def SecondPiolaStress(self, F, p=None, deviatoric=False):
import dolfin
from pulse import kinematics
material = self.material
I = kinematics.SecondOrderIdentity(F)
f0 = material.f0
f0f0 = dolfin.outer(f0, f0)
I1 = dolfin.variable(material.active.I1(F))
I4f = dolfin.variable(material.active.I4(F))
Fe = material.active.Fe(F)
Fa = material.active.Fa
Ce = Fe.T * Fe
# fe = Fe*f0
# fefe = dolfin.outer(fe, fe)
# Elastic volume ratio
J = dolfin.variable(dolfin.det(Fe))
# Active volume ration
Ja = dolfin.det(Fa)
dim = self.geometry.dim()
Ce_bar = pow(J, -2.0 / float(dim)) * Ce
w1 = material.W_1(I1, diff=1, dim=dim)
w4f = material.W_4(I4f, diff=1)
# Total Stress
S_bar = Ja * (2 * w1 * I + 2 * w4f * f0f0) * dolfin.inv(Fa).T
if material.is_isochoric:
# Deviatoric
Dev_S_bar = S_bar - (1.0 / 3.0) * dolfin.inner(
S_bar, Ce_bar) * dolfin.inv(Ce_bar)
S_mat = J**(-2.0 / 3.0) * Dev_S_bar
else:
S_mat = S_bar
# Volumetric
if p is None or deviatoric:
S_vol = dolfin.zero((dim, dim))
else:
psi_vol = material.compressibility(p, J)
S_vol = J * dolfin.diff(psi_vol, J) * dolfin.inv(Ce)
# Active stress
wactive = material.active.Wactive(F, diff=1)
eta = material.active.eta
S_active = wactive * (f0f0 + eta * (I - f0f0))
S = S_mat + S_vol + S_active
return S
def _build_residuals(V, dx, phi, omega, Mu, Sigma, convections, voltages):
#class OuterBoundary(SubDomain):
# def inside(self, x, on_boundary):
# return on_boundary and abs(x[0]) > DOLFIN_EPS
#boundaries = FacetFunction('size_t', mesh)
#boundaries.set_all(0)
#outer_boundary = OuterBoundary()
#outer_boundary.mark(boundaries, 1)
#ds = Measure('ds')[boundaries]
r = Expression('x[0]', degree=1, domain=V.mesh())
subdomain_indices = Mu.keys()
#u = TrialFunction(V)
v = TestFunction(V)
r_r = zero() * dx(0)
for i in subdomain_indices:
r_r += 1.0 / (Mu[i] * r) * dot(grad(r * phi[0]), grad(r * v)) * 2 * pi * dx(i) \
- omega * Sigma[i] * phi[1] * v * 2 * pi * r * dx(i)
# convections
for i, conv in convections.items():
r_r += dot(conv, grad(r * phi[0])) * v * 2 * pi * dx(i)
# rhs
for i, voltage in voltages.items():
r_r -= Sigma[i] * voltage.real * v * dx(i)
## boundaries
#r_r += 1.0/Mu[i] * phi[0] * v * 2*pi*ds(1)
# imaginary part
r_i = zero() * dx(0)
for i in subdomain_indices:
r_i += 1.0 / (Mu[i] * r) * dot(grad(r * phi[1]), grad(r * v)) * 2 * pi * dx(i) \
+ omega * Sigma[i] * phi[0] * v * 2 * pi * r * dx(i)
# convections
for i, conv in convections.items():
r_i += dot(conv, grad(r * phi[1])) * v * 2 * pi * dx(i)
# rhs
for i, voltage in voltages.items():
r_r -= Sigma[i] * voltage.imag * v * dx(i)
## boundaries
#r_i += 1.0/Mu[i] * phi[1] * v * 2*pi*ds(1)
return r_r, r_i
def _residual_strong(dx, v, phi, mu, sigma, omega, conv, voltages):
'''Get the residual in strong form, projected onto V.
