def choosing_element(V, degree):
cell_geometry = V.mesh().ufl_cell()
if cell_geometry == quadrilateral:
T = UFCQuadrilateral()
raise ValueError(
"Point interpolation for quads implemented somewhere else.")
elif cell_geometry == triangle:
T = UFCTriangle()
elif cell_geometry == tetrahedron:
T = UFCTetrahedron()
else:
raise ValueError("Unrecognized cell geometry.")
if V.ufl_element().family() == "Kong-Mulder-Veldhuizen":
element = KMV(T, degree)
elif V.ufl_element().family() == "Lagrange":
element = CG(T, degree)
elif V.ufl_element().family() == "Discontinuous Lagrange":
element = DG(T, degree)
else:
raise ValueError("Function space not yet supported.")
return element
from __future__ import absolute_import, print_function, division
import pytest
import numpy as np
from FIAT.reference_element import UFCInterval, UFCTriangle, UFCTetrahedron
from FIAT.reference_element import FiredrakeQuadrilateral, TensorProductCell
from tsfc.fem import make_cell_facet_jacobian
interval = UFCInterval()
triangle = UFCTriangle()
quadrilateral = FiredrakeQuadrilateral()
tetrahedron = UFCTetrahedron()
interval_x_interval = TensorProductCell(interval, interval)
triangle_x_interval = TensorProductCell(triangle, interval)
quadrilateral_x_interval = TensorProductCell(quadrilateral, interval)
@pytest.mark.parametrize(
('cell', 'cell_facet_jacobian'),
[(interval, [[], []]), (triangle, [[-1, 1], [0, 1], [1, 0]]),
(quadrilateral, [[0, 1], [0, 1], [1, 0], [1, 0]]),
(tetrahedron, [[-1, -1, 1, 0, 0, 1], [0, 0, 1, 0, 0, 1],
[1, 0, 0, 0, 0, 1], [1, 0, 0, 1, 0, 0]])])
def test_cell_facet_jacobian(cell, cell_facet_jacobian):
facet_dim = cell.get_spatial_dimension() - 1
for facet_number in range(len(cell.get_topology()[facet_dim])):
actual = make_cell_facet_jacobian(cell, facet_dim, facet_number)
expected = np.reshape(cell_facet_jacobian[facet_number], actual.shape)
assert np.allclose(expected, actual)
from FIAT.gauss_legendre import GaussLegendre # noqa: F401
from FIAT.gauss_lobatto_legendre import GaussLobattoLegendre # noqa: F401
from FIAT.restricted import RestrictedElement # noqa: F401
from FIAT.tensor_product import TensorProductElement # noqa: F401
from FIAT.tensor_product import FlattenedDimensions # noqa: F401
from FIAT.hdivcurl import Hdiv, Hcurl # noqa: F401
from FIAT.argyris import Argyris, QuinticArgyris # noqa: F401
from FIAT.hermite import CubicHermite # noqa: F401
from FIAT.morley import Morley # noqa: F401
from FIAT.bubble import Bubble
from FIAT.enriched import EnrichedElement # noqa: F401
from FIAT.nodal_enriched import NodalEnrichedElement
I = UFCInterval() # noqa: E741
T = UFCTriangle()
S = UFCTetrahedron()
def test_basis_derivatives_scaling():
"Regression test for issue #9"
class Interval(ReferenceElement):
def __init__(self, a, b):
verts = ((a, ), (b, ))
edges = {0: (0, 1)}
topology = {0: {0: (0, ), 1: (1, )}, 1: edges}
super(Interval, self).__init__(LINE, verts, topology)
random.seed(42)
for i in range(26):
a = 1000.0 * (random.random() - 0.5)
def _tetrahedron_scheme(degree):
"""Return a quadrature scheme on a tetrahedron of specified
degree. Falls back on canonical rule for higher orders"""
if degree == 0 or degree == 1:
# Scheme from Zienkiewicz and Taylor, 1 point, degree of precision 1
x = array([[1.0/4.0, 1.0/4.0, 1.0/4.0]])
w = array([1.0/6.0])
elif degree == 2:
# Scheme from Zienkiewicz and Taylor, 4 points, degree of precision 2
a, b = 0.585410196624969, 0.138196601125011
x = array([[a, b, b],
[b, a, b],
[b, b, a],
[b, b, b]])
w = arange(4, dtype=float64)
w[:] = 1.0/24.0
elif degree == 3:
# Scheme from Zienkiewicz and Taylor, 5 points, degree of precision 3
# Note: this scheme has a negative weight
x = array([[0.2500000000000000, 0.2500000000000000, 0.2500000000000000],
[0.5000000000000000, 0.1666666666666666, 0.1666666666666666],
[0.1666666666666666, 0.5000000000000000, 0.1666666666666666],
