def comparison_plot(f, u, Omega, filename='tmp.pdf',
plot_title='', ymin=None, ymax=None,
u_legend='approximation'):
"""Compare f(x) and u(x) for x in Omega in a plot."""
x = sm.Symbol('x')
print 'f:', f
f = sm.lambdify([x], f, modules="numpy")
u = sm.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
exact = f(xcoor)
approx = u(xcoor)
plot(xcoor, approx, '-')
hold('on')
plot(xcoor, exact, '-')
legend([u_legend, 'exact'])
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def approximate(f, symbolic=False, d=1, N_e=4, Omega=[0, 1], filename='tmp'):
phi = basis(d)
print 'phi basis (reference element):\n', phi
nodes, elements = mesh_uniform(N_e, d, Omega, symbolic)
A, b = assemble(nodes, elements, phi, f, symbolic=symbolic)
print 'nodes:', nodes
print 'elements:', elements
print 'A:\n', A
print 'b:\n', b
print sp.latex(A, mode='plain')
#print sp.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
print 'Plain interpolation/collocation:'
x = sp.Symbol('x')
f = sp.lambdify([x], f, modules='numpy')
try:
f_at_nodes = [f(xc) for xc in nodes]
except NameError as e:
raise NameError('numpy does not support special function:\n%s' % e)
print f_at_nodes
if not symbolic and filename is not None:
xf = np.linspace(Omega[0], Omega[1], 10001)
U = np.asarray(c)
xu, u = u_glob(U, elements, nodes)
plt.plot(xu, u, '-',
xf, f(xf), '--')
plt.legend(['u', 'f'])
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
def run_solvers_and_check_amplitudes(solvers,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
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)
file_name = 'Amplitudes' # initialize filename for plot
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
solver_name = \
'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
file_name = file_name + '_' + solver_name
minima, maxima = minmax(t, u[:, 0])
a = amplitudes(minima, maxima)
plt.plot(range(len(a)), a, '-', label=solver_name)
plt.hold('on')
plt.xlabel('Number of periods')
plt.ylabel('Amplitude (absolute value)')
plt.legend(loc='upper left')
plt.savefig(file_name + '.png')
plt.savefig(file_name + '.pdf')
plt.show()
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
V = 0.2
m = 4
b = 0.2
s = lambda u: 2 * u
F = lambda t: 0
w = np.sqrt(2. / 4) # approx freq
dt = 0.9 * 2 / w # longest possible time step
w = 0.5
P = 2 * pi / w
T = 4 * P
from vib import solver as solver2
import scitools.std as plt
for k in range(4):
u2, t2 = solver2(I, V, m, b, s, F, dt, T, 'quadratic')
u, v, t = solver(I, V, m, b, s, F, dt, T, 'quadratic')
plt.figure()
plt.plot(t, u, 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', 'centered scheme'])
plt.title('dt=%.3g' % dt)
raw_input()
plt.savefig('tmp_%d' % k + '.png')
plt.savefig('tmp_%d' % k + '.pdf')
dt /= 2
def plot(self, include_exact=True, plt=None):
"""
Add solver.u curve to the plotting object plt,
and include the exact solution if include_exact is True.
This plot function can be called several times (if
the solver object has computed new solutions).
"""
if plt is None:
import scitools.std as plt # can use matplotlib as well
plt.plot(self.solver.t, self.solver.u, '--o')
plt.hold('on')
theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
name = theta2name.get(self.solver.theta, '')
legends = ['numerical %s' % name]
if include_exact:
t_e = linspace(0, self.problem.T, 1001)
u_e = self.problem.exact_solution(t_e)
plt.plot(t_e, u_e, 'b-')
legends.append('exact')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title('theta=%g, dt=%g' %
(self.solver.theta, self.solver.dt))
plt.savefig('%s_%g.png' % (name, self.solver.dt))
return plt
def run_solvers_and_check_amplitudes(solvers, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
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)
file_name = 'Amplitudes' # initialize filename for plot
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
solver_name = \
'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
file_name = file_name + '_' + solver_name
minima, maxima = minmax(t, u[:,0])
a = amplitudes(minima, maxima)
plt.plot(range(len(a)), a, '-', label=solver_name)
plt.hold('on')
plt.xlabel('Number of periods')
plt.ylabel('Amplitude (absolute value)')
plt.legend(loc='upper left')
plt.savefig(file_name + '.png')
plt.savefig(file_name + '.pdf')
plt.show()
def u_P1():
"""
Plot P1 basis functions and a resulting u to
illustrate what it means to use finite elements.
"""
import matplotlib.pyplot as plt
x = [0, 1.5, 2.5, 3.5, 4]
phi = [np.zeros(len(x)) for i in range(len(x)-2)]
for i in range(len(phi)):
phi[i][i+1] = 1
#u = 5*x*np.exp(-0.25*x**2)*(4-x)
u = [0, 8, 5, 4, 0]
for i in range(len(phi)):
plt.plot(x, phi[i], 'r-') #, label=r'$\varphi_%d$' % i)
plt.text(x[i+1], 1.2, r'$\varphi_%d$' % i)
plt.plot(x, u, 'b-', label='$u$')
plt.legend(loc='upper left')
plt.axis([0, x[-1], 0, 9])
plt.savefig('u_example_P1.png')
plt.savefig('u_example_P1.pdf')
# Mark elements
for xi in x[1:-1]:
plt.plot([xi, xi], [0, 9], 'm--')
# Mark nodes
#plt.plot(x, np.zeros(len(x)), 'ro', markersize=4)
plt.savefig('u_example_P1_welms.png')
plt.savefig('u_example_P1_welms.pdf')
plt.show()
def u_P1():
"""
Plot P1 basis functions and a resulting u to
illustrate what it means to use finite elements.
"""
import matplotlib.pyplot as plt
x = [0, 1.5, 2.5, 3.5, 4]
phi = [np.zeros(len(x)) for i in range(len(x)-2)]
for i in range(len(phi)):
phi[i][i+1] = 1
#u = 5*x*np.exp(-0.25*x**2)*(4-x)
u = [0, 8, 5, 4, 0]
for i in range(len(phi)):
plt.plot(x, phi[i], 'r-') #, label=r'$\varphi_%d$' % i)
plt.text(x[i+1], 1.2, r'$\varphi_%d$' % i)
plt.plot(x, u, 'b-', label='$u$')
plt.legend(loc='upper left')
plt.axis([0, x[-1], 0, 9])
plt.savefig('tmp_u_P1.png')
plt.savefig('tmp_u_P1.pdf')
# Mark elements
for xi in x[1:-1]:
plt.plot([xi, xi], [0, 9], 'm--')
# Mark nodes
#plt.plot(x, np.zeros(len(x)), 'ro', markersize=4)
plt.savefig('tmp_u_P1_welms.png')
plt.savefig('tmp_u_P1_welms.pdf')
plt.show()
def plot(self, include_exact=True, plt=None):
"""
Add solver.u curve to scitools plotting object plt,
and include the exact solution if include_exact is True.
This plot function can be called several times (if
the solver object has computed new solutions).
