system = [
vx,
-alpha * np.abs(v) * vx,
vy,
-2 - alpha * np.abs(v) * vy,
]
return system
Nt = int(round(T / dt))
t_mesh = np.linspace(0, Nt * dt, Nt + 1)
solver = odespy.RK4(f)
solver.set_initial_condition(ic)
u, t = solver.solve(t_mesh, terminate=lambda u, t, n: u[n][2] < 0)
x = u[:, 0]
y = u[:, 2]
return x, y, t
def run_solvers_and_plot(solvers, rhs, T, dt, title='', filename='tmp'):
"""
Run solvers from the `solvers` list and plot solution curves in
the same figure.
"""
Nt = int(round(T / float(dt)))
t_mesh = np.linspace(0, T, Nt + 1)
t_fine = np.linspace(0, T, 8 * Nt + 1) # used for very accurate solution
legends = []
for solver in solvers:
solver.set_initial_condition([rhs.I, 0])
u, t = solver.solve(t_mesh)
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
if len(t_mesh) <= 50:
plt.plot(t, u[:, 0]) # markers by default
else:
plt.plot(t, u[:, 0], '-2') # no markers
plt.hold('on')
legends.append(solver_name)
# Compare with RK4 on a much finer mesh
solver_exact = odespy.RK4(rhs)
solver_exact.set_initial_condition([rhs.I, 0])
u_e, t_e = solver_exact.solve(t_fine)
plt.plot(t_e, u_e[:, 0], '-') # avoid markers by spec. line type
legends.append('RK4, dt=%g' % (t_fine[1] - t_fine[0]))
plt.legend(legends, loc='lower left')
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plotfilestem = '_'.join(legends)
plt.savefig('%s.png' % filename)
plt.savefig('%s.pdf' % filename)
def simulate(alpha, beta, gamma, delta, T=20):
def f(u, t, alpha, beta, gamma):
H, L = u
return [alpha * H - L * H, beta * L * H - gamma * L]
t = np.linspace(0, T, 1501)
solver = odespy.RK4(f, f_args=(alpha, beta, gamma))
solver.set_initial_condition([1, delta])
u, t = solver.solve(t)
H = u[:, 0]
L = u[:, 1]
plt.figure()
global fig_counter
fig_counter += 1
plt.plot(t, H, t, L)
plt.legend(['H', 'L'])
plt.title(r'$\alpha=%g$, $\beta=%g$, $\gamma=%g$, $\delta=%g$' %
(alpha, beta, gamma, delta))
plt.savefig('tmp%d.png' % fig_counter)
plt.savefig('tmp%d.pdf' % fig_counter)
return H, L, t
def biochemical_solver(alpha, beta, epsilon, T, dt=0.1):
def f(u, t):
Q, P, S, E = u
# Consistency checks
conservation1 = abs(Q/(alpha*epsilon) + E - 1)
conservation2 = abs(alpha*S + Q + P - alpha)
tol = 1E-14
if conservation1 > tol or conservation2 > tol:
print 't=%g *** conservations:' % t, \
conservation1, conservation2
if Q < 0:
print 't=%g *** Q=%g < 0' % (t, Q)
if P < 0:
print 't=%g *** P=%g < 0' % (t, P)
if S < 0 or S > 1:
print 't=%g *** S=%g' % (t, S)
if E < 0 or S > 1:
print 't=%g *** E=%g' % (t, E)
return [
alpha*(E*S - Q),
beta*Q,
-E*S + (1-beta/alpha)*Q,
(-E*S + Q)/epsilon,
]
import numpy as np
Nt = int(round(T/dt))
t_mesh = np.linspace(0, Nt*dt, Nt+1)
solver = odespy.RK4(f)
solver.set_initial_condition([0, 0, 1, 1])
u, t = solver.solve(t_mesh)
Q = u[:,0]
P = u[:,1]
S = u[:,2]
E = u[:,3]
return Q, P, S, E, t
def demo():
from numpy import pi, linspace, cos
omega = 2
P = 2 * pi / omega
dt = P / 20
T = 40 * P
X_0 = 2
RK4 = odespy.RK4(f, f_args=[omega])
RK4.set_initial_condition([X_0, 0])
N_t = int(round(T / dt))
u, t = RK4.solve(linspace(0, T, N_t + 1))
u, v = u[:, 0], u[:, 1]
# Last p periods
p = 10
m = p * 20
fig = plt.figure()
l1, l2 = plt.plot(t[-m:], u[-m:], 'b-', t[-m:],
X_0 * cos(omega * t)[-m:], 'r--')
fig.legend((l1, l2), ('numerical', 'exact'), 'lower left')
plt.xlabel('t')
plt.show()
plt.savefig('tmp.pdf')
plt.savefig('tmp.png')
"""As exde1.py, but y and x as names instead of u and t."""