[0.1666666666666666, 0.1666666666666666, 0.5000000000000000],
[0.1666666666666666, 0.1666666666666666, 0.1666666666666666]])
w = arange(5, dtype=float64)
w[0] = -0.8
w[1:5] = 0.45
w = w/6.0
elif degree == 4:
# Keast rule, 14 points, degree of precision 4
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST5)
x = array([[0.0000000000000000, 0.5000000000000000, 0.5000000000000000],
[0.5000000000000000, 0.0000000000000000, 0.5000000000000000],
[0.5000000000000000, 0.5000000000000000, 0.0000000000000000],
[0.5000000000000000, 0.0000000000000000, 0.0000000000000000],
[0.0000000000000000, 0.5000000000000000, 0.0000000000000000],
[0.0000000000000000, 0.0000000000000000, 0.5000000000000000],
[0.6984197043243866, 0.1005267652252045, 0.1005267652252045],
[0.1005267652252045, 0.1005267652252045, 0.1005267652252045],
[0.1005267652252045, 0.1005267652252045, 0.6984197043243866],
[0.1005267652252045, 0.6984197043243866, 0.1005267652252045],
[0.0568813795204234, 0.3143728734931922, 0.3143728734931922],
[0.3143728734931922, 0.3143728734931922, 0.3143728734931922],
[0.3143728734931922, 0.3143728734931922, 0.0568813795204234],
[0.3143728734931922, 0.0568813795204234, 0.3143728734931922]])
w = arange(14, dtype=float64)
w[0:6] = 0.0190476190476190
w[6:10] = 0.0885898247429807
w[10:14] = 0.1328387466855907
w = w/6.0
elif degree == 5:
# Keast rule, 15 points, degree of precision 5
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST6)
x = array([[0.2500000000000000, 0.2500000000000000, 0.2500000000000000],
[0.0000000000000000, 0.3333333333333333, 0.3333333333333333],
[0.3333333333333333, 0.3333333333333333, 0.3333333333333333],
[0.3333333333333333, 0.3333333333333333, 0.0000000000000000],
[0.3333333333333333, 0.0000000000000000, 0.3333333333333333],
[0.7272727272727273, 0.0909090909090909, 0.0909090909090909],
[0.0909090909090909, 0.0909090909090909, 0.0909090909090909],
[0.0909090909090909, 0.0909090909090909, 0.7272727272727273],
[0.0909090909090909, 0.7272727272727273, 0.0909090909090909],
[0.4334498464263357, 0.0665501535736643, 0.0665501535736643],
[0.0665501535736643, 0.4334498464263357, 0.0665501535736643],
[0.0665501535736643, 0.0665501535736643, 0.4334498464263357],
[0.0665501535736643, 0.4334498464263357, 0.4334498464263357],
[0.4334498464263357, 0.0665501535736643, 0.4334498464263357],
[0.4334498464263357, 0.4334498464263357, 0.0665501535736643]])
w = arange(15, dtype=float64)
w[0] = 0.1817020685825351
w[1:5] = 0.0361607142857143
w[5:9] = 0.0698714945161738
w[9:15] = 0.0656948493683187
w = w/6.0
elif degree == 6:
# Keast rule, 24 points, degree of precision 6
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST7)
x = array([[0.3561913862225449, 0.2146028712591517, 0.2146028712591517],
[0.2146028712591517, 0.2146028712591517, 0.2146028712591517],
[0.2146028712591517, 0.2146028712591517, 0.3561913862225449],
[0.2146028712591517, 0.3561913862225449, 0.2146028712591517],
[0.8779781243961660, 0.0406739585346113, 0.0406739585346113],
[0.0406739585346113, 0.0406739585346113, 0.0406739585346113],
[0.0406739585346113, 0.0406739585346113, 0.8779781243961660],
[0.0406739585346113, 0.8779781243961660, 0.0406739585346113],
[0.0329863295731731, 0.3223378901422757, 0.3223378901422757],
[0.3223378901422757, 0.3223378901422757, 0.3223378901422757],
[0.3223378901422757, 0.3223378901422757, 0.0329863295731731],
[0.3223378901422757, 0.0329863295731731, 0.3223378901422757],
[0.2696723314583159, 0.0636610018750175, 0.0636610018750175],
[0.0636610018750175, 0.2696723314583159, 0.0636610018750175],
[0.0636610018750175, 0.0636610018750175, 0.2696723314583159],
[0.6030056647916491, 0.0636610018750175, 0.0636610018750175],
[0.0636610018750175, 0.6030056647916491, 0.0636610018750175],
[0.0636610018750175, 0.0636610018750175, 0.6030056647916491],
[0.0636610018750175, 0.2696723314583159, 0.6030056647916491],
[0.2696723314583159, 0.6030056647916491, 0.0636610018750175],