"""
if plt is None:
import scitools.std as plt
plt.plot(self.solver.t, self.solver.u, '--o')
plt.hold('on')
theta = self.solver.get('theta')
theta2name = {0: 'FE', 1: 'BE', 0.5: 'CN'}
name = theta2name.get(theta, '')
legends = ['numerical %s' % name]
if include_exact:
t_e = np.linspace(0, self.problem.get('T'), 1001)
u_e = self.problem.exact_solution(t_e)
plt.plot(t_e, u_e, 'b-')
legends.append('exact')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
dt = self.solver.get('dt')
plt.title('theta=%g, dt=%g' % (theta, dt))
plt.savefig('%s_%g.png' % (name, dt))
return plt
def demo():
"""
Demonstrate difference between Euler-Cromer and the
scheme for the corresponding 2nd-order ODE.
"""
I = 1.2
V = 0.2
m = 4
b = 0.2
s = lambda u: 2 * u
F = lambda t: 0
w = np.sqrt(2.0 / 4) # approx freq
dt = 0.9 * 2 / w # longest possible time step
w = 0.5
P = 2 * pi / w
T = 4 * P
from vib import solver as solver2
import scitools.std as plt
for k in range(4):
u2, t2 = solver2(I, V, m, b, s, F, dt, T, "quadratic")
u, v, t = solver(I, V, m, b, s, F, dt, T, "quadratic")
plt.figure()
plt.plot(t, u, "r-", t2, u2, "b-")
plt.legend(["Euler-Cromer", "centered scheme"])
plt.title("dt=%.3g" % dt)
raw_input()
plt.savefig("tmp_%d" % k + ".png")
plt.savefig("tmp_%d" % k + ".pdf")
dt /= 2
def run_solvers_and_plot(solvers, rhs, T, dt, title=''):
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 = []
solver_exact = odespy.RK4(rhs)
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.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('exact (RK4, dt=%g)' % (t_fine[1]-t_fine[0]))
plt.legend(legends, loc='upper right')
plt.xlabel('t'); plt.ylabel('u')
plt.title(title)
plotfilestem = '_'.join(legends)
plt.savefig('tmp_%s.png' % plotfilestem)
plt.savefig('tmp_%s.pdf' % plotfilestem)
def run_solvers_and_plot(solvers, rhs, T, dt, title=''):
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 = []
solver_exact = odespy.RK4(rhs)
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.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('exact (RK4, dt=%g)' % (t_fine[1] - t_fine[0]))
plt.legend(legends, loc='upper right')
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plotfilestem = '_'.join(legends)
plt.savefig('tmp_%s.png' % plotfilestem)
plt.savefig('tmp_%s.pdf' % plotfilestem)
def comparison_plot(u, Omega, u_e=None, filename='tmp.eps',
plot_title='', ymin=None, ymax=None):
x = sp.Symbol('x')
u = sp.lambdify([x], u, modules="numpy")
if len(Omega) != 2:
raise ValueError('Omega=%s must be an interval (2-list)' % str(Omega))
# When doing symbolics, Omega can easily contain symbolic expressions,
# assume .evalf() will work in that case to obtain numerical
# expressions, which then must be converted to float before calling
# linspace below
if not isinstance(Omega[0], (int,float)):
Omega[0] = float(Omega[0].evalf())
if not isinstance(Omega[1], (int,float)):
Omega[1] = float(Omega[1].evalf())
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
# Vectorized functions expressions does not work with
# lambdify'ed functions without the modules="numpy"
approx = u(xcoor)
plot(xcoor, approx)
legends = ['approximation']
if u_e is not None:
exact = u_e(xcoor)
hold('on')
plot(xcoor, exact)
legends = ['exact']
legend(legends)
title(plot_title)
xlabel('x')
if ymin is not None and ymax is not None:
axis([xcoor[0], xcoor[-1], ymin, ymax])
savefig(filename)
def visualize(list_of_curves, legends, title='', filename='tmp'):
"""Plot list of curves: (u, t)."""
for u, t in list_of_curves:
plt.plot(t, u)
plt.hold('on')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def visualize(list_of_curves, legends, title='', filename='tmp'):
"""Plot list of curves: (u, t)."""
for u, t in list_of_curves:
plt.plot(t, u)
plt.hold('on')
plt.legend(legends)
plt.xlabel('t')
plt.ylabel('u')
plt.title(title)
plt.savefig(filename + '.png')
plt.savefig(filename + '.pdf')
plt.show()
def comparison_plot(f, u, Omega, filename='tmp.pdf'):
x = sm.Symbol('x')
f = sm.lambdify([x], f, modules="numpy")
u = sm.lambdify([x], u, modules="numpy")
resolution = 401 # no of points in plot
xcoor = linspace(Omega[0], Omega[1], resolution)
exact = f(xcoor)
approx = u(xcoor)
plot(xcoor, approx)
hold('on')
plot(xcoor, exact)
legend(['approximation', 'exact'])
savefig(filename)
def plot_empirical_freq_and_amplitude(u, t):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure()
from math import pi
w = 2 * pi / p
plt.plot(range(len(p)), w, 'r-')
plt.hold('on')
plt.plot(range(len(a)), a, 'b-')
ymax = 1.1 * max(w.max(), a.max())
ymin = 0.9 * min(w.min(), a.min())
plt.axis([0, max(len(p), len(a)), ymin, ymax])
plt.legend(['estimated frequency', 'estimated amplitude'],
loc='upper right')
return len(maxima)
def plot_empirical_freq_and_amplitude(u, t):
minima, maxima = minmax(t, u)
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure()
from math import pi
w = 2*pi/p
plt.plot(range(len(p)), w, 'r-')
plt.hold('on')
plt.plot(range(len(a)), a, 'b-')
ymax = 1.1*max(w.max(), a.max())
ymin = 0.9*min(w.min(), a.min())
plt.axis([0, max(len(p), len(a)), ymin, ymax])
plt.legend(['estimated frequency', 'estimated amplitude'],
loc='upper right')
return len(maxima)
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 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 approximate(f, symbolic=False, d=1, N_e=4,
Omega=[0, 1], filename='tmp'):
phi = basis(d)
print 'phi basis (reference element):\n', phi
nodes, elements = mesh_uniform(N_e, d, Omega, symbolic)
A, b = assemble(nodes, elements, phi, f, symbolic=symbolic)
print 'nodes:', nodes
print 'elements:', elements
print 'A:\n', A
print 'b:\n', b
print sym.latex(A, mode='plain')
#print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
c = [c[i,0] for i in range(c.shape[0])]
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
print 'Plain interpolation/collocation:'
x = sym.Symbol('x')
f = sym.lambdify([x], f, modules='numpy')
try:
f_at_nodes = [f(xc) for xc in nodes]
except NameError as e:
raise NameError('numpy does not support special function:\n%s' % e)
print f_at_nodes
if not symbolic and filename is not None:
xf = np.linspace(Omega[0], Omega[1], 10001)
U = np.asarray(c) # c is a plain array
xu, u = u_glob(U, elements, nodes)
import scitools.std as plt
#import matplotlib.pyplot as plt
plt.plot(xu, u, '-',
xf, f(xf), '--')
plt.legend(['u', 'f'])
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def approximate(f, symbolic=False, d=1, N_e=4, Omega=[0, 1], filename="tmp"):
phi = basis(d)
print "phi basis (reference element):\n", phi
nodes, elements = mesh_uniform(N_e, d, Omega, symbolic)
A, b = assemble(nodes, elements, phi, f, symbolic=symbolic)
print "nodes:", nodes
print "elements:", elements
print "A:\n", A
print "b:\n", b
print sym.latex(A, mode="plain")
# print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
c = [c[i, 0] for i in range(c.shape[0])]
else:
c = np.linalg.solve(A, b)
print "c:\n", c
print "Plain interpolation/collocation:"
x = sym.Symbol("x")
f = sym.lambdify([x], f, modules="numpy")
try:
f_at_nodes = [f(xc) for xc in nodes]
except NameError as e:
raise NameError("numpy does not support special function:\n%s" % e)
print f_at_nodes
if not symbolic and filename is not None:
xf = np.linspace(Omega[0], Omega[1], 10001)
U = np.asarray(c) # c is a plain array
xu, u = u_glob(U, elements, nodes)
import scitools.std as plt
# import matplotlib.pyplot as plt
plt.plot(xu, u, "-", xf, f(xf), "--")
plt.legend(["u", "f"])
plt.savefig(filename + ".pdf")
plt.savefig(filename + ".png")
return c
def run_solver_and_plot(solver, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
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)
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
from vib_undamped import solver
u2, t2 = solver(I, w, dt, T)
plt.plot(t, u[:,1], 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', '2nd-order ODE'])
plt.xlabel('t'); plt.ylabel('u')
plt.savefig('tmp1.png'); plt.savefig('tmp1.pdf')
def u_sines():
"""
Plot sine basis functions and a resulting u to
illustrate what it means to use global basis functions.