def myrhs(y, x):
return -y
import odespy
solver = odespy.RK4(myrhs)
solver.set_initial_condition(1)
import numpy
# Make sure integration interval [0, L] is large enough
N = 50
L = 10
x_points = numpy.linspace(0, L, N + 1)
def terminate(y, x, stepnumber):
tol = 0.001
return True if abs(y[stepnumber]) < tol else False
y, x = solver.solve(x_points, terminate)
print 'Final y(x=%g)=%g' % (x[-1], y[-1])
from matplotlib.pyplot import *
plot(x, y)
show()
import odespy, numpy, time
w = 2 * numpy.pi
n = 600 # no of periods
r = 40 # resolution of each period
tp = numpy.linspace(0, n, n * r + 1)
solvers = [
odespy.Vode(f,
complex_valued=True,
atol=1E-7,
rtol=1E-6,
adams_or_bdf='adams'),
odespy.RK4(f, complex_valued=True),
odespy.RKFehlberg(f, complex_valued=True, atol=1E-7, rtol=1E-6)
]
cpu = []
for solver in solvers:
solver.set_initial_condition(1 + 0j)
t0 = time.clock()
solver.solve(tp)
t1 = time.clock()
cpu.append(t1 - t0)
# Compare solutions at the end point:
exact = numpy.exp(1j * w * tp).real[-1]
min_cpu = min(cpu)
cpu = [c / min_cpu for c in cpu] # normalize
print 'Exact: u(%g)=%g' % (tp[-1], exact)
#f = RHS(b=0.4, A_F=1, w_F=2)
f = RHS(b=0.4, A_F=0, w_F=2)
f = RHS(b=0.4, A_F=1, w_F=np.pi)
f = RHS(b=0.4, A_F=20,
w_F=2 * np.pi) # qualitatively wrong FE, almost ok BE, smaller T
#f = RHS(b=0.4, A_F=20, w_F=0.5*np.pi) # cool, FE almost there, BE good
# Define different sets of experiments
solvers_theta = [
odespy.ForwardEuler(f),
# Implicit methods must use Newton solver to converge
odespy.BackwardEuler(f, nonlinear_solver='Newton'),
odespy.CrankNicolson(f, nonlinear_solver='Newton'),
]
solvers_RK = [odespy.RK2(f), odespy.RK4(f)]
solvers_accurate = [
odespy.RK4(f),
odespy.CrankNicolson(f, nonlinear_solver='Newton'),
odespy.DormandPrince(f, atol=0.001, rtol=0.02)
]
solvers_CN = [odespy.CrankNicolson(f, nonlinear_solver='Newton')]
if __name__ == '__main__':
timesteps_per_period = 20
solver_collection = 'theta'
num_periods = 1
try:
# Example: python vib_odespy.py 30 accurate 50
timesteps_per_period = int(sys.argv[1])
solver_collection = sys.argv[2]
self.period = 2 * pi / self.freq
def __call__(self, u, t):
theta, omega = u
c = self.c
return [omega, -c * theta]
problem = Problem(c=1, Theta=pi / 4)
import odespy
solvers = [
odespy.ThetaRule(problem, theta=0), # Forward Euler
odespy.ThetaRule(problem, theta=0.5), # Midpoint
odespy.ThetaRule(problem, theta=1), # Backward Euler
odespy.RK4(problem),
odespy.RK2(problem),
odespy.MidpointIter(problem, max_iter=5, eps_iter=0.01),
odespy.Leapfrog(problem),
odespy.LeapfrogFiltered(problem),
]
theta_exact = lambda t: problem.Theta * numpy.cos(sqrt(problem.c) * t)
import sys
try:
num_periods = int(sys.argv[1])
except IndexError:
num_periods = 8 # default
T = num_periods * problem.period # final time
self.a = a
self.R = R
self.A = A
def f(self, u, t):
a, R = self.a, self.R # short form
return a * u * (1 - u / R)
def u_exact(self, t):
a, R, A = self.a, self.R, self.A # short form
return R * A * exp(a * t) / (R + A * (exp(a * t) - 1))
import odespy
problem = Logistic(a=2, R=1E+5, A=1)
solver = odespy.RK4(problem.f)
solver.set_initial_condition(problem.A)
T = 10 # end of simulation
N = 30 # no of time steps
time_points = linspace(0, T, N + 1)
u, t = solver.solve(time_points)
from matplotlib.pyplot import *
plot(t, u, 'r-', t, problem.u_exact(t), 'bo')
savefig('tmppng')
savefig('tmp.pdf')
show()
