def newton(f, x0, maxiter, restol, lintol):
'''find x such that f(x) = 0, starting at x0'''
for i in log.range('newton', maxiter + 1):
if i > 0:
# solve system linearised around `x`, solution in `direction`
J = f(x, jacobian=True)
direction = -J.solve(residual, tol=lintol)
xprev = x
residual_norm_prev = residual_norm
# line search, stop if there is convergence w.r.t. iteration `i-1`
for j in range(4):
x = xprev + (0.5)**j * direction
residual = f(x)
residual_norm = numpy.linalg.norm(residual, 2)
if residual_norm < residual_norm_prev:
break
# else: no convergence, repeat and reduce step size by one half
else:
log.warning('divergence')
else:
# before first iteration, compute initial residual
x = x0
residual = f(x)
residual_norm = numpy.linalg.norm(residual, 2)
log.info('residual norm: {:.5e}'.format(residual_norm))
if residual_norm < restol:
break
else:
raise ValueError(
'newton did not converge in {} iterations'.format(maxiter))
return x
def __init__(self, g, order, f=None, **kwargs):
if order < 6:
log.warning('''
Warning, for this method a Gauss-scheme of at least
order 6 is recommended.
''')
super().__init__(g, order, **kwargs)
if isinstance(f, go.TensorGridObject):
if len(f) != 2:
raise NotImplementedError
if not (f.knotvector <= g.knotvector).all():
log.warning('''
Warning, using a control mapping whose knotvector is
not a subset of the target GridObject knotvector
or that does not have the same periodicity properties
is not recommended, since Gaussian quadrature is
ill-defined in this case.
''')
self._f = f.toscipy()
else:
'''
So far, we only allow for instantiations of the class
``go.TensorGridObject`` as control mapping.
XXX: allow for other means of instantiation, for instance
via a list / dictionary implementing functions for the
control mapping and all its relevant derivatives.
'''
raise NotImplementedError
self._tablulate_control_mapping()
def from_parent(cls, parent, *key):
assert isinstance(parent, cls)
if len(key) == 1:
side, = key
d = parent.__dict__.copy()
d['domain'] = d['domain'].boundary[side]
# flatten appropriate axis
bnames = parent.domain._bnames
sidetovalue = dict(zip(bnames, range(len(bnames))))
deleteindex = sidetovalue[side] // 2
deleteaxis = d['axesnames'][deleteindex]
d['knotvector'] = np.delete(d['knotvector'], deleteindex)
geom = d['geom']
d[ 'geom' ] = function.stack( [ geom[i] for i in range( len( parent ) ) \
if not i == deleteindex ] )
d['axesnames'] = tuple(
filter(lambda x: x != deleteaxis, d['axesnames']))
if len(d['knotvector']) == 0:
log.warning(
'The GridObject has not been implemented with an empty knotvector yet'
)
raise NotImplementedError
ret = cls(**d)
ret.x = parent[side]
for name in ret.domain._bnames:
ret._cons[ret.index.boundary(name).flatten] = ret[name]
return ret
ret = parent
for side in key:
ret = ret.from_parent(ret, side)
return ret
def smoothen_vector(vec0,
dt,
method='finite_difference',
stop={'T': 0.01},
miniter=10):
"""
Smoothen vec[1: -1] using `method' until stopping criterion has been reached,
while vec[0] and vec[-1] are held fixed.