"""
import matplotlib.pyplot as plt
x = np.linspace(0, 4, 1001)
psi0 = np.sin(2*np.pi/4*x)
psi1 = np.sin(2*np.pi*x)
psi2 = np.sin(2*np.pi*4*x)
u = 4*psi0 - 0.5*psi1 - 0*psi2
plt.plot(x, psi0, 'r-', label=r"$\psi_0$")
plt.plot(x, psi1, 'g-', label=r"$\psi_1$")
#plt.plot(x, psi2, label=r"$\psi_2$")
plt.plot(x, u, 'b-', label=r"u")
plt.legend()
plt.savefig('tmp_u_sines.pdf')
plt.savefig('tmp_u_sines.png')
plt.show()
def fe_basis_function_figure(d,
target_elm=[1],
N_e=3,
derivative=0,
filename='tmp.pdf',
labels=False):
"""
Draw all basis functions (or their derivative), of degree d,
associated with element target_elm (may be list of elements).
Add a mesh with N_e elements.
"""
nodes, elements = mesh_uniform(N_e, d)
"""
x = 1.1
print locate_element_vectorized(x, elements, nodes)
print locate_element_scalar(x, elements, nodes)
x = 0.1, 0.4, 0.8
print locate_element_vectorized(x, elements, nodes)
"""
if isinstance(target_elm, int):
target_elm = [target_elm] # wrap in list
# Draw the basis functions for element 1
phi_drawn = [] # list of already drawn phi functions
ymin = ymax = 0
for e in target_elm:
for i in elements[e]:
if not i in phi_drawn:
x, y = phi_glob(i, elements, nodes, derivative=derivative)
if x is None and y is None:
return # abort
ymax = max(ymax, max(y))
ymin = min(ymin, min(y))
plt.plot(x, y, '-')
plt.hold('on')
if labels:
if plt.backend == 'gnuplot':
if derivative == 0:
plt.legend(r'basis func. %d' % i)
else:
plt.legend(r'derivative of basis func. %d' % i)
elif plt.backend == 'matplotlib':
if derivative == 0:
plt.legend(r'\varphi_%d' % i)
else:
plt.legend(r"\varphi_%d'(x)" % i)
phi_drawn.append(i)
plt.axis([nodes[0], nodes[-1], ymin - 0.1, ymax + 0.1])
plot_fe_mesh(nodes, elements, element_marker=[ymin - 0.1, ymax + 0.1])
plt.hold('off')
plt.savefig(filename)
def u_sines():
"""
Plot sine basis functions and a resulting u to
illustrate what it means to use global basis functions.
"""
import matplotlib.pyplot as plt
x = np.linspace(0, 4, 1001)
psi0 = np.sin(2*np.pi/4*x)
psi1 = np.sin(2*np.pi*x)
psi2 = np.sin(2*np.pi*4*x)
#u = 4*psi0 - 0.5*psi1 - 0*psi2
u = 4*psi0 - 0.5*psi1
plt.plot(x, psi0, 'r-', label=r"$\psi_0$")
plt.plot(x, psi1, 'g-', label=r"$\psi_1$")
#plt.plot(x, psi2, label=r"$\psi_2$")
plt.plot(x, u, 'b-', label=r"$u=4\psi_0 - \frac{1}{2}\psi_1$")
plt.legend()
plt.savefig('u_example_sin.pdf')
plt.savefig('u_example_sin.png')
plt.show()
def fe_basis_function_figure(d, target_elm=[1], n_e=3,
derivative=0, filename='tmp.pdf',
labels=False):
"""
Draw all basis functions (or their derivative), of degree d,
associated with element target_elm (may be list of elements).
Add a mesh with n_e elements.
"""
nodes, elements = mesh_uniform(n_e, d)
"""
x = 1.1
print locate_element_vectorized(x, elements, nodes)
print locate_element_scalar(x, elements, nodes)
x = 0.1, 0.4, 0.8
print locate_element_vectorized(x, elements, nodes)
"""
if isinstance(target_elm, int):
target_elm = [target_elm] # wrap in list
# Draw the basis functions for element 1
phi_drawn = [] # list of already drawn phi functions
ymin = ymax = 0
for e in target_elm:
for i in elements[e]:
if not i in phi_drawn:
x, y = phi_glob(i, elements, nodes,
derivative=derivative)
if x is None and y is None:
return # abort
ymax = max(ymax, max(y))
ymin = min(ymin, min(y))
plt.plot(x, y, '-')
plt.hold('on')
if labels:
if plt.backend == 'gnuplot':
if derivative == 0:
plt.legend(r'basis function no. %d' % i)
else:
plt.legend(r'derivative of basis function no. %d' % i)
elif plt.backend == 'matplotlib':
if derivative == 0:
plt.legend(r'\varphi_%d' % i)
else:
plt.legend(r"\varphi_%d'(x)" % i)
phi_drawn.append(i)
plt.axis([nodes[0], nodes[-1], ymin-0.1, ymax+0.1])
plot_fe_mesh(nodes, elements, element_marker=[ymin-0.1, ymax+0.1])
plt.hold('off')
plt.savefig(filename)
def run_solver_and_plot(solver,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
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)
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([0, I])
u, t = solver.solve(t_mesh)
from vib_undamped import solver
u2, t2 = solver(I, w, dt, T)
plt.plot(t, u[:, 1], 'r-', t2, u2, 'b-')
plt.legend(['Euler-Cromer', '2nd-order ODE'])
plt.xlabel('t')
plt.ylabel('u')
plt.savefig('tmp1.png')
plt.savefig('tmp1.pdf')
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
t_fine = np.linspace(0, T, 10001)
u_e = I * np.exp(-a * t_fine)
plt.plot(t_fine, u_e, '-') # avoid markers by specifying line type
legends.append('exact')
plt.legend(legends)
plt.title('Time step: %g' % dt)
plt.savefig('odespy1_dt_%g.png' % dt)
plt.savefig('odespy1_dt_%g.pdf' % dt)
plt.show()
def approximate(f, symbolic=False, d=1, N_e=4, numint=None,
Omega=[0, 1], collocation=False, filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a symbolic expression f (with x
as independent variable) on a domain Omega. N_e is the
number of elements.