plt.savefig('vib_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.figure(4)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated amplitudes')
plt.savefig('vib_%d_%d_a.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.png' % (timesteps_per_period, num_periods))
# Define different sets of experiments
solvers_theta = [
odespy.ForwardEuler(f),
# Implicit methods must use Newton solver to converge
odespy.BackwardEuler(f, nonlinear_solver='Newton'),
odespy.CrankNicolson(f, nonlinear_solver='Newton'),
]
solvers_RK = [odespy.RK2(f), odespy.RK4(f)]
solvers_accurate = [odespy.RK4(f),
odespy.CrankNicolson(f, nonlinear_solver='Newton'),
odespy.DormandPrince(f, atol=0.001, rtol=0.02)]
solvers_CN = [odespy.CrankNicolson(f, nonlinear_solver='Newton')]
solvers_EC = [odespy.EulerCromer(f)]
if __name__ == '__main__':
# Default values
timesteps_per_period = 20
solver_collection = 'theta'
num_periods = 1
# Override from command line
try:
# Example: python vib_undamped_odespy.py 30 accurate 50
timesteps_per_period = int(sys.argv[1])
self.period = 2 * pi / self.freq
def f(self, u, t):
theta, omega = u
c = self.c
return [omega, -c * theta]
problem = Problem(c=1, Theta=pi / 4)
import odespy
solvers = [
odespy.ThetaRule(problem.f, theta=0), # Forward Euler
odespy.ThetaRule(problem.f, theta=0.5), # Midpoint method
odespy.ThetaRule(problem.f, theta=1), # Backward Euler
odespy.RK4(problem.f),
odespy.MidpointIter(problem.f, max_iter=2, eps_iter=0.01),
odespy.LeapfrogFiltered(problem.f),
]
N_per_period = 20
T = 3 * problem.period # final time
import numpy
import matplotlib.pyplot as plt
legends = []
for solver in solvers:
solver_name = str(solver) # short description of solver
print solver_name
solver.set_initial_condition([problem.Theta, 0])
def f(u, t, a=1, R=1):
return a * u * (1 - u / R)
A = 1
import odespy, numpy
from matplotlib.pyplot import plot, hold, show, axis
solver = odespy.RK4(f, f_kwargs=dict(a=2, R=1E+5))
# Split time domain into subdomains and
# integrate the ODE in each subdomain
T = [0, 1, 4, 8, 12] # subdomain boundaries
N_tot = 30 # total no of time steps
dt = float(T[-1]) / N_tot # time step, kept fixed
u = []
t = [] # collectors for u and t in each domain
for i in range(len(T) - 1):
T_interval = T[i + 1] - T[i]
N = int(round(T_interval / dt))
time_points = numpy.linspace(T[i], T[i + 1], N + 1)
solver.set_initial_condition(A) # at time_points[0]
print 'Solving in [%s, %s] with %d intervals' % \
(T[i], T[i+1], N)
ui, ti = solver.solve(time_points)
A = ui[-1] # newest ui value is next initial condition
def model_free_lift(duration,
bal_mass, bal_diam, bal_mat='rubber',
bal_gas='helium',
payload=0,
plot_show=False, plot_save_as='',
show_debug_msg=False,
):
""" Моделирование процесса свободного подъёма для шара с полезной нагрузкой
Функция создаёт изображение-график, если в аргументе <plot_save_as> для
него передано имя
---------------------------------------------------------------------------
:param duration: продолжительность моделируемого процесса, с
:param bal_mass: масса метеошара, кг
:param bal_diam: диаметр метеошара в состоянии без растяжения, м
:param bal_mat: наименование материала метеошара. Варианты:
{'rubber', }
(см пакет <material>)
:param bal_gas: наименование наполняющего газа метеошара. Варианты:
{'helium', }
(см пакет <gas>)
:param payload: полезная нагрузка, поднимаемая метеошаром, кг
:param plot_show: True/False отображение графика процесса
:param plot_save_as: путь к сохраняемому файлу графика, с расширением.
