from sympy import symbols, legendre, Array, diff
import numpy as np
from FIAT.finite_element import FiniteElement
from FIAT.dual_set import make_entity_closure_ids
from FIAT.polynomial_set import mis
from FIAT.reference_element import compute_unflattening_map, flatten_reference_cube
x, y, z = symbols('x y z')
variables = (x, y, z)
leg = legendre
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def triangular_number(n):
return int((n+1)*n/2)
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def choose_ijk_total(degree):
top = 1
for i in range(1, 2 + degree + 1):
top = i * top
bottom = 1
for i in range(1, degree + 1):
bottom = i * bottom
return int(top / (2 * bottom))
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class TrimmedSerendipity(FiniteElement):
def __init__(self, ref_el, degree, mapping):
if degree < 1:
raise Exception("Trimmed serendipity elements only valid for k >= 1")
flat_el = flatten_reference_cube(ref_el)
dim = flat_el.get_spatial_dimension()
self.fdim = dim
if dim != 3:
if dim != 2:
raise Exception("Trimmed serendipity elements only valid for dimensions 2 and 3")
flat_topology = flat_el.get_topology()
entity_ids = {}
cur = 0
for top_dim, entities in flat_topology.items():
entity_ids[top_dim] = {}
for entity in entities:
entity_ids[top_dim][entity] = []
# 3-d case.
if dim == 3:
entity_ids[3] = {}
for j in sorted(flat_topology[2]):
entity_ids[2][j] = list(range(cur, cur + triangular_number(degree)))
cur = cur + triangular_number(degree)
interior_ids = 0
for k in range(2, degree):
interior_ids = interior_ids + 3 * choose_ijk_total(k - 2)
if (degree > 1):
interior_tilde_ids = 3
for k in range(1, degree - 1):
interior_tilde_ids = interior_tilde_ids + 3
if (degree == 4):
interior_tilde_ids += choose_ijk_total(degree - 2) - (degree - 1) - (degree - 1) + 1
if (degree > 4):
interior_tilde_ids += choose_ijk_total(degree - 2) - (degree - 1) - (degree - 1) + 1
if degree == 1:
interior_tilde_ids = 0
entity_ids[3][0] = list(range(cur, cur + interior_ids + interior_tilde_ids))
cur = cur + interior_ids + interior_tilde_ids
else:
for j in sorted(flat_topology[1]):
entity_ids[1][j] = list(range(cur, cur + degree))
cur = cur + degree
if (degree >= 2):
entity_ids[2][0] = list(range(cur, cur + 2*triangular_number(degree - 2) + degree))
cur += 2*triangular_number(degree - 2) + degree
formdegree = dim - 1
entity_closure_ids = make_entity_closure_ids(flat_el, entity_ids)
super().__init__(ref_el=ref_el,
dual=None,
order=degree,
formdegree=formdegree,
mapping=mapping)
topology = ref_el.get_topology()
unflattening_map = compute_unflattening_map(topology)
unflattened_entity_ids = {}
unflattened_entity_closure_ids = {}
for dim, entities in sorted(topology.items()):
unflattened_entity_ids[dim] = {}
unflattened_entity_closure_ids[dim] = {}
for dim, entities in sorted(flat_topology.items()):
for entity in entities:
unflat_dim, unflat_entity = unflattening_map[(dim, entity)]
unflattened_entity_ids[unflat_dim][unflat_entity] = entity_ids[dim][entity]
unflattened_entity_closure_ids[unflat_dim][unflat_entity] = entity_closure_ids[dim][entity]
self.entity_ids = unflattened_entity_ids
self.entity_closure_ids = unflattened_entity_closure_ids
self._degree = degree
self.flat_el = flat_el
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def degree(self):
return self._degree
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def get_nodal_basis(self):
raise NotImplementedError("get_nodal_basis not implemented for trimmed serendipity")
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def get_dual_set(self):
raise NotImplementedError("get_dual_set is not implemented for trimmed serendipity")
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def get_coeffs(self):
raise NotImplementedError("get_coeffs not implemented for trimmed serendipity")
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def tabulate(self, order, points, entity=None):
if entity is None:
entity = (self.ref_el.get_dimension(), 0)
entity_dim, entity_id = entity
transform = self.ref_el.get_entity_transform(entity_dim, entity_id)
points = transform(points)
phivals = {}
for o in range(order+1):
alphas = mis(self.fdim, o)
for alpha in alphas:
try:
polynomials = self.basis[alpha]
except KeyError:
zr = tuple([0] * self.fdim)
polynomials = diff(self.basis[zr], *zip(variables, alpha))
self.basis[alpha] = polynomials
T = np.zeros((len(polynomials[:, 0]), self.fdim, len(points)))
for i in range(len(points)):
subs = {v: points[i][k] for k, v in enumerate(variables[:self.fdim])}
for ell in range(self.fdim):
for j, f in enumerate(polynomials[:, ell]):
T[j, ell, i] = f.evalf(subs=subs)
phivals[alpha] = T
return phivals
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def entity_dofs(self):
"""Return the map of topological entities to degrees of
freedom for the finite element."""
return self.entity_ids
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def entity_closure_dofs(self):
"""Return the map of topological entities to degrees of
freedom on the closure of those entities for the finite element."""
return self.entity_closure_ids
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def value_shape(self):
return (self.fdim,)
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def dmats(self):
raise NotImplementedError
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def get_num_members(self, arg):
raise NotImplementedError
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def space_dimension(self):
return int(len(self.basis[tuple([0] * self.fdim)])/self.fdim)
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class TrimmedSerendipityDiv(TrimmedSerendipity):
def __init__(self, ref_el, degree):
if degree < 1:
raise Exception("Trimmed serendipity face elements only valid for k >= 1")
flat_el = flatten_reference_cube(ref_el)
dim = flat_el.get_spatial_dimension()
if dim != 2:
if dim != 3:
raise Exception("Trimmed serendipity face elements only valid for dimensions 2 and 3")
verts = flat_el.get_vertices()
dx = ((verts[-1][0] - x)/(verts[-1][0] - verts[0][0]), (x - verts[0][0])/(verts[-1][0] - verts[0][0]))
dy = ((verts[-1][1] - y)/(verts[-1][1] - verts[0][1]), (y - verts[0][1])/(verts[-1][1] - verts[0][1]))
x_mid = 2*x-(verts[-1][0] + verts[0][0])
y_mid = 2*y-(verts[-1][1] + verts[0][1])
try:
dz = ((verts[-1][2] - z)/(verts[-1][2] - verts[0][2]), (z - verts[0][2])/(verts[-1][2] - verts[0][2]))
z_mid = 2*z-(verts[-1][2] + verts[0][2])
except IndexError:
dz = None
z_mid = None
if dim == 3:
FL = f_lambda_2_3d(degree, dx, dy, dz, x_mid, y_mid, z_mid)
if (degree > 1):
IL = I_lambda_2_3d(degree, dx, dy, dz, x_mid, y_mid, z_mid)
else:
IL = ()
Sminus_list = FL + IL
self.basis = {(0, 0, 0): Array(Sminus_list)}
super().__init__(ref_el=ref_el, degree=degree, mapping="contravariant piola")
else:
# Put all 2 dimensional stuff here.
if degree < 1:
raise Exception("Trimmed serendipity face elements only valid for k >= 1")
EL = e_lambda_1_2d_part_one(degree, dx, dy, x_mid, y_mid)
if degree >= 2:
FL = trimmed_f_lambda_2d(degree, dx, dy, x_mid, y_mid)
else:
FL = ()
Sminus_list = EL + FL
Sminus_list = [[-a[1], a[0]] for a in Sminus_list]
self.basis = {(0, 0): Array(Sminus_list)}
super().__init__(ref_el=ref_el, degree=degree, mapping="contravariant piola")
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def f_lambda_2_3d(degree, dx, dy, dz, x_mid, y_mid, z_mid):
FL = tuple([(-leg(j, y_mid) * leg(k, z_mid) * a, 0, 0)
for a in dx for k in range(0, degree) for j in range(0, degree - k)] +
[(0, leg(j, x_mid) * leg(k, z_mid) * b, 0)
for b in dy for k in range(0, degree) for j in range(0, degree - k)] +
[(0, 0, -leg(j, x_mid) * leg(k, y_mid) * c)
for c in dz for k in range(0, degree) for j in range(0, degree - k)])
return FL
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def I_lambda_2_3d_pieces(current_deg, dx, dy, dz, x_mid, y_mid, z_mid):
assert current_deg > 1, 'invalid for i = 1'
ILpiece = tuple([])
for j in range(0, current_deg - 1):
for k in range(0, current_deg - 1 - j):
ILpiece += tuple([(0, 0, -leg(j, x_mid) * leg(k, y_mid) * leg(current_deg - 2 - j - k, z_mid) *
dz[0] * dz[1])] +
[(0, -leg(j, x_mid) * leg(k, y_mid) * leg(current_deg - 2 - j - k, z_mid) *
dy[0] * dy[1], 0)] +
[(-leg(j, x_mid) * leg(k, y_mid) * leg(current_deg - 2 - j - k, z_mid) * dx[0] *
dx[1], 0, 0)])
return ILpiece
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def I_lambda_2_3d_tilde(degree, dx, dy, dz, x_mid, y_mid, z_mid):
assert degree > 1, 'invalid for i = 1'
IL_tilde = tuple([(0, 0, leg(degree - 2, z_mid) * dz[0] * dz[1])] +
[(0, leg(degree - 2, y_mid) * dy[0] * dy[1], 0)] +
[(leg(degree - 2, x_mid) * dx[0] * dx[1], 0, 0)])
IL_tilde += tuple([(leg(degree - j - 2, x_mid) * leg(j, y_mid) * dx[0] * dx[1], leg(degree - j - 1, x_mid) *
leg(j - 1, y_mid) * dy[0] * dy[1], 0) for j in range(1, degree - 1)] +
[(leg(degree - j - 2, x_mid) * leg(j, z_mid) * dx[0] * dx[1], 0, leg(degree - j - 1, x_mid) *
leg(j - 1, z_mid) * dz[0] * dz[1]) for j in range(1, degree - 1)] +
[(0, leg(degree - j - 2, y_mid) * leg(j, z_mid) * dy[0] * dy[1], leg(degree - j - 1, y_mid) *
leg(j - 1, z_mid) * dz[0] * dz[1]) for j in range(1, degree - 1)])
for k in range(1, degree - 2):
for l in range(1, degree - 1 - k):
j = degree - 2 - k - l
IL_tilde += tuple([(-leg(j, x_mid) * leg(k, y_mid) * leg(l, z_mid) * dx[0] * dx[1],
leg(j + 1, x_mid) * leg(k - 1, y_mid) * leg(l, z_mid) * dy[0] * dy[1],
-leg(j + 1, x_mid) * leg(k, y_mid) * leg(l - 1, z_mid) * dz[0] * dz[1])])
return IL_tilde
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def I_lambda_2_3d(degree, dx, dy, dz, x_mid, y_mid, z_mid):
IL = tuple([])
for j in range(2, degree):
IL += I_lambda_2_3d_pieces(j, dx, dy, dz, x_mid, y_mid, z_mid)
IL += I_lambda_2_3d_tilde(degree, dx, dy, dz, x_mid, y_mid, z_mid)
return IL
# Everything for 2-d should work already.
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def e_lambda_1_2d_part_one(deg, dx, dy, x_mid, y_mid):
EL = tuple(
[(0, -leg(j, y_mid) * dx[0]) for j in range(deg)] +
[(0, -leg(j, y_mid) * dx[1]) for j in range(deg)] +
[(-leg(j, x_mid)*dy[0], 0) for j in range(deg)] +
[(-leg(j, x_mid)*dy[1], 0) for j in range(deg)])
return EL
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def e_lambda_tilde_1_2d_part_two(deg, dx, dy, x_mid, y_mid):
ELTilde = tuple([(-leg(deg, x_mid) * dy[0],
-leg(deg-1, x_mid) * dx[0] * dx[1] / (deg+1))] +
[(-leg(deg, x_mid) * dy[1],
leg(deg-1, x_mid) * dx[0] * dx[1] / (deg+1))] +
[(-leg(deg-1, y_mid) * dy[0] * dy[1] / (deg+1),
-leg(deg, y_mid) * dx[0])] +
[(leg(deg-1, y_mid) * dy[0] * dy[1] / (deg+1),
-leg(deg, y_mid) * dx[1])])
return ELTilde
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def e_lambda_1_2d(deg, dx, dy, x_mid, y_mid):
EL = e_lambda_1_2d_part_one(deg, dx, dy, x_mid, y_mid)
ELTilde = e_lambda_tilde_1_2d_part_two(deg, dx, dy, x_mid, y_mid)
result = EL + ELTilde
return result
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def determine_f_lambda_portions_2d(deg):
if (deg < 2):
DegsOfIteration = []
else:
DegsOfIteration = []
for i in range(2, deg):
DegsOfIteration += [i]
return DegsOfIteration
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def f_lambda_1_2d_pieces(current_deg, dx, dy, x_mid, y_mid):
if (current_deg == 2):
FLpiece = [(leg(0, x_mid) * leg(0, y_mid) * dy[0] * dy[1], 0)]
FLpiece += [(0, leg(0, x_mid) * leg(0, y_mid) * dx[0] * dx[1])]
else:
target_power = current_deg - 2
FLpiece = tuple([])
for j in range(0, target_power + 1):
k = target_power - j
FLpiece += tuple([(leg(j, x_mid) * leg(k, y_mid) * dy[0] * dy[1], 0)])
FLpiece += tuple([(0, leg(j, x_mid) * leg(k, y_mid) * dx[0] * dx[1])])
return FLpiece
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def f_lambda_1_2d_trim(deg, dx, dy, x_mid, y_mid):
DegsOfIteration = determine_f_lambda_portions_2d(deg)
FL = []
for i in DegsOfIteration:
FL += f_lambda_1_2d_pieces(i, dx, dy, x_mid, y_mid)
return tuple(FL)
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def f_lambda_1_2d_tilde(deg, dx, dy, x_mid, y_mid):
FLTilde = tuple([])
FLTilde += tuple([(leg(deg - 2, y_mid)*dy[0]*dy[1], 0)])
FLTilde += tuple([(0, leg(deg - 2, x_mid)*dx[0]*dx[1])])
for k in range(1, deg - 1):
FLTilde += tuple([(leg(k, x_mid) * leg(deg - k - 2, y_mid) * dy[0] * dy[1], -leg(k - 1, x_mid) * leg(deg - k - 1, y_mid) * dx[0] * dx[1])])
return tuple(FLTilde)
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def F_lambda_1_2d(deg, dx, dy, x_mid, y_mid):
FL = []
for k in range(2, deg):
for j in range(k-1):
FL += [(0, leg(j, x_mid)*leg(k-2-j, y_mid)*dx[0]*dx[1])]
FL += [(leg(k-2-j, x_mid)*leg(j, y_mid)*dy[0]*dy[1], 0)]
return tuple(FL)
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def trimmed_f_lambda_2d(deg, dx, dy, x_mid, y_mid):
FL = F_lambda_1_2d(deg, dx, dy, x_mid, y_mid)
FLT = f_lambda_1_2d_tilde(deg, dx, dy, x_mid, y_mid)
result = FL + FLT
return result