symbolic=True implies symbolic expressions in the
calculations, while symbolic=False means numerical
computing.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
numint_name = numint # save name
if symbolic:
if numint == 'Trapezoidal':
numint = [[sym.S(-1), sym.S(1)], [sym.S(1), sym.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sym.S(-1), sym.S(0), sym.S(1)],
[sym.Rational(1,3), sym.Rational(4,3), sym.Rational(1,3)]]
elif numint == 'Midpoint':
numint = [[sym.S(0)], [sym.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1/sym.sqrt(3), 1/sym.sqrt(3)], [sym.S(1), sym.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[-sym.sqrt(sym.Rational(3,5)), 0,
sym.sqrt(sym.Rational(3,5))],
[sym.Rational(5,9), sym.Rational(8,9),
sym.Rational(5,9)]]
elif numint is not None:
print 'Numerical rule %s is not supported for symbolic computing' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[ 0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. , 0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f,
symbolic=symbolic, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sym.latex(A, mode='plain')
#print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
c = np.asarray([c[i,0] for i in range(c.shape[0])])
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
x = sym.Symbol('x')
f = sym.lambdify([x], f, modules='numpy')
if collocation and not symbolic:
print 'Plain interpolation/collocation:'
# Should use vertices, but compute all nodes!
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(c, vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def approximate(f, d, N_e, numint, Omega=[0,1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a Python functionn f on a domain
Omega. N_e is the number of elements.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
from math import sqrt
numint_name = numint # save name
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104,
0.86113631],
[ 0.34785485, 0.65214515, 0.65214515,
0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. ,
0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889,
0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
sys.exit(1)
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f,
numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
c = np.linalg.solve(A, b)
print 'c:\n', c
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(np.asarray(c), vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def f(u, t):
return -a * u
def exact_solution(t):
return I * np.exp(-a * t)
I = 1
a = 2
T = 5
tol = float(sys.argv[1])
solver = odespy.DormandPrince(f, atol=tol, rtol=0.1 * tol)
N = 1 # just one step - let the scheme find its intermediate points
t_mesh = np.linspace(0, T, N + 1)
t_fine = np.linspace(0, T, 10001)
solver.set_initial_condition(I)
u, t = solver.solve(t_mesh)
# u and t will only consist of [I, u^N] and [0,T]
# solver.u_all and solver.t_all contains all computed points
plt.plot(solver.t_all, solver.u_all, 'ko')
plt.hold('on')
plt.plot(t_fine, exact_solution(t_fine), 'b-')
plt.legend(['tol=%.0E' % tol, 'exact'])
plt.savefig('tmp_odespy_adaptive.png')
plt.show()
def estimate(truncation_error, T, N_0, m, makeplot=True):
"""
Compute the truncation error in a problem with one independent
variable, using m meshes, and estimate the convergence
rate of the truncation error.
The user-supplied function truncation_error(dt, N) computes
the truncation error on a uniform mesh with N intervals of
length dt::
R, t, R_a = truncation_error(dt, N)
where R holds the truncation error at points in the array t,
and R_a are the corresponding theoretical truncation error
values (None if not available).
The truncation_error function is run on a series of meshes
with 2**i*N_0 intervals, i=0,1,...,m-1.
The values of R and R_a are restricted to the coarsest mesh.
and based on these data, the convergence rate of R (pointwise)
and time-integrated R can be estimated empirically.
"""
N = [2**i*N_0 for i in range(m)]
R_I = np.zeros(m) # time-integrated R values on various meshes
R = [None]*m # time series of R restricted to coarsest mesh
R_a = [None]*m # time series of R_a restricted to coarsest mesh
dt = np.zeros(m)
legends_R = []; legends_R_a = [] # all legends of curves
for i in range(m):
dt[i] = T/float(N[i])
R[i], t, R_a[i] = truncation_error(dt[i], N[i])
R_I[i] = np.sqrt(dt[i]*np.sum(R[i]**2))
if i == 0:
t_coarse = t # the coarsest mesh
stride = N[i]/N_0
R[i] = R[i][::stride] # restrict to coarsest mesh
R_a[i] = R_a[i][::stride]
if makeplot:
plt.figure(1)
plt.plot(t_coarse, R[i], log='y')
legends_R.append('N=%d' % N[i])
plt.hold('on')
plt.figure(2)
plt.plot(t_coarse, R_a[i] - R[i], log='y')
plt.hold('on')
legends_R_a.append('N=%d' % N[i])
if makeplot:
plt.figure(1)
plt.xlabel('time')
plt.ylabel('pointwise truncation error')
plt.legend(legends_R)
plt.savefig('R_series.png')
plt.savefig('R_series.pdf')
plt.figure(2)
plt.xlabel('time')
plt.ylabel('pointwise error in estimated truncation error')
plt.legend(legends_R_a)
plt.savefig('R_error.png')
plt.savefig('R_error.pdf')
# Convergence rates
r_R_I = convergence_rates(dt, R_I)
print 'R integrated in time; r:',
print ' '.join(['%.1f' % r for r in r_R_I])
R = np.array(R) # two-dim. numpy array
r_R = [convergence_rates(dt, R[:,n])[-1]
for n in range(len(t_coarse))]
# Plot convergence rates
if makeplot:
plt.figure()
plt.plot(t_coarse, r_R)
plt.xlabel('time')
plt.ylabel('r')
plt.axis([t_coarse[0], t_coarse[-1], 0, 2.5])
plt.title('Pointwise rate $r$ in truncation error $\sim\Delta t^r$')
plt.savefig('R_rate_series.png')
plt.savefig('R_rate_series.pdf')
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
if scaling == 1:
v = gamma
a = 1
elif scaling == 2:
v = 1
a = 1.0/gamma
b = 0.5*beta**2
L = 1.0
ymin = 0
# Need gloal to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta*np.exp(-b*(x - v*t)**2)
import time
import scitools.std as plt
plot_arrays = []
def process_u(u, x, t, n):
global ymax
if animate:
plt.plot(x, u, 'r-',
x, f(x, t[n])/delta, 'b-',
axis=[0, L, ymin, ymax], title='t=%f' % t[n],
xlabel='x', ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt/10.0
if abs(t[n] - 0.2) < tol or abs(t[n] - 0.5) < tol:
plot_arrays.append((u.copy(), f(x, t[n])/delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(
I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x, u, 'r-', x, f, 'b--', axis=[x[0], x[-1], 0, ymax],
xlabel='$x$', ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend(['$u,\\ t=0.2$', '$f/%g,\\ t=0.2$' % delta,
'$u,\\ t=0.5$', '$f/%g,\\ t=0.5$' % delta])
filename = 'tmp1_gamma%g_s%d' % (gamma, scaling)
s = 'diffusion' if scaling == 1 else 'source'
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma)
+ 'scaling=%s' % s)
plt.savefig(filename + '.pdf'); plt.savefig(filename + '.png')
return cpu
def approximate(f, symbolic=False, d=1, N_e=4, numint=None,
Omega=[0, 1], filename='tmp'):
if symbolic:
if numint == 'Trapezoidal':
numint = [[sp.S(-1), sp.S(1)], [sp.S(1), sp.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sp.S(-1), sp.S(0), sp.S(1)],
[sp.Rational(1,3), sp.Rational(4,3), sp.Rational(1,3)]]
elif numint == 'Midpoint':
numint = [[sp.S(0)], [sp.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1/sp.sqrt(3), 1/sp.sqrt(3)], [sp.S(1), sp.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[-sp.sqrt(sp.Rational(3,5)), 0,
sp.sqrt(sp.Rational(3,5))],
[sp.Rational(5,9), sp.Rational(8,9),
sp.Rational(5,9)]]
elif numint is not None:
print 'Numerical rule %s is not supported' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint is not None:
print 'Numerical rule %s is not supported' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
print 'phi basis (reference element):\n', phi
A, b = assemble(vertices, cells, dof_map, phi, f,
symbolic=symbolic, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sp.latex(A, mode='plain')
#print sp.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
if not symbolic:
print 'Plain interpolation/collocation:'
x = sp.Symbol('x')
f = sp.lambdify([x], f, modules='numpy')
try:
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
except Exception as e:
print 'could not evaluate f numerically:'
print e
# else: nodes are symbolic so f(nodes[i]) only makes sense
# in the non-symbolic case
if not symbolic and filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint
x_u, u = u_glob(np.asarray(c), vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def run_solvers_and_plot(solvers, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
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)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Compute energy
dt = t[1] - t[0]
E = 0.5*((u[2:,0] - u[:-2,0])/(2*dt))**2 + 0.5*w**2*u[1:-1,0]**2
# Compute error in energy
E0 = 0.5*0**2 + 0.5*w**2*I**2
e_E = E - E0
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
print '*** Relative max error in energy for %s [0,%g] with dt=%g: %.3E' % (solver_name, t[-1], dt, np.abs(e_E).max()/E0)
# Make plots
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.__class__.__name__)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:,0], u[:,1]) # markers by default
else:
plt.plot(u[:,0], u[:,1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:,0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2*np.pi/p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I*np.cos(w*t_fine)
v_e = -w*I*np.sin(w*t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
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))
plt.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)'); plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.eps' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vib_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_p.eps' % (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.eps' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_a.eps' % (timesteps_per_period, num_periods))
t_mesh = np.linspace(0, T, Nt + 1)
t_fine = np.linspace(0, T, 10001)
solver.set_initial_condition(u0)
u, t = solver.solve(t_mesh)
# u and t will only consist of [I, u^Nt] and [0,T], i.e. 2 values
# each, while solver.u_all and solver.t_all contain all computed
# points. solver.u_all is a list with arrays, one array (with 2
# values) for each point in time.
u_adaptive = np.array(solver.u_all)
# For comparison, we solve also with simple FDM method
import sys, os
sys.path.insert(0, os.path.join(os.pardir, 'src-vib'))
from vib_undamped import solver as simple_solver
Nt_simple = len(solver.t_all)
dt = float(T) / Nt_simple
u_simple, t_simple = simple_solver(I, w, dt, T)
# Compare in plot: adaptive, constant dt, exact
plt.plot(solver.t_all, u_adaptive[:, 0], 'k-')
plt.hold('on')
plt.plot(t_simple, u_simple, 'r--')
plt.plot(t_fine, u_exact(t_fine), 'b-')
plt.legend(['tol=%.0E' % tol, 'u simple', 'exact'])
plt.savefig('tmp_odespy_adaptive.png')
plt.savefig('tmp_odespy_adaptive.pdf')
plt.show()
raw_input()
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
if __name__ == '__main__':
alpha_values = [0, 0.05, 0.1]
for alpha in alpha_values:
u, t = simulate(alpha, 0, 6, 60)
plt.plot(t, u)
plt.hold('on')
plt.legend([r'$\alpha=%g$' % alpha for alpha in alpha_values])
plt.xlabel(r'$\bar t$')
plt.ylabel(r'$\bar u$')
plt.savefig('tmp.png')
plt.savefig('tmp.pdf')
plt.show()
raw_input() # for scitools' matplotlib engine
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
gamma = float(gamma) # avoid integer division
if scaling == 'a':
v = gamma
a = 1
L = 1.0
b = 0.5 * beta**2
elif scaling == 'b':
v = 1
a = 1.0 / gamma
L = 1.0
b = 0.5 * beta**2
elif scaling == 'c':
v = 1
a = beta / gamma
L = beta
b = 0.5
elif scaling == 'd':
# PDE: u_t = gamma**(-1)u_xx + gamma**(-1)*delta*f
v = 1
a = 1.0 / gamma
L = 1.0
b = 0.5 * beta**2
delta *= 1.0 / gamma
ymin = 0
# Need global ymax to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta * np.exp(-b * (x - v * t)**2)
import time
import scitools.std as plt
plot_arrays = []
if scaling == 'c':
plot_times = [0.2 * beta, 0.5 * beta]
else:
plot_times = [0.2, 0.5]
def process_u(u, x, t, n):
"""
Animate u, and store arrays in plot_arrays if
t coincides with chosen times for plotting (plot_times).
"""
global ymax
if animate:
plt.plot(x,
u,
'r-',
x,
f(x, t[n]) / delta,
'b-',
axis=[0, L, ymin, ymax],
title='t=%f' % t[n],
xlabel='x',
ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt / 10.0
if abs(t[n] - plot_times[0]) < tol or \
abs(t[n] - plot_times[1]) < tol:
plot_arrays.append((u.copy(), f(x, t[n]) / delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
if scaling == 'c':
T = 0.5 * beta
else:
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x,
u,
'r-',
x,
f,
'b--',
axis=[x[0], x[-1], 0, ymax],
xlabel='$x$',
ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend([
'$u,\\ t=%g$' % plot_times[0],
'$f/%g,\\ t=%g$' % (delta, plot_times[0]),
'$u,\\ t=%g$' % plot_times[1],
'$f/%g,\\ t=%g$' % (delta, plot_times[1])
])
filename = 'tmp1_gamma%g_%s' % (gamma, scaling)
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma) +
'scaling=%s' % scaling)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return cpu
T = num_periods * P
t = np.linspace(0, T, time_steps_per_period * num_periods + 1)
import odespy
def f(u, t, alpha):
# Note the sequence of unknowns: v, u (v=du/dt)
v, u = u
return [-alpha * np.sign(v) - u, v]
solver = odespy.RK4(f, f_args=[alpha])
solver.set_initial_condition([beta, 1]) # sequence must match f
uv, t = solver.solve(t)
u = uv[:, 1] # recall sequence in f: v, u
v = uv[:, 0]
return u, t
if __name__ == "__main__":
alpha_values = [0, 0.05, 0.1]
for alpha in alpha_values:
u, t = simulate(alpha, 0, 6, 60)
plt.plot(t, u)
plt.hold("on")
plt.legend([r"$\alpha=%g$" % alpha for alpha in alpha_values])
plt.xlabel(r"$\bar t$")
plt.ylabel(r"$\bar u$")
plt.savefig("tmp.png")
plt.savefig("tmp.pdf")
plt.show()
raw_input() # for scitools' matplotlib engine
import sys
#import matplotlib.pyplot as plt
import scitools.std as plt
def f(u, t):
return -a*u
def u_exact(t):
return I*np.exp(-a*t)
I = 1; a = 2; T = 5
tol = float(sys.argv[1])
solver = odespy.DormandPrince(f, atol=tol, rtol=0.1*tol)
Nt = 1 # just one step - let the scheme find its intermediate points
t_mesh = np.linspace(0, T, Nt+1)
t_fine = np.linspace(0, T, 10001)
solver.set_initial_condition(I)
u, t = solver.solve(t_mesh)
# u and t will only consist of [I, u^Nt] and [0,T]
# solver.u_all and solver.t_all contains all computed points
plt.plot(solver.t_all, solver.u_all, 'ko')
plt.hold('on')
plt.plot(t_fine, u_exact(t_fine), 'b-')
plt.legend(['tol=%.0E' % tol, 'exact'])
plt.savefig('tmp_odespy_adaptive.png')
plt.show()
def approximate(f, d, N_e, numint, Omega=[0, 1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a Python functionn f on a domain
Omega. N_e is the number of elements.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
from math import sqrt
numint_name = numint # save name
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1. / 3, 4. / 3, 1. / 3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1 / sqrt(3), 1 / sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3. / 5), 0, sqrt(3. / 5)], [5. / 9, 8. / 9, 5. / 9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0., 0.53846931, 0.90617985],
[0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported '\
'for numerical computing' % numint
sys.exit(1)
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e]) - 1) for e in range(N_e)]
A, b = assemble(vertices, cells, dof_map, phi, f, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
c = np.linalg.solve(A, b)
print 'c:\n', c
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
title += ', integration: %s' % numint_name
x_u, u, _ = u_glob(np.asarray(c),
vertices,
cells,
dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-', x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
gamma = float(gamma) # avoid integer division
if scaling == 'a':
v = gamma
a = 1
L = 1.0
b = 0.5*beta**2
elif scaling == 'b':
v = 1
a = 1.0/gamma
L = 1.0
b = 0.5*beta**2
elif scaling == 'c':
v = 1
a = beta/gamma
L = beta
b = 0.5
elif scaling == 'd':
# PDE: u_t = gamma**(-1)u_xx + gamma**(-1)*delta*f
v = 1
a = 1.0/gamma
L = 1.0
b = 0.5*beta**2
delta *= 1.0/gamma
ymin = 0
# Need global ymax to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta*np.exp(-b*(x - v*t)**2)
import time
import scitools.std as plt
plot_arrays = []
if scaling == 'c':
plot_times = [0.2*beta, 0.5*beta]
else:
plot_times = [0.2, 0.5]
def process_u(u, x, t, n):
"""
Animate u, and store arrays in plot_arrays if
t coincides with chosen times for plotting (plot_times).