'' - пустая строка - не сохранять изображение
:param show_debug_msg: отображать отладочные сообщения выводом print()
---------------------------------------------------------------------------
:return: exit_status:
0 - успешное завршение
1 - некорректные входные данные
2 - ошибка создания объекта метеошара
3 - ошибка интегрирования системы ОДУ
4 - ошибка создания графика решения
Примеры вызова:
>>> model_free_lift(duration=180*60,
... bal_mass=3.0,
... bal_diam=2.164,
... payload=1.05,
... plot_save_as='doctest.png',
... plot_show=True,
... show_debug_msg=True)
Called function:
model_free_lift(
duration=10800,
bal_mass=3.0,
bal_diam=2.164,
bal_mat=rubber,
bal_gas=helium,
payload=1.05,
plot_show=True,
plot_save_as=doctest.png,
show_debug_msg=True)
RK4 terminated at t=7018
Successfull end.
0
>>> kwargs = {'duration': 180*60,
... 'bal_mass': 3.0,
... 'bal_diam':2.164,
... 'payload': 1.05,
... 'plot_save_as': 'doctest.png'}
>>> model_free_lift(**kwargs)
RK4 terminated at t=7018
0
"""
# Send inputs to log
if show_debug_msg:
log_msg = "Called function:\n" \
" model_free_lift(\n" \
" duration={0},\n" \
" bal_mass={1},\n" \
" bal_diam={2},\n" \
" bal_mat={3},\n" \
" bal_gas={4},\n" \
" payload={5},\n" \
" plot_show={6},\n" \
" plot_save_as={7},\n" \
" show_debug_msg={8})".\
format(duration, bal_mass, bal_diam, bal_mat, bal_gas,
payload, plot_show, plot_save_as, show_debug_msg)
print(log_msg)
# Create ODE-function for model
def odefun(y, time, balloon, payload):
""" Дифф. закон Ньютона -
уравнение процесса свободного подъёма для шара с полезной нагрузкой
В форме Коши: dy/dt = f(y, t)
Внимание! Условие разрыва шара должно проверяться извне функции.
"""
alt = y[0]
vel = y[1]
# Model equations
f_sum_bal = balloon.get_forces_sum(alt, vel)
f_sum_pl = - payload*const.g
f_sum = f_sum_bal + f_sum_pl
mass = balloon.get_mass() + payload
return [vel, f_sum/mass]
def terminator(y, t, step_no):
# Функция, останавливающая интегрирование при разрыве метеошара
h = y[step_no][0]
tf = balloon.is_burst(h)
