def test_TFE_2Dx1D_scalar_quad():
T = UFCInterval()
P1 = Lagrange(T, 1)
P1_DG = DiscontinuousLagrange(T, 1)
elt = TensorProductElement(TensorProductElement(P1, P1_DG), P1)
assert elt.value_shape() == ()
tab = elt.tabulate(1, [(0.1, 0.2, 0.3)])
tA = P1.tabulate(1, [(0.1, )])
tB = P1_DG.tabulate(1, [(0.2, )])
tC = P1.tabulate(1, [(0.3, )])
for da, db, dc in [[(0, ), (0, ), (0, )], [(1, ), (0, ), (0, )],
[(0, ), (1, ), (0, )], [(0, ), (0, ), (1, )]]:
dd = da + db + dc
assert np.isclose(tab[dd][0][0],
tA[da][0][0] * tB[db][0][0] * tC[dc][0][0])
assert np.isclose(tab[dd][1][0],
tA[da][0][0] * tB[db][0][0] * tC[dc][1][0])
assert np.isclose(tab[dd][2][0],
tA[da][0][0] * tB[db][1][0] * tC[dc][0][0])
assert np.isclose(tab[dd][3][0],
tA[da][0][0] * tB[db][1][0] * tC[dc][1][0])
assert np.isclose(tab[dd][4][0],
tA[da][1][0] * tB[db][0][0] * tC[dc][0][0])
assert np.isclose(tab[dd][5][0],
tA[da][1][0] * tB[db][0][0] * tC[dc][1][0])
assert np.isclose(tab[dd][6][0],
tA[da][1][0] * tB[db][1][0] * tC[dc][0][0])
assert np.isclose(tab[dd][7][0],
tA[da][1][0] * tB[db][1][0] * tC[dc][1][0])
def test_flattened_against_tpe_quad():
T = UFCInterval()
P1 = Lagrange(T, 1)
tpe_quad = TensorProductElement(P1, P1)
flattened_quad = FlattenedDimensions(tpe_quad)
assert tpe_quad.value_shape() == ()
tpe_tab = tpe_quad.tabulate(1, [(0.1, 0.2)])
flattened_tab = flattened_quad.tabulate(1, [(0.1, 0.2)])
for da, db in [[(0,), (0,)], [(1,), (0,)], [(0,), (1,)]]:
dc = da + db
assert np.isclose(tpe_tab[dc][0][0], flattened_tab[dc][0][0])
assert np.isclose(tpe_tab[dc][1][0], flattened_tab[dc][1][0])
assert np.isclose(tpe_tab[dc][2][0], flattened_tab[dc][2][0])
assert np.isclose(tpe_tab[dc][3][0], flattened_tab[dc][3][0])
def test_flattened_against_tpe_quad():
T = UFCInterval()
P1 = Lagrange(T, 1)
tpe_quad = TensorProductElement(P1, P1)
flattened_quad = FlattenedDimensions(tpe_quad)
assert tpe_quad.value_shape() == ()
tpe_tab = tpe_quad.tabulate(1, [(0.1, 0.2)])
flattened_tab = flattened_quad.tabulate(1, [(0.1, 0.2)])
for da, db in [[(0, ), (0, )], [(1, ), (0, )], [(0, ), (1, )]]:
dc = da + db
assert np.isclose(tpe_tab[dc][0][0], flattened_tab[dc][0][0])
assert np.isclose(tpe_tab[dc][1][0], flattened_tab[dc][1][0])
assert np.isclose(tpe_tab[dc][2][0], flattened_tab[dc][2][0])
assert np.isclose(tpe_tab[dc][3][0], flattened_tab[dc][3][0])
def test_TFE_1Dx1D_scalar():
T = UFCInterval()
P1_DG = DiscontinuousLagrange(T, 1)
P2 = Lagrange(T, 2)
elt = TensorProductElement(P1_DG, P2)
assert elt.value_shape() == ()
tab = elt.tabulate(1, [(0.1, 0.2)])
tabA = P1_DG.tabulate(1, [(0.1,)])
tabB = P2.tabulate(1, [(0.2,)])
for da, db in [[(0,), (0,)], [(1,), (0,)], [(0,), (1,)]]:
dc = da + db
assert np.isclose(tab[dc][0][0], tabA[da][0][0]*tabB[db][0][0])
assert np.isclose(tab[dc][1][0], tabA[da][0][0]*tabB[db][1][0])
assert np.isclose(tab[dc][2][0], tabA[da][0][0]*tabB[db][2][0])
assert np.isclose(tab[dc][3][0], tabA[da][1][0]*tabB[db][0][0])
assert np.isclose(tab[dc][4][0], tabA[da][1][0]*tabB[db][1][0])
assert np.isclose(tab[dc][5][0], tabA[da][1][0]*tabB[db][2][0])
def test_TFE_1Dx1D_scalar():
T = UFCInterval()
P1_DG = DiscontinuousLagrange(T, 1)
P2 = Lagrange(T, 2)
elt = TensorProductElement(P1_DG, P2)
assert elt.value_shape() == ()
tab = elt.tabulate(1, [(0.1, 0.2)])
tabA = P1_DG.tabulate(1, [(0.1, )])
tabB = P2.tabulate(1, [(0.2, )])
for da, db in [[(0, ), (0, )], [(1, ), (0, )], [(0, ), (1, )]]:
dc = da + db
assert np.isclose(tab[dc][0][0], tabA[da][0][0] * tabB[db][0][0])
assert np.isclose(tab[dc][1][0], tabA[da][0][0] * tabB[db][1][0])
assert np.isclose(tab[dc][2][0], tabA[da][0][0] * tabB[db][2][0])
assert np.isclose(tab[dc][3][0], tabA[da][1][0] * tabB[db][0][0])
assert np.isclose(tab[dc][4][0], tabA[da][1][0] * tabB[db][1][0])
assert np.isclose(tab[dc][5][0], tabA[da][1][0] * tabB[db][2][0])
def test_flattened_against_tpe_hex():
T = UFCInterval()
P1 = Lagrange(T, 1)
tpe_quad = TensorProductElement(P1, P1)
tpe_hex = TensorProductElement(tpe_quad, P1)
flattened_quad = FlattenedDimensions(tpe_quad)
flattened_hex = FlattenedDimensions(TensorProductElement(flattened_quad, P1))
assert tpe_quad.value_shape() == ()
tpe_tab = tpe_hex.tabulate(1, [(0.1, 0.2, 0.3)])
flattened_tab = flattened_hex.tabulate(1, [(0.1, 0.2, 0.3)])
for da, db, dc in [[(0,), (0,), (0,)], [(1,), (0,), (0,)], [(0,), (1,), (0,)], [(0,), (0,), (1,)]]:
dd = da + db + dc
assert np.isclose(tpe_tab[dd][0][0], flattened_tab[dd][0][0])
assert np.isclose(tpe_tab[dd][1][0], flattened_tab[dd][1][0])
assert np.isclose(tpe_tab[dd][2][0], flattened_tab[dd][2][0])
assert np.isclose(tpe_tab[dd][3][0], flattened_tab[dd][3][0])
assert np.isclose(tpe_tab[dd][4][0], flattened_tab[dd][4][0])
assert np.isclose(tpe_tab[dd][5][0], flattened_tab[dd][5][0])
assert np.isclose(tpe_tab[dd][6][0], flattened_tab[dd][6][0])
assert np.isclose(tpe_tab[dd][7][0], flattened_tab[dd][7][0])
def test_TFE_2Dx1D_scalar_quad():
T = UFCInterval()
P1 = Lagrange(T, 1)
P1_DG = DiscontinuousLagrange(T, 1)
elt = TensorProductElement(TensorProductElement(P1, P1_DG), P1)
assert elt.value_shape() == ()
tab = elt.tabulate(1, [(0.1, 0.2, 0.3)])
tA = P1.tabulate(1, [(0.1,)])
tB = P1_DG.tabulate(1, [(0.2,)])
tC = P1.tabulate(1, [(0.3,)])
for da, db, dc in [[(0,), (0,), (0,)], [(1,), (0,), (0,)], [(0,), (1,), (0,)], [(0,), (0,), (1,)]]:
dd = da + db + dc
assert np.isclose(tab[dd][0][0], tA[da][0][0]*tB[db][0][0]*tC[dc][0][0])
assert np.isclose(tab[dd][1][0], tA[da][0][0]*tB[db][0][0]*tC[dc][1][0])
assert np.isclose(tab[dd][2][0], tA[da][0][0]*tB[db][1][0]*tC[dc][0][0])
assert np.isclose(tab[dd][3][0], tA[da][0][0]*tB[db][1][0]*tC[dc][1][0])
assert np.isclose(tab[dd][4][0], tA[da][1][0]*tB[db][0][0]*tC[dc][0][0])
assert np.isclose(tab[dd][5][0], tA[da][1][0]*tB[db][0][0]*tC[dc][1][0])
assert np.isclose(tab[dd][6][0], tA[da][1][0]*tB[db][1][0]*tC[dc][0][0])
assert np.isclose(tab[dd][7][0], tA[da][1][0]*tB[db][1][0]*tC[dc][1][0])
def test_flattened_against_tpe_hex():
T = UFCInterval()
P1 = Lagrange(T, 1)
tpe_quad = TensorProductElement(P1, P1)
tpe_hex = TensorProductElement(tpe_quad, P1)
flattened_quad = FlattenedDimensions(tpe_quad)
flattened_hex = FlattenedDimensions(
TensorProductElement(flattened_quad, P1))
assert tpe_quad.value_shape() == ()
tpe_tab = tpe_hex.tabulate(1, [(0.1, 0.2, 0.3)])
flattened_tab = flattened_hex.tabulate(1, [(0.1, 0.2, 0.3)])
for da, db, dc in [[(0, ), (0, ), (0, )], [(1, ), (0, ), (0, )],
[(0, ), (1, ), (0, )], [(0, ), (0, ), (1, )]]:
dd = da + db + dc
assert np.isclose(tpe_tab[dd][0][0], flattened_tab[dd][0][0])
assert np.isclose(tpe_tab[dd][1][0], flattened_tab[dd][1][0])
assert np.isclose(tpe_tab[dd][2][0], flattened_tab[dd][2][0])
assert np.isclose(tpe_tab[dd][3][0], flattened_tab[dd][3][0])
assert np.isclose(tpe_tab[dd][4][0], flattened_tab[dd][4][0])
assert np.isclose(tpe_tab[dd][5][0], flattened_tab[dd][5][0])
assert np.isclose(tpe_tab[dd][6][0], flattened_tab[dd][6][0])
assert np.isclose(tpe_tab[dd][7][0], flattened_tab[dd][7][0])
def Hdiv(element):
if not isinstance(element, TensorProductElement):
raise NotImplementedError
if element.A.get_formdegree() is None or element.B.get_formdegree(
) is None:
raise ValueError(
"form degree of sub-element was None (not set during initialisation), Hdiv cannot be done without this information"
)
formdegree = element.A.get_formdegree() + element.B.get_formdegree()
if formdegree != element.get_reference_element().get_spatial_dimension(
) - 1:
raise ValueError("Tried to use Hdiv on a non-(n-1)-form element")
newelement = TensorProductElement(element.A,
element.B) # make a copy to return
# redefine value_shape()
def value_shape(self):
"Return the value shape of the finite element functions."
return (self.get_reference_element().get_spatial_dimension(), )
newelement.value_shape = types.MethodType(value_shape, newelement)
# store old _mapping
newelement._oldmapping = newelement._mapping
# redefine _mapping
newelement._mapping = "contravariant piola"
# store formdegree
newelement.formdegree = formdegree
# redefine tabulate
newelement.old_tabulate = newelement.tabulate
def tabulate(self, order, points, entity=None):
"""Return tabulated values of derivatives up to given order of
basis functions at given points."""
# don't duplicate what the old function does fine...
old_result = self.old_tabulate(order, points, entity)
new_result = {}
sd = self.get_reference_element().get_spatial_dimension()
for alpha in old_result.keys():
temp_old = old_result[alpha]
if self._oldmapping == "affine":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[1]),
dtype=temp_old.dtype)
# both constituents affine, i.e., they were 0 forms or n-forms.
# to sum to n-1, we must have "0-form on an interval" crossed
# with something discontinuous.
# look for the (continuous) 0-form, and put the value there
if self.A.get_formdegree() == 0:
# first element, so (-x, 0, ...)
# Sign flip to ensure that a positive value of the node
# means a vector field having a direction "to the left"
# relative to direction in which the nodes are placed on an
# edge in case of higher-order schemes.
# This is required for unstructured quadrilateral meshes.
temp[:, 0, :] = -temp_old[:, :]
elif self.B.get_formdegree() == 0:
# second element, so (..., 0, x)
temp[:, -1, :] = temp_old[:, :]
else:
raise Exception("Hdiv affine/affine form degrees broke")
elif self._oldmapping == "contravariant piola":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[2]),
dtype=temp_old.dtype)
Asd = self.A.get_reference_element().get_spatial_dimension()
# one component is affine, one is contravariant piola
# the affine one must be an n-form, hence discontinuous
# this component/these components get zeroed out
if element.A.mapping()[0] == "contravariant piola":
# first element, so (x1, ..., xn, 0, ...)
temp[:, :Asd, :] = temp_old[:, :, :]
elif element.B.mapping()[0] == "contravariant piola":
# second element, so (..., 0, x1, ..., xn)
temp[:, Asd:, :] = temp_old[:, :, :]
else:
raise ValueError(
"Hdiv contravariant piola couldn't find an existing ConPi subelement"
)
elif self._oldmapping == "covariant piola":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[2]),
dtype=temp_old.dtype)
# one component is affine, one is covariant piola
# the affine one must be an n-form, hence discontinuous
# this component/these components get zeroed out
# the remaining part gets perped
if element.A.mapping()[0] == "covariant piola":
Asd = self.A.get_reference_element().get_spatial_dimension(
)
if not Asd == 2:
raise ValueError(
"Must be 2d shape to automatically convert covariant to contravariant"
)
temp_perp = numpy.zeros(temp_old.shape,
dtype=temp_old.dtype)
# first element, so (x2, -x1, 0, ...)
temp_perp[:, 0, :] = temp_old[:, 1, :]
temp_perp[:, 1, :] = -temp_old[:, 0, :]
temp[:, :Asd, :] = temp_perp[:, :, :]
elif element.B.mapping()[0] == "covariant piola":
Bsd = self.B.get_reference_element().get_spatial_dimension(
)
if not Bsd == 2:
raise ValueError(
"Must be 2d shape to automatically convert covariant to contravariant"
)
temp_perp = numpy.zeros(temp_old.shape,
dtype=temp_old.dtype)
# second element, so (..., 0, x2, -x1)
temp_perp[:, 0, :] = temp_old[:, 1, :]
temp_perp[:, 1, :] = -temp_old[:, 0, :]
temp[:, Asd:, :] = temp_old[:, :, :]
else:
raise ValueError(
"Hdiv covariant piola couldn't find an existing CovPi subelement"
)
new_result[alpha] = temp
return new_result
newelement.tabulate = types.MethodType(tabulate, newelement)
# splat any PointEvaluation functionals.
# they become a nasty mix of internal and external component DOFs
if newelement._oldmapping == "affine":
oldnodes = newelement.dual.nodes
newnodes = []
for node in oldnodes:
if isinstance(node, functional.PointEvaluation):
newnodes.append(
functional.Functional(None, None, None, {}, "Undefined"))
else:
newnodes.append(node)
newelement.dual.nodes = newnodes
return newelement
def Hdiv(element):
if not isinstance(element, TensorProductElement):
raise NotImplementedError
if element.A.get_formdegree() is None or element.B.get_formdegree() is None:
raise ValueError("form degree of sub-element was None (not set during initialisation), Hdiv cannot be done without this information")
formdegree = element.A.get_formdegree() + element.B.get_formdegree()
if formdegree != element.get_reference_element().get_spatial_dimension() - 1:
raise ValueError("Tried to use Hdiv on a non-(n-1)-form element")
newelement = TensorProductElement(element.A, element.B) # make a copy to return
# redefine value_shape()
def value_shape(self):
"Return the value shape of the finite element functions."
return (self.get_reference_element().get_spatial_dimension(),)
newelement.value_shape = types.MethodType(value_shape, newelement)
# store old _mapping
newelement._oldmapping = newelement._mapping
# redefine _mapping
newelement._mapping = "contravariant piola"
# store formdegree
newelement.formdegree = formdegree
# redefine tabulate
newelement.old_tabulate = newelement.tabulate
def tabulate(self, order, points, entity=None):
"""Return tabulated values of derivatives up to given order of
basis functions at given points."""
# don't duplicate what the old function does fine...
old_result = self.old_tabulate(order, points, entity)
new_result = {}
sd = self.get_reference_element().get_spatial_dimension()
for alpha in old_result.keys():
temp_old = old_result[alpha]
if self._oldmapping == "affine":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[1]), dtype=temp_old.dtype)
# both constituents affine, i.e., they were 0 forms or n-forms.
# to sum to n-1, we must have "0-form on an interval" crossed
# with something discontinuous.
# look for the (continuous) 0-form, and put the value there
if self.A.get_formdegree() == 0:
# first element, so (-x, 0, ...)
# Sign flip to ensure that a positive value of the node
# means a vector field having a direction "to the left"
# relative to direction in which the nodes are placed on an
# edge in case of higher-order schemes.
# This is required for unstructured quadrilateral meshes.
temp[:, 0, :] = -temp_old[:, :]
elif self.B.get_formdegree() == 0:
# second element, so (..., 0, x)
temp[:, -1, :] = temp_old[:, :]
else:
raise Exception("Hdiv affine/affine form degrees broke")
elif self._oldmapping == "contravariant piola":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[2]), dtype=temp_old.dtype)
Asd = self.A.get_reference_element().get_spatial_dimension()
# one component is affine, one is contravariant piola
# the affine one must be an n-form, hence discontinuous
# this component/these components get zeroed out
if element.A.mapping()[0] == "contravariant piola":
# first element, so (x1, ..., xn, 0, ...)
temp[:, :Asd, :] = temp_old[:, :, :]
elif element.B.mapping()[0] == "contravariant piola":
# second element, so (..., 0, x1, ..., xn)
temp[:, Asd:, :] = temp_old[:, :, :]
else:
raise ValueError("Hdiv contravariant piola couldn't find an existing ConPi subelement")
elif self._oldmapping == "covariant piola":
temp = numpy.zeros((temp_old.shape[0], sd, temp_old.shape[2]), dtype=temp_old.dtype)
# one component is affine, one is covariant piola
# the affine one must be an n-form, hence discontinuous
# this component/these components get zeroed out
# the remaining part gets perped
if element.A.mapping()[0] == "covariant piola":
Asd = self.A.get_reference_element().get_spatial_dimension()
if not Asd == 2:
raise ValueError("Must be 2d shape to automatically convert covariant to contravariant")
temp_perp = numpy.zeros(temp_old.shape, dtype=temp_old.dtype)
# first element, so (x2, -x1, 0, ...)
temp_perp[:, 0, :] = temp_old[:, 1, :]
temp_perp[:, 1, :] = -temp_old[:, 0, :]
temp[:, :Asd, :] = temp_perp[:, :, :]
elif element.B.mapping()[0] == "covariant piola":
Bsd = self.B.get_reference_element().get_spatial_dimension()
if not Bsd == 2:
raise ValueError("Must be 2d shape to automatically convert covariant to contravariant")
temp_perp = numpy.zeros(temp_old.shape, dtype=temp_old.dtype)
# second element, so (..., 0, x2, -x1)
temp_perp[:, 0, :] = temp_old[:, 1, :]
temp_perp[:, 1, :] = -temp_old[:, 0, :]
temp[:, Asd:, :] = temp_old[:, :, :]
else:
raise ValueError("Hdiv covariant piola couldn't find an existing CovPi subelement")
new_result[alpha] = temp
return new_result
newelement.tabulate = types.MethodType(tabulate, newelement)
# splat any PointEvaluation functionals.
# they become a nasty mix of internal and external component DOFs
if newelement._oldmapping == "affine":
oldnodes = newelement.dual.nodes
newnodes = []
for node in oldnodes:
if isinstance(node, functional.PointEvaluation):
newnodes.append(functional.Functional(None, None, None, {}, "Undefined"))
else:
newnodes.append(node)
newelement.dual.nodes = newnodes
return newelement