"""
global ymax
if animate:
plt.plot(x, u, 'r-',
x, f(x, t[n])/delta, 'b-',
axis=[0, L, ymin, ymax], title='t=%f' % t[n],
xlabel='x', ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt/10.0
if abs(t[n] - plot_times[0]) < tol or \
abs(t[n] - plot_times[1]) < tol:
plot_arrays.append((u.copy(), f(x, t[n])/delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
if scaling == 'c':
T = 0.5*beta
else:
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(
I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x, u, 'r-', x, f, 'b--', axis=[x[0], x[-1], 0, ymax],
xlabel='$x$', ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend(['$u,\\ t=%g$' % plot_times[0],
'$f/%g,\\ t=%g$' % (delta, plot_times[0]),
'$u,\\ t=%g$' % plot_times[1],
'$f/%g,\\ t=%g$' % (delta, plot_times[1])])
filename = 'tmp1_gamma%g_%s' % (gamma, scaling)
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma)
+ 'scaling=%s' % scaling)
plt.savefig(filename + '.pdf'); plt.savefig(filename + '.png')
return cpu
solver.set_initial_condition(u0)
u, t = solver.solve(t_mesh)
# u and t will only consist of [I, u^Nt] and [0,T], i.e. 2 values
# each, while solver.u_all and solver.t_all contain all computed
# points. solver.u_all is a list with arrays, one array (with 2
# values) for each point in time.
u_adaptive = np.array(solver.u_all)
# For comparison, we solve also with simple FDM method
Nt_simple = len(solver.t_all) # get no of time steps from ad. meth.
t_simple = np.linspace(0, T, Nt_simple+1) # create equal steps
dt = t_simple[1] - t_simple[0] # constant time step
u_simple = np.zeros(Nt_simple+1)
u_simple[0] = I
u_simple[1] = u_simple[0] - 0.5*dt**2*w**2*u_simple[0]
for n in range(1, Nt_simple):
u_simple[n+1] = 2*u_simple[n] - u_simple[n-1] - \
dt**2*w**2*u_simple[n]
plt.plot(solver.t_all, u_adaptive[:,0], 'k-')
plt.hold('on')
plt.plot(t_simple, u_simple, 'r--')
plt.plot(t_fine, u_exact(t_fine), 'b-')
plt.legend(['tol=%.0E' % tol, 'u simple', 'exact'])
plt.savefig('tmp_odespy_adaptive.png')
plt.savefig('tmp_odespy_adaptive.pdf')
plt.show()
def run_solvers_and_plot(solvers,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
P = 2 * np.pi / w # one period
dt = P / timesteps_per_period
N = num_periods * timesteps_per_period
T = N * dt
t_mesh = np.linspace(0, T, N + 1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Make plots
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.__class__.__name__)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:, 0], u[:, 1]) # markers by default
else:
plt.plot(u[:, 0], u[:, 1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:, 0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2 * np.pi / p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I * np.cos(w * t_fine)
v_e = -w * I * np.sin(w * t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
plt.legend(legends, loc='upper left')
plt.xlabel('t')
plt.ylabel('u')
plt.title('Time step: %g' % dt)
plt.savefig('vb_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.eps' % (timesteps_per_period, num_periods))
plt.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)')
plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vb_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.eps' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.figure(4)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated amplitudes')
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
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
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
t_fine = np.linspace(0, T, 10001)
u_e = I*np.exp(-a*t_fine)
plt.plot(t_fine, u_e, '-') # avoid markers by specifying line type
legends.append('exact')
plt.legend(legends)
plt.title('Time step: %g' % dt)
plt.savefig('odespy1_dt_%g.png' % dt)
plt.savefig('odespy1_dt_%g.pdf' % dt)
plt.show()
def approximate(f, symbolic=False, d=1, N_e=4, numint=None,
Omega=[0, 1], filename='tmp'):
"""
Compute the finite element approximation, using Lagrange
elements of degree d, to a symbolic expression f (with x
as independent variable) on a domain Omega. N_e is the
number of elements.
symbolic=True implies symbolic expressions in the
calculations, while symbolic=False means numerical
computing.
numint is the name of the numerical integration rule
(Trapezoidal, Simpson, GaussLegendre2, GaussLegendre3,
GaussLegendre4, etc.). numint=None implies exact
integration.
