if PDE.Dirichlet:
U = PDE.unknown_dirichlet
else:
U = PDE.unknown
# ...
from pigasus.fem.utils import function
# ...
def func(U, x, y):
_U = U.evaluate()
return [4. * exp(_U)]
# ...
F = function(func, fields=[U])
list_L2, list_H1 = PDE.solve(F,
u0=None,
maxiter=50,
rtol=1.e-6,
verbose=True)
print("norm using Picard ", PDE.norm(exact=u_exact))
# ...
if PLOT:
fig = plt.figure()
plt.subplot(121, aspect='equal')
U.fast_plot()
print ">>> Solving using Picard <<<"
# ...
if PDE.Dirichlet:
U = PDE.unknown_dirichlet
else:
U = PDE.unknown
# ...
from pigasus.fem.utils import function
# ...
def func(U,x,y):
_U = U.evaluate()
return [4. * exp (_U)]
# ...
F = function(func, fields=[U])
list_L2, list_H1 = PDE.solve(F, u0=None, maxiter=50, rtol=1.e-6,
verbose=True)
print "norm using Picard ", PDE.norm(exact=u_exact)
# ...
if PLOT:
fig = plt.figure()
plt.subplot(121, aspect='equal')
U.fast_plot() ; plt.colorbar(orientation='horizontal') ; plt.title('$u_h$')
# plot error evolution
plt.subplot(122)
def solve(self, rho0, rho1, c_rho=None, u0=None, maxiter=100, rtol=1.e-6, rtol2=1.e-6 \
, verbose=False, update=False):
"""
solves the nonlinear poisson equation using PIcard algorithm
rho0:
the initial density
rho1:
the new density
u0:
this is the initial value for u. Default: all B-splines coeff = 0
maxiter:
the maximum number of iterations for the Picard algorithm. Default 100
rtol:
the relative tolerance. Default 1.e-6
verbose:
True => print the error for each iteration
Returns:
The residual error (as a numpy array)
"""
PDE = self
V = self.space
# print ">> solve-Monge Ampere"
# ... compute the ratio int rho1 / int rho0
if c_rho is None:
# assembly the stifness matrix and bc terms
# print "Entering assembly"
PDE_picard.assembly(self, update=update)
# print "Leaving assembly"
if self.Dirichlet:
U = self.unknown_dirichlet
else:
U = self.unknown
U.reset()
u_sqr = lambda x,y : [sqrt(rho0(x,y))]
C0 = 1./PDE.norm(exact=u_sqr)**2
u_sqr = lambda x,y : [sqrt(rho1(x,y))]
C1 = 1./PDE.norm(exact=u_sqr)**2
c_rho = C0/C1
# ...
self.rho0 = rho0
self.rho1 = rho1
self.c_rho = c_rho
# ...
from pigasus.fem.utils import function
def _F (U,x,y):
D = U.evaluate(nderiv=2, parametric=False)
_U = D[0,0,:]
Udx = D[0,1,:]
Udy = D[0,2,:]
Udxx = D[0,3,:]
Udxy = D[0,4,:]
Udyy = D[0,5,:]
f_values = c_rho * rho0(x,y) / rho1 (Udx,Udy)
return [-sqrt ( Udxx**2 + Udyy**2 + 2 * Udxy**2 + 2 * f_values)]
# ...
func_F = function(_F, fields=[U])
# # ...
# def F(U,x,y):
#
# # ...
# D = U.evaluate(nderiv=2, parametric=False)
#
# _U = D[0,0,:]
# Udx = D[0,1,:]
# Udy = D[0,2,:]
# Udxx = D[0,3,:]
# Udxy = D[0,4,:]
# Udyy = D[0,5,:]
#
# f_values = c_rho * rho0(x,y) / rho1 (Udx,Udy)
# _F = - np.sqrt ( Udxx**2 + Udyy**2 + 2 * Udxy**2 + 2 * f_values )
#
# return [_F]
# # ...
return PDE_picard.solve( self, func_F \
, u0=u0, maxiter=maxiter, rtol=rtol, rtol2=rtol2 \
, verbose=verbose, update=update)
def create_function(U):
return function(func_field, fields=[U])