Parameters
----------
vec0 : to-be-smoothened vector
dt : (initial) timestep, is reduced if the convergence criterion is reached
before #iterations >= ``miniter``
method : smoothing method
stop = {
'T': (finite difference) if t > T, terminate
'maxiter': (finite difference) if i > maxiter, terminate
'vec': if not stop[ 'vec' ]( vec ) terminate
}
miniter : minimum number of iterations
"""
t, i = 0, 0
vec = vec0.copy()
d = ChainMap(stop, {'T': np.inf, 'maxiter': 100, 'vec': lambda x: True})
if method == 'finite_difference':
N = len(vec)
dx = 1 / (N - 1)
fac = dt / (dx**2)
A = sparse.diags([(-fac * np.ones(N - 2)).tolist() + [0], [1] +
((1 + 2 * fac) * np.ones(N - 2)).tolist() + [1],
[0] + (-fac * np.ones(N - 2)).tolist()],
[-1, 0, 1]).tocsc()
A = sparse.linalg.splu(A)
while True:
if not all([t < d['T'], i < d['maxiter'], d['vec'](vec)]):
if i <= miniter: # timestep too big
log.info('Initial timestep too big, reducing to {}'.format(
dt / 10))
return smoothen_vector(vec0,
dt / 10,
method=method,
stop=stop)
break
vec = A.solve(vec)
t += dt
i += 1
else:
raise "Unknown method '{}'".format(method)
if d['vec'](vec):
log.warning('Failed to reach the termination criterion')
else:
log.info('Criterion reached at t={} in {} iterations'.format(t, i))
return vec
def __init__(self, g, order, absc=None):
# any problems with ``g`` will be caught by the parent __init__
super().__init__(g, order)
self._dindices = self._g.dofindices
# The Jacobian-Determinant is evaluated in
# the tensor-product points of xi and eta
# as a constraint to warrant bijectivity of the result
if isinstance(absc, int):
# if an integer is passed to absc, we construct
# abscissae that correspond to and ``absc``-order
# Gauss quadrature scheme over all elements
absc = fsol.make_quadrature(g, absc)[0]
self._absc = absc
if absc is not None:
assert all(
all( [aux.isincreasing(x_), x_[0] >= 0, x_[-1] <= 1] )
for x_ in absc ), \
'Invalid constraint abscissae received.'
_bsplev = lambda i, **kwargs: sparse.csr_matrix(
self._splev(i, self._absc, **kwargs).ravel()[:, None])
structure = sparse.hstack([_bsplev(i) for i in range(self._N)])
self._w_xi = sparse.hstack(
[_bsplev(i, dx=1) for i in range(self._N)])
self._w_eta = sparse.hstack(
[_bsplev(i, dy=1) for i in range(self._N)])
# sparsity structure of the constraint jacobian
self.jacobianstructure = \
sparse.hstack( [ structure, structure ] ).\
tolil()[:, self._dindices ].nonzero()
log.info('Constraint jacobian sparsity pattern stored.')
if not (self.constraint(self._g[self._dindices]) >= 0).all():
log.warning('''
Warning, the initial guess corresponding to the passed
GridObject is not feasible. Unless another initial guess
is specified, the optimizer will most likely fail.
''')
def main(
nelems: 'number of elements' = 20,
epsilon: 'epsilon, 0 for automatic (based on nelems)' = 0,
timestep: 'time step' = .01,
maxtime: 'end time' = 1.,
theta: 'contact angle (degrees)' = 90,
init: 'initial condition (random/bubbles)' = 'random',
figures: 'create figures' = True,
):
mineps = 1. / nelems
if not epsilon:
log.info('setting epsilon={}'.format(mineps))
epsilon = mineps
elif epsilon < mineps:
log.warning('epsilon under crititical threshold: {} < {}'.format(
epsilon, mineps))
# create namespace
ns = function.Namespace()
ns.epsilon = epsilon
ns.ewall = .5 * numpy.cos(theta * numpy.pi / 180)
# construct mesh
xnodes = ynodes = numpy.linspace(0, 1, nelems + 1)
domain, ns.x = mesh.rectilinear([xnodes, ynodes])
# prepare bases
ns.cbasis, ns.mubasis = function.chain(
[domain.basis('spline', degree=2),
domain.basis('spline', degree=2)])
# polulate namespace
ns.c = 'cbasis_n ?lhs_n'
ns.c0 = 'cbasis_n ?lhs0_n'
ns.mu = 'mubasis_n ?lhs_n'
ns.f = '(6 c0 - 2 c0^3 - 4 c) / epsilon^2'
# construct initial condition
if init == 'random':
numpy.random.seed(0)
lhs0 = numpy.random.normal(0, .5, ns.cbasis.shape)
elif init == 'bubbles':
R1 = .25
R2 = numpy.sqrt(.5) * R1 # area2 = .5 * area1
ns.cbubble1 = function.tanh(
(R1 - function.norm2(ns.x -
(.5 + R2 / numpy.sqrt(2) + .8 * ns.epsilon)))
/ ns.epsilon)
ns.cbubble2 = function.tanh(
(R2 - function.norm2(ns.x -
(.5 - R1 / numpy.sqrt(2) - .8 * ns.epsilon)))
/ ns.epsilon)
sqr = domain.integral('(c - cbubble1 - cbubble2 - 1)^2 + mu^2' @ ns,
geometry=ns.x,
degree=4)
lhs0 = solver.optimize('lhs', sqr)
else:
raise Exception('unknown init %r' % init)
# construct residual
res = domain.integral(
'epsilon^2 cbasis_n,k mu_,k + mubasis_n (mu + f) - mubasis_n,k c_,k'
@ ns,
geometry=ns.x,
degree=4)
res -= domain.boundary.integral('mubasis_n ewall' @ ns,
geometry=ns.x,
degree=4)
inertia = domain.integral('cbasis_n c' @ ns, geometry=ns.x, degree=4)
# solve time dependent problem
nsteps = numeric.round(maxtime / timestep)
makeplots = MakePlots(domain, nsteps,
ns) if figures else lambda *args: None
for istep, lhs in log.enumerate(
'timestep',
solver.impliciteuler('lhs',
target0='lhs0',
residual=res,
inertia=inertia,
timestep=timestep,
lhs0=lhs0)):
makeplots(lhs)
if istep == nsteps:
break
return lhs0, lhs
def _tablulate_control_mapping(self):
'''
Tabulate the control mapping and all its required derivatives
in the quadrature points.