"""
numint_name = numint # save name
if symbolic:
if numint == 'Trapezoidal':
numint = [[sym.S(-1), sym.S(1)], [sym.S(1), sym.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sym.S(-1), sym.S(0), sym.S(1)],
[sym.Rational(1,3), sym.Rational(4,3), sym.Rational(1,3)]]
elif numint == 'Midpoint':
numint = [[sym.S(0)], [sym.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1/sym.sqrt(3), 1/sym.sqrt(3)], [sym.S(1), sym.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[-sym.sqrt(sym.Rational(3,5)), 0,
sym.sqrt(sym.Rational(3,5))],
[sym.Rational(5,9), sym.Rational(8,9),
sym.Rational(5,9)]]
elif numint is not None:
print 'Numerical rule %s is not supported for symbolic computing' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1./3, 4./3, 1./3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1/sqrt(3), 1/sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3./5), 0, sqrt(3./5)],
[5./9, 8./9, 5./9]]
elif numint == 'GaussLegendre4':
numint = [[-0.86113631, -0.33998104, 0.33998104, 0.86113631],
[ 0.34785485, 0.65214515, 0.65214515, 0.34785485]]
elif numint == 'GaussLegendre5':
numint = [[-0.90617985, -0.53846931, -0. , 0.53846931, 0.90617985],
[ 0.23692689, 0.47862867, 0.56888889, 0.47862867, 0.23692689]]
elif numint is not None:
print 'Numerical rule %s is not supported for numerical computing' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e])-1) for e in range(N_e)]
print 'phi basis (reference element):\n', phi
A, b = assemble(vertices, cells, dof_map, phi, f,
symbolic=symbolic, numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sym.latex(A, mode='plain')
#print sym.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
if not symbolic:
print 'Plain interpolation/collocation:'
x = sym.Symbol('x')
f = sym.lambdify([x], f, modules='numpy')
try:
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
except Exception as e:
print 'could not evaluate f numerically:'
print e
# else: nodes are symbolic so f(nodes[i]) only makes sense
# in the non-symbolic case
if filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint_name
x_u, u = u_glob(np.asarray(c), vertices, cells, dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
import scitools.std as plt
plt.plot(x_u, u, '-',
x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
import scitools.std as plt
import sys
import numpy as np
import matplotlib.pyplot as plt
x, y = np.loadtxt('volt.txt', delimiter=',', unpack=True)
plt.figure(1)
plt.plot(x, y, '*', linewidth=1) # avoid markers by spec. line type
#plt.xlim([0.0, 10])
#plt.ylim([0.0, 2])
plt.legend(['Force-- Tip'],
loc='upper right',
prop={"family": "Times New Roman"})
plt.xlabel('Sampled time')
plt.ylabel('$Force /m$')
plt.savefig('volt1v.png')
plt.savefig('volt1v.pdf')
plt.hold(True)
def approximate(f, symbolic=False, d=1, N_e=4, numint=None, Omega=[0, 1], filename="tmp"):
if symbolic:
if numint == "Trapezoidal":
numint = [[sm.S(-1), sm.S(1)], [sm.S(1), sm.S(1)]] # sympy integers
elif numint == "Simpson":
numint = [[sm.S(-1), sm.S(0), sm.S(1)], [sm.Rational(1, 3), sm.Rational(4, 3), sm.Rational(1, 3)]]
elif numint == "Midpoint":
numint = [[sm.S(0)], [sm.S(2)]]
elif numint == "GaussLegendre2":
numint = [[-1 / sm.sqrt(3), 1 / sm.sqrt(3)], [sm.S(1), sm.S(1)]]
elif numint == "GaussLegendre3":
numint = [
[-sm.sqrt(sm.Rational(3, 5)), 0, sm.sqrt(sm.Rational(3, 5))],
[sm.Rational(5, 9), sm.Rational(8, 9), sm.Rational(5, 9)],
]
elif numint is not None:
print "Numerical rule %s is not supported" % numint
numint = None
else:
if numint == "Trapezoidal":
numint = [[-1, 1], [1, 1]]
elif numint == "Simpson":
numint = [[-1, 0, 1], [1.0 / 3, 4.0 / 3, 1.0 / 3]]
elif numint == "Midpoint":
numint = [[0], [2]]
elif numint == "GaussLegendre2":
numint = [[-1 / sqrt(3), 1 / sqrt(3)], [1, 1]]
elif numint == "GaussLegendre3":
numint = [[-sqrt(3.0 / 5), 0, sqrt(3.0 / 5)], [5.0 / 9, 8.0 / 9, 5.0 / 9]]
elif numint is not None:
print "Numerical rule %s is not supported" % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e]) - 1) for e in range(N_e)]
print "phi basis (reference element):\n", phi
A, b = assemble(vertices, cells, dof_map, phi, f, symbolic=symbolic, numint=numint)
print "cells:", cells
print "vertices:", vertices
print "dof_map:", dof_map
print "A:\n", A
print "b:\n", b
# print sm.latex(A, mode='plain')
# print sm.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print "c:\n", c
if not symbolic:
print "Plain interpolation/collocation:"
x = sm.Symbol("x")
f = sm.lambdify([x], f, modules="numpy")
try:
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
except Exception as e:
print "could not evaluate f numerically:"
print e
# else: nodes are symbolic so f(nodes[i]) only makes sense
# in the non-symbolic case
if not symbolic and filename is not None:
title = "P%d, N_e=%d" % (d, N_e)
if numint is None:
title += ", exact integration"
else:
title += ", integration: %s" % numint
x_u, u = u_glob(np.asarray(c), vertices, cells, dof_map, resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
plt.plot(x_u, u, "-", x_f, f(x_f), "--")
plt.legend(["u", "f"])
plt.title(title)
plt.savefig(filename + ".pdf")
plt.savefig(filename + ".png")
return c
def run_solvers_and_plot(solvers,
timesteps_per_period=20,
num_periods=1,
I=1,
w=2 * np.pi):
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)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Compute energy
dt = t[1] - t[0]
E = 0.5 * ((u[2:, 0] - u[:-2, 0]) /
(2 * dt))**2 + 0.5 * w**2 * u[1:-1, 0]**2
# Compute error in energy
E0 = 0.5 * 0**2 + 0.5 * w**2 * I**2
e_E = E - E0
solver_name = 'CrankNicolson' if solver.__class__.__name__ == \
'MidpointImplicit' else solver.__class__.__name__
print '*** Relative max error in energy for %s [0,%g] with dt=%g: %.3E' % (
solver_name, t[-1], dt, np.abs(e_E).max() / E0)
# Make plots
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)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:, 0], u[:, 1]) # markers by default
else:
plt.plot(u[:, 0], u[:, 1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:, 0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2 * np.pi / p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I * np.cos(w * t_fine)
v_e = -w * I * np.sin(w * t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
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.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)')
plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vib_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vib_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vib_%d_%d_p.pdf' % (timesteps_per_period, num_periods))
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))
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
if __name__ == '__main__':
Theta_values_degrees = [1, 20, 45, 60]
for Theta_degrees in Theta_values_degrees:
Theta = Theta_degrees * np.pi / 180
theta, t = simulate(Theta, 6, 60)
plt.plot(t, theta)
plt.hold('on')
plt.legend([r'$\Theta=%g$' % Theta for Theta in Theta_values_degrees],
loc='lower left')
plt.xlabel(r'$\bar t$')
plt.ylabel(r'$\bar\theta$')
plt.savefig('tmp.png')
plt.savefig('tmp.pdf')
plt.show()
raw_input()
def approximate(f,
symbolic=False,
d=1,
N_e=4,
numint=None,
Omega=[0, 1],
filename='tmp'):
if symbolic:
if numint == 'Trapezoidal':
numint = [[sm.S(-1), sm.S(1)], [sm.S(1),
sm.S(1)]] # sympy integers
elif numint == 'Simpson':
numint = [[sm.S(-1), sm.S(0), sm.S(1)],
[
sm.Rational(1, 3),
sm.Rational(4, 3),
sm.Rational(1, 3)
]]
elif numint == 'Midpoint':
numint = [[sm.S(0)], [sm.S(2)]]
elif numint == 'GaussLegendre2':
numint = [[-1 / sm.sqrt(3), 1 / sm.sqrt(3)], [sm.S(1), sm.S(1)]]
elif numint == 'GaussLegendre3':
numint = [[
-sm.sqrt(sm.Rational(3, 5)), 0,
sm.sqrt(sm.Rational(3, 5))
], [sm.Rational(5, 9),
sm.Rational(8, 9),
sm.Rational(5, 9)]]
elif numint is not None:
print 'Numerical rule %s is not supported' % numint
numint = None
else:
if numint == 'Trapezoidal':
numint = [[-1, 1], [1, 1]]
elif numint == 'Simpson':
numint = [[-1, 0, 1], [1. / 3, 4. / 3, 1. / 3]]
elif numint == 'Midpoint':
numint = [[0], [2]]
elif numint == 'GaussLegendre2':
numint = [[-1 / sqrt(3), 1 / sqrt(3)], [1, 1]]
elif numint == 'GaussLegendre3':
numint = [[-sqrt(3. / 5), 0, sqrt(3. / 5)],
[5. / 9, 8. / 9, 5. / 9]]
elif numint is not None:
print 'Numerical rule %s is not supported' % numint
numint = None
vertices, cells, dof_map = mesh_uniform(N_e, d, Omega, symbolic)