'''
r = Expression('x[0]', degree=1, cell=triangle)
R = [zero() * dx(0),
zero() * dx(0)]
subdomain_indices = mu.keys()
for i in subdomain_indices:
# diffusion, reaction
R_r = - div(1 / (mu[i] * r) * grad(r * phi[0])) \
- sigma[i] * omega * phi[1]
R_i = - div(1 / (mu[i] * r) * grad(r * phi[1])) \
+ sigma[i] * omega * phi[0]
# convection
if i in conv:
R_r += dot(conv[i], 1 / r * grad(r * phi[0]))
R_i += dot(conv[i], 1 / r * grad(r * phi[1]))
# right-hand side
if i in voltages:
R_r -= sigma[i] * voltages[i].real / (2 * pi * r)
R_i -= sigma[i] * voltages[i].imag / (2 * pi * r)
R[0] += R_r * v * dx(i)
R[1] += R_i * v * dx(i)
return R
def form_rhs(self, state, w):
"""
Returns the right-hand-side contribution of the effective field
for Alouges-type integration schemes.
*Arguments*
state (:class:`State`)
The simulation state.
w (:class:`dolfin.TestFunction`)
The test function used in the Alouges integrator.
*Returns*
:class:`dolfin.Form`
the form contribution for the RHS
"""
return zero()
def form_rhs(self, state, w):
"""
Returns the right-hand-side contribution of the effective field
for Alouges-type integration schemes.
*Arguments*
state (:class:`State`)
The simulation state.
w (:class:`dolfin.TestFunction`)
The test function used in the Alouges integrator.
*Returns*
:class:`dolfin.Form`
the form contribution for the RHS
"""
return zero()
def form_lhs(self, state, w, dt_v):
"""
Returns the left-hand-side contribution of the effective field
for Alouges-type integration schemes.
*Arguments*
state (:class:`State`)
The simulation state.
w (:class:`dolfin.TestFunction`)
The test function used in the Alouges integrator.
dt_v (:class:`dolfin.TrialFunction`)
The trial function multiplied with the timestep and theta.
*Returns*
:class:`dolfin.Form`
the form contribution for the LHS
"""
return zero()
def form_lhs(self, state, w, dt_v):
"""
Returns the left-hand-side contribution of the effective field
for Alouges-type integration schemes.
*Arguments*
state (:class:`State`)
The simulation state.
w (:class:`dolfin.TestFunction`)
The test function used in the Alouges integrator.
dt_v (:class:`dolfin.TrialFunction`)
The trial function multiplied with the timestep and theta.
*Returns*
:class:`dolfin.Form`
the form contribution for the LHS
"""
return zero()
def _error_estimator(dx, phi, mu, sigma, omega, conv, voltages):
'''Simple error estimator from
A posteriori error estimation and adaptive mesh-refinement techniques;
R. Verfürth;
Journal of Computational and Applied Mathematics;
Volume 50, Issues 1-3, 20 May 1994, Pages 67-83;
<https://www.sciencedirect.com/science/article/pii/0377042794902909>.
The strong PDE is
- div(1/(mu r) grad(rphi)) + <u, 1/r grad(rphi)> + i sigma omega phi
= sigma v_k / (2 pi r).
'''
from dolfin import cells
mesh = phi.function_space().mesh()
# Assemble the cell-wise residual in DG space
DG = FunctionSpace(mesh, 'DG', 0)
# get residual in DG
v = TestFunction(DG)
R = _residual_strong(dx, v, phi, mu, sigma, omega, conv, voltages)
r_r = assemble(R[0])
r_i = assemble(R[1])
r = r_r * r_r + r_i * r_i
visualize = True
if visualize:
# Plot the cell-wise residual
u = TrialFunction(DG)
a = zero() * dx(0)
subdomain_indices = mu.keys()
for i in subdomain_indices:
a += u * v * dx(i)
A = assemble(a)
R2 = Function(DG)
solve(A, R2.vector(), r)
plot(R2, title='||R||^2')
interactive()
K = r.array()
info('%r' % K)
h = numpy.array([c.diameter() for c in cells(mesh)])
eta = h * numpy.sqrt(K)
return eta
def solve_maxwell(V, dx,
Mu, Sigma, # dictionaries
omega,
f_list, # list of dictionaries
convections, # dictionary
bcs=None,
tol=1.0e-12,
compute_residuals=True,
verbose=False
):
'''Solve the complex-valued time-harmonic Maxwell system in 2D cylindrical
coordinates.