XXX: see if we can make this look prettier
(possibly using some symbolic approach).
'''
# by the time this is called, _f and _quad have been set
# in the __init__ method
_f = self._f
_q = self._quad
# all required first and second derivatives
X_xi = _f(*_q, dx=1)
X_xi_eta = _f(*_q, dx=1, dy=1)
X_xi_xi = _f(*_q, dx=2)
X_eta = _f(*_q, dy=1)
X_eta_eta = _f(*_q, dy=2)
_0 = (Ellipsis, 0)
_1 = (Ellipsis, 1)
# Jacobian determinant and its derivatives
jacdet = X_xi[_0] * X_eta[_1] - X_eta[_0] * X_xi[_1]
jacsqrt = jacdet**2
jacdet_xi = X_xi_xi[_0] * X_eta[_1] + X_xi[_0] * X_xi_eta[_1] \
- X_xi_eta[_0] * X_xi[_1] - X_eta[_0] * X_xi_xi[_1]
jacdet_eta = X_xi_eta[_0] * X_eta[_1] + X_xi[_0] * X_eta_eta[_1] \
- X_eta_eta[_0] * X_xi[_1] - X_eta[_0] * X_xi_eta[_1]
if not (jacdet > 0).all():
log.warning('''
Warning, the control mapping is defective on at least one
quadrature point. Proceeding with this control mapping may
lead to failure of convergence and / or a defective mapping.
''')
# metric tensor of the control mapping divided by the Jacobian determinant
self._G11 = (X_xi**2).sum(-1) / jacdet
self._G12 = (X_xi * X_eta).sum(-1) / jacdet
self._G22 = (X_eta**2).sum(-1) / jacdet
# derivatives of _G11
self._G11_xi = \
2 * ( X_xi * X_xi_xi ).sum(-1) / jacdet \
- jacdet_xi * (X_xi ** 2).sum(-1) / jacsqrt
self._G11_eta = \
2 * ( X_xi * X_xi_eta ).sum(-1) / jacdet \
- jacdet_eta * (X_xi ** 2).sum(-1) / jacsqrt
# derivatives of _G12
self._G12_xi = \
(X_xi_xi * X_eta + X_xi * X_xi_eta).sum(-1) / jacdet \
- jacdet_xi * (X_xi * X_eta).sum(-1) / jacsqrt
self._G12_eta = \
(X_xi_eta * X_eta + X_xi * X_eta_eta).sum(-1) / jacdet \
- jacdet_eta * (X_xi * X_eta).sum(-1) / jacsqrt
# derivatives of _G22
self._G22_xi = \
2 * ( X_eta * X_xi_eta ).sum(-1) / jacdet \
- jacdet_xi * (X_eta ** 2).sum(-1) / jacsqrt
self._G22_eta = \
2 * ( X_eta * X_eta_eta ).sum(-1) / jacdet \
- jacdet_eta * (X_eta ** 2).sum(-1) / jacsqrt
log.info('The control mapping has been tabulated.')
def inverse(self):
''' Return exact inverse of self '''
log.warning(
'Computing the exact inverse is slow, use approximate_inverse instead'
)
return self.__class__(pynverse.inversefunc(self._f, domain=[0, 1]))
def InterpolatedUnivariateSpline(pc, k=3, c0_thresh=None, **scipyargs):
if isinstance(pc, np.ndarray):
pc = PointCloud(pc)
assert isinstance(pc, PointCloud)
if k > len(pc) + 1:
log.warning(
'Warning, the number of points is too small for a {}-th order interpolations, \
it will be clipped.'.format(k))
k = min(k, len(pc) + 1)
if c0_thresh is None:
periodic = pc.periodic
def f(x):
verts, points = pc.verts, pc.points
if periodic: # repeat points to acquire (nearly) continuous derivative at x = 0
verts = np.concatenate([verts - 1, verts, verts + 1])
points = np.vstack([points] * 3)
splines = (interpolate.InterpolatedUnivariateSpline(verts,
i,
k=k,
**scipyargs)
for i in points.T)
return np.stack([s(x % 1 if periodic else x) for s in splines],
axis=1)
return UnivariateFunction(f, periodic=periodic)
else:
c0_indices = pc.c0_indices(thresh=c0_thresh)
if len(c0_indices) == 0:
return InterpolatedUnivariateSpline(pc,
c0_thresh=None,
k=k,
**scipyargs)
shift_amount = 0
if pc.periodic: # roll pointcloud to first C0-index, then it can be treated as an unperiodic Pointcloud
shift_amount = pc.verts[c0_indices[0]]
pc = pc.roll(-c0_indices[0])
c0_indices = (
c0_indices -
c0_indices[0])[1:] # first index is gonna be added anyways
c0_indices = [0] + c0_indices.tolist() + [
pc._points.shape[0] - 1
] # pc._points because in the periodic case we repeat the last
funcs = [
InterpolatedUnivariateSpline(pc._points[i:j + 1],
c0_thresh=None,
k=k,
**scipyargs)
for i, j in zip(c0_indices[:-1], c0_indices[1:])
]
return UnivariateFunction.stack_multiple(
funcs, pc._verts[c0_indices[1:-1]],
periodic=pc.periodic).shift(shift_amount)
def toPointCloud(self, x, verts=None):
if self._periodic:
if x[-1] != x[0]:
log.warning( 'Warning, a periodic function is evaluated over' + \
' a non-periodic set of points, this is usually a bug' )
return PointCloud(self(x), verts=verts)
def main(
nelems: 'number of elements' = 20,
epsilon: 'epsilon, 0 for automatic (based on nelems)' = 0,
timestep: 'time step' = .01,
maxtime: 'end time' = 1.,
theta: 'contact angle (degrees)' = 90,
init: 'initial condition (random/bubbles)' = 'random',
withplots: 'create plots' = True,
):
mineps = 1. / nelems
if not epsilon:
log.info('setting epsilon=%f' % mineps)
epsilon = mineps
elif epsilon < mineps:
log.warning('epsilon under crititical threshold: %f < %f' %
(epsilon, mineps))
ewall = .5 * numpy.cos(theta * numpy.pi / 180)
# construct mesh
xnodes = ynodes = numpy.linspace(0, 1, nelems + 1)
domain, geom = mesh.rectilinear([xnodes, ynodes])
# prepare bases
cbasis, mubasis = function.chain(
[domain.basis('spline', degree=2),
domain.basis('spline', degree=2)])
# define mixing energy and splitting: F' = f_p - f_n
F = lambda c_: (.5 / epsilon**2) * (c_**2 - 1)**2
f_p = lambda c_: (1. / epsilon**2) * 4 * c_
f_n = lambda c_: (1. / epsilon**2) * (6 * c_ - 2 * c_**3)
# prepare matrix
A = function.outer( cbasis ) \
+ (timestep*epsilon**2) * function.outer( cbasis.grad(geom), mubasis.grad(geom) ).sum(-1) \
+ function.outer( mubasis, mubasis - f_p(cbasis) ) \
- function.outer( mubasis.grad(geom), cbasis.grad(geom) ).sum(-1)
matrix = domain.integrate(A, geometry=geom, ischeme='gauss4')
# prepare wall energy right hand side
rhs0 = domain.boundary.integrate(mubasis * ewall,
geometry=geom,
ischeme='gauss4')