return tf
# Check inputs.
# -----------------------------------------------------------
try:
# Confirm input numeric types
duration = float(duration)
if duration <= 0:
raise ValueError('duration must be positive')
bal_mass = float(bal_mass)
if bal_mass <= 0:
raise ValueError('bal_mass must be positive')
bal_diam = float(bal_diam)
if bal_diam <= 0:
raise ValueError('bal_diam must be positive')
payload = float(payload)
if payload < 0:
raise ValueError('payload must be non-negative')
# Define material
if not (bal_mat in material.__all__):
raise ValueError("Unknown balloon material name ", bal_mat)
# Define gas
if not (bal_gas in gas.__all__):
raise ValueError("Unknown gas name ", bal_gas)
except ValueError as err:
print(err, file=sys.stderr)
return 1
# Create balloon instance
# -----------------------------------------------------------
try:
balloon = BalloonStatic(
bal_mass=bal_mass,
bal_diam=bal_diam,
bal_mat=material.BY_NAME[bal_mat],
gas=gas.BY_NAME[bal_gas])
except Exception as err:
print(err, file=sys.stderr)
return 2
# Шаг детализации процесса по времени, с
tstep = duration/100.0 if duration < 100.0 else 1
time_points = np.arange(0, duration, tstep)
# solve the DEs
# -----------------------------------------------------------
try:
# Create solver (Runge-Kutta, 4th order)
solver = odespy.RK4(
odefun,
f_args=(balloon, payload),
rtol=1E-2)
solver.set_initial_condition([0, 0])
y_sln, time = solver.solve(
time_points,
terminate=terminator)
alt = y_sln[:, 0]
vel = y_sln[:, 1]
# Максимальная высота
alt_max = max(alt)
# Момент достижения макс. высоты
time_alt_max = time[np.where(alt == alt_max)][0]
# Индекс высшей точки в массиве
ind_max = np.argmax(alt)
# Скрыть некорректные точки расчёта после разрыва шара
time = time[:ind_max]
alt = alt[:ind_max]
vel = vel[:ind_max]
except Exception as err:
print(err, file=sys.stderr)
return 3
# Plot
# -----------------------------------------------------------
if plot_show or plot_save_as:
try:
# масштаб отображения
scale_v = 1.0 # скорости
scale_h = 1.0/1000.0 # высоты
scale_t = 1.0/60.0 # времени
time *= scale_t
time_alt_max *= scale_t
alt *= scale_h
alt_max *= scale_h
vel *= scale_v
# Построение графика
matplotlib.rcParams['font.size'] = 14
matplotlib.rcParams['font.family'] = utils.get_font()
matplotlib.rcParams["axes.grid"] = True
plt.figure()
plt.plot(
time, alt, 'b-', linewidth=2.0, label=u'Высота, км')
plt.plot(
time, vel, 'r-', linewidth=2.0, label=u'Скорость, м/с')
plt.xlabel(u'Время, мин')
matplotlib.pyplot.ylim([0.0, alt_max*1.2])
plt.legend(loc='upper left', shadow=True)
plt.title(
u"Полёт метеошара\n"
u"(m = {0} кг; Ø = {1} м, нагрузка {2} кг)\n"
u"Макс. высота {3} км достигнута спустя {4} мин".format(
balloon.bal_mass, balloon.d0, payload,
round(alt_max), round(time_alt_max)
))
if plot_save_as:
plt.savefig(plot_save_as, bbox_inches='tight')
if plot_show:
plt.show()
plt.close()
except Exception as err:
print(err, file=sys.stderr)
return 4
else:
warnings.warn("All output's disabled.")
if show_debug_msg:
print('Successfull end.')