:param V: function space for potentials
:param dx: measure
:param omega: current frequency
:type omega: float
:param f_list: list of right-hand sides
:param convections: convection terms by subdomains
:type convections: dictionary
:param bcs: Dirichlet boundary conditions
:param tol: solver tolerance
:type tol: float
:param verbose: solver verbosity
:type verbose: boolean
:rtype: list of functions
'''
# For the exact solution of the magnetic scalar potential, see
# <http://www.physics.udel.edu/~jim/PHYS809_10F/Class_Notes/Class_26.pdf>.
# Here, the value of \phi along the rotational axis is specified as
#
# phi(z) = 2 pi I / c * (z/|z| - z/sqrt(z^2 + a^2))
#
# where 'a' is the radius of the coil. This expression contradicts what is
# specified by [Chaboudez97]_ who claim that phi=0 is the natural value
# at the symmetry axis.
#
# For more analytic expressions, see
#
# Simple Analytic Expressions for the Magnetic Field of a Circular
# Current Loop;
# James Simpson, John Lane, Christopher Immer, and Robert Youngquist;
# <http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20010038494_2001057024.pdf>.
#
# Check if boundary conditions on phi are explicitly provided.
if not bcs:
# Create Dirichlet boundary conditions.
# In the cylindrically symmetric formulation, the magnetic vector
# potential is given by
#
# A = e^{i omega t} phi(r,z) e_{theta}.
#
# It is natural to demand phi=0 along the symmetry axis r=0 to avoid
# discontinuities there.
# Also, this makes sure that the system is well-defined (see comment
# below).
#
def xzero(x, on_boundary):
return on_boundary and abs(x[0]) < DOLFIN_EPS
bcs = DirichletBC(V * V, (0.0, 0.0), xzero)
#
# Concerning the boundary conditions for the rest of the system:
# At the other boundaries, it is not uncommon (?) to set so-called
# impedance boundary conditions; see, e.g.,
#
# Chaboudez et al.,
# Numerical Modeling in Induction Heating for Axisymmetric
# Geometries,
# IEEE Transactions on Magnetics, vol. 33, no. 1, Jan 1997,
# <http://www.esi-group.com/products/casting/publications/Articles_PDF/InductionaxiIEEE97.pdf>.
#
# or
#
# <ftp://ftp.math.ethz.ch/pub/sam-reports/reports/reports2010/2010-39.pdf>.
#
# TODO review those, references don't seem to be too accurate
# Those translate into Robin-type boundary conditions (and are in fact
# sometimes called that, cf.
# https://en.wikipedia.org/wiki/Robin_boundary_condition).
# The classical reference is
#
# Impedance boundary conditions for imperfectly conducting
# surfaces,
# T.B.A. Senior,
# <http://link.springer.com/content/pdf/10.1007/BF02920074>.
#
#class OuterBoundary(SubDomain):
# def inside(self, x, on_boundary):
# return on_boundary and abs(x[0]) > DOLFIN_EPS
#boundaries = FacetFunction('size_t', mesh)
#boundaries.set_all(0)
#outer_boundary = OuterBoundary()
#outer_boundary.mark(boundaries, 1)
#ds = Measure('ds')[boundaries]
##n = FacetNormal(mesh)
##a += - 1.0/Mu[i] * dot(grad(r*ur), n) * vr * ds(1) \
## - 1.0/Mu[i] * dot(grad(r*ui), n) * vi * ds(1)
##L += - 1.0/Mu[i] * 1.0 * vr * ds(1) \
## - 1.0/Mu[i] * 1.0 * vi * ds(1)
## This is -n.grad(r u) = u:
#a += 1.0/Mu[i] * ur * vr * ds(1) \
# + 1.0/Mu[i] * ui * vi * ds(1)
# Create the system matrix, preconditioner, and the right-hand sides.
# For preconditioners, there are two approaches. The first one, described
# in
#
# Algebraic Multigrid for Complex Symmetric Systems;
# D. Lahaye, H. De Gersem, S. Vandewalle, and K. Hameyer;
# <https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=877730>
#
# doesn't work too well here.
# The matrix P, created in _build_system(), provides a better alternative.
# For more details, see documentation in _build_system().
#
A, P, b_list, M, W = _build_system(V, dx,
Mu, Sigma, # dictionaries
omega,
f_list, # list of dicts
convections, # dict
bcs
)
#from matplotlib import pyplot as pp
#rows, cols, values = M.data()
#from scipy.sparse import csr_matrix
#M_matrix = csr_matrix((values, cols, rows))
##from matplotlib import pyplot as pp
###pp.spy(M_matrix, precision=1e-3, marker='.', markersize=5)
##pp.spy(M_matrix)
##pp.show()
## colormap
#cmap = pp.cm.gray_r
#M_dense = M_matrix.todense()
#from matplotlib.colors import LogNorm
#im = pp.imshow(abs(M_dense), cmap=cmap, interpolation='nearest', norm=LogNorm())
##im = pp.imshow(abs(M_dense), cmap=cmap, interpolation='nearest')
##im = pp.imshow(abs(A_r), cmap=cmap, interpolation='nearest')
##im = pp.imshow(abs(A_i), cmap=cmap, interpolation='nearest')
#pp.colorbar()
#pp.show()
#exit()
#print A
#rows, cols, values = A.data()
#from scipy.sparse import csr_matrix
#A_matrix = csr_matrix((values, cols, rows))
###pp.spy(A_matrix, precision=1e-3, marker='.', markersize=5)
##pp.spy(A_matrix)
##pp.show()
## colormap
#cmap = pp.cm.gray_r
#A_dense = A_matrix.todense()
##A_r = A_dense[0::2][0::2]
##A_i = A_dense[1::2][0::2]
#cmap.set_bad('r')
##im = pp.imshow(abs(A_dense), cmap=cmap, interpolation='nearest', norm=LogNorm())
#im = pp.imshow(abs(A_dense), cmap=cmap, interpolation='nearest')
##im = pp.imshow(abs(A_r), cmap=cmap, interpolation='nearest')
##im = pp.imshow(abs(A_i), cmap=cmap, interpolation='nearest')
#pp.colorbar()
#pp.show()
# prepare solver
solver = KrylovSolver('gmres', 'amg')
solver.set_operators(A, P)
# The PDE for A has huge coefficients (order 10^8) all over. Hence, if
# relative residual is set to 10^-6, the actual residual will still be of
# the order 10^2. While this isn't too bad (after all the equations are
# upscaled by a large factor), one can choose a very small relative
# tolerance here to get a visually pleasing residual norm.
solver.parameters['relative_tolerance'] = 1.0e-12
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['maximum_iterations'] = 100
solver.parameters['report'] = verbose
solver.parameters['monitor_convergence'] = verbose
phi_list = []
for k, b in enumerate(b_list):
with Message('Computing coil ring %d/%d...' % (k + 1, len(b_list))):
# Define goal functional for adaptivity.
# Adaptivity not working for subdomains, cf.
# https://bugs.launchpad.net/dolfin/+bug/872105.
#(phi_r, phi_i) = split(phi)
#M = (phi_r*phi_r + phi_i*phi_i) * dx(2)
phi_list.append(Function(W))
phi_list[-1].rename('phi%d' % k, 'phi%d' % k)
solver.solve(phi_list[-1].vector(), b)
## Adaptive mesh refinement.
#_adaptive_mesh_refinement(dx,
# phi_list[-1],
# Mu, Sigma, omega,
# convections,
# f_list[k]
# )
#exit()
if compute_residuals:
# Sanity check: Compute residuals.
# This is quite the good test that we haven't messed up
# real/imaginary in the above formulation.
r_r, r_i = _build_residuals(V, dx, phi_list[-1],
omega, Mu, Sigma,
convections, voltages
)
def xzero(x, on_boundary):
return on_boundary and abs(x[0]) < DOLFIN_EPS
subdomain_indices = Mu.keys()
# Solve an FEM problem to get the corresponding residual function
# out.
# This is exactly what we need here! :)
u = TrialFunction(V)
v = TestFunction(V)
a = zero() * dx(0)
for i in subdomain_indices:
a += u * v * dx(i)
# TODO don't hard code the boundary conditions like this
R_r = Function(V)
solve(a == r_r, R_r,
bcs=DirichletBC(V, 0.0, xzero)
)
# TODO don't hard code the boundary conditions like this
R_i = Function(V)
solve(a == r_i, R_i,
bcs=DirichletBC(V, 0.0, xzero)
)
nrm_r = norm(R_r)
info('||r_r|| = %e' % nrm_r)
nrm_i = norm(R_i)
info('||r_i|| = %e' % nrm_i)
res_norm = sqrt(nrm_r * nrm_r + nrm_i * nrm_i)
info('||r|| = %e' % res_norm)
plot(R_r, title='R_r')
plot(R_i, title='R_i')
interactive()
#exit()
return phi_list