# construct initial condition
if init == 'random':
numpy.random.seed(0)
c = cbasis.dot(numpy.random.normal(0, .5, cbasis.shape))
elif init == 'bubbles':
R1 = .25
R2 = numpy.sqrt(.5) * R1 # area2 = .5 * area1
c = 1 + function.tanh( (R1-function.norm2(geom-(.5+R2/numpy.sqrt(2)+.8*epsilon)))/epsilon ) \
+ function.tanh( (R2-function.norm2(geom-(.5-R1/numpy.sqrt(2)-.8*epsilon)))/epsilon )
else:
raise Exception('unknown init %r' % init)
# prepare plotting
nsteps = numeric.round(maxtime / timestep)
makeplots = MakePlots(domain, geom, nsteps, video=withplots
== 'video') if withplots else lambda *args: None
# start time stepping
for istep in log.range('timestep', nsteps):
Emix = F(c)
Eiface = .5 * (c.grad(geom)**2).sum(-1)
Ewall = (abs(ewall) + ewall * c)
b = cbasis * c - mubasis * f_n(c)
rhs, total, energy_mix, energy_iface = domain.integrate(
[b, c, Emix, Eiface], geometry=geom, ischeme='gauss4')
energy_wall = domain.boundary.integrate(Ewall,
geometry=geom,
ischeme='gauss4')
log.user('concentration {}, energy {}'.format(
total, energy_mix + energy_iface + energy_wall))
lhs = matrix.solve(rhs0 + rhs, tol=1e-12, restart=999)
c = cbasis.dot(lhs)
mu = mubasis.dot(lhs)
makeplots(c, mu, energy_mix, energy_iface, energy_wall)
return lhs, energy_mix, energy_iface, energy_wall
def main(
nelems: 'number of elements' = 20,
epsilon: 'epsilon, 0 for automatic (based on nelems)' = 0,
timestep: 'time step' = .01,
maxtime: 'end time' = 1.,
theta: 'contact angle (degrees)' = 90,
init: 'initial condition (random/bubbles)' = 'random',
withplots: 'create plots' = True,
):
mineps = 1./nelems
if not epsilon:
log.info('setting epsilon={}'.format(mineps))
epsilon = mineps
elif epsilon < mineps:
log.warning('epsilon under crititical threshold: {} < {}'.format(epsilon, mineps))
# create namespace
ns = function.Namespace()
ns.epsilon = epsilon
ns.ewall = .5 * numpy.cos( theta * numpy.pi / 180 )
# construct mesh
xnodes = ynodes = numpy.linspace(0,1,nelems+1)
domain, ns.x = mesh.rectilinear( [ xnodes, ynodes ] )
# prepare bases
ns.cbasis, ns.mubasis = function.chain([
domain.basis('spline', degree=2),
domain.basis('spline', degree=2)
])
# polulate namespace
ns.c = 'cbasis_n ?lhs_n'
ns.c0 = 'cbasis_n ?lhs0_n'
ns.mu = 'mubasis_n ?lhs_n'
ns.f = '(6 c0 - 2 c0^3 - 4 c) / epsilon^2'
# construct initial condition
if init == 'random':
numpy.random.seed(0)
lhs0 = numpy.random.normal(0, .5, ns.cbasis.shape)
elif init == 'bubbles':
R1 = .25
R2 = numpy.sqrt(.5) * R1 # area2 = .5 * area1
ns.cbubble1 = function.tanh((R1-function.norm2(ns.x-(.5+R2/numpy.sqrt(2)+.8*ns.epsilon)))/ns.epsilon)
ns.cbubble2 = function.tanh((R2-function.norm2(ns.x-(.5-R1/numpy.sqrt(2)-.8*ns.epsilon)))/ns.epsilon)
sqr = domain.integral('(c - cbubble1 - cbubble2 - 1)^2 + mu^2' @ ns, geometry=ns.x, degree=4)
lhs0 = solver.optimize('lhs', sqr)
else:
raise Exception( 'unknown init %r' % init )
# construct residual
res = domain.integral('epsilon^2 cbasis_n,k mu_,k + mubasis_n (mu + f) - mubasis_n,k c_,k' @ ns, geometry=ns.x, degree=4)
res -= domain.boundary.integral('mubasis_n ewall' @ ns, geometry=ns.x, degree=4)
inertia = domain.integral('cbasis_n c' @ ns, geometry=ns.x, degree=4)
# solve time dependent problem
nsteps = numeric.round(maxtime/timestep)
makeplots = MakePlots(domain, nsteps, ns) if withplots else lambda *args: None
for istep, lhs in log.enumerate('timestep', solver.impliciteuler('lhs', target0='lhs0', residual=res, inertia=inertia, timestep=timestep, lhs0=lhs0)):
makeplots(lhs)
if istep == nsteps:
break
return lhs0, lhs