return 0
line_color = ['k', 'm', 'b', 'r']
figure(figsize=(20, 8))
hold('on')
LNWDT = 4
FNT = 18
rcParams['lines.linewidth'] = LNWDT
rcParams['font.size'] = FNT
# computing and plotting
# smooth ball with drag
for i in range(0, N2):
z0 = np.zeros(4)
z0[2] = v0 * np.cos(angle[i])
z0[3] = v0 * np.sin(angle[i])
solver = odespy.RK4(f)
solver.set_initial_condition(z0)
z, t = solver.solve(time)
plot(z[:, 0], z[:, 1], ':', color=line_color[i])
legends.append('angle=' + str(alfa[i]) + ', smooth ball')
# golf ball with drag
for i in range(0, N2):
z0 = np.zeros(4)
z0[2] = v0 * np.cos(angle[i])
z0[3] = v0 * np.sin(angle[i])
solver = odespy.RK4(f2)
solver.set_initial_condition(z0)
z, t = solver.solve(time)
plot(z[:, 0], z[:, 1], '-.', color=line_color[i])
legends.append('angle=' + str(alfa[i]) + ', golf ball')
def f(u, t):
return a * u * (1 - u / R)
a = 2
R = 1E+5
A = 1
import odespy
solver = odespy.RK4(f)
solver.set_initial_condition(A)
from numpy import linspace, exp
T = 10 # end of simulation
N = 30 # no of time steps
time_points = linspace(0, T, N + 1)
u, t = solver.solve(time_points)
def u_exact(t):
return R * A * exp(a * t) / (R + A * (exp(a * t) - 1))
from matplotlib.pyplot import *
plot(t, u, 'r-', t, u_exact(t), 'bo')
savefig('tmppng')
savefig('tmp.pdf')
show()
dist = cp.J(dist_a, dist_I) # joint multivariate dist
P, norms = cp.orth_ttr(2, dist, retall=True)
variable_a, variable_I = cp.variable(2)
PP = cp.outer(P, P)
E_aPP = cp.E(variable_a * PP, dist)
E_IP = cp.E(variable_I * P, dist)
def right_hand_side(c, x): # c' = right_hand_side(c, x)
return -np.dot(E_aPP, c) / norms # -M*c
initial_condition = E_IP / norms
solver = odespy.RK4(right_hand_side)
solver.set_initial_condition(initial_condition)
x = np.linspace(0, 10, 1000)
c = solver.solve(x)[0]
u_hat = cp.dot(P, c)
#Rosenblat transformation using point collocation
def u(x, a, I):
return I * np.exp(-a * x)
dist_R = cp.J(cp.Normal(), cp.Normal())
C = [[1, 0.5], [0.5, 1]]
mu = [0, 0]
def f(u, t):
return -a * u
I = 1
a = 2
T = 6
dt = float(sys.argv[1]) if len(sys.argv) >= 2 else 0.75
Nt = int(round(T / dt))
t_mesh = np.linspace(0, Nt * dt, Nt + 1)
solvers = [
odespy.RK2(f),
odespy.RK3(f),
odespy.RK4(f),
# BackwardEuler must use Newton solver to converge
# (Picard is default and leads to divergence)
odespy.BackwardEuler(f, nonlinear_solver='Newton')
]
legends = []
for solver in solvers:
solver.set_initial_condition(I)
u, t = solver.solve(t_mesh)
plt.plot(t, u)
plt.hold('on')
legends.append(solver.__class__.__name__)
# Compare with exact solution plotted on a very fine mesh
def fblasius(y, x):
"""ODE-system for the Blasius-equation"""
return [y[1],y[2], -y[0]*y[2]]
import odespy
solvers=[]
solvers.append(odespy.RK4(fblasius))
solvers.append(odespy.RK2(fblasius))
solvers.append(odespy.RK3(fblasius))
from numpy import linspace, exp
xmin = 0
xmax = 5.75
N = 150 # no x-values
xspan = linspace(xmin, xmax, N+1)
smin=0.1
smax=0.8
Ns=30
srange = linspace(smin,smax,Ns)
from matplotlib.pyplot import *
#change some default values to make plots more readable on the screen
LNWDT=5; FNT=25
matplotlib.rcParams['lines.linewidth'] = LNWDT; matplotlib.rcParams['font.size'] = FNT
figure()
legends=[]
linet=['r-',':','.','-.','--']
k4 = func(z[i, :] + k3 * dt, t + dt) # predictor step 4
z[i + 1, :] = z[i, :] + dt / 6.0 * (k1 + 2.0 * k2 + 2.0 * k3 + k4
) # Corrector step
return z
# Main program starts here
from numpy import linspace
T = 10 # end of simulation
N = 20 # no of time steps
time = linspace(0, T, N + 1)
solvers = []
solvers.append(odespy.RK3(f2))
solvers.append(odespy.RK4(f2))
legends = []
z0 = np.zeros(2)
z0[0] = 2.0
for i, solver in enumerate(solvers):
solver.set_initial_condition(z0)
z, t = solver.solve(time)
plot(t, z[:, 1])
legends.append(str(solver))
scheme_list = [euler, heun, rk4_own]
for scheme in scheme_list:
