def _prepare_output(self, function, cg):
from firedrake import FunctionSpace, VectorFunctionSpace, \
TensorFunctionSpace, Function, Projector, Interpolator
name = function.name()
# Need to project/interpolate?
# If space is linear and continuity of output space matches
# continuity of current space, then we can just use the
# input function.
if is_linear(function.function_space()) and \
is_dg(function.function_space()) == (not cg) and \
is_cg(function.function_space()) == cg:
return OFunction(array=get_array(function),
name=name, function=function)
# OK, let's go and do it.
if cg:
family = "Lagrange"
else:
family = "Discontinuous Lagrange"
output = self._output_functions.get(function)
if output is None:
# Build appropriate space for output function.
shape = function.ufl_shape
if len(shape) == 0:
V = FunctionSpace(function.ufl_domain(), family, 1)
elif len(shape) == 1:
if numpy.prod(shape) > 3:
raise ValueError("Can't write vectors with more than 3 components")
V = VectorFunctionSpace(function.ufl_domain(), family, 1,
dim=shape[0])
elif len(shape) == 2:
if numpy.prod(shape) > 9:
raise ValueError("Can't write tensors with more than 9 components")
V = TensorFunctionSpace(function.ufl_domain(), family, 1,
shape=shape)
else:
raise ValueError("Unsupported shape %s" % (shape, ))
output = Function(V)
self._output_functions[function] = output
if self.project:
projector = self._mappers.get(function)
if projector is None:
projector = Projector(function, output)
self._mappers[function] = projector
projector.project()
else:
interpolator = self._mappers.get(function)
if interpolator is None:
interpolator = Interpolator(function, output)
self._mappers[function] = interpolator
interpolator.interpolate()
return OFunction(array=get_array(output), name=name, function=output)
def _prepare_output(self, function, cg):
from firedrake import FunctionSpace, VectorFunctionSpace, \
TensorFunctionSpace, Function, Projector, Interpolator
name = function.name()
# Need to project/interpolate?
# If space is linear and continuity of output space matches
# continuity of current space, then we can just use the
# input function.
if is_linear(function.function_space()) and \
is_dg(function.function_space()) == (not cg) and \
is_cg(function.function_space()) == cg:
return OFunction(array=get_array(function),
name=name, function=function)
# OK, let's go and do it.
if cg:
family = "Lagrange"
else:
family = "Discontinuous Lagrange"
output = self._output_functions.get(function)
if output is None:
# Build appropriate space for output function.
shape = function.ufl_shape
if len(shape) == 0:
V = FunctionSpace(function.ufl_domain(), family, 1)
elif len(shape) == 1:
if numpy.prod(shape) > 3:
raise ValueError("Can't write vectors with more than 3 components")
V = VectorFunctionSpace(function.ufl_domain(), family, 1,
dim=shape[0])
elif len(shape) == 2:
if numpy.prod(shape) > 9:
raise ValueError("Can't write tensors with more than 9 components")
V = TensorFunctionSpace(function.ufl_domain(), family, 1,
shape=shape)
else:
raise ValueError("Unsupported shape %s" % (shape, ))
output = Function(V)
self._output_functions[function] = output
if self.project:
projector = self._mappers.get(function)
if projector is None:
projector = Projector(function, output)
self._mappers[function] = projector
projector.project()
else:
interpolator = self._mappers.get(function)
if interpolator is None:
interpolator = Interpolator(function, output)
self._mappers[function] = interpolator
interpolator.interpolate()
return OFunction(array=get_array(output), name=name, function=output)
def _prepare_output(self, function, max_elem):
from firedrake import FunctionSpace, VectorFunctionSpace, \
TensorFunctionSpace, Function, Projector, Interpolator
name = function.name()
# Need to project/interpolate?
# If space is not the max element, we can do so.
if function.ufl_element == max_elem:
return OFunction(array=get_array(function),
name=name,
function=function)
# OK, let's go and do it.
shape = function.ufl_shape
output = self._output_functions.get(function)
if output is None:
# Build appropriate space for output function.
shape = function.ufl_shape
if len(shape) == 0:
V = FunctionSpace(function.ufl_domain(), max_elem)
elif len(shape) == 1:
if numpy.prod(shape) > 3:
raise ValueError(
"Can't write vectors with more than 3 components")
V = VectorFunctionSpace(function.ufl_domain(),
max_elem,
dim=shape[0])
elif len(shape) == 2:
if numpy.prod(shape) > 9:
raise ValueError(
"Can't write tensors with more than 9 components")
V = TensorFunctionSpace(function.ufl_domain(),
max_elem,
shape=shape)
else:
raise ValueError("Unsupported shape %s" % (shape, ))
output = Function(V)
self._output_functions[function] = output
if self.project:
projector = self._mappers.get(function)
if projector is None:
projector = Projector(function, output)
self._mappers[function] = projector
projector.project()
else:
interpolator = self._mappers.get(function)
if interpolator is None:
interpolator = Interpolator(function, output)
self._mappers[function] = interpolator
interpolator.interpolate()
return OFunction(array=get_array(output), name=name, function=output)
class Boundary_Recoverer(object):
"""
An object that performs a `recovery` process at the domain
boundaries that has second order accuracy. This is necessary
because the :class:`Averager` object does not recover a field
with sufficient accuracy at the boundaries.
The strategy is to minimise the curvature of the function in
the boundary cells, subject to the constraints of conserved
mass and continuity on the interior facets. The quickest way
to perform this is by using the analytic solution and a parloop.
Currently this is only implemented for the (DG0, DG1, CG1)
set of spaces, and only on a `PeriodicIntervalMesh` or
'PeriodicUnitIntervalMesh` that has been extruded.
:arg v_CG1: the continuous function after the first recovery
is performed. Should be in CG1. This is correct
on the interior of the domain.
:arg v_DG1: the function to be output. Should be in DG1.
:arg method: a Boundary_Method Enum object.
:arg coords_to_adjust: a DG1 field containing 1 at locations of
coords that must be adjusted to give they
effective coords.
"""
def __init__(self,
v_CG1,
v_DG1,
method=Boundary_Method.physics,
coords_to_adjust=None):
self.v_DG1 = v_DG1
self.v_CG1 = v_CG1
self.v_DG1_old = Function(v_DG1.function_space())
self.coords_to_adjust = coords_to_adjust
self.method = method
mesh = v_CG1.function_space().mesh()
VDG0 = FunctionSpace(mesh, "DG", 0)
VCG1 = FunctionSpace(mesh, "CG", 1)
if VDG0.extruded:
cell = mesh._base_mesh.ufl_cell().cellname()
DG1_hori_elt = FiniteElement("DG", cell, 1, variant="equispaced")
DG1_vert_elt = FiniteElement("DG",
interval,
1,
variant="equispaced")
DG1_element = TensorProductElement(DG1_hori_elt, DG1_vert_elt)
else:
cell = mesh.ufl_cell().cellname()
DG1_element = FiniteElement("DG", cell, 1, variant="equispaced")
VDG1 = FunctionSpace(mesh, DG1_element)
self.num_ext = Function(VDG0)
# check function spaces of functions
if self.method == Boundary_Method.dynamics:
if v_CG1.function_space() != VCG1:
raise NotImplementedError(
"This boundary recovery method requires v1 to be in CG1.")
if v_DG1.function_space() != VDG1:
raise NotImplementedError(
"This boundary recovery method requires v_out to be in DG1."
)
# check whether mesh is valid
if mesh.topological_dimension() == 2:
# if mesh is extruded then we're fine, but if not needs to be quads
if not VDG0.extruded and mesh.ufl_cell().cellname(
) != 'quadrilateral':
raise NotImplementedError(
'For 2D meshes this recovery method requires that elements are quadrilaterals'
)
elif mesh.topological_dimension() == 3:
# assume that 3D mesh is extruded
if mesh._base_mesh.ufl_cell().cellname() != 'quadrilateral':
raise NotImplementedError(
'For 3D extruded meshes this recovery method requires a base mesh with quadrilateral elements'
)
elif mesh.topological_dimension() != 1:
raise NotImplementedError(
'This boundary recovery is implemented only on certain classes of mesh.'
)
if coords_to_adjust is None:
raise ValueError(
'Need coords_to_adjust field for dynamics boundary methods'
)
elif self.method == Boundary_Method.physics:
# check that mesh is valid -- must be an extruded mesh
if not VDG0.extruded:
raise NotImplementedError(
'The physics boundary method only works on extruded meshes'
)
# base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
w_hori = FiniteElement("DG", cell, 0, variant="equispaced")
w_vert = FiniteElement("CG", interval, 1, variant="equispaced")
# build element
theta_element = TensorProductElement(w_hori, w_vert)
# spaces
Vtheta = FunctionSpace(mesh, theta_element)
Vtheta_broken = FunctionSpace(mesh, BrokenElement(theta_element))
if v_CG1.function_space() != Vtheta:
raise ValueError(
"This boundary recovery method requires v_CG1 to be in DG0xCG1 TensorProductSpace."
)
if v_DG1.function_space() != Vtheta_broken:
raise ValueError(
"This boundary recovery method requires v_DG1 to be in the broken DG0xCG1 TensorProductSpace."
)
else:
raise ValueError(
"Boundary method should be a Boundary Method Enum object.")
VuDG1 = VectorFunctionSpace(VDG0.mesh(), DG1_element)
x = SpatialCoordinate(VDG0.mesh())
self.interpolator = Interpolator(self.v_CG1, self.v_DG1)
if self.method == Boundary_Method.dynamics:
# STRATEGY
# obtain a coordinate field for all the nodes
self.act_coords = Function(VuDG1).project(x) # actual coordinates
self.eff_coords = Function(VuDG1).project(
x) # effective coordinates
self.output = Function(VDG1)
shapes = {
"nDOFs":
self.v_DG1.function_space().finat_element.space_dimension(),
"dim":
np.prod(VuDG1.shape, dtype=int)
}
num_ext_domain = ("{{[i]: 0 <= i < {nDOFs}}}").format(**shapes)
num_ext_instructions = ("""
<float64> SUM_EXT = 0
for i
SUM_EXT = SUM_EXT + EXT_V1[i]
end
NUM_EXT[0] = SUM_EXT
""")
coords_domain = ("{{[i, j, k, ii, jj, kk, ll, mm, iii, kkk]: "
"0 <= i < {nDOFs} and "
"0 <= j < {nDOFs} and 0 <= k < {dim} and "
"0 <= ii < {nDOFs} and 0 <= jj < {nDOFs} and "
"0 <= kk < {dim} and 0 <= ll < {dim} and "
"0 <= mm < {dim} and 0 <= iii < {nDOFs} and "
"0 <= kkk < {dim}}}").format(**shapes)
coords_insts = (
"""
<float64> sum_V1_ext = 0
<int> index = 100
<float64> dist = 0.0
<float64> max_dist = 0.0
<float64> min_dist = 0.0
"""
# only do adjustment in cells with at least one DOF to adjust
"""
if NUM_EXT[0] > 0
"""
# find the maximum distance between DOFs in this cell, to serve as starting point for finding min distances
"""
for i
for j
dist = 0.0
for k
dist = dist + pow(ACT_COORDS[i,k] - ACT_COORDS[j,k], 2.0)
end
dist = pow(dist, 0.5) {{id=sqrt_max_dist, dep=*}}
max_dist = fmax(dist, max_dist) {{id=max_dist, dep=sqrt_max_dist}}
end
end
"""
# loop through cells and find which ones to adjust
"""
for ii
if EXT_V1[ii] > 0.5
"""
# find closest interior node
"""
min_dist = max_dist
index = 100
for jj
if EXT_V1[jj] < 0.5
dist = 0.0
for kk
dist = dist + pow(ACT_COORDS[ii,kk] - ACT_COORDS[jj,kk], 2)
end
dist = pow(dist, 0.5)
if dist <= min_dist
index = jj
end
min_dist = fmin(min_dist, dist)
for ll
EFF_COORDS[ii,ll] = 0.5 * (ACT_COORDS[ii,ll] + ACT_COORDS[index,ll])
end
end
end
else
"""
# for DOFs that aren't exterior, use the original coordinates
"""
for mm
EFF_COORDS[ii, mm] = ACT_COORDS[ii, mm]
end
end
end
else
"""
# for interior elements, just use the original coordinates
"""
for iii
for kkk
EFF_COORDS[iii, kkk] = ACT_COORDS[iii, kkk]
end
end
end
""").format(**shapes)
_num_ext_kernel = (num_ext_domain, num_ext_instructions)
_eff_coords_kernel = (coords_domain, coords_insts)
self.gaussian_elimination_kernel = kernels.GaussianElimination(
VDG1)
# find number of external DOFs per cell
par_loop(_num_ext_kernel,
dx, {
"NUM_EXT": (self.num_ext, WRITE),
"EXT_V1": (self.coords_to_adjust, READ)
},
is_loopy_kernel=True)
# find effective coordinates
logger.warning(
'Finding effective coordinates for boundary recovery. This could give unexpected results for deformed meshes over very steep topography.'
)
par_loop(_eff_coords_kernel,
dx, {
"EFF_COORDS": (self.eff_coords, WRITE),
"ACT_COORDS": (self.act_coords, READ),
"NUM_EXT": (self.num_ext, READ),
"EXT_V1": (self.coords_to_adjust, READ)
},
is_loopy_kernel=True)
elif self.method == Boundary_Method.physics:
self.bottom_kernel = kernels.PhysicsRecoveryBottom()
self.top_kernel = kernels.PhysicsRecoveryTop()
def apply(self):
self.interpolator.interpolate()
if self.method == Boundary_Method.physics:
self.bottom_kernel.apply(self.v_DG1, self.v_CG1)
self.top_kernel.apply(self.v_DG1, self.v_CG1)
else:
self.v_DG1_old.assign(self.v_DG1)
self.gaussian_elimination_kernel.apply(self.v_DG1_old, self.v_DG1,
self.act_coords,
self.eff_coords,
self.num_ext)
class Recoverer(object):
"""
An object that 'recovers' a field from a low order space
(e.g. DG0) into a higher order space (e.g. CG1). This encompasses
the process of interpolating first to a the right space before
using the :class:`Averager` object, and also automates the
boundary recovery process. If no boundary method is specified,
this simply performs the action of the :class: `Averager`.
:arg v_in: the :class:`ufl.Expr` or
:class:`.Function` to project. (e.g. a VDG0 function)
:arg v_out: :class:`.Function` to put the result in. (e.g. a CG1 function)
:arg VDG: optional :class:`.FunctionSpace`. If not None, v_in is interpolated
to this space first before recovery happens.
:arg boundary_method: an Enum object, .
"""
def __init__(self, v_in, v_out, VDG=None, boundary_method=None):
# check if v_in is valid
if isinstance(v_in, expression.Expression) or not isinstance(
v_in, (ufl.core.expr.Expr, function.Function)):
raise ValueError(
"Can only recover UFL expression or Functions not '%s'" %
type(v_in))
self.v_in = v_in
self.v_out = v_out
self.V = v_out.function_space()
if VDG is not None:
self.v = Function(VDG)
self.interpolator = Interpolator(v_in, self.v)
else:
self.v = v_in
self.interpolator = None
self.VDG = VDG
self.boundary_method = boundary_method
self.averager = Averager(self.v, self.v_out)
# check boundary method options are valid
if boundary_method is not None:
if boundary_method != Boundary_Method.dynamics and boundary_method != Boundary_Method.physics:
raise ValueError(
"Boundary method must be a Boundary_Method Enum object.")
if VDG is None:
raise ValueError(
"If boundary_method is specified, VDG also needs specifying."
)
# now specify things that we'll need if we are doing boundary recovery
if boundary_method == Boundary_Method.physics:
# check dimensions
if self.V.value_size != 1:
raise ValueError(
'This method only works for scalar functions.')
self.boundary_recoverer = Boundary_Recoverer(
self.v_out, self.v, method=Boundary_Method.physics)
else:
mesh = self.V.mesh()
# this ensures we get the pure function space, not an indexed function space
V0 = FunctionSpace(mesh,
self.v_in.function_space().ufl_element())
VCG1 = FunctionSpace(mesh, "CG", 1)
if V0.extruded:
cell = mesh._base_mesh.ufl_cell().cellname()
DG1_hori_elt = FiniteElement("DG",
cell,
1,
variant="equispaced")
DG1_vert_elt = FiniteElement("DG",
interval,
1,
variant="equispaced")
DG1_element = TensorProductElement(DG1_hori_elt,
DG1_vert_elt)
else:
cell = mesh.ufl_cell().cellname()
DG1_element = FiniteElement("DG",
cell,
1,
variant="equispaced")
VDG1 = FunctionSpace(mesh, DG1_element)
if self.V.value_size == 1:
coords_to_adjust = find_coords_to_adjust(V0, VDG1)
self.boundary_recoverer = Boundary_Recoverer(
self.v_out,
self.v,
coords_to_adjust=coords_to_adjust,
method=Boundary_Method.dynamics)
else:
VuDG1 = VectorFunctionSpace(mesh, DG1_element)
coords_to_adjust = find_coords_to_adjust(V0, VuDG1)
# now, break the problem down into components
v_scalars = []
v_out_scalars = []
self.boundary_recoverers = []
self.project_to_scalars_CG = []
self.extra_averagers = []
coords_to_adjust_list = []
for i in range(self.V.value_size):
v_scalars.append(Function(VDG1))
v_out_scalars.append(Function(VCG1))
coords_to_adjust_list.append(
Function(VDG1).project(coords_to_adjust[i]))
self.project_to_scalars_CG.append(
Projector(self.v_out[i], v_out_scalars[i]))
self.boundary_recoverers.append(
Boundary_Recoverer(
v_out_scalars[i],
v_scalars[i],
method=Boundary_Method.dynamics,
coords_to_adjust=coords_to_adjust_list[i]))
# need an extra averager that works on the scalar fields rather than the vector one
self.extra_averagers.append(
Averager(v_scalars[i], v_out_scalars[i]))
# the boundary recoverer needs to be done on a scalar fields
# so need to extract component and restore it after the boundary recovery is done
self.interpolate_to_vector = Interpolator(
as_vector(v_out_scalars), self.v_out)
def project(self):
"""
Perform the fully specified recovery.
"""
if self.interpolator is not None:
self.interpolator.interpolate()
self.averager.project()
if self.boundary_method is not None:
if self.V.value_size > 1:
for i in range(self.V.value_size):
self.project_to_scalars_CG[i].project()
self.boundary_recoverers[i].apply()
self.extra_averagers[i].project()
self.interpolate_to_vector.interpolate()
else:
self.boundary_recoverer.apply()
self.averager.project()
return self.v_out
class StochasticFunctions(object):
"""
The stochastic functions object.
:arg prognostic_variables: a PrognosticVariables object.
:arg simulation_parameters: a dictionary of the simulation parameters.
"""
def __init__(self, prognostic_variables, simulation_parameters):
mesh = simulation_parameters['mesh'][-1]
x, = SpatialCoordinate(mesh)
Ld = simulation_parameters['Ld'][-1]
self.scheme = simulation_parameters['scheme'][-1]
self.dt = simulation_parameters['dt'][-1]
self.num_Xis = simulation_parameters['num_Xis'][-1]
self.Xi_family = simulation_parameters['Xi_family'][-1]
self.dXi = prognostic_variables.dXi
self.dWs = [Constant(0.0) for dw in range(self.num_Xis)]
self.dW_nums = prognostic_variables.dW_nums
self.Xi_functions = []
self.nXi_updates = simulation_parameters['nXi_updates'][-1]
self.smooth_t = simulation_parameters['smooth_t'][-1]
self.fixed_dW = simulation_parameters['fixed_dW'][-1]
if self.smooth_t is not None and self.nXi_updates > 1:
raise ValueError('Prescribing forcing and including multiple Xi updates are not compatible.')
if self.smooth_t is not None or self.fixed_dW is not None:
print('WARNING: Remember to change sigma to sigma * sqrt(dt) with the prescribed forcing option or the fixed_dW option.')
seed = simulation_parameters['seed'][-1]
np.random.seed(seed)
# make sure sigma is a Constant
if self.num_Xis != 0:
if isinstance(simulation_parameters['sigma'][-1], Constant):
self.sigma = simulation_parameters['sigma'][-1]
else:
self.sigma = Constant(simulation_parameters['sigma'][-1])
else:
self.sigma = Constant(0.0)
self.pure_xi_list = prognostic_variables.pure_xi_list
self.pure_xi_x_list = prognostic_variables.pure_xi_x_list
self.pure_xi_xx_list = prognostic_variables.pure_xi_xx_list
self.pure_xi_xxx_list = prognostic_variables.pure_xi_xxx_list
self.pure_xi_xxxx_list = prognostic_variables.pure_xi_xxxx_list
for xi in range(self.num_Xis):
self.pure_xi_list.append(Function(self.dXi.function_space()))
self.pure_xi_x_list.append(Function(self.dXi.function_space()))
self.pure_xi_xx_list.append(Function(self.dXi.function_space()))
self.pure_xi_xxx_list.append(Function(self.dXi.function_space()))
self.pure_xi_xxxx_list.append(Function(self.dXi.function_space()))
if self.Xi_family == 'sines':
for n in range(self.num_Xis):
if (n+1) % 2 == 1:
self.Xi_functions.append(self.sigma * sin(2*(n+1)*pi*x/Ld))
else:
self.Xi_functions.append(self.sigma * cos(2*(n+1)*pi*x/Ld))
elif self.Xi_family == 'double_sines':
for n in range(self.num_Xis):
if (n+1) % 2 == 1:
self.Xi_functions.append(self.sigma * sin(4*(n+1)*pi*x/Ld))
else:
self.Xi_functions.append(self.sigma * cos(4*(n+1)*pi*x/Ld))
elif self.Xi_family == 'high_freq_sines':
for n in range(self.num_Xis):
if (n+1) % 2 == 1:
self.Xi_functions.append(self.sigma * sin((2*(n+1)+10)*pi*x/Ld))
else:
self.Xi_functions.append(self.sigma * cos((2*(n+1)+10)*pi*x/Ld))
elif self.Xi_family == 'gaussians':
for n in range(self.num_Xis):
self.Xi_functions.append(self.sigma * 0.5*self.num_Xis*exp(-((x-Ld*(n+1)/(self.num_Xis +1.0))/2.)**2))
elif self.Xi_family == 'quadratic':
if self.num_Xis > 1:
raise NotImplementedError('Quadratic Xi not yet implemented for more than one Xi')
else:
self.Xi_functions.append(32/(Ld*Ld)*conditional(x > Ld/4,
conditional(x > 3*Ld/8,
conditional(x > 5*Ld/8,
conditional(x < 3*Ld/4,
self.sigma * (x - 3*Ld/4)**2,
0.0),
(x-Ld/2)**2+Ld**2/32),
(x-Ld/4)**2),
0.0))
elif self.Xi_family == 'proper_peak':
if self.num_Xis > 1:
raise NotImplementedError('Quadratic Xi not yet implemented for more than one Xi')
else:
self.Xi_functions.append(self.sigma * 0.5*2/(exp(x-Ld/2)+exp(-x+Ld/2)))
elif self.Xi_family == 'constant':
if self.num_Xis > 1:
raise NotImplementedError('Constant Xi not yet implemented for more than one Xi')
else:
self.Xi_functions.append(self.sigma * (sin(0*pi*x/Ld)+1))
else:
raise NotImplementedError('Xi_family %s not implemented' % self.Xi_family)
# make lists of functions for xi_x, xi_xx and xi_xxx
if self.scheme in ['hydrodynamic', 'LASCH_hydrodynamic']:
self.dXi_x = prognostic_variables.dXi_x
self.dXi_xx = prognostic_variables.dXi_xx
self.Xi_x_functions = []
self.Xi_xx_functions = []
for Xi_expr in self.Xi_functions:
Xi_x_function = Function(self.dXi_x.function_space())
Xi_xx_function = Function(self.dXi_xx.function_space())
phi_x = TestFunction(self.dXi_x.function_space())
phi_xx = TestFunction(self.dXi_xx.function_space())
Xi_x_eqn = phi_x * Xi_x_function * dx + phi_x.dx(0) * Xi_expr * dx
Xi_xx_eqn = phi_xx * Xi_xx_function * dx + phi_xx.dx(0) * Xi_x_function * dx
Xi_x_problem = NonlinearVariationalProblem(Xi_x_eqn, Xi_x_function)
Xi_xx_problem = NonlinearVariationalProblem(Xi_xx_eqn, Xi_xx_function)
Xi_x_solver = NonlinearVariationalSolver(Xi_x_problem)
Xi_xx_solver = NonlinearVariationalSolver(Xi_xx_problem)
# for some reason these solvers don't work for constant Xi functions
# so just manually make the derivatives be zero
if self.Xi_family == 'constant':
Xi_x_function.interpolate(0.0*x)
Xi_xx_function.interpolate(0.0*x)
else:
Xi_x_solver.solve()
Xi_xx_solver.solve()
self.Xi_x_functions.append(Xi_x_function)
self.Xi_xx_functions.append(Xi_xx_function)
# now make a master xi
Xi_expr = 0.0*x
for dW, Xi_function, pure_xi, pure_xi_x, pure_xi_xx, pure_xi_xxx, pure_xi_xxxx in zip(self.dWs, self.Xi_functions, self.pure_xi_list, self.pure_xi_x_list, self.pure_xi_xx_list, self.pure_xi_xxx_list, self.pure_xi_xxxx_list):
Xi_expr += dW * Xi_function
if self.scheme in ['upwind', 'LASCH']:
pure_xi.interpolate(as_vector([Xi_function]))
pure_xi_x.project(as_vector([Xi_function.dx(0)]))
CG1 = FunctionSpace(mesh, "CG", 1)
psi = TestFunction(CG1)
xixx_scalar = Function(CG1)
xixx_eqn = psi * xixx_scalar * dx + psi.dx(0) * Xi_function.dx(0) * dx
prob = NonlinearVariationalProblem(xixx_eqn, xixx_scalar)
solver = NonlinearVariationalSolver(prob)
solver.solve()
pure_xi_xx.interpolate(as_vector([xixx_scalar]))
else:
pure_xi.interpolate(Xi_function)
# I guess we can't take the gradient of constants
if self.Xi_family != 'constant':
pure_xi_x.project(Xi_function.dx(0))
pure_xi_xx.project(pure_xi_x.dx(0))
pure_xi_xxx.project(pure_xi_xx.dx(0))
pure_xi_xxxx.project(pure_xi_xxx.dx(0))
if self.scheme in ['upwind', 'LASCH']:
self.dXi_interpolator = Interpolator(as_vector([Xi_expr]), self.dXi)
else:
self.dXi_interpolator = Interpolator(Xi_expr, self.dXi)
if self.scheme in ['hydrodynamic', 'LASCH_hydrodynamic']:
# initialise blank expressions
Xi_x_expr = 0.0*x
Xi_xx_expr = 0.0*x
# make full expressions by adding all dW * Xi_xs
for dW, Xi_x_function, Xi_xx_function in zip(self.dWs, self.Xi_x_functions, self.Xi_xx_functions):
Xi_x_expr += dW * Xi_x_function
Xi_xx_expr += dW * Xi_xx_function
self.dXi_x_interpolator = Interpolator(Xi_x_expr, self.dXi_x)
self.dXi_xx_interpolator = Interpolator(Xi_xx_expr, self.dXi_xx)
def update(self, t):
"""
Updates the Xi function for the next time step.
"""
if self.num_Xis > 0:
# Try to calculate dW numbers separately
# For nXi_updates > 1 then the ordering of calls to np.random.randn
# needs to be equivalent to that for the corresponding smaller dt
if self.nXi_updates > 1:
self.dW_nums[:] = 0.0
for j in range(self.nXi_updates):
for i in range(self.num_Xis):
self.dW_nums[i] += np.random.randn() * np.sqrt(self.dt/self.nXi_updates)
else:
for i in range(self.num_Xis):
if self.smooth_t is not None:
self.dW_nums[i] = self.smooth_t(t)
elif self.fixed_dW is not None:
self.dW_nums[i] = self.fixed_dW * np.sqrt(self.dt)
else:
self.dW_nums[i] = np.random.randn() * np.sqrt(self.dt)
# This is to ensure we stick close to what the original code did
[dw.assign(dw_num) for dw, dw_num in zip(self.dWs, self.dW_nums)]
self.dXi_interpolator.interpolate()
if self.scheme in ['hydrodynamic', 'LASCH_hydrodynamic']:
self.dXi_x_interpolator.interpolate()
self.dXi_xx_interpolator.interpolate()
else:
pass
class Advection(object, metaclass=ABCMeta):
"""
Base class for advection schemes.
:arg state: :class:`.State` object.
:arg field: field to be advected
:arg equation: :class:`.Equation` object, specifying the equation
that field satisfies
:arg solver_parameters: solver_parameters
:arg limiter: :class:`.Limiter` object.
:arg options: :class:`.AdvectionOptions` object
"""
def __init__(self, state, field, equation=None, *, solver_parameters=None,
limiter=None):
if equation is not None:
self.state = state
self.field = field
self.equation = equation
# get ubar from the equation class
self.ubar = self.equation.ubar
self.dt = self.state.timestepping.dt
# get default solver options if none passed in
if solver_parameters is None:
self.solver_parameters = equation.solver_parameters
else:
self.solver_parameters = solver_parameters
if logger.isEnabledFor(DEBUG):
self.solver_parameters["ksp_monitor_true_residual"] = True
self.limiter = limiter
if hasattr(equation, "options"):
self.discretisation_option = equation.options.name
self._setup(state, field, equation.options)
else:
self.discretisation_option = None
self.fs = field.function_space()
# setup required functions
self.dq = Function(self.fs)
self.q1 = Function(self.fs)
def _setup(self, state, field, options):
if options.name in ["embedded_dg", "recovered"]:
self.fs = options.embedding_space
self.xdg_in = Function(self.fs)
self.xdg_out = Function(self.fs)
self.x_projected = Function(field.function_space())
parameters = {'ksp_type': 'cg',
'pc_type': 'bjacobi',
'sub_pc_type': 'ilu'}
self.Projector = Projector(self.xdg_out, self.x_projected,
solver_parameters=parameters)
if options.name == "recovered":
# set up the necessary functions
self.x_in = Function(field.function_space())
x_rec = Function(options.recovered_space)
x_brok = Function(options.broken_space)
# set up interpolators and projectors
self.x_rec_projector = Recoverer(self.x_in, x_rec, VDG=self.fs, boundary_method=options.boundary_method) # recovered function
self.x_brok_projector = Projector(x_rec, x_brok) # function projected back
self.xdg_interpolator = Interpolator(self.x_in + x_rec - x_brok, self.xdg_in)
if self.limiter is not None:
self.x_brok_interpolator = Interpolator(self.xdg_out, x_brok)
self.x_out_projector = Recoverer(x_brok, self.x_projected)
def pre_apply(self, x_in, discretisation_option):
"""
Extra steps to advection if using an embedded method,
which might be either the plain embedded method or the
recovered space advection scheme.
:arg x_in: the input set of prognostic fields.
:arg discretisation option: string specifying which scheme to use.
"""
if discretisation_option == "embedded_dg":
try:
self.xdg_in.interpolate(x_in)
except NotImplementedError:
self.xdg_in.project(x_in)
elif discretisation_option == "recovered":
self.x_in.assign(x_in)
self.x_rec_projector.project()
self.x_brok_projector.project()
self.xdg_interpolator.interpolate()
def post_apply(self, x_out, discretisation_option):
"""
The projection steps, returning a field to its original space
for an embedded DG advection scheme. For the case of the
recovered scheme, there are two options dependent on whether
the scheme is limited or not.
:arg x_out: the outgoing field.
:arg discretisation_option: string specifying which option to use.
"""
if discretisation_option == "embedded_dg":
self.Projector.project()
elif discretisation_option == "recovered":
if self.limiter is not None:
self.x_brok_interpolator.interpolate()
self.x_out_projector.project()
else:
self.Projector.project()
x_out.assign(self.x_projected)
@abstractproperty
def lhs(self):
return self.equation.mass_term(self.equation.trial)
@abstractproperty
def rhs(self):
return self.equation.mass_term(self.q1) - self.dt*self.equation.advection_term(self.q1)
def update_ubar(self, xn, xnp1, alpha):
un = xn.split()[0]
unp1 = xnp1.split()[0]
self.ubar.assign(un + alpha*(unp1-un))
@cached_property
def solver(self):
# setup solver using lhs and rhs defined in derived class
problem = LinearVariationalProblem(self.lhs, self.rhs, self.dq)
solver_name = self.field.name()+self.equation.__class__.__name__+self.__class__.__name__
return LinearVariationalSolver(problem, solver_parameters=self.solver_parameters, options_prefix=solver_name)
@abstractmethod
def apply(self, x_in, x_out):
"""
Function takes x as input, computes L(x) as defined by the equation,
and returns x_out as output.
:arg x: :class:`.Function` object, the input Function.
:arg x_out: :class:`.Function` object, the output Function.
"""
pass
class Condensation(Physics):
"""
The process of condensation of water vapour
into liquid water and evaporation of liquid
water into water vapour, with the associated
latent heat changes. The parametrization follows
that used in Bryan and Fritsch (2002).
:arg state: :class:`.State.` object.
:arg iterations: number of iterations to do
of condensation scheme per time step.
"""
def __init__(self, state, iterations=1):
super().__init__(state)
self.iterations = iterations
# obtain our fields
self.theta = state.fields('theta')
self.water_v = state.fields('water_v')
self.water_c = state.fields('water_c')
rho = state.fields('rho')
try:
rain = state.fields('rain')
water_l = self.water_c + rain
except NotImplementedError:
water_l = self.water_c
# declare function space
Vt = self.theta.function_space()
# make rho variables
# we recover rho into theta space
if state.vertical_degree == 0 and state.horizontal_degree == 0:
boundary_method = Boundary_Method.physics
else:
boundary_method = None
Vt_broken = FunctionSpace(state.mesh, BrokenElement(Vt.ufl_element()))
rho_averaged = Function(Vt)
self.rho_recoverer = Recoverer(rho,
rho_averaged,
VDG=Vt_broken,
boundary_method=boundary_method)
# define some parameters as attributes
dt = state.timestepping.dt
R_d = state.parameters.R_d
cp = state.parameters.cp
cv = state.parameters.cv
c_pv = state.parameters.c_pv
c_pl = state.parameters.c_pl
c_vv = state.parameters.c_vv
R_v = state.parameters.R_v
# make useful fields
Pi = thermodynamics.pi(state.parameters, rho_averaged, self.theta)
T = thermodynamics.T(state.parameters,
self.theta,
Pi,
r_v=self.water_v)
p = thermodynamics.p(state.parameters, Pi)
L_v = thermodynamics.Lv(state.parameters, T)
R_m = R_d + R_v * self.water_v
c_pml = cp + c_pv * self.water_v + c_pl * water_l
c_vml = cv + c_vv * self.water_v + c_pl * water_l
# use Teten's formula to calculate w_sat
w_sat = thermodynamics.r_sat(state.parameters, T, p)
# make appropriate condensation rate
dot_r_cond = ((self.water_v - w_sat) / (dt * (1.0 +
((L_v**2.0 * w_sat) /
(cp * R_v * T**2.0)))))
# make cond_rate function, that needs to be the same for all updates in one time step
cond_rate = Function(Vt)
# adjust cond rate so negative concentrations don't occur
self.lim_cond_rate = Interpolator(
conditional(dot_r_cond < 0,
max_value(dot_r_cond, -self.water_c / dt),
min_value(dot_r_cond, self.water_v / dt)), cond_rate)
# tell the prognostic fields what to update to
self.water_v_new = Interpolator(self.water_v - dt * cond_rate, Vt)
self.water_c_new = Interpolator(self.water_c + dt * cond_rate, Vt)
self.theta_new = Interpolator(
self.theta * (1.0 + dt * cond_rate *
(cv * L_v / (c_vml * cp * T) - R_v * cv * c_pml /
(R_m * cp * c_vml))), Vt)
def apply(self):
self.rho_recoverer.project()
for i in range(self.iterations):
self.lim_cond_rate.interpolate()
self.theta.assign(self.theta_new.interpolate())
self.water_v.assign(self.water_v_new.interpolate())
self.water_c.assign(self.water_c_new.interpolate())
class Evaporation(Physics):
"""
The process of evaporation of rain into water vapour
with the associated latent heat change. This
parametrization comes from Klemp and Wilhelmson (1978).
:arg state: :class:`.State.` object.
"""
def __init__(self, state):
super().__init__(state)
# obtain our fields
self.theta = state.fields('theta')
self.water_v = state.fields('water_v')
self.rain = state.fields('rain')
rho = state.fields('rho')
try:
water_c = state.fields('water_c')
water_l = self.rain + water_c
except NotImplementedError:
water_l = self.rain
# declare function space
Vt = self.theta.function_space()
# make rho variables
# we recover rho into theta space
if state.vertical_degree == 0 and state.horizontal_degree == 0:
boundary_method = Boundary_Method.physics
else:
boundary_method = None
Vt_broken = FunctionSpace(state.mesh, BrokenElement(Vt.ufl_element()))
rho_averaged = Function(Vt)
self.rho_recoverer = Recoverer(rho,
rho_averaged,
VDG=Vt_broken,
boundary_method=boundary_method)
# define some parameters as attributes
dt = state.timestepping.dt
R_d = state.parameters.R_d
cp = state.parameters.cp
cv = state.parameters.cv
c_pv = state.parameters.c_pv
c_pl = state.parameters.c_pl
c_vv = state.parameters.c_vv
R_v = state.parameters.R_v
# make useful fields
Pi = thermodynamics.pi(state.parameters, rho_averaged, self.theta)
T = thermodynamics.T(state.parameters,
self.theta,
Pi,
r_v=self.water_v)
p = thermodynamics.p(state.parameters, Pi)
L_v = thermodynamics.Lv(state.parameters, T)
R_m = R_d + R_v * self.water_v
c_pml = cp + c_pv * self.water_v + c_pl * water_l
c_vml = cv + c_vv * self.water_v + c_pl * water_l
# use Teten's formula to calculate w_sat
w_sat = thermodynamics.r_sat(state.parameters, T, p)
# expression for ventilation factor
a = Constant(1.6)
b = Constant(124.9)
c = Constant(0.2046)
C = a + b * (rho_averaged * self.rain)**c
# make appropriate condensation rate
f = Constant(5.4e5)
g = Constant(2.55e6)
h = Constant(0.525)
dot_r_evap = (((1 - self.water_v / w_sat) * C *
(rho_averaged * self.rain)**h) /
(rho_averaged * (f + g / (p * w_sat))))
# make evap_rate function, needs to be the same for all updates in one time step
evap_rate = Function(Vt)
# adjust evap rate so negative rain doesn't occur
self.lim_evap_rate = Interpolator(
conditional(
dot_r_evap < 0, 0.0,
conditional(self.rain < 0.0, 0.0,
min_value(dot_r_evap, self.rain / dt))), evap_rate)
# tell the prognostic fields what to update to
self.water_v_new = Interpolator(self.water_v + dt * evap_rate, Vt)
self.rain_new = Interpolator(self.rain - dt * evap_rate, Vt)
self.theta_new = Interpolator(
self.theta * (1.0 - dt * evap_rate *
(cv * L_v / (c_vml * cp * T) - R_v * cv * c_pml /
(R_m * cp * c_vml))), Vt)
def apply(self):
self.rho_recoverer.project()
self.lim_evap_rate.interpolate()
self.theta.assign(self.theta_new.interpolate())
self.water_v.assign(self.water_v_new.interpolate())
self.rain.assign(self.rain_new.interpolate())
class Coalescence(Physics):
"""
The process of the coalescence of cloud
droplets to form rain droplets. These
parametrizations come from Klemp and
Wilhelmson (1978).
:arg state: :class:`.State.` object.
:arg accretion: Boolean which determines
whether the accretion
process is used.
:arg accumulation: Boolean which determines
whether the accumulation
process is used.
"""
def __init__(self, state, accretion=True, accumulation=True):
super().__init__(state)
# obtain our fields
self.water_c = state.fields('water_c')
self.rain = state.fields('rain')
# declare function space
Vt = self.water_c.function_space()
# define some parameters as attributes
dt = state.timestepping.dt
k_1 = Constant(0.001) # accretion rate in 1/s
k_2 = Constant(2.2) # accumulation rate in 1/s
a = Constant(0.001) # min cloud conc in kg/kg
b = Constant(0.875) # power for rain in accumulation
# make default rates to be zero
accr_rate = Constant(0.0)
accu_rate = Constant(0.0)
if accretion:
accr_rate = k_1 * (self.water_c - a)
if accumulation:
accu_rate = k_2 * self.water_c * self.rain**b
# make coalescence rate function, that needs to be the same for all updates in one time step
coalesce_rate = Function(Vt)
# adjust coalesce rate using min_value so negative cloud concentration doesn't occur
self.lim_coalesce_rate = Interpolator(
conditional(
self.rain < 0.0, # if rain is negative do only accretion
conditional(accr_rate < 0.0, 0.0,
min_value(accr_rate, self.water_c / dt)),
# don't turn rain back into cloud
conditional(
accr_rate + accu_rate < 0.0,
0.0,
# if accretion rate is negative do only accumulation
conditional(
accr_rate < 0.0, min_value(accu_rate,
self.water_c / dt),
min_value(accr_rate + accu_rate, self.water_c / dt)))),
coalesce_rate)
# tell the prognostic fields what to update to
self.water_c_new = Interpolator(self.water_c - dt * coalesce_rate, Vt)
self.rain_new = Interpolator(self.rain + dt * coalesce_rate, Vt)
def apply(self):
self.lim_coalesce_rate.interpolate()
self.rain.assign(self.rain_new.interpolate())
self.water_c.assign(self.water_c_new.interpolate())
class Condensation(Physics):
"""
The process of condensation of water vapour
into liquid water and evaporation of liquid
water into water vapour, with the associated
latent heat changes.
:arg state: :class:`.State.` object.
"""
def __init__(self, state):
super(Condensation, self).__init__(state)
# obtain our fields
self.theta = state.fields('theta')
self.water_v = state.fields('water_v')
self.water_c = state.fields('water_c')
rho = state.fields('rho')
# declare function space
Vt = self.theta.function_space()
param = self.state.parameters
# define some parameters as attributes
dt = self.state.timestepping.dt
R_d = param.R_d
p_0 = param.p_0
kappa = param.kappa
cp = param.cp
cv = param.cv
c_pv = param.c_pv
c_pl = param.c_pl
c_vv = param.c_vv
R_v = param.R_v
L_v0 = param.L_v0
T_0 = param.T_0
w_sat1 = param.w_sat1
w_sat2 = param.w_sat2
w_sat3 = param.w_sat3
w_sat4 = param.w_sat4
# make useful fields
Pi = ((R_d * rho * self.theta / p_0)**(kappa / (1.0 - kappa)))
T = Pi * self.theta * R_d / (R_d + self.water_v * R_v)
p = p_0 * Pi**(1.0 / kappa)
L_v = L_v0 - (c_pl - c_pv) * (T - T_0)
R_m = R_d + R_v * self.water_v
c_pml = cp + c_pv * self.water_v + c_pl * self.water_c
c_vml = cv + c_vv * self.water_v + c_pl * self.water_c
# use Teten's formula to calculate w_sat
w_sat = (w_sat1 / (p * exp(w_sat2 * (T - T_0) /
(T - w_sat3)) - w_sat4))
# make appropriate condensation rate
dot_r_cond = ((self.water_v - w_sat) / (dt * (1.0 +
((L_v**2.0 * w_sat) /
(cp * R_v * T**2.0)))))
# make cond_rate function, that needs to be the same for all updates in one time step
self.cond_rate = Function(Vt)
# adjust cond rate so negative concentrations don't occur
self.lim_cond_rate = Interpolator(
conditional(dot_r_cond < 0,
max_value(dot_r_cond, -self.water_c / dt),
min_value(dot_r_cond, self.water_v / dt)),
self.cond_rate)
# tell the prognostic fields what to update to
self.water_v_new = Interpolator(self.water_v - dt * self.cond_rate, Vt)
self.water_c_new = Interpolator(self.water_c + dt * self.cond_rate, Vt)
self.theta_new = Interpolator(
self.theta * (1.0 + dt * self.cond_rate *
(cv * L_v / (c_vml * cp * T) - R_v * cv * c_pml /
(R_m * cp * c_vml))), Vt)
def apply(self):
self.lim_cond_rate.interpolate()
self.theta.assign(self.theta_new.interpolate())
self.water_v.assign(self.water_v_new.interpolate())
self.water_c.assign(self.water_c_new.interpolate())
class TimeDiscretisation(object, metaclass=ABCMeta):
"""
Base class for time discretisation schemes.
:arg state: :class:`.State` object.
:arg field: field to be evolved
:arg equation: :class:`.Equation` object, specifying the equation
that field satisfies
:arg solver_parameters: solver_parameters
:arg limiter: :class:`.Limiter` object.
:arg options: :class:`.DiscretisationOptions` object
"""
def __init__(self, state, field_name=None, solver_parameters=None,
limiter=None, options=None):
self.state = state
self.field_name = field_name
self.dt = self.state.dt
self.limiter = limiter
self.options = options
if options is not None:
self.discretisation_option = options.name
else:
self.discretisation_option = None
# get default solver options if none passed in
if solver_parameters is None:
self.solver_parameters = {'ksp_type': 'cg',
'pc_type': 'bjacobi',
'sub_pc_type': 'ilu'}
else:
self.solver_parameters = solver_parameters
if logger.isEnabledFor(DEBUG):
self.solver_parameters["ksp_monitor_true_residual"] = None
def setup(self, equation, uadv=None, apply_bcs=True, *active_labels):
self.residual = equation.residual
if self.field_name is not None:
self.idx = equation.field_names.index(self.field_name)
self.fs = self.state.fields(self.field_name).function_space()
self.residual = self.residual.label_map(
lambda t: t.get(prognostic) == self.field_name,
lambda t: Term(
split_form(t.form)[self.idx].form,
t.labels),
drop)
bcs = equation.bcs[self.field_name]
else:
self.field_name = equation.field_name
self.fs = equation.function_space
self.idx = None
if type(self.fs.ufl_element()) is MixedElement:
bcs = [bc for _, bcs in equation.bcs.items() for bc in bcs]
else:
bcs = equation.bcs[self.field_name]
if len(active_labels) > 0:
self.residual = self.residual.label_map(
lambda t: any(t.has_label(time_derivative, *active_labels)),
map_if_false=drop)
options = self.options
# -------------------------------------------------------------------- #
# Routines relating to transport
# -------------------------------------------------------------------- #
if hasattr(self.options, 'ibp'):
self.replace_transport_term()
self.replace_transporting_velocity(uadv)
# -------------------------------------------------------------------- #
# Wrappers for embedded / recovery methods
# -------------------------------------------------------------------- #
if self.discretisation_option in ["embedded_dg", "recovered"]:
# construct the embedding space if not specified
if options.embedding_space is None:
V_elt = BrokenElement(self.fs.ufl_element())
self.fs = FunctionSpace(self.state.mesh, V_elt)
else:
self.fs = options.embedding_space
self.xdg_in = Function(self.fs)
self.xdg_out = Function(self.fs)
if self.idx is None:
self.x_projected = Function(equation.function_space)
else:
self.x_projected = Function(self.state.fields(self.field_name).function_space())
new_test = TestFunction(self.fs)
parameters = {'ksp_type': 'cg',
'pc_type': 'bjacobi',
'sub_pc_type': 'ilu'}
# -------------------------------------------------------------------- #
# Make boundary conditions
# -------------------------------------------------------------------- #
if not apply_bcs:
self.bcs = None
elif self.discretisation_option in ["embedded_dg", "recovered"]:
# Transfer boundary conditions onto test function space
self.bcs = [DirichletBC(self.fs, bc.function_arg, bc.sub_domain) for bc in bcs]
else:
self.bcs = bcs
# -------------------------------------------------------------------- #
# Modify test function for SUPG methods
# -------------------------------------------------------------------- #
if self.discretisation_option == "supg":
# construct tau, if it is not specified
dim = self.state.mesh.topological_dimension()
if options.tau is not None:
# if tau is provided, check that is has the right size
tau = options.tau
assert as_ufl(tau).ufl_shape == (dim, dim), "Provided tau has incorrect shape!"
else:
# create tuple of default values of size dim
default_vals = [options.default*self.dt]*dim
# check for directions is which the space is discontinuous
# so that we don't apply supg in that direction
if is_cg(self.fs):
vals = default_vals
else:
space = self.fs.ufl_element().sobolev_space()
if space.name in ["HDiv", "DirectionalH"]:
vals = [default_vals[i] if space[i].name == "H1"
else 0. for i in range(dim)]
else:
raise ValueError("I don't know what to do with space %s" % space)
tau = Constant(tuple([
tuple(
[vals[j] if i == j else 0. for i, v in enumerate(vals)]
) for j in range(dim)])
)
self.solver_parameters = {'ksp_type': 'gmres',
'pc_type': 'bjacobi',
'sub_pc_type': 'ilu'}
test = TestFunction(self.fs)
new_test = test + dot(dot(uadv, tau), grad(test))
if self.discretisation_option is not None:
# replace the original test function with one defined on
# the embedding space, as this is the space where the
# the problem will be solved
self.residual = self.residual.label_map(
all_terms,
map_if_true=replace_test_function(new_test))
if self.discretisation_option == "embedded_dg":
if self.limiter is None:
self.x_out_projector = Projector(self.xdg_out, self.x_projected,
solver_parameters=parameters)
else:
self.x_out_projector = Recoverer(self.xdg_out, self.x_projected)
if self.discretisation_option == "recovered":
# set up the necessary functions
self.x_in = Function(self.state.fields(self.field_name).function_space())
x_rec = Function(options.recovered_space)
x_brok = Function(options.broken_space)
# set up interpolators and projectors
self.x_rec_projector = Recoverer(self.x_in, x_rec, VDG=self.fs, boundary_method=options.boundary_method) # recovered function
self.x_brok_projector = Projector(x_rec, x_brok) # function projected back
self.xdg_interpolator = Interpolator(self.x_in + x_rec - x_brok, self.xdg_in)
if self.limiter is not None:
self.x_brok_interpolator = Interpolator(self.xdg_out, x_brok)
self.x_out_projector = Recoverer(x_brok, self.x_projected)
else:
self.x_out_projector = Projector(self.xdg_out, self.x_projected)
# setup required functions
self.dq = Function(self.fs)
self.q1 = Function(self.fs)
def pre_apply(self, x_in, discretisation_option):
"""
Extra steps to discretisation if using an embedded method,
which might be either the plain embedded method or the
recovered space scheme.
:arg x_in: the input set of prognostic fields.
:arg discretisation option: string specifying which scheme to use.
"""
if discretisation_option == "embedded_dg":
try:
self.xdg_in.interpolate(x_in)
except NotImplementedError:
self.xdg_in.project(x_in)
elif discretisation_option == "recovered":
self.x_in.assign(x_in)
self.x_rec_projector.project()
self.x_brok_projector.project()
self.xdg_interpolator.interpolate()
def post_apply(self, x_out, discretisation_option):
"""
The projection steps, returning a field to its original space
for an embedded DG scheme. For the case of the
recovered scheme, there are two options dependent on whether
the scheme is limited or not.
:arg x_out: the outgoing field.
:arg discretisation_option: string specifying which option to use.
"""
if discretisation_option == "recovered" and self.limiter is not None:
self.x_brok_interpolator.interpolate()
self.x_out_projector.project()
x_out.assign(self.x_projected)
@abstractproperty
def lhs(self):
l = self.residual.label_map(
lambda t: t.has_label(time_derivative),
map_if_true=replace_subject(self.dq, self.idx),
map_if_false=drop)
return l.form
@abstractproperty
def rhs(self):
r = self.residual.label_map(
all_terms,
map_if_true=replace_subject(self.q1, self.idx))
r = r.label_map(
lambda t: t.has_label(time_derivative),
map_if_false=lambda t: -self.dt*t)
return r.form
def replace_transport_term(self):
"""
This routine allows the default transport term to be replaced with a
different one, specified through the transport options.
This is necessary because when the prognostic equations are declared,
the whole transport
"""
# Extract transport term of equation
old_transport_term_list = self.residual.label_map(
lambda t: t.has_label(transport), map_if_false=drop)
# If there are more transport terms, extract only the one for this variable
if len(old_transport_term_list.terms) > 1:
raise NotImplementedError('Cannot replace transport terms when there are more than one')
# Then we should only have one transport term
old_transport_term = old_transport_term_list.terms[0]
# If the transport term has an ibp label, then it could be replaced
if old_transport_term.has_label(ibp_label) and hasattr(self.options, 'ibp'):
# Do the options specify a different ibp to the old transport term?
if old_transport_term.labels['ibp'] != self.options.ibp:
# Set up a new transport term
field = self.state.fields(self.field_name)
test = TestFunction(self.fs)
# Set up new transport term (depending on the type of transport equation)
if old_transport_term.labels['transport'] == TransportEquationType.advective:
new_transport_term = advection_form(self.state, test, field, ibp=self.options.ibp)
elif old_transport_term.labels['transport'] == TransportEquationType.conservative:
new_transport_term = continuity_form(self.state, test, field, ibp=self.options.ibp)
else:
raise NotImplementedError(f'Replacement of transport term not implemented yet for {old_transport_term.labels["transport"]}')
# Finally, drop the old transport term and add the new one
self.residual = self.residual.label_map(
lambda t: t.has_label(transport), map_if_true=drop)
self.residual += subject(new_transport_term, field)
def replace_transporting_velocity(self, uadv):
# replace the transporting velocity in any terms that contain it
if any([t.has_label(transporting_velocity) for t in self.residual]):
assert uadv is not None
if uadv == "prognostic":
self.residual = self.residual.label_map(
lambda t: t.has_label(transporting_velocity),
map_if_true=lambda t: Term(ufl.replace(
t.form, {t.get(transporting_velocity): split(t.get(subject))[0]}), t.labels)
)
else:
self.residual = self.residual.label_map(
lambda t: t.has_label(transporting_velocity),
map_if_true=lambda t: Term(ufl.replace(
t.form, {t.get(transporting_velocity): uadv}), t.labels)
)
self.residual = transporting_velocity.update_value(self.residual, uadv)
@cached_property
def solver(self):
# setup solver using lhs and rhs defined in derived class
problem = NonlinearVariationalProblem(self.lhs-self.rhs, self.dq, bcs=self.bcs)
solver_name = self.field_name+self.__class__.__name__
return NonlinearVariationalSolver(problem, solver_parameters=self.solver_parameters, options_prefix=solver_name)
@abstractmethod
def apply(self, x_in, x_out):
"""
Function takes x as input, computes L(x) as defined by the equation,
and returns x_out as output.
:arg x: :class:`.Function` object, the input Function.
:arg x_out: :class:`.Function` object, the output Function.
"""
pass
class Boundary_Recoverer(object):
"""
An object that performs a `recovery` process at the domain
boundaries that has second order accuracy. This is necessary
because the :class:`Averager` object does not recover a field
with sufficient accuracy at the boundaries.
The strategy is to minimise the curvature of the function in
the boundary cells, subject to the constraints of conserved
mass and continuity on the interior facets. The quickest way
to perform this is by using the analytic solution and a parloop.
Currently this is only implemented for the (DG0, DG1, CG1)
set of spaces, and only on a `PeriodicIntervalMesh` or
'PeriodicUnitIntervalMesh` that has been extruded.
:arg v_CG1: the continuous function after the first recovery
is performed. Should be in CG1. This is correct
on the interior of the domain.
:arg v_DG1: the function to be output. Should be in DG1.
:arg method: a Boundary_Method Enum object.
:arg coords_to_adjust: a DG1 field containing 1 at locations of
coords that must be adjusted to give they
effective coords.
"""
def __init__(self,
v_CG1,
v_DG1,
method=Boundary_Method.physics,
coords_to_adjust=None):
self.v_DG1 = v_DG1
self.v_CG1 = v_CG1
self.v_DG1_old = Function(v_DG1.function_space())
self.coords_to_adjust = coords_to_adjust
self.method = method
mesh = v_CG1.function_space().mesh()
VDG0 = FunctionSpace(mesh, "DG", 0)
VCG1 = FunctionSpace(mesh, "CG", 1)
VDG1 = FunctionSpace(mesh, "DG", 1)
self.num_ext = Function(VDG0)
# check function spaces of functions
if self.method == Boundary_Method.dynamics:
if v_CG1.function_space() != VCG1:
raise NotImplementedError(
"This boundary recovery method requires v1 to be in CG1.")
if v_DG1.function_space() != VDG1:
raise NotImplementedError(
"This boundary recovery method requires v_out to be in DG1."
)
# check whether mesh is valid
if mesh.topological_dimension() == 2:
# if mesh is extruded then we're fine, but if not needs to be quads
if not VDG0.extruded and mesh.ufl_cell().cellname(
) != 'quadrilateral':
raise NotImplementedError(
'For 2D meshes this recovery method requires that elements are quadrilaterals'
)
elif mesh.topological_dimension() == 3:
# assume that 3D mesh is extruded
if mesh._base_mesh.ufl_cell().cellname() != 'quadrilateral':
raise NotImplementedError(
'For 3D extruded meshes this recovery method requires a base mesh with quadrilateral elements'
)
elif mesh.topological_dimension() != 1:
raise NotImplementedError(
'This boundary recovery is implemented only on certain classes of mesh.'
)
if coords_to_adjust is None:
raise ValueError(
'Need coords_to_adjust field for dynamics boundary methods'
)
elif self.method == Boundary_Method.physics:
# check that mesh is valid -- must be an extruded mesh
if not VDG0.extruded:
raise NotImplementedError(
'The physics boundary method only works on extruded meshes'
)
# base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
w_hori = FiniteElement("DG", cell, 0)
w_vert = FiniteElement("CG", interval, 1)
# build element
theta_element = TensorProductElement(w_hori, w_vert)
# spaces
Vtheta = FunctionSpace(mesh, theta_element)
Vtheta_broken = FunctionSpace(mesh, BrokenElement(theta_element))
if v_CG1.function_space() != Vtheta:
raise ValueError(
"This boundary recovery method requires v_CG1 to be in DG0xCG1 TensorProductSpace."
)
if v_DG1.function_space() != Vtheta_broken:
raise ValueError(
"This boundary recovery method requires v_DG1 to be in the broken DG0xCG1 TensorProductSpace."
)
else:
raise ValueError(
"Boundary method should be a Boundary Method Enum object.")
VuDG1 = VectorFunctionSpace(VDG0.mesh(), "DG", 1)
x = SpatialCoordinate(VDG0.mesh())
self.interpolator = Interpolator(self.v_CG1, self.v_DG1)
if self.method == Boundary_Method.dynamics:
# STRATEGY
# obtain a coordinate field for all the nodes
VuDG1 = VectorFunctionSpace(mesh, "DG", 1)
self.act_coords = Function(VuDG1).project(x) # actual coordinates
self.eff_coords = Function(VuDG1).project(
x) # effective coordinates
shapes = {
"nDOFs":
self.v_DG1.function_space().finat_element.space_dimension(),
"dim":
np.prod(VuDG1.shape, dtype=int)
}
num_ext_domain = ("{{[i]: 0 <= i < {nDOFs}}}").format(**shapes)
num_ext_instructions = ("""
<float64> SUM_EXT = 0
for i
SUM_EXT = SUM_EXT + EXT_V1[i]
end
NUM_EXT[0] = SUM_EXT
""")
coords_domain = ("{{[i, j, k, ii, jj, kk, ll, mm, iii, kkk]: "
"0 <= i < {nDOFs} and "
"0 <= j < {nDOFs} and 0 <= k < {dim} and "
"0 <= ii < {nDOFs} and 0 <= jj < {nDOFs} and "
"0 <= kk < {dim} and 0 <= ll < {dim} and "
"0 <= mm < {dim} and 0 <= iii < {nDOFs} and "
"0 <= kkk < {dim}}}").format(**shapes)
coords_insts = (
"""
<float64> sum_V1_ext = 0
<int> index = 100
<float64> dist = 0.0
<float64> max_dist = 0.0
<float64> min_dist = 0.0
"""
# only do adjustment in cells with at least one DOF to adjust
"""
if NUM_EXT[0] > 0
"""
# find the maximum distance between DOFs in this cell, to serve as starting point for finding min distances
"""
for i
for j
dist = 0.0
for k
dist = dist + pow(ACT_COORDS[i,k] - ACT_COORDS[j,k], 2.0)
end
dist = pow(dist, 0.5) {{id=sqrt_max_dist, dep=*}}
max_dist = fmax(dist, max_dist) {{id=max_dist, dep=sqrt_max_dist}}
end
end
"""
# loop through cells and find which ones to adjust
"""
for ii
if EXT_V1[ii] > 0.5
"""
# find closest interior node
"""
min_dist = max_dist
index = 100
for jj
if EXT_V1[jj] < 0.5
dist = 0.0
for kk
dist = dist + pow(ACT_COORDS[ii,kk] - ACT_COORDS[jj,kk], 2)
end
dist = pow(dist, 0.5)
if dist <= min_dist
index = jj
end
min_dist = fmin(min_dist, dist)
for ll
EFF_COORDS[ii,ll] = 0.5 * (ACT_COORDS[ii,ll] + ACT_COORDS[index,ll])
end
end
end
else
"""
# for DOFs that aren't exterior, use the original coordinates
"""
for mm
EFF_COORDS[ii, mm] = ACT_COORDS[ii, mm]
end
end
end
else
"""
# for interior elements, just use the original coordinates
"""
for iii
for kkk
EFF_COORDS[iii, kkk] = ACT_COORDS[iii, kkk]
end
end
end
""").format(**shapes)
elimin_domain = (
"{{[i, ii_loop, jj_loop, kk, ll_loop, mm, iii_loop, kkk_loop, iiii, iiiii]: "
"0 <= i < {nDOFs} and 0 <= ii_loop < {nDOFs} and "
"ii_loop + 1 <= jj_loop < {nDOFs} and ii_loop <= kk < {nDOFs} and "
"ii_loop + 1 <= ll_loop < {nDOFs} and ii_loop <= mm < {nDOFs} + 1 and "
"0 <= iii_loop < {nDOFs} and {nDOFs} - iii_loop <= kkk_loop < {nDOFs} + 1 and "
"0 <= iiii < {nDOFs} and 0 <= iiiii < {nDOFs}}}").format(
**shapes)
elimin_insts = (
"""
<int> ii = 0
<int> jj = 0
<int> ll = 0
<int> iii = 0
<int> jjj = 0
<int> i_max = 0
<float64> A_max = 0.0
<float64> temp_f = 0.0
<float64> temp_A = 0.0
<float64> c = 0.0
<float64> f[{nDOFs}] = 0.0
<float64> a[{nDOFs}] = 0.0
<float64> A[{nDOFs},{nDOFs}] = 0.0
"""
# We are aiming to find the vector a that solves A*a = f, for matrix A and vector f.
# This is done by performing row operations (swapping and scaling) to obtain A in upper diagonal form.
# N.B. several for loops must be executed in numerical order (loopy does not necessarily do this).
# For these loops we must manually iterate the index.
"""
if NUM_EXT[0] > 0.0
"""
# only do Gaussian elimination for elements with effective coordinates
"""
for i
"""
# fill f with the original field values and A with the effective coordinate values
"""
f[i] = DG1_OLD[i]
A[i,0] = 1.0
A[i,1] = EFF_COORDS[i,0]
if {nDOFs} > 3
A[i,2] = EFF_COORDS[i,1]
A[i,3] = EFF_COORDS[i,0]*EFF_COORDS[i,1]
if {nDOFs} > 7
A[i,4] = EFF_COORDS[i,{dim}-1]
A[i,5] = EFF_COORDS[i,0]*EFF_COORDS[i,{dim}-1]
A[i,6] = EFF_COORDS[i,1]*EFF_COORDS[i,{dim}-1]
A[i,7] = EFF_COORDS[i,0]*EFF_COORDS[i,1]*EFF_COORDS[i,{dim}-1]
end
end
end
"""
# now loop through rows/columns of A
"""
for ii_loop
A_max = fabs(A[ii,ii])
i_max = ii
"""
# loop to find the largest value in the ith column
# set i_max as the index of the row with this largest value.
"""
jj = ii + 1
for jj_loop
if fabs(A[jj,ii]) > A_max
i_max = jj
end
A_max = fmax(A_max, fabs(A[jj,ii]))
jj = jj + 1
end
"""
# if the max value in the ith column isn't in the ith row, we must swap the rows
"""
if i_max != ii
"""
# swap the elements of f
"""
temp_f = f[ii] {{id=set_temp_f, dep=*}}
f[ii] = f[i_max] {{id=set_f_imax, dep=set_temp_f}}
f[i_max] = temp_f {{id=set_f_ii, dep=set_f_imax}}
"""
# swap the elements of A
# N.B. kk runs from ii to (nDOFs-1) as elements below diagonal should be 0
"""
for kk
temp_A = A[ii,kk] {{id=set_temp_A, dep=*}}
A[ii, kk] = A[i_max, kk] {{id=set_A_ii, dep=set_temp_A}}
A[i_max, kk] = temp_A {{id=set_A_imax, dep=set_A_ii}}
end
end
"""
# scale the rows below the ith row
"""
ll = ii + 1
for ll_loop
if ll > ii
"""
# find scaling factor
"""
c = - A[ll,ii] / A[ii,ii]
"""
# N.B. mm runs from ii to (nDOFs-1) as elements below diagonal should be 0
"""
for mm
A[ll, mm] = A[ll, mm] + c * A[ii,mm]
end
f[ll] = f[ll] + c * f[ii]
end
ll = ll + 1
end
ii = ii + 1
end
"""
# do back substitution of upper diagonal A to obtain a
"""
iii = 0
for iii_loop
"""
# jjj starts at the bottom row and works upwards
"""
jjj = {nDOFs} - iii - 1 {{id=assign_jjj, dep=*}}
a[jjj] = f[jjj] {{id=set_a, dep=assign_jjj}}
for kkk_loop
a[jjj] = a[jjj] - A[jjj,kkk_loop] * a[kkk_loop]
end
a[jjj] = a[jjj] / A[jjj,jjj]
iii = iii + 1
end
"""
# Having found a, this gives us the coefficients for the Taylor expansion with the actual coordinates.
"""
for iiii
if {nDOFs} == 2
DG1[iiii] = a[0] + a[1]*ACT_COORDS[iiii,0]
elif {nDOFs} == 4
DG1[iiii] = a[0] + a[1]*ACT_COORDS[iiii,0] + a[2]*ACT_COORDS[iiii,1] + a[3]*ACT_COORDS[iiii,0]*ACT_COORDS[iiii,1]
elif {nDOFs} == 8
DG1[iiii] = a[0] + a[1]*ACT_COORDS[iiii,0] + a[2]*ACT_COORDS[iiii,1] + a[3]*ACT_COORDS[iiii,0]*ACT_COORDS[iiii,1] + a[4]*ACT_COORDS[iiii,{dim}-1] + a[5]*ACT_COORDS[iiii,0]*ACT_COORDS[iiii,{dim}-1] + a[6]*ACT_COORDS[iiii,1]*ACT_COORDS[iiii,{dim}-1] + a[7]*ACT_COORDS[iiii,0]*ACT_COORDS[iiii,1]*ACT_COORDS[iiii,{dim}-1]
end
end
"""
# if element is not external, just use old field values.
"""
else
for iiiii
DG1[iiiii] = DG1_OLD[iiiii]
end
end
""").format(**shapes)
_num_ext_kernel = (num_ext_domain, num_ext_instructions)
_eff_coords_kernel = (coords_domain, coords_insts)
self._gaussian_elimination_kernel = (elimin_domain, elimin_insts)
# find number of external DOFs per cell
par_loop(_num_ext_kernel,
dx, {
"NUM_EXT": (self.num_ext, WRITE),
"EXT_V1": (self.coords_to_adjust, READ)
},
is_loopy_kernel=True)
# find effective coordinates
logger.warning(
'Finding effective coordinates for boundary recovery. This could give unexpected results for deformed meshes over very steep topography.'
)
par_loop(_eff_coords_kernel,
dx, {
"EFF_COORDS": (self.eff_coords, WRITE),
"ACT_COORDS": (self.act_coords, READ),
"NUM_EXT": (self.num_ext, READ),
"EXT_V1": (self.coords_to_adjust, READ)
},
is_loopy_kernel=True)
elif self.method == Boundary_Method.physics:
top_bottom_domain = ("{[i]: 0 <= i < 1}")
bottom_instructions = ("""
DG1[0] = 2 * CG1[0] - CG1[1]
DG1[1] = CG1[1]
""")
top_instructions = ("""
DG1[0] = CG1[0]
DG1[1] = -CG1[0] + 2 * CG1[1]
""")
self._bottom_kernel = (top_bottom_domain, bottom_instructions)
self._top_kernel = (top_bottom_domain, top_instructions)
def apply(self):
self.interpolator.interpolate()
if self.method == Boundary_Method.physics:
par_loop(self._bottom_kernel,
dx,
args={
"DG1": (self.v_DG1, WRITE),
"CG1": (self.v_CG1, READ)
},
is_loopy_kernel=True,
iterate=ON_BOTTOM)
par_loop(self._top_kernel,
dx,
args={
"DG1": (self.v_DG1, WRITE),
"CG1": (self.v_CG1, READ)
},
is_loopy_kernel=True,
iterate=ON_TOP)
else:
self.v_DG1_old.assign(self.v_DG1)
par_loop(self._gaussian_elimination_kernel,
dx, {
"DG1_OLD": (self.v_DG1_old, READ),
"DG1": (self.v_DG1, WRITE),
"ACT_COORDS": (self.act_coords, READ),
"EFF_COORDS": (self.eff_coords, READ),
"NUM_EXT": (self.num_ext, READ)
},
is_loopy_kernel=True)
class Boundary_Recoverer(object):
"""
An object that performs a `recovery` process at the domain
boundaries that has second order accuracy. This is necessary
because the :class:`Averager` object does not recover a field
with sufficient accuracy at the boundaries.
The strategy is to minimise the curvature of the function in
the boundary cells, subject to the constraints of conserved
mass and continuity on the interior facets. The quickest way
to perform this is by using the analytic solution and a parloop.
Currently this is only implemented for the (DG0, DG1, CG1)
set of spaces, and only on a `PeriodicIntervalMesh` or
'PeriodicUnitIntervalMesh` that has been extruded.
:arg v_CG1: the continuous function after the first recovery
is performed. Should be in CG1. This is correct
on the interior of the domain.
:arg v_DG1: the function to be output. Should be in DG1.
:arg method: a Boundary_Method Enum object.
:arg eff_coords: the effective coordinates of the iniital recovery.
This must be provided for the dynamics Boundary_Method.
"""
def __init__(self,
v_CG1,
v_DG1,
method=Boundary_Method.physics,
eff_coords=None):
self.v_DG1 = v_DG1
self.v_CG1 = v_CG1
self.v_DG1_old = Function(v_DG1.function_space())
self.eff_coords = eff_coords
self.method = method
mesh = v_CG1.function_space().mesh()
DG0 = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
if DG0.extruded:
cell = mesh._base_mesh.ufl_cell().cellname()
DG1_hori_elt = FiniteElement("DG", cell, 1, variant="equispaced")
DG1_vert_elt = FiniteElement("DG",
interval,
1,
variant="equispaced")
DG1_element = TensorProductElement(DG1_hori_elt, DG1_vert_elt)
else:
cell = mesh.ufl_cell().cellname()
DG1_element = FiniteElement("DG", cell, 1, variant="equispaced")
DG1 = FunctionSpace(mesh, DG1_element)
self.num_ext = find_domain_boundaries(mesh)
# check function spaces of functions
if self.method == Boundary_Method.dynamics:
if v_CG1.function_space() != CG1:
raise NotImplementedError(
"This boundary recovery method requires v1 to be in CG1.")
if v_DG1.function_space() != DG1:
raise NotImplementedError(
"This boundary recovery method requires v_out to be in DG1."
)
if eff_coords is None:
raise ValueError(
'Need eff_coords field for dynamics boundary methods')
elif self.method == Boundary_Method.physics:
# check that mesh is valid -- must be an extruded mesh
if not DG0.extruded:
raise NotImplementedError(
'The physics boundary method only works on extruded meshes'
)
# base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
w_hori = FiniteElement("DG", cell, 0, variant="equispaced")
w_vert = FiniteElement("CG", interval, 1, variant="equispaced")
# build element
theta_element = TensorProductElement(w_hori, w_vert)
# spaces
Vtheta = FunctionSpace(mesh, theta_element)
Vtheta_broken = FunctionSpace(mesh, BrokenElement(theta_element))
if v_CG1.function_space() != Vtheta:
raise ValueError(
"This boundary recovery method requires v_CG1 to be in DG0xCG1 TensorProductSpace."
)
if v_DG1.function_space() != Vtheta_broken:
raise ValueError(
"This boundary recovery method requires v_DG1 to be in the broken DG0xCG1 TensorProductSpace."
)
else:
raise ValueError(
"Boundary method should be a Boundary Method Enum object.")
vec_DG1 = VectorFunctionSpace(DG0.mesh(), DG1_element)
x = SpatialCoordinate(DG0.mesh())
self.interpolator = Interpolator(self.v_CG1, self.v_DG1)
if self.method == Boundary_Method.dynamics:
# STRATEGY
# obtain a coordinate field for all the nodes
self.act_coords = Function(vec_DG1).project(
x) # actual coordinates
self.eff_coords = eff_coords # effective coordinates
self.output = Function(DG1)
self.on_exterior = find_domain_boundaries(mesh)
self.gaussian_elimination_kernel = kernels.GaussianElimination(DG1)
elif self.method == Boundary_Method.physics:
self.bottom_kernel = kernels.PhysicsRecoveryBottom()
self.top_kernel = kernels.PhysicsRecoveryTop()
def apply(self):
self.interpolator.interpolate()
if self.method == Boundary_Method.physics:
self.bottom_kernel.apply(self.v_DG1, self.v_CG1)
self.top_kernel.apply(self.v_DG1, self.v_CG1)
else:
self.v_DG1_old.assign(self.v_DG1)
self.gaussian_elimination_kernel.apply(self.v_DG1_old, self.v_DG1,
self.act_coords,
self.eff_coords,
self.num_ext)