def create_quadrature_monkey_patched(ref_el, degree, scheme="default"):
"""Monkey patched FIAT.quadrature_schemes.create_quadrature()
for implementing lumped scheme"""
# Our "special" scheme
if scheme == "lumped":
if isinstance(ref_el, UFCTriangle) and degree == 2:
x = array([[0.0, 0.0], [1.0, 0.0], [0.0, 1.0], [0.5, 0.0],
[0.5, 0.5], [0.0, 0.5]])
w = array([
1 / 12.,
] * 6)
return QuadratureRule(ref_el, x, w)
elif isinstance(ref_el, UFCTetrahedron) and degree == 2:
x = array([[0.0, 0.0, 0.0], [1.0, 0.0, 0.0], [0.0, 1.0, 0.0],
[0.5, 0.0, 0.0], [0.5, 0.5, 0.0], [0.0, 0.5, 0.0],
[0.5, 0.0, 0.5], [0.5, 0.5, 0.5], [0.0, 0.5, 0.5],
[0.0, 0.0, 1.0]])
w = array([
1 / 60.,
] * 10)
return QuadratureRule(ref_el, x, w)
raise NotImplementedError(
"Scheme {} of degree {} on {} not implemented".format(
scheme, degree, ref_el))
# Fallback to FIAT's normal operation
return create_quadrature(ref_el, degree, scheme=scheme)
def __init__(self, ref_el):
entity_ids = {}
nodes = []
cur = 0
# make nodes by getting points
# need to do this dimension-by-dimension, facet-by-facet
top = ref_el.get_topology()
verts = ref_el.get_vertices()
sd = ref_el.get_spatial_dimension()
if ref_el.get_shape() != TRIANGLE:
raise ValueError("Bell only defined on triangles")
pd = functional.PointDerivative
# get jet at each vertex
entity_ids[0] = {}
for v in sorted(top[0]):
nodes.append(functional.PointEvaluation(ref_el, verts[v]))
# first derivatives
for i in range(sd):
alpha = [0] * sd
alpha[i] = 1
nodes.append(pd(ref_el, verts[v], alpha))
# second derivatives
alphas = [[2, 0], [1, 1], [0, 2]]
for alpha in alphas:
nodes.append(pd(ref_el, verts[v], alpha))
entity_ids[0][v] = list(range(cur, cur + 6))
cur += 6
# we need an edge quadrature rule for the moment
from FIAT.quadrature_schemes import create_quadrature
from FIAT.jacobi import eval_jacobi
rline = ufc_simplex(1)
q1d = create_quadrature(rline, 8)
q1dpts = q1d.get_points()
leg4_at_qpts = eval_jacobi(0, 0, 4, 2.0 * q1dpts - 1)
imond = functional.IntegralMomentOfNormalDerivative
entity_ids[1] = {}
for e in sorted(top[1]):
entity_ids[1][e] = [18 + e]
nodes.append(imond(ref_el, e, q1d, leg4_at_qpts))
entity_ids[2] = {0: []}
super(BellDualSet, self).__init__(nodes, ref_el, entity_ids)
def __init__(self, ref_el):
entity_ids = {}
nodes = []
cur = 0
# make nodes by getting points
# need to do this dimension-by-dimension, facet-by-facet
top = ref_el.get_topology()
verts = ref_el.get_vertices()
sd = ref_el.get_spatial_dimension()
if ref_el.get_shape() != TRIANGLE:
raise ValueError("Bell only defined on triangles")
pd = functional.PointDerivative
# get jet at each vertex
entity_ids[0] = {}
for v in sorted(top[0]):
nodes.append(functional.PointEvaluation(ref_el, verts[v]))
# first derivatives
for i in range(sd):
alpha = [0] * sd
alpha[i] = 1
nodes.append(pd(ref_el, verts[v], alpha))
# second derivatives
alphas = [[2, 0], [1, 1], [0, 2]]
for alpha in alphas:
nodes.append(pd(ref_el, verts[v], alpha))
entity_ids[0][v] = list(range(cur, cur + 6))
cur += 6
# we need an edge quadrature rule for the moment
from FIAT.quadrature_schemes import create_quadrature
from FIAT.jacobi import eval_jacobi
rline = ufc_simplex(1)
q1d = create_quadrature(rline, 8)
q1dpts = q1d.get_points()
leg4_at_qpts = eval_jacobi(0, 0, 4, 2.0*q1dpts - 1)
imond = functional.IntegralMomentOfNormalDerivative
entity_ids[1] = {}
for e in sorted(top[1]):
entity_ids[1][e] = [18+e]
nodes.append(imond(ref_el, e, q1d, leg4_at_qpts))
entity_ids[2] = {0: []}
super(BellDualSet, self).__init__(nodes, ref_el, entity_ids)
def test_bernstein_2nd_derivatives():
ref_el = ufc_simplex(2)
degree = 3
elem = Bernstein(ref_el, degree)
rule = create_quadrature(ref_el, degree)
points = rule.get_points()
actual = elem.tabulate(2, points)
assert numpy.allclose(D02, actual[(0, 2)])
assert numpy.allclose(D11, actual[(1, 1)])
assert numpy.allclose(D20, actual[(2, 0)])
def test_bernstein_2nd_derivatives():
ref_el = ufc_simplex(2)
degree = 3
elem = Bernstein(ref_el, degree)
rule = create_quadrature(ref_el, degree)
points = rule.get_points()
actual = elem.tabulate(2, points)
assert numpy.allclose(D02, actual[(0, 2)])
assert numpy.allclose(D11, actual[(1, 1)])
assert numpy.allclose(D20, actual[(2, 0)])
def entity_support_dofs(elem, entity_dim):
"""Return the map of entity id to the degrees of freedom for which the
corresponding basis functions take non-zero values
:arg elem: FIAT finite element
:arg entity_dim: Dimension of the cell subentity.
"""
if not hasattr(elem, "_entity_support_dofs"):
elem._entity_support_dofs = {}
cache = elem._entity_support_dofs
try:
return cache[entity_dim]
except KeyError:
pass
ref_el = elem.get_reference_element()
dim = ref_el.get_spatial_dimension()
entity_cell = ref_el.construct_subelement(entity_dim)
quad = create_quadrature(entity_cell, max(2 * elem.degree(), 1))
weights = quad.get_weights()
eps = 1.e-8 # Is this a safe value?
result = {}
for f in elem.entity_dofs()[entity_dim].keys():
entity_transform = ref_el.get_entity_transform(entity_dim, f)
points = list(map(entity_transform, quad.get_points()))
# Integrate the square of the basis functions on the facet.
vals = numpy.double(elem.tabulate(0, points)[(0, ) * dim])
# Ints contains the square of the basis functions
# integrated over the facet.
if elem.value_shape():
# Vector-valued functions.
ints = numpy.dot(numpy.einsum("...ij,...ij->...j", vals, vals),
weights)
else:
ints = numpy.dot(vals**2, weights)
result[f] = [dof for dof, i in enumerate(ints) if i > eps]
cache[entity_dim] = result
return result
def entity_support_dofs(elem, entity_dim):
"""Return the map of entity id to the degrees of freedom for which the
corresponding basis functions take non-zero values
:arg elem: FIAT finite element
:arg entity_dim: Dimension of the cell subentity.
"""
if not hasattr(elem, "_entity_support_dofs"):
elem._entity_support_dofs = {}
cache = elem._entity_support_dofs
try:
return cache[entity_dim]
except KeyError:
pass
ref_el = elem.get_reference_element()
dim = ref_el.get_spatial_dimension()
entity_cell = ref_el.construct_subelement(entity_dim)
quad = create_quadrature(entity_cell, max(2*elem.degree(), 1))
weights = quad.get_weights()
eps = 1.e-8 # Is this a safe value?
result = {}
for f in elem.entity_dofs()[entity_dim].keys():
entity_transform = ref_el.get_entity_transform(entity_dim, f)
points = list(map(entity_transform, quad.get_points()))
# Integrate the square of the basis functions on the facet.
vals = numpy.double(elem.tabulate(0, points)[(0,) * dim])
# Ints contains the square of the basis functions
# integrated over the facet.
if elem.value_shape():
# Vector-valued functions.
ints = numpy.dot(numpy.einsum("...ij,...ij->...j", vals, vals), weights)
else:
ints = numpy.dot(vals**2, weights)
result[f] = [dof for dof, i in enumerate(ints) if i > eps]
cache[entity_dim] = result
return result