def set_fc(self,delta):
""" interpolate the exact solitary wave with offset x0 + delta onto a function in Function Space V
"""
# Interpolate R onto the function
self.setRx(self.x0+delta)
self.fc.interpolate(self.R)
# scale R by h_on_delta
fx = self.fc.vector()
fx *= self.h_on_delta
#get numpy array for radial function and interpolate using sinc
rmesh = fx.array(); # get numpy array
phi = sincinterp_e(self.rr, self.ff - 1.0, rmesh) + 1.0
# reload phi onto function
for i in range(fx.size()):
fx[i] = phi[i]
def set_fc(self, delta):
""" interpolate the exact solitary wave with offset x0 + delta onto a function in Function Space V
"""
# Interpolate R onto the function
self.setRx(self.x0 + delta)
self.fc.interpolate(self.R)
# scale R by h_on_delta
fx = self.fc.vector()
fx *= self.h_on_delta
#get numpy array for radial function and interpolate using sinc
rmesh = fx.array()
# get numpy array
phi = sincinterp_e(self.rr, self.ff - 1.0, rmesh) + 1.0
# reload phi onto function
for i in range(fx.size()):
fx[i] = phi[i]
def ProjectSolitaryWave(V, x0, c, n, m, d, h_on_delta, N=150):
""" Calculate the projection of a Solitary Wave onto a function space V
x0: origin of solitary wave on domain
c: wave speed
n: permeability exponent
m: bulk viscosity exponent
d: wave dimension
h_on_delta: scale length in compaction lengths
N: number of collocation points for sinc method
"""
# initialize Solitary Wave profile
rr, ff = wave_profile(c, n, m, d, N)
# initialize the radial Expression function
dim = V.mesh().geometry().dim()
R = radial_expression(d, dim)
R.x0 = x0[0]
if dim >= 2:
R.x1 = x0[1]
if dim == 3:
R.x2 = x0[2]
# project r onto Function f
f = project(R, V)
# scale R by h_on_delta
fx = f.vector()
fx *= h_on_delta
#get numpy array for radial function and interpolate using sinc
rmesh = fx.array()
# get numpy array
phi = sincinterp_e(rr, ff - 1.0, rmesh) + 1.0
# reload phi onto function
for i in range(fx.size()):
fx[i] = phi[i]
return f
def ProjectSolitaryWave(V,x0,c,n,m,d,h_on_delta,N=150):
""" Calculate the projection of a Solitary Wave onto a function space V
x0: origin of solitary wave on domain
c: wave speed
n: permeability exponent
m: bulk viscosity exponent
d: wave dimension
h_on_delta: scale length in compaction lengths
N: number of collocation points for sinc method
"""
# initialize Solitary Wave profile
rr, ff = wave_profile(c,n,m,d,N)
# initialize the radial Expression function
dim = V.mesh().geometry().dim()
R = radial_expression(d,dim)
R.x0 = x0[0]
if dim >= 2:
R.x1 = x0[1]
if dim == 3:
R.x2 = x0[2]
# project r onto Function f
f = project(R,V)
# scale R by h_on_delta
fx = f.vector()
fx *= h_on_delta
#get numpy array for radial function and interpolate using sinc
rmesh = fx.array(); # get numpy array
phi = sincinterp_e(rr, ff - 1.0, rmesh) + 1.0
# reload phi onto function
for i in range(fx.size()):
fx[i] = phi[i]
return f
def eval(self,r):
""" use sincinterp_e to return porosity at radius r"""
f = sincinterp_e(self.r, self.f - 1.0, r) + 1.0
return f
def solwave_m1(c, n, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the general d-dimensional
solitary wave equation with n > 1, and m=1
Inputs: c - soliton parameter,
n - first nonlinearity,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
# Iterate in soliton parameter, if neccessary, to achieve a good
# guess. This may be replaced with another algorithm if a better
# initial guess for the 1D soliton is available.
if verbose:
print " iterating in c"
# Initial value for delta c. This can be tuned as needed.
delta_c = .5 * n / c
# Loop will terminate if the delta c value gets too small, this may be
# adjusted as needed
delta_c_min = 1.e-6
c0 = n
d0 = 1.
while c0 < c and delta_c > delta_c_min:
c1 = min([c0 + delta_c, c])
decay = sqrt(1. - n / c1)
h = sqrt((pi * dist) / (decay * M))
r1 = sinc_eo.points(M, h)
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c1, 'n': n, 'd': d0,
'D2': D2, 'D1': D1, 'D1_r': D1_r, 'IN': IN1}
try:
# uguess = spline(r0, u0, r1)
uguess = sinc_eo.sincinterp_e(r0,u0,r1)
except:
uguess = 3. * (1. - n / c1) / cosh(.5 * decay * r1)**2
if verbose:
print ' computing with c = ', c1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v,params), uguess, \
# fprime = lambda v:solwave_m1_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-10:
c0 = c1
u0 = u1
r0 = r1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_c"
else:
print " solver failed to converge, adjusting delta_c"
print " solver err = ", ier
delta_c = delta_c / 2.
u = u0
r = r0
# Iterate in dimension, if neccessary
if d > 1.:
if verbose:
print " iterating in d"
delta_dim = (n - 1.) / 10.
# Loop will terminate if the delta dim value gets too
# small, this can be adjusted as desired
delta_dim_min = 1.e-6
d0 = 1.
u0 = u
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v, params), u0, \
# fprime = lambda v:solwave_m1_eq_jac(v, params),
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
def solwave_m1(c, n, d, M, dist = pi / 2., xtol = 1.e-12, verbose = False):
"""
Compute the collocation points and values for the general d-dimensional
solitary wave equation with n > 1, and m=1
Inputs: c - soliton parameter,
n - first nonlinearity,
d - dimension,
M - No. of points,
[dist = pi/2] - distance from the x axis to the nearest
singularity in the complex plane,
[xtol = 1.e-12] - solver tolerance,
[verbose=True/False]
Outputs: r, f - collocation points and the soliton
"""
# Iterate in soliton parameter, if neccessary, to achieve a good
# guess. This may be replaced with another algorithm if a better
# initial guess for the 1D soliton is available.
if verbose:
print " iterating in c"
# Initial value for delta c. This can be tuned as needed.
delta_c = .5 * n / c
# Loop will terminate if the delta c value gets too small, this may be
# adjusted as needed
delta_c_min = 1.e-6
c0 = n
d0 = 1.
while c0 < c and delta_c > delta_c_min:
c1 = min([c0 + delta_c, c])
decay = sqrt(1. - n / c1)
h = sqrt((pi * dist) / (decay * M))
r1 = sinc_eo.points(M, h)
D2 = sinc_eo.D2_e(M, h)
D1 = sinc_eo.D1_eo(M, h)
D1_r = sinc_eo.D1_x_e(M, h)
IN1 = sinc_eo.IN1_oe(M, h)
params = {'c': c1, 'n': n, 'd': d0,
'D2': D2, 'D1': D1, 'D1_r': D1_r, 'IN': IN1}
try:
# uguess = spline(r0, u0, r1)
uguess = sinc_eo.sincinterp_e(r0,u0,r1)
except:
uguess = 3. * (1. - n / c1) / cosh(.5 * decay * r1)**2
if verbose:
print ' computing with c = ', c1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v,params), uguess, \
# fprime = lambda v:solwave_m1_eq_jac(v, params), \
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-10:
c0 = c1
u0 = u1
r0 = r1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_c"
else:
print " solver failed to converge, adjusting delta_c"
print " solver err = ", ier
delta_c = delta_c / 2.
u = u0
r = r0
# Iterate in dimension, if neccessary
if d > 1.:
if verbose:
print " iterating in d"
delta_dim = (n - 1.) / 10.
# Loop will terminate if the delta dim value gets too
# small, this can be adjusted as desired
delta_dim_min = 1.e-6
d0 = 1.
u0 = u
while d0 < d and delta_dim > delta_dim_min:
d1 = min([d0 + delta_dim, d])
params['d'] = d1
if verbose:
print ' computing with d = ', d1
u1, infodict, ier, mesg = \
fsolve(lambda v: solwave_m1_eq(v, params), u0, \
# fprime = lambda v:solwave_m1_eq_jac(v, params),
xtol= xtol,
full_output=1)
if ier == 1 and np.max(np.abs(u1)) > 1.e-8:
d0 = d1
u0 = u1
else:
if verbose:
if np.max(np.abs(u1)) <= 1.e-8:
print " converged to the zero solution, adjusteing delta_dim"
else:
print " solver failed to converge, adjusting delta_dim"
print " solver err = ", ier
delta_dim = delta_dim / 2.
u = u0
if d0 < d:
print " Failed to converge to the soliton solution, last value of d =", d0
f = 0.0
else:
f = 1. + u
return r, f
