def build_hdiv_space(self, family, degree):
if self.extruded_mesh:
if not self._initialised_base_spaces:
self.build_base_spaces(family, degree)
Vh_elt = HDiv(TensorProductElement(self.S1, self.T1))
Vt_elt = TensorProductElement(self.S2, self.T0)
Vv_elt = HDiv(Vt_elt)
V_elt = Vh_elt + Vv_elt
else:
cell = self.mesh.ufl_cell().cellname()
V_elt = FiniteElement(family, cell, degree + 1)
return FunctionSpace(self.mesh, V_elt, name='HDiv')
def test_physics_recovery_kernels(boundary):
m = IntervalMesh(3, 3)
mesh = ExtrudedMesh(m, layers=3, layer_height=1.0)
cell = m.ufl_cell().cellname()
hori_elt = FiniteElement("DG", cell, 0)
vert_elt = FiniteElement("CG", interval, 1)
theta_elt = TensorProductElement(hori_elt, vert_elt)
Vt = FunctionSpace(mesh, theta_elt)
Vt_brok = FunctionSpace(mesh, BrokenElement(theta_elt))
initial_field = Function(Vt)
true_field = Function(Vt_brok)
new_field = Function(Vt_brok)
initial_field, true_field = setup_values(boundary, initial_field,
true_field)
kernel = kernels.PhysicsRecoveryTop(
) if boundary == "top" else kernels.PhysicsRecoveryBottom()
kernel.apply(new_field, initial_field)
tolerance = 1e-12
index = 11 if boundary == "top" else 6
assert abs(true_field.dat.data[index] - new_field.dat.data[index]) < tolerance, \
"Value at %s from physics recovery is not correct" % boundary
def build_theta_space(self, degree):
assert self.extruded_mesh
if not self._initialised_base_spaces:
cell = self.mesh._base_mesh.ufl_cell().cellname()
self.S2 = FiniteElement("DG", cell, degree)
self.T0 = FiniteElement("CG", interval, degree + 1)
V_elt = TensorProductElement(self.S2, self.T0)
return FunctionSpace(self.mesh, V_elt, name='Vtheta')
def _build_spaces(self, mesh, vertical_degree, horizontal_degree, family):
"""
Build:
velocity space self.V2,
pressure space self.V3,
temperature space self.Vt,
mixed function space self.W = (V2,V3,Vt)
"""
self.spaces = SpaceCreator()
if vertical_degree is not None:
# horizontal base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
S1 = FiniteElement(family, cell, horizontal_degree+1)
S2 = FiniteElement("DG", cell, horizontal_degree)
# vertical base spaces
T0 = FiniteElement("CG", interval, vertical_degree+1)
T1 = FiniteElement("DG", interval, vertical_degree)
# build spaces V2, V3, Vt
V2h_elt = HDiv(TensorProductElement(S1, T1))
V2t_elt = TensorProductElement(S2, T0)
V3_elt = TensorProductElement(S2, T1)
V2v_elt = HDiv(V2t_elt)
V2_elt = V2h_elt + V2v_elt
V0 = self.spaces("HDiv", mesh, V2_elt)
V1 = self.spaces("DG", mesh, V3_elt)
V2 = self.spaces("HDiv_v", mesh, V2t_elt)
self.Vv = self.spaces("Vv", mesh, V2v_elt)
self.W = MixedFunctionSpace((V0, V1, V2))
else:
cell = mesh.ufl_cell().cellname()
V1_elt = FiniteElement(family, cell, horizontal_degree+1)
V0 = self.spaces("HDiv", mesh, V1_elt)
V1 = self.spaces("DG", mesh, "DG", horizontal_degree)
self.W = MixedFunctionSpace((V0, V1))
def __init__(self, space):
"""
Initialise limiter
:arg space: the space in which the transported variables lies.
It should be a form of the DG1xCG2 space.
"""
if not space.extruded:
raise ValueError(
'The Theta Limiter can only be used on an extruded mesh')
# check that horizontal degree is 1 and vertical degree is 2
sub_elements = space.ufl_element().sub_elements()
if (sub_elements[0].family() not in ['Discontinuous Lagrange', 'DQ']
or sub_elements[1].family() != 'Lagrange'
or space.ufl_element().degree() != (1, 2)):
raise ValueError(
'Theta Limiter should only be used with the DG1xCG2 space')
# Transport will happen in broken form of Vtheta
mesh = space.mesh()
self.Vt_brok = FunctionSpace(mesh, BrokenElement(space.ufl_element()))
# Create equispaced DG1 space needed for limiting
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")
CG2_vert_elt = FiniteElement("CG", interval, 2)
DG1_element = TensorProductElement(DG1_hori_elt, DG1_vert_elt)
Vt_element = TensorProductElement(DG1_hori_elt, CG2_vert_elt)
DG1_equispaced = FunctionSpace(mesh, DG1_element)
Vt_equispaced = FunctionSpace(mesh, Vt_element)
Vt_brok_equispaced = FunctionSpace(
mesh, BrokenElement(Vt_equispaced.ufl_element()))
self.vertex_limiter = VertexBasedLimiter(DG1_equispaced)
self.field_hat = Function(Vt_brok_equispaced)
self.field_old = Function(Vt_brok_equispaced)
self.field_DG1 = Function(DG1_equispaced)
self._limit_midpoints_kernel = LimitMidpoints(Vt_brok_equispaced)
def get_functionspace(mesh,
h_family,
h_degree,
v_family=None,
v_degree=None,
vector=False,
hdiv=False,
variant=None,
v_variant=None,
**kwargs):
cell_dim = mesh.cell_dimension()
print(cell_dim)
assert cell_dim in [2, (2, 1), (1, 1)], 'Unsupported cell dimension'
hdiv_families = [
'RT',
'RTF',
'RTCF',
'RAVIART-THOMAS',
'BDM',
'BDMF',
'BDMCF',
'BREZZI-DOUGLAS-MARINI',
]
if variant is None:
if h_family.upper() in hdiv_families:
if h_family in ['RTCF', 'BDMCF']:
variant = 'equispaced'
else:
variant = 'integral'
else:
print("var = equi")
variant = 'equispaced'
if v_variant is None:
v_variant = 'equispaced'
if cell_dim == (2, 1) or (1, 1):
if v_family is None:
v_family = h_family
if v_degree is None:
v_degree = h_degree
h_cell, v_cell = mesh.ufl_cell().sub_cells()
h_elt = FiniteElement(h_family, h_cell, h_degree, variant=variant)
v_elt = FiniteElement(v_family, v_cell, v_degree, variant=v_variant)
elt = TensorProductElement(h_elt, v_elt)
if hdiv:
elt = ufl.HDiv(elt)
else:
elt = FiniteElement(h_family,
mesh.ufl_cell(),
h_degree,
variant=variant)
constructor = VectorFunctionSpace if vector else FunctionSpace
return constructor(mesh, elt, **kwargs)
def build_dg_space(self, degree, variant=None):
if self.extruded_mesh:
if not self._initialised_base_spaces or self.T1.degree(
) != degree or self.T1.variant() != variant:
cell = self.mesh._base_mesh.ufl_cell().cellname()
S2 = FiniteElement("DG", cell, degree, variant=variant)
T1 = FiniteElement("DG", interval, degree, variant=variant)
else:
S2 = self.S2
T1 = self.T1
V_elt = TensorProductElement(S2, T1)
else:
cell = self.mesh.ufl_cell().cellname()
V_elt = FiniteElement("DG", cell, degree, variant=variant)
name = f'DG{degree}_equispaced' if variant == 'equispaced' else f'DG{degree}'
return FunctionSpace(self.mesh, V_elt, name=name)
def correct_eff_coords(eff_coords):
"""
Correct the effective coordinates calculated by simply averaging
which will not be correct at periodic boundaries.
:arg eff_coords: the effective coordinates in vec_DG1 space.
"""
mesh = eff_coords.function_space().mesh()
vec_CG1 = VectorFunctionSpace(mesh, "CG", 1)
if vec_CG1.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")
vec_DG1 = VectorFunctionSpace(mesh, DG1_element)
x = SpatialCoordinate(mesh)
if eff_coords.function_space() != vec_DG1:
raise ValueError('eff_coords needs to be in the vector DG1 space')
# obtain different coords in DG1
DG1_coords = Function(vec_DG1).interpolate(x)
CG1_coords_from_DG1 = Function(vec_CG1)
averager = Averager(DG1_coords, CG1_coords_from_DG1)
averager.project()
DG1_coords_from_averaged_CG1 = Function(vec_DG1).interpolate(
CG1_coords_from_DG1)
DG1_coords_diff = Function(vec_DG1).interpolate(
DG1_coords - DG1_coords_from_averaged_CG1)
# interpolate coordinates, adjusting those different coordinates
adjusted_coords = Function(vec_DG1)
adjusted_coords.interpolate(eff_coords + DG1_coords_diff)
return adjusted_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 __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 get_latlon_mesh(mesh):
coords_orig = mesh.coordinates
coords_fs = coords_orig.function_space()
if coords_fs.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_elt = TensorProductElement(DG1_hori_elt, DG1_vert_elt)
else:
cell = mesh.ufl_cell().cellname()
DG1_elt = FiniteElement("DG", cell, 1, variant="equispaced")
vec_DG1 = VectorFunctionSpace(mesh, DG1_elt)
coords_dg = Function(vec_DG1).interpolate(coords_orig)
coords_latlon = Function(vec_DG1)
shapes = {"nDOFs": vec_DG1.finat_element.space_dimension(), 'dim': 3}
radius = np.min(
np.sqrt(coords_dg.dat.data[:, 0]**2 + coords_dg.dat.data[:, 1]**2 +
coords_dg.dat.data[:, 2]**2))
# lat-lon 'x' = atan2(y, x)
coords_latlon.dat.data[:, 0] = np.arctan2(coords_dg.dat.data[:, 1],
coords_dg.dat.data[:, 0])
# lat-lon 'y' = asin(z/sqrt(x^2 + y^2 + z^2))
coords_latlon.dat.data[:, 1] = np.arcsin(
coords_dg.dat.data[:, 2] /
np.sqrt(coords_dg.dat.data[:, 0]**2 + coords_dg.dat.data[:, 1]**2 +
coords_dg.dat.data[:, 2]**2))
# our vertical coordinate is radius - the minimum radius
coords_latlon.dat.data[:,
2] = np.sqrt(coords_dg.dat.data[:, 0]**2 +
coords_dg.dat.data[:, 1]**2 +
coords_dg.dat.data[:, 2]**2) - radius
# We need to ensure that all points in a cell are on the same side of the branch cut in longitude coords
# This kernel amends the longitude coords so that all longitudes in one cell are close together
kernel = op2.Kernel(
"""
#define PI 3.141592653589793
#define TWO_PI 6.283185307179586
void splat_coords(double *coords) {{
double max_diff = 0.0;
double diff = 0.0;
for (int i=0; i<{nDOFs}; i++) {{
for (int j=0; j<{nDOFs}; j++) {{
diff = coords[i*{dim}] - coords[j*{dim}];
if (fabs(diff) > max_diff) {{
max_diff = diff;
}}
}}
}}
if (max_diff > PI) {{
for (int i=0; i<{nDOFs}; i++) {{
if (coords[i*{dim}] < 0) {{
coords[i*{dim}] += TWO_PI;
}}
}}
}}
}}
""".format(**shapes), "splat_coords")
op2.par_loop(kernel, coords_latlon.cell_set,
coords_latlon.dat(op2.RW, coords_latlon.cell_node_map()))
return Mesh(coords_latlon)
def find_eff_coords(V0):
"""
Takes a function in a field V0 and returns the effective coordinates,
in a vector DG1 space, of a recovery into a CG1 field. This is for use with the
Boundary_Recoverer, as it facilitates the Gaussian elimination used to get
second-order recovery at boundaries.
If V0 is a vector function space, this returns an array of coordinates for
each component.
:arg V0: the original function space.
"""
mesh = V0.mesh()
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")
vec_CG1 = VectorFunctionSpace(mesh, "CG", 1)
vec_DG1 = VectorFunctionSpace(mesh, DG1_element)
x = SpatialCoordinate(mesh)
if V0.ufl_element().value_size() > 1:
eff_coords_list = []
V0_coords_list = []
# treat this separately for each component
for i in range(V0.ufl_element().value_size()):
# fill an d-dimensional list with i-th coordinate
x_list = [x[i] for j in range(V0.ufl_element().value_size())]
# the i-th element in V0_coords_list is a vector with all components the i-th coord
ith_V0_coords = Function(V0).project(as_vector(x_list))
V0_coords_list.append(ith_V0_coords)
for i in range(V0.ufl_element().value_size()):
# slice through V0_coords_list to obtain the coords of the DOFs for that component
x_list = [V0_coords[i] for V0_coords in V0_coords_list]
# average these to find effective coords in CG1
V0_coords_in_DG1 = Function(vec_DG1).interpolate(as_vector(x_list))
eff_coords_in_CG1 = Function(vec_CG1)
eff_coords_averager = Averager(V0_coords_in_DG1, eff_coords_in_CG1)
eff_coords_averager.project()
# obtain these in DG1
eff_coords_in_DG1 = Function(vec_DG1).interpolate(
eff_coords_in_CG1)
eff_coords_list.append(correct_eff_coords(eff_coords_in_DG1))
return eff_coords_list
else:
# find the coordinates at DOFs in V0
vec_V0 = VectorFunctionSpace(mesh, V0.ufl_element())
V0_coords = Function(vec_V0).project(x)
# average these to find effective coords in CG1
V0_coords_in_DG1 = Function(vec_DG1).interpolate(V0_coords)
eff_coords_in_CG1 = Function(vec_CG1)
eff_coords_averager = Averager(V0_coords_in_DG1, eff_coords_in_CG1)
eff_coords_averager.project()
# obtain these in DG1
eff_coords_in_DG1 = Function(vec_DG1).interpolate(eff_coords_in_CG1)
return correct_eff_coords(eff_coords_in_DG1)
def test_3D_cartesian_recovery(geometry, element, mesh, expr):
family = "RTCF" if element == "quadrilateral" else "BDM"
# horizontal base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
u_hori = FiniteElement(family, cell, 1)
w_hori = FiniteElement("DG", cell, 0)
# vertical base spaces
u_vert = FiniteElement("DG", interval, 0)
w_vert = FiniteElement("CG", interval, 1)
# build elements
u_element = HDiv(TensorProductElement(u_hori, u_vert))
w_element = HDiv(TensorProductElement(w_hori, w_vert))
theta_element = TensorProductElement(w_hori, w_vert)
v_element = u_element + w_element
# DG1
DG1_hori = FiniteElement("DG", cell, 1, variant="equispaced")
DG1_vert = FiniteElement("DG", interval, 1, variant="equispaced")
DG1_elt = TensorProductElement(DG1_hori, DG1_vert)
DG1 = FunctionSpace(mesh, DG1_elt)
vec_DG1 = VectorFunctionSpace(mesh, DG1_elt)
# spaces
DG0 = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
Vt = FunctionSpace(mesh, theta_element)
Vt_brok = FunctionSpace(mesh, BrokenElement(theta_element))
Vu = FunctionSpace(mesh, v_element)
vec_CG1 = VectorFunctionSpace(mesh, "CG", 1)
# our actual theta and rho and v
rho_CG1_true = Function(CG1).interpolate(expr)
theta_CG1_true = Function(CG1).interpolate(expr)
v_CG1_true = Function(vec_CG1).interpolate(as_vector([expr, expr, expr]))
rho_Vt_true = Function(Vt).interpolate(expr)
# make the initial fields by projecting expressions into the lowest order spaces
rho_DG0 = Function(DG0).interpolate(expr)
rho_CG1 = Function(CG1)
theta_Vt = Function(Vt).interpolate(expr)
theta_CG1 = Function(CG1)
v_Vu = Function(Vu).project(as_vector([expr, expr, expr]))
v_CG1 = Function(vec_CG1)
rho_Vt = Function(Vt)
# make the recoverers and do the recovery
rho_recoverer = Recoverer(rho_DG0, rho_CG1, VDG=DG1, boundary_method=Boundary_Method.dynamics)
theta_recoverer = Recoverer(theta_Vt, theta_CG1, VDG=DG1, boundary_method=Boundary_Method.dynamics)
v_recoverer = Recoverer(v_Vu, v_CG1, VDG=vec_DG1, boundary_method=Boundary_Method.dynamics)
rho_Vt_recoverer = Recoverer(rho_DG0, rho_Vt, VDG=Vt_brok, boundary_method=Boundary_Method.physics)
rho_recoverer.project()
theta_recoverer.project()
v_recoverer.project()
rho_Vt_recoverer.project()
rho_diff = errornorm(rho_CG1, rho_CG1_true) / norm(rho_CG1_true)
theta_diff = errornorm(theta_CG1, theta_CG1_true) / norm(theta_CG1_true)
v_diff = errornorm(v_CG1, v_CG1_true) / norm(v_CG1_true)
rho_Vt_diff = errornorm(rho_Vt, rho_Vt_true) / norm(rho_Vt_true)
tolerance = 1e-7
error_message = ("""
Incorrect recovery for {variable} with {boundary} boundary method
on {geometry} 3D Cartesian domain with {element} elements
""")
assert rho_diff < tolerance, error_message.format(variable='rho', boundary='dynamics',
geometry=geometry, element=element)
assert v_diff < tolerance, error_message.format(variable='v', boundary='dynamics',
geometry=geometry, element=element)
assert theta_diff < tolerance, error_message.format(variable='rho', boundary='dynamics',
geometry=geometry, element=element)
assert rho_Vt_diff < tolerance, error_message.format(variable='rho', boundary='physics',
geometry=geometry, element=element)
def setup_3d_recovery(dirname):
L = 100.
H = 10.
W = 1.
deltax = L / 5.
deltay = W / 5.
deltaz = H / 5.
nlayers = int(H / deltaz)
ncolumnsx = int(L / deltax)
ncolumnsy = int(W / deltay)
m = RectangleMesh(ncolumnsx, ncolumnsy, L, W, quadrilateral=True)
mesh = ExtrudedMesh(m, layers=nlayers, layer_height=H / nlayers)
x, y, z = SpatialCoordinate(mesh)
# horizontal base spaces
cell = mesh._base_mesh.ufl_cell().cellname()
u_hori = FiniteElement("RTCF", cell, 1)
w_hori = FiniteElement("DG", cell, 0)
# vertical base spaces
u_vert = FiniteElement("DG", interval, 0)
w_vert = FiniteElement("CG", interval, 1)
# build elements
u_element = HDiv(TensorProductElement(u_hori, u_vert))
w_element = HDiv(TensorProductElement(w_hori, w_vert))
theta_element = TensorProductElement(w_hori, w_vert)
v_element = u_element + w_element
# spaces
VDG0 = FunctionSpace(mesh, "DG", 0)
VCG1 = FunctionSpace(mesh, "CG", 1)
VDG1 = FunctionSpace(mesh, "DG", 1)
Vt = FunctionSpace(mesh, theta_element)
Vt_brok = FunctionSpace(mesh, BrokenElement(theta_element))
Vu = FunctionSpace(mesh, v_element)
VuCG1 = VectorFunctionSpace(mesh, "CG", 1)
VuDG1 = VectorFunctionSpace(mesh, "DG", 1)
# set up initial conditions
np.random.seed(0)
expr = np.random.randn(
) + np.random.randn() * x + np.random.randn() * y + np.random.randn(
) * z + np.random.randn() * x * y + np.random.randn(
) * x * z + np.random.randn() * y * z + np.random.randn() * x * y * z
# our actual theta and rho and v
rho_CG1_true = Function(VCG1).interpolate(expr)
theta_CG1_true = Function(VCG1).interpolate(expr)
v_CG1_true = Function(VuCG1).interpolate(as_vector([expr, expr, expr]))
rho_Vt_true = Function(Vt).interpolate(expr)
# make the initial fields by projecting expressions into the lowest order spaces
rho_DG0 = Function(VDG0).interpolate(expr)
rho_CG1 = Function(VCG1)
theta_Vt = Function(Vt).interpolate(expr)
theta_CG1 = Function(VCG1)
v_Vu = Function(Vu).project(as_vector([expr, expr, expr]))
v_CG1 = Function(VuCG1)
rho_Vt = Function(Vt)
# make the recoverers and do the recovery
rho_recoverer = Recoverer(rho_DG0,
rho_CG1,
VDG=VDG1,
boundary_method=Boundary_Method.dynamics)
theta_recoverer = Recoverer(theta_Vt,
theta_CG1,
VDG=VDG1,
boundary_method=Boundary_Method.dynamics)
v_recoverer = Recoverer(v_Vu,
v_CG1,
VDG=VuDG1,
boundary_method=Boundary_Method.dynamics)
rho_Vt_recoverer = Recoverer(rho_DG0,
rho_Vt,
VDG=Vt_brok,
boundary_method=Boundary_Method.physics)
rho_recoverer.project()
theta_recoverer.project()
v_recoverer.project()
rho_Vt_recoverer.project()
rho_diff = errornorm(rho_CG1, rho_CG1_true) / norm(rho_CG1_true)
theta_diff = errornorm(theta_CG1, theta_CG1_true) / norm(theta_CG1_true)
v_diff = errornorm(v_CG1, v_CG1_true) / norm(v_CG1_true)
rho_Vt_diff = errornorm(rho_Vt, rho_Vt_true) / norm(rho_Vt_true)
return (rho_diff, theta_diff, v_diff, rho_Vt_diff)
def find_coords_to_adjust(V0, DG1):
"""
This function finds the coordinates that need to be adjusted
for the recovery at the boundary. These are assigned by a 1,
while all coordinates to be left unchanged are assigned a 0.
This field is returned as a DG1 field.
Fields can be scalar or vector.
:arg V0: the space of the original field (before recovery).
:arg DG1: a DG1 space, in which the boundary recovery will happen.
"""
# check that spaces are correct
mesh = DG1.mesh()
if DG1.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")
scalar_DG1 = FunctionSpace(mesh, DG1_element)
vector_DG1 = VectorFunctionSpace(mesh, DG1_element)
# check DG1 field is correct
if type(DG1.ufl_element()) == VectorElement:
if DG1 != vector_DG1:
raise ValueError(
'The function space entered as vector DG1 is not vector DG1.')
elif DG1 != scalar_DG1:
raise ValueError('The function space entered as DG1 is not DG1.')
# STRATEGY
# We need to pass the boundary recoverer a field denoting the location
# of nodes on the boundary, which denotes the coordinates to adjust to be new effective
# coords. This field will be 1 for these coords and 0 otherwise.
# How do we do this?
# 1. Obtain a DG1 field which is 1 at all exterior DOFs by applying Dirichlet
# boundary conditions. i.e. for cells in the bottom right corner of a domain:
# ------- 0 ------- 0 ------- 1
# | | ||
# | | ||
# | | ||
# ======= 1 ======= 1 ======= 1
# 2. Obtain a field in DG1 that is 1 at exterior DOFs adjacent to the exterior
# DOFs of V0 (i.e. the original space). For V0=DG0 there will be no exterior
# DOFs, but could be if velocity is in RT or if there is a temperature space.
# This is done by applying topological boundary conditions to a field in V0,
# before interpolating these into DG1.
# For instance, marking V0 DOFs with x, for rho and theta spaces this would give
# ------- 0 ------- 0 ------- 0 ---x--- 0 ---x--- 0 ---x--- 0
# | | || | | ||
# x | x | x || | | ||
# | | || | | ||
# ======= 0 ======= 0 ======= 0 ===x=== 1 ===x=== 1 ===x=== 1
# 3. Obtain a field that is 1 at corners in 2D or along edges in 3D.
# We do this by using that corners in 2D and edges in 3D are intersections
# of edges/faces respectively. In 2D, this means that if a field which is 1 on a
# horizontal edge is summed with a field that is 1 on a vertical edge, the
# corner value will be 2. Subtracting the exterior DG1 field from step 1 leaves
# a field that is 1 in the corner. This is generalised to 3D.
# ------- 0 ------- 0 ------- 0 ------- 1 ------- 0 ------- 1 ------- 0 ------- 0
# | || | || | || | ||
# | || + | || - | || = | ||
# | || | || | || | ||
# ======= 1 ======= 1 ======= 0 ======= 1 ======= 1 ======= 1 ======= 0 ======= 1
# 4. The field of coords to be adjusted is then found by the following formula:
# f1 + f3 - f2
# where f1, f2 and f3 are the DG1 fields obtained from steps 1, 2 and 3.
# make DG1 field with 1 at all exterior coords
all_ext_in_DG1 = Function(DG1)
bcs = [DirichletBC(DG1, Constant(1.0), "on_boundary", method="geometric")]
if DG1.extruded:
bcs.append(DirichletBC(DG1, Constant(1.0), "top", method="geometric"))
bcs.append(
DirichletBC(DG1, Constant(1.0), "bottom", method="geometric"))
for bc in bcs:
bc.apply(all_ext_in_DG1)
# make DG1 field with 1 at coords surrounding exterior coords of V0
# first do topological BCs to get V0 function which is 1 at DOFs on edges
all_ext_in_V0 = Function(V0)
bcs = [DirichletBC(V0, Constant(1.0), "on_boundary", method="topological")]
if V0.extruded:
bcs.append(DirichletBC(V0, Constant(1.0), "top", method="topological"))
bcs.append(
DirichletBC(V0, Constant(1.0), "bottom", method="topological"))
for bc in bcs:
bc.apply(all_ext_in_V0)
if DG1.value_size > 1:
# for vector valued functions, DOFs aren't pointwise evaluation. We break into components and use a conditional interpolation to get values of 1
V0_ext_in_DG1_components = []
for i in range(DG1.value_size):
V0_ext_in_DG1_components.append(
Function(scalar_DG1).interpolate(
conditional(abs(all_ext_in_V0[i]) > 0.0, 1.0, 0.0)))
V0_ext_in_DG1 = Function(DG1).project(
as_vector(V0_ext_in_DG1_components))
else:
# for scalar functions (where DOFs are pointwise evaluation) we can simply interpolate to get these values
V0_ext_in_DG1 = Function(DG1).interpolate(all_ext_in_V0)
corners_in_DG1 = Function(DG1)
if DG1.mesh().topological_dimension() == 2:
if DG1.extruded:
DG1_ext_hori = Function(DG1)
DG1_ext_vert = Function(DG1)
hori_bcs = [
DirichletBC(DG1, Constant(1.0), "top", method="geometric"),
DirichletBC(DG1, Constant(1.0), "bottom", method="geometric")
]
vert_bc = DirichletBC(DG1,
Constant(1.0),
"on_boundary",
method="geometric")
for bc in hori_bcs:
bc.apply(DG1_ext_hori)
vert_bc.apply(DG1_ext_vert)
corners_in_DG1.assign(DG1_ext_hori + DG1_ext_vert - all_ext_in_DG1)
else:
# we don't know whether its periodic or in how many directions
DG1_ext_x = Function(DG1)
DG1_ext_y = Function(DG1)
x_bcs = [
DirichletBC(DG1, Constant(1.0), 1, method="geometric"),
DirichletBC(DG1, Constant(1.0), 2, method="geometric")
]
y_bcs = [
DirichletBC(DG1, Constant(1.0), 3, method="geometric"),
DirichletBC(DG1, Constant(1.0), 4, method="geometric")
]
# there is no easy way to know if the mesh is periodic or in which
# directions, so we must use a try here
# LookupError is the error for asking for a boundary number that doesn't exist
try:
for bc in x_bcs:
bc.apply(DG1_ext_x)
except LookupError:
pass
try:
for bc in y_bcs:
bc.apply(DG1_ext_y)
except LookupError:
pass
corners_in_DG1.assign(DG1_ext_x + DG1_ext_y - all_ext_in_DG1)
elif DG1.mesh().topological_dimension() == 3:
DG1_vert_x = Function(DG1)
DG1_vert_y = Function(DG1)
DG1_hori = Function(DG1)
x_bcs = [
DirichletBC(DG1, Constant(1.0), 1, method="geometric"),
DirichletBC(DG1, Constant(1.0), 2, method="geometric")
]
y_bcs = [
DirichletBC(DG1, Constant(1.0), 3, method="geometric"),
DirichletBC(DG1, Constant(1.0), 4, method="geometric")
]
hori_bcs = [
DirichletBC(DG1, Constant(1.0), "top", method="geometric"),
DirichletBC(DG1, Constant(1.0), "bottom", method="geometric")
]
# there is no easy way to know if the mesh is periodic or in which
# directions, so we must use a try here
# LookupError is the error for asking for a boundary number that doesn't exist
try:
for bc in x_bcs:
bc.apply(DG1_vert_x)
except LookupError:
pass
try:
for bc in y_bcs:
bc.apply(DG1_vert_y)
except LookupError:
pass
for bc in hori_bcs:
bc.apply(DG1_hori)
corners_in_DG1.assign(DG1_vert_x + DG1_vert_y + DG1_hori -
all_ext_in_DG1)
# we now combine the different functions. We use max_value to avoid getting 2s or 3s at corners/edges
# we do this component-wise because max_value only works component-wise
if DG1.value_size > 1:
coords_to_correct_components = []
for i in range(DG1.value_size):
coords_to_correct_components.append(
Function(scalar_DG1).interpolate(
max_value(corners_in_DG1[i],
all_ext_in_DG1[i] - V0_ext_in_DG1[i])))
coords_to_correct = Function(DG1).project(
as_vector(coords_to_correct_components))
else:
coords_to_correct = Function(DG1).interpolate(
max_value(corners_in_DG1, all_ext_in_DG1 - V0_ext_in_DG1))
return coords_to_correct
def setup_hori_limiters(dirname):
# declare grid shape
L = 400.
H = L
ncolumns = int(L / 10.)
nlayers = ncolumns
# make mesh
m = PeriodicIntervalMesh(ncolumns, L)
mesh = ExtrudedMesh(m, layers=nlayers, layer_height=(H / nlayers))
x, z = SpatialCoordinate(mesh)
fieldlist = ['u']
timestepping = TimesteppingParameters(dt=1.0, maxk=4, maxi=1)
output = OutputParameters(dirname=dirname + "/limiting_hori",
dumpfreq=5,
dumplist=['u'],
perturbation_fields=['theta0', 'theta1'])
parameters = CompressibleParameters()
state = State(mesh,
vertical_degree=1,
horizontal_degree=1,
family="CG",
timestepping=timestepping,
output=output,
parameters=parameters,
fieldlist=fieldlist)
# make elements
# v is continuous in vertical, h is horizontal
cell = mesh._base_mesh.ufl_cell().cellname()
DG0_element = FiniteElement("DG", cell, 0)
CG1_element = FiniteElement("CG", cell, 1)
DG1_element = FiniteElement("DG", cell, 1)
CG2_element = FiniteElement("CG", cell, 2)
V0_element = TensorProductElement(DG0_element, CG1_element)
V1_element = TensorProductElement(DG1_element, CG2_element)
# spaces
Vpsi = FunctionSpace(mesh, "CG", 2)
VDG1 = FunctionSpace(mesh, "DG", 1)
VCG1 = FunctionSpace(mesh, "CG", 1)
V0 = FunctionSpace(mesh, V0_element)
V1 = FunctionSpace(mesh, V1_element)
V0_brok = FunctionSpace(mesh, BrokenElement(V0.ufl_element()))
V0_spaces = (VDG1, VCG1, V0_brok)
# declare initial fields
u0 = state.fields("u")
theta0 = state.fields("theta0", V0)
theta1 = state.fields("theta1", V1)
# make a gradperp
gradperp = lambda u: as_vector([-u.dx(1), u.dx(0)])
# Isentropic background state
Tsurf = 300.
thetab = Constant(Tsurf)
theta_b1 = Function(V1).interpolate(thetab)
theta_b0 = Function(V0).interpolate(thetab)
# set up bubble
xc = 200.
zc = 200.
rc = 100.
theta_expr = conditional(
sqrt((x - xc)**2.0) < rc,
conditional(sqrt((z - zc)**2.0) < rc, Constant(2.0), Constant(0.0)),
Constant(0.0))
theta_pert1 = Function(V1).interpolate(theta_expr)
theta_pert0 = Function(V0).interpolate(theta_expr)
# set up velocity field
u_max = Constant(10.0)
psi_expr = -u_max * z
psi0 = Function(Vpsi).interpolate(psi_expr)
u0.project(gradperp(psi0))
theta0.interpolate(theta_b0 + theta_pert0)
theta1.interpolate(theta_b1 + theta_pert1)
state.initialise([('u', u0), ('theta1', theta1), ('theta0', theta0)])
state.set_reference_profiles([('theta1', theta_b1), ('theta0', theta_b0)])
# set up advection schemes
thetaeqn1 = EmbeddedDGAdvection(state, V1, equation_form="advective")
thetaeqn0 = EmbeddedDGAdvection(state,
V0,
equation_form="advective",
recovered_spaces=V0_spaces)
# build advection dictionary
advected_fields = []
advected_fields.append(('u', NoAdvection(state, u0, None)))
advected_fields.append(('theta1',
SSPRK3(state,
theta1,
thetaeqn1,
limiter=ThetaLimiter(thetaeqn1))))
advected_fields.append(('theta0',
SSPRK3(state,
theta0,
thetaeqn0,
limiter=VertexBasedLimiter(VDG1))))
# build time stepper
stepper = AdvectionDiffusion(state, advected_fields)
return stepper, 40.0
def setup_limiters(dirname, direction, grid_params, ic_params):
# declare grid shape
L, H, ncolumns, nlayers = grid_params
# make mesh
m = PeriodicIntervalMesh(ncolumns, L)
mesh = ExtrudedMesh(m, layers=nlayers, layer_height=(H / nlayers))
x, z = SpatialCoordinate(mesh)
fieldlist = ['u']
timestepping = TimesteppingParameters(dt=1.0)
output = OutputParameters(dirname=dirname,
dumpfreq=5,
dumplist=['u'],
perturbation_fields=['theta0', 'theta1'])
parameters = CompressibleParameters()
state = State(mesh,
vertical_degree=1,
horizontal_degree=1,
family="CG",
timestepping=timestepping,
output=output,
parameters=parameters,
fieldlist=fieldlist)
# make elements
# v is continuous in vertical, h is horizontal
cell = mesh._base_mesh.ufl_cell().cellname()
DG0_element = FiniteElement("DG", cell, 0)
CG1_element = FiniteElement("CG", cell, 1)
DG1_element = FiniteElement("DG", cell, 1)
CG2_element = FiniteElement("CG", cell, 2)
V0_element = TensorProductElement(DG0_element, CG1_element)
V1_element = TensorProductElement(DG1_element, CG2_element)
# spaces
VDG1 = FunctionSpace(mesh, "DG", 1)
VCG1 = FunctionSpace(mesh, "CG", 1)
V0 = FunctionSpace(mesh, V0_element)
V1 = FunctionSpace(mesh, V1_element)
V0_brok = FunctionSpace(mesh, BrokenElement(V0.ufl_element()))
# declare initial fields
u0 = state.fields("u")
theta0 = state.fields("theta0", V0)
theta1 = state.fields("theta1", V1)
Tsurf, xc, zc, rc, u_max = ic_params
# Isentropic background state
thetab = Constant(Tsurf)
theta_b1 = Function(V1).interpolate(thetab)
theta_b0 = Function(V0).interpolate(thetab)
# set up bubble
theta_expr = conditional(
sqrt((x - xc)**2.0) < rc,
conditional(sqrt((z - zc)**2.0) < rc, Constant(2.0), Constant(0.0)),
Constant(0.0))
theta_pert1 = Function(V1).interpolate(theta_expr)
theta_pert0 = Function(V0).interpolate(theta_expr)
if direction == "horizontal":
Vpsi = FunctionSpace(mesh, "CG", 2)
psi_expr = -u_max * z
psi0 = Function(Vpsi).interpolate(psi_expr)
gradperp = lambda u: as_vector([-u.dx(1), u.dx(0)])
u0.project(gradperp(psi0))
elif direction == "vertical":
u0.project(as_vector([0, -u_max]))
theta0.interpolate(theta_b0 + theta_pert0)
theta1.interpolate(theta_b1 + theta_pert1)
state.initialise([('u', u0), ('theta1', theta1), ('theta0', theta0)])
state.set_reference_profiles([('theta1', theta_b1), ('theta0', theta_b0)])
# set up advection schemes
dg_opts = EmbeddedDGOptions()
recovered_opts = RecoveredOptions(embedding_space=VDG1,
recovered_space=VCG1,
broken_space=V0_brok)
thetaeqn1 = EmbeddedDGAdvection(state,
V1,
equation_form="advective",
options=dg_opts)
thetaeqn0 = EmbeddedDGAdvection(state,
V0,
equation_form="advective",
options=recovered_opts)
# build advection dictionary
advected_fields = []
advected_fields.append(
('theta1', SSPRK3(state, theta1, thetaeqn1, limiter=ThetaLimiter(V1))))
advected_fields.append(('theta0',
SSPRK3(state,
theta0,
thetaeqn0,
limiter=VertexBasedLimiter(VDG1))))
# build time stepper
stepper = AdvectionDiffusion(state, advected_fields)
return stepper, 40.0
def setup_limiters(dirname):
dt = 0.01
Ld = 1.
tmax = 0.2
rotations = 0.1
m = PeriodicIntervalMesh(20, Ld)
mesh = ExtrudedMesh(m, layers=20, layer_height=(Ld / 20))
output = OutputParameters(
dirname=dirname,
dumpfreq=1,
dumplist=['u', 'chemical', 'moisture_higher', 'moisture_lower'])
parameters = CompressibleParameters()
timestepping = TimesteppingParameters(dt=dt, maxk=4, maxi=1)
fieldlist = [
'u', 'rho', 'theta', 'chemical', 'moisture_higher', 'moisture_lower'
]
diagnostic_fields = []
state = State(mesh,
vertical_degree=1,
horizontal_degree=1,
family="CG",
timestepping=timestepping,
output=output,
parameters=parameters,
fieldlist=fieldlist,
diagnostic_fields=diagnostic_fields)
x, z = SpatialCoordinate(mesh)
Vr = state.spaces("DG")
Vt = state.spaces("HDiv_v")
Vpsi = FunctionSpace(mesh, "CG", 2)
cell = mesh._base_mesh.ufl_cell().cellname()
DG0_element = FiniteElement("DG", cell, 0)
CG1_element = FiniteElement("CG", interval, 1)
Vt0_element = TensorProductElement(DG0_element, CG1_element)
Vt0 = FunctionSpace(mesh, Vt0_element)
Vt0_brok = FunctionSpace(mesh, BrokenElement(Vt0_element))
VCG1 = FunctionSpace(mesh, "CG", 1)
u = state.fields("u", dump=True)
chemical = state.fields("chemical", Vr, dump=True)
moisture_higher = state.fields("moisture_higher", Vt, dump=True)
moisture_lower = state.fields("moisture_lower", Vt0, dump=True)
x_lower = 2 * Ld / 5
x_upper = 3 * Ld / 5
z_lower = 6 * Ld / 10
z_upper = 8 * Ld / 10
bubble_expr_1 = conditional(
x > x_lower,
conditional(
x < x_upper,
conditional(z > z_lower, conditional(z < z_upper, 1.0, 0.0), 0.0),
0.0), 0.0)
bubble_expr_2 = conditional(
x > z_lower,
conditional(
x < z_upper,
conditional(z > x_lower, conditional(z < x_upper, 1.0, 0.0), 0.0),
0.0), 0.0)
chemical.assign(1.0)
moisture_higher.assign(280.)
chem_pert_1 = Function(Vr).interpolate(bubble_expr_1)
chem_pert_2 = Function(Vr).interpolate(bubble_expr_2)
moist_h_pert_1 = Function(Vt).interpolate(bubble_expr_1)
moist_h_pert_2 = Function(Vt).interpolate(bubble_expr_2)
moist_l_pert_1 = Function(Vt0).interpolate(bubble_expr_1)
moist_l_pert_2 = Function(Vt0).interpolate(bubble_expr_2)
chemical.assign(chemical + chem_pert_1 + chem_pert_2)
moisture_higher.assign(moisture_higher + moist_h_pert_1 + moist_h_pert_2)
moisture_lower.assign(moisture_lower + moist_l_pert_1 + moist_l_pert_2)
# set up solid body rotation for advection
# we do this slightly complicated stream function to make the velocity 0 at edges
# thus we avoid odd effects at boundaries
xc = Ld / 2
zc = Ld / 2
r = sqrt((x - xc)**2 + (z - zc)**2)
omega = rotations * 2 * pi / tmax
r_out = 9 * Ld / 20
r_in = 2 * Ld / 5
A = omega * r_in / (2 * (r_in - r_out))
B = -omega * r_in * r_out / (r_in - r_out)
C = omega * r_in**2 * r_out / (r_in - r_out) / 2
psi_expr = conditional(
r < r_in, omega * r**2 / 2,
conditional(r < r_out, A * r**2 + B * r + C,
A * r_out**2 + B * r_out + C))
psi = Function(Vpsi).interpolate(psi_expr)
gradperp = lambda v: as_vector([-v.dx(1), v.dx(0)])
u.project(gradperp(psi))
state.initialise([('u', u), ('chemical', chemical),
('moisture_higher', moisture_higher),
('moisture_lower', moisture_lower)])
# set up advection schemes
dg_opts = EmbeddedDGOptions()
recovered_opts = RecoveredOptions(embedding_space=Vr,
recovered_space=VCG1,
broken_space=Vt0_brok,
boundary_method=Boundary_Method.dynamics)
chemeqn = AdvectionEquation(state, Vr, equation_form="advective")
moisteqn_higher = EmbeddedDGAdvection(state,
Vt,
equation_form="advective",
options=dg_opts)
moisteqn_lower = EmbeddedDGAdvection(state,
Vt0,
equation_form="advective",
options=recovered_opts)
# build advection dictionary
advected_fields = []
advected_fields.append(('chemical',
SSPRK3(state,
chemical,
chemeqn,
limiter=VertexBasedLimiter(Vr))))
advected_fields.append(('moisture_higher',
SSPRK3(state,
moisture_higher,
moisteqn_higher,
limiter=ThetaLimiter(Vt))))
advected_fields.append(('moisture_lower',
SSPRK3(state,
moisture_lower,
moisteqn_lower,
limiter=VertexBasedLimiter(Vr))))
# build time stepper
stepper = AdvectionDiffusion(state, advected_fields)
return stepper, tmax
def initialize(self, pc):
""" Set up the problem context. Takes the original
mixed problem and transforms it into the equivalent
hybrid-mixed system.
A KSP object is created for the Lagrange multipliers
on the top/bottom faces of the mesh cells.
"""
from firedrake import (FunctionSpace, Function, Constant,
FiniteElement, TensorProductElement,
TrialFunction, TrialFunctions, TestFunction,
DirichletBC, interval, MixedElement,
BrokenElement)
from firedrake.assemble import (allocate_matrix,
create_assembly_callable)
from firedrake.formmanipulation import split_form
from ufl.algorithms.replace import replace
from ufl.cell import TensorProductCell
# Extract PC context
prefix = pc.getOptionsPrefix() + "vert_hybridization_"
_, P = pc.getOperators()
self.ctx = P.getPythonContext()
if not isinstance(self.ctx, ImplicitMatrixContext):
raise ValueError(
"The python context must be an ImplicitMatrixContext")
test, trial = self.ctx.a.arguments()
V = test.function_space()
mesh = V.mesh()
# Magically determine which spaces are vector and scalar valued
for i, Vi in enumerate(V):
# Vector-valued spaces will have a non-empty value_shape
if Vi.ufl_element().value_shape():
self.vidx = i
else:
self.pidx = i
Vv = V[self.vidx]
Vp = V[self.pidx]
# Create the space of approximate traces in the vertical.
# NOTE: Technically a hack since the resulting space is technically
# defined in cell interiors, however the degrees of freedom will only
# be geometrically defined on edges. Arguments will only be used in
# surface integrals
deg, _ = Vv.ufl_element().degree()
# Assumes a tensor product cell (quads, triangular-prisms, cubes)
if not isinstance(Vp.ufl_element().cell(), TensorProductCell):
raise NotImplementedError(
"Currently only implemented for tensor product discretizations"
)
# Only want the horizontal cell
cell, _ = Vp.ufl_element().cell()._cells
DG = FiniteElement("DG", cell, deg)
CG = FiniteElement("CG", interval, 1)
Vv_tr_element = TensorProductElement(DG, CG)
Vv_tr = FunctionSpace(mesh, Vv_tr_element)
# Break the spaces
broken_elements = MixedElement(
[BrokenElement(Vi.ufl_element()) for Vi in V])
V_d = FunctionSpace(mesh, broken_elements)
# Set up relevant functions
self.broken_solution = Function(V_d)
self.broken_residual = Function(V_d)
self.trace_solution = Function(Vv_tr)
self.unbroken_solution = Function(V)
self.unbroken_residual = Function(V)
# Set up transfer kernels to and from the broken velocity space
# NOTE: Since this snippet of code is used in a couple places in
# in Gusto, might be worth creating a utility function that is
# is importable and just called where needed.
shapes = {
"i": Vv.finat_element.space_dimension(),
"j": np.prod(Vv.shape, dtype=int)
}
weight_kernel = """
for (int i=0; i<{i}; ++i)
for (int j=0; j<{j}; ++j)
w[i*{j} + j] += 1.0;
""".format(**shapes)
self.weight = Function(Vv)
par_loop(weight_kernel, dx, {"w": (self.weight, INC)})
# Averaging kernel
self.average_kernel = """
for (int i=0; i<{i}; ++i)
for (int j=0; j<{j}; ++j)
vec_out[i*{j} + j] += vec_in[i*{j} + j]/w[i*{j} + j];
""".format(**shapes)
# Original mixed operator replaced with "broken" arguments
arg_map = {test: TestFunction(V_d), trial: TrialFunction(V_d)}
Atilde = Tensor(replace(self.ctx.a, arg_map))
gammar = TestFunction(Vv_tr)
n = FacetNormal(mesh)
sigma = TrialFunctions(V_d)[self.vidx]
# Again, assumes tensor product structure. Why use this if you
# don't have some form of vertical extrusion?
Kform = gammar('+') * jump(sigma, n=n) * dS_h
# Here we deal with boundary conditions
if self.ctx.row_bcs:
# Find all the subdomains with neumann BCS
# These are Dirichlet BCs on the vidx space
neumann_subdomains = set()
for bc in self.ctx.row_bcs:
if bc.function_space().index == self.pidx:
raise NotImplementedError(
"Dirichlet conditions for scalar variable not supported. Use a weak bc."
)
if bc.function_space().index != self.vidx:
raise NotImplementedError(
"Dirichlet bc set on unsupported space.")
# append the set of sub domains
subdom = bc.sub_domain
if isinstance(subdom, str):
neumann_subdomains |= set([subdom])
else:
neumann_subdomains |= set(as_tuple(subdom, int))
# separate out the top and bottom bcs
extruded_neumann_subdomains = neumann_subdomains & {
"top", "bottom"
}
neumann_subdomains = neumann_subdomains - extruded_neumann_subdomains
integrand = gammar * dot(sigma, n)
measures = []
trace_subdomains = []
for subdomain in sorted(extruded_neumann_subdomains):
measures.append({"top": ds_t, "bottom": ds_b}[subdomain])
trace_subdomains.extend(
sorted({"top", "bottom"} - extruded_neumann_subdomains))
measures.extend((ds(sd) for sd in sorted(neumann_subdomains)))
markers = [int(x) for x in mesh.exterior_facets.unique_markers]
dirichlet_subdomains = set(markers) - neumann_subdomains
trace_subdomains.extend(sorted(dirichlet_subdomains))
for measure in measures:
Kform += integrand * measure
else:
trace_subdomains = ["top", "bottom"]
trace_bcs = [
DirichletBC(Vv_tr, Constant(0.0), subdomain)
for subdomain in trace_subdomains
]
# Make a SLATE tensor from Kform
K = Tensor(Kform)
# Assemble the Schur complement operator and right-hand side
self.schur_rhs = Function(Vv_tr)
self._assemble_Srhs = create_assembly_callable(
K * Atilde.inv * AssembledVector(self.broken_residual),
tensor=self.schur_rhs,
form_compiler_parameters=self.ctx.fc_params)
mat_type = PETSc.Options().getString(prefix + "mat_type", "aij")
schur_comp = K * Atilde.inv * K.T
self.S = allocate_matrix(schur_comp,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type,
options_prefix=prefix)
self._assemble_S = create_assembly_callable(
schur_comp,
tensor=self.S,
bcs=trace_bcs,
form_compiler_parameters=self.ctx.fc_params,
mat_type=mat_type)
self._assemble_S()
self.S.force_evaluation()
Smat = self.S.petscmat
nullspace = self.ctx.appctx.get("vert_trace_nullspace", None)
if nullspace is not None:
nsp = nullspace(Vv_tr)
Smat.setNullSpace(nsp.nullspace(comm=pc.comm))
# Set up the KSP for the system of Lagrange multipliers
trace_ksp = PETSc.KSP().create(comm=pc.comm)
trace_ksp.setOptionsPrefix(prefix)
trace_ksp.setOperators(Smat)
trace_ksp.setUp()
trace_ksp.setFromOptions()
self.trace_ksp = trace_ksp
split_mixed_op = dict(split_form(Atilde.form))
split_trace_op = dict(split_form(K.form))
# Generate reconstruction calls
self._reconstruction_calls(split_mixed_op, split_trace_op)
def test_limit_midpoints(profile):
# ------------------------------------------------------------------------ #
# Set up meshes and spaces
# ------------------------------------------------------------------------ #
m = IntervalMesh(3, 3)
mesh = ExtrudedMesh(m, layers=1, layer_height=3.0)
cell = m.ufl_cell().cellname()
DG_hori_elt = FiniteElement("DG", cell, 1, variant='equispaced')
DG_vert_elt = FiniteElement("DG", interval, 1, variant='equispaced')
Vt_vert_elt = FiniteElement("CG", interval, 2)
DG_elt = TensorProductElement(DG_hori_elt, DG_vert_elt)
theta_elt = TensorProductElement(DG_hori_elt, Vt_vert_elt)
Vt_brok = FunctionSpace(mesh, BrokenElement(theta_elt))
DG1 = FunctionSpace(mesh, DG_elt)
new_field = Function(Vt_brok)
init_field = Function(Vt_brok)
DG1_field = Function(DG1)
# ------------------------------------------------------------------------ #
# Initial conditions
# ------------------------------------------------------------------------ #
_, z = SpatialCoordinate(mesh)
if profile == 'undershoot':
# A quadratic whose midpoint is lower than the top and bottom values
init_expr = (80. / 9.) * z**2 - 20. * z + 300.
elif profile == 'overshoot':
# A quadratic whose midpoint is higher than the top and bottom values
init_expr = (-80. / 9.) * z**2 + (100. / 3) * z + 300.
elif profile == 'linear':
# Linear profile which must be unchanged
init_expr = (20. / 3.) * z + 300.
else:
raise NotImplementedError
# Linear DG field has the same values at top and bottom as quadratic
DG_expr = (20. / 3.) * z + 300.
init_field.interpolate(init_expr)
DG1_field.interpolate(DG_expr)
# ------------------------------------------------------------------------ #
# Apply kernel
# ------------------------------------------------------------------------ #
kernel = kernels.LimitMidpoints(Vt_brok)
kernel.apply(new_field, DG1_field, init_field)
# ------------------------------------------------------------------------ #
# Check values
# ------------------------------------------------------------------------ #
tol = 1e-12
assert np.max(new_field.dat.data) <= np.max(init_field.dat.data) + tol, \
'LimitMidpoints kernel is giving an overshoot'
assert np.min(new_field.dat.data) >= np.min(init_field.dat.data) - tol, \
'LimitMidpoints kernel is giving an undershoot'
if profile == 'linear':
assert np.allclose(init_field.dat.data, new_field.dat.data), \
'For a profile with no maxima or minima, the LimitMidpoints ' + \
'kernel should leave the field unchanged'
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 __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 __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')
# check that function spaces are valid
sub_elements = v_CG1.function_space().ufl_element().sub_elements()
if (sub_elements[0].family() not in ['Discontinuous Lagrange', 'DQ']
or sub_elements[1].family() != 'Lagrange'
or v_CG1.function_space().ufl_element().degree() != (0, 1)):
raise ValueError("This boundary recovery method requires v_CG1 to be in DG0xCG1 TensorProductSpace.")
brok_elt = v_DG1.function_space().ufl_element()
if (brok_elt.degree() != (0, 1)
or (type(brok_elt) is not BrokenElement
and (brok_elt.sub_elements[0].family() not in ['Discontinuous Lagrange', 'DQ']
or brok_elt.sub_elements[1].family() != 'Discontinuous Lagrange'))):
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()