# phi is a list where phi[e] holds the basis in cell no e
# (this is required by assemble, which can work with
# meshes with different types of elements).
# len(dof_map[e]) is the number of nodes in cell e,
# and the degree of the polynomial is len(dof_map[e])-1
phi = [basis(len(dof_map[e]) - 1) for e in range(N_e)]
print 'phi basis (reference element):\n', phi
A, b = assemble(vertices,
cells,
dof_map,
phi,
f,
symbolic=symbolic,
numint=numint)
print 'cells:', cells
print 'vertices:', vertices
print 'dof_map:', dof_map
print 'A:\n', A
print 'b:\n', b
#print sm.latex(A, mode='plain')
#print sm.latex(b, mode='plain')
if symbolic:
c = A.LUsolve(b)
else:
c = np.linalg.solve(A, b)
print 'c:\n', c
if not symbolic:
print 'Plain interpolation/collocation:'
x = sm.Symbol('x')
f = sm.lambdify([x], f, modules='numpy')
try:
f_at_vertices = [f(xc) for xc in vertices]
print f_at_vertices
except Exception as e:
print 'could not evaluate f numerically:'
print e
# else: nodes are symbolic so f(nodes[i]) only makes sense
# in the non-symbolic case
if not symbolic and filename is not None:
title = 'P%d, N_e=%d' % (d, N_e)
if numint is None:
title += ', exact integration'
else:
title += ', integration: %s' % numint
x_u, u = u_glob(np.asarray(c),
vertices,
cells,
dof_map,
resolution_per_element=51)
x_f = np.linspace(Omega[0], Omega[1], 10001) # mesh for f
plt.plot(x_u, u, '-', x_f, f(x_f), '--')
plt.legend(['u', 'f'])
plt.title(title)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return c
def run(gamma, beta=10, delta=40, scaling=1, animate=False):
"""Run the scaled model for welding."""
if scaling == 1:
v = gamma
a = 1
elif scaling == 2:
v = 1
a = 1.0 / gamma
b = 0.5 * beta**2
L = 1.0
ymin = 0
# Need gloal to be able change ymax in closure process_u
global ymax
ymax = 1.2
I = lambda x: 0
f = lambda x, t: delta * np.exp(-b * (x - v * t)**2)
import time
import scitools.std as plt
plot_arrays = []
def process_u(u, x, t, n):
global ymax
if animate:
plt.plot(x,
u,
'r-',
x,
f(x, t[n]) / delta,
'b-',
axis=[0, L, ymin, ymax],
title='t=%f' % t[n],
xlabel='x',
ylabel='u and f/%g' % delta)
if t[n] == 0:
time.sleep(1)
plot_arrays.append(x)
dt = t[1] - t[0]
tol = dt / 10.0
if abs(t[n] - 0.2) < tol or abs(t[n] - 0.5) < tol:
plot_arrays.append((u.copy(), f(x, t[n]) / delta))
if u.max() > ymax:
ymax = u.max()
Nx = 100
D = 10
T = 0.5
u_L = u_R = 0
theta = 1.0
cpu = solver(I, a, f, L, Nx, D, T, theta, u_L, u_R, user_action=process_u)
x = plot_arrays[0]
plt.figure()
for u, f in plot_arrays[1:]:
plt.plot(x,
u,
'r-',
x,
f,
'b--',
axis=[x[0], x[-1], 0, ymax],
xlabel='$x$',
ylabel=r'$u, \ f/%g$' % delta)
plt.hold('on')
plt.legend([
'$u,\\ t=0.2$',
'$f/%g,\\ t=0.2$' % delta, '$u,\\ t=0.5$',
'$f/%g,\\ t=0.5$' % delta
])
filename = 'tmp1_gamma%g_s%d' % (gamma, scaling)
s = 'diffusion' if scaling == 1 else 'source'
plt.title(r'$\beta = %g,\ \gamma = %g,\ $' % (beta, gamma) +
'scaling=%s' % s)
plt.savefig(filename + '.pdf')
plt.savefig(filename + '.png')
return cpu
def run_solvers_and_plot(solvers, timesteps_per_period=20,
num_periods=1, I=1, w=2*np.pi):
P = 2*np.pi/w # one period
dt = P/timesteps_per_period
N = num_periods*timesteps_per_period
T = N*dt
t_mesh = np.linspace(0, T, N+1)
legends = []
for solver in solvers:
solver.set(f_kwargs={'w': w})
solver.set_initial_condition([I, 0])
u, t = solver.solve(t_mesh)
# Make plots
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.__class__.__name__)
plt.figure(2)
if len(t_mesh) <= 50:
plt.plot(u[:,0], u[:,1]) # markers by default
else:
plt.plot(u[:,0], u[:,1], '-2') # no markers
plt.hold('on')
if num_periods > 20:
minima, maxima = minmax(t, u[:,0])
p = periods(maxima)
a = amplitudes(minima, maxima)
plt.figure(3)
plt.plot(range(len(p)), 2*np.pi/p, '-')
plt.hold('on')
plt.figure(4)
plt.plot(range(len(a)), a, '-')
plt.hold('on')
# Compare with exact solution plotted on a very fine mesh
t_fine = np.linspace(0, T, 10001)
u_e = I*np.cos(w*t_fine)
v_e = -w*I*np.sin(w*t_fine)
if num_periods < 80:
plt.figure(1)
plt.plot(t_fine, u_e, '-') # avoid markers by spec. line type
legends.append('exact')
plt.legend(legends, loc='upper left')
plt.xlabel('t'); plt.ylabel('u')
plt.title('Time step: %g' % dt)
plt.savefig('vb_%d_%d_u.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_u.eps' % (timesteps_per_period, num_periods))
plt.figure(2)
plt.plot(u_e, v_e, '-') # avoid markers by spec. line type
plt.legend(legends, loc='lower right')
plt.xlabel('u(t)'); plt.ylabel('v(t)')
plt.title('Time step: %g' % dt)
plt.savefig('vb_%d_%d_pp.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.pdf' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_pp.eps' % (timesteps_per_period, num_periods))
del legends[-1] # fig 3 and 4 does not have exact value
if num_periods > 20:
plt.figure(3)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated periods')
plt.savefig('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_p.eps' % (timesteps_per_period, num_periods))
plt.figure(4)
plt.legend(legends, loc='center right')
plt.title('Empirically estimated amplitudes')
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.png' % (timesteps_per_period, num_periods))
plt.savefig('vb_%d_%d_a.eps' % (timesteps_per_period, num_periods))