[0.6030056647916491, 0.0636610018750175, 0.2696723314583159],
[0.0636610018750175, 0.6030056647916491, 0.2696723314583159],
[0.2696723314583159, 0.0636610018750175, 0.6030056647916491],
[0.6030056647916491, 0.2696723314583159, 0.0636610018750175]])
w = arange(24, dtype=float64)
w[0:4] = 0.0399227502581679
w[4:8] = 0.0100772110553207
w[8:12] = 0.0553571815436544
w[12:24] = 0.0482142857142857
w = w/6.0
else:
# Get canonical scheme
return _fiat_scheme(UFCTetrahedron(), degree)
# Return scheme
return QuadratureRule(UFCTetrahedron(), x, w)
from FIAT import finite_element, polynomial_set, dual_set, functional
from FIAT.reference_element import (Point, DefaultLine, UFCInterval,
UFCQuadrilateral, UFCHexahedron,
UFCTriangle, UFCTetrahedron,
make_affine_mapping,
flatten_reference_cube)
from FIAT.P0 import P0Dual
import numpy as np
hypercube_simplex_map = {
Point(): Point(),
DefaultLine(): DefaultLine(),
UFCInterval(): UFCInterval(),
UFCQuadrilateral(): UFCTriangle(),
UFCHexahedron(): UFCTetrahedron()
}
class DPC0(finite_element.CiarletElement):
def __init__(self, ref_el):
flat_el = flatten_reference_cube(ref_el)
poly_set = polynomial_set.ONPolynomialSet(
hypercube_simplex_map[flat_el], 0)
dual = P0Dual(ref_el)
degree = 0
formdegree = ref_el.get_spatial_dimension() # n-form
super(DPC0, self).__init__(poly_set=poly_set,
dual=dual,
order=degree,
ref_el=ref_el,
def tetrahedron():
return UFCTetrahedron()
def _kmv_lump_scheme(ref_el, degree):
"""Specialized quadrature schemes for P < 6 for KMV simplical elements."""
sd = ref_el.get_spatial_dimension()
# set the unit element
if sd == 2:
T = UFCTriangle()
elif sd == 3:
T = UFCTetrahedron()
else:
raise ValueError("Dimension not supported")
if degree == 1:
x = ref_el.vertices
w = arange(sd + 1, dtype=float64)
if sd == 2:
w[:] = 1.0 / 6.0
elif sd == 3:
w[:] = 1.0 / 24.0
else:
raise ValueError("Dimension not supported")
elif degree == 2:
if sd == 2:
x = list(ref_el.vertices)
for e in range(3):
x.extend(ref_el.make_points(1, e, 2)) # edge midpoints
x.extend(ref_el.make_points(2, 0, 3)) # barycenter
w = arange(7, dtype=float64)
w[0:3] = 1.0 / 40.0
w[3:6] = 1.0 / 15.0
w[6] = 9.0 / 40.0
elif sd == 3:
x = list(ref_el.vertices)
x.extend([
(0.0, 0.50, 0.50),
(0.50, 0.0, 0.50),
(0.50, 0.50, 0.0),
(0.0, 0.0, 0.50),
(0.0, 0.50, 0.0),
(0.50, 0.0, 0.0),
])
# in facets
x.extend([
(0.33333333333333337, 0.3333333333333333, 0.3333333333333333),
(0.0, 0.3333333333333333, 0.3333333333333333),
(0.3333333333333333, 0.0, 0.3333333333333333),
(0.3333333333333333, 0.3333333333333333, 0.0),
])
# in the cell
x.extend([(1 / 4, 1 / 4, 1 / 4)])
w = arange(15, dtype=float64)
w[0:4] = 17.0 / 5040.0
w[4:10] = 2.0 / 315.0
w[10:14] = 9.0 / 560.0
w[14] = 16.0 / 315.0
else:
raise ValueError("Dimension not supported")
elif degree == 3:
if sd == 2:
alpha = 0.2934695559090401
beta = 0.2073451756635909
x = list(ref_el.vertices)
x.extend([
(1 - alpha, alpha),
(alpha, 1 - alpha),
(0.0, 1 - alpha),
(0.0, alpha),
(alpha, 0.0),
(1 - alpha, 0.0),
] # edge points
)
x.extend([(beta, beta), (1 - 2 * beta, beta),
(beta, 1 - 2 * beta)]) # points in center of cell
w = arange(12, dtype=float64)
w[0:3] = 0.007436456512410291
w[3:9] = 0.02442084061702551
w[9:12] = 0.1103885289202054
elif sd == 3:
x = list(ref_el.vertices)
x.extend([
(0, 0.685789657581967, 0.314210342418033),
(0, 0.314210342418033, 0.685789657581967),
(0.314210342418033, 0, 0.685789657581967),
(0.685789657581967, 0, 0.314210342418033),
(0.685789657581967, 0.314210342418033, 0.0),
(0.314210342418033, 0.685789657581967, 0.0),
(0, 0, 0.685789657581967),
(0, 0, 0.314210342418033),
(0, 0.314210342418033, 0.0),
(0, 0.685789657581967, 0.0),
(0.314210342418033, 0, 0.0),
(0.685789657581967, 0, 0.0),
]) # 12 points on edges of facets (0-->1-->2)
x.extend(
[
(0.21548220313557542, 0.5690355937288492,
0.21548220313557542),
(0.21548220313557542, 0.21548220313557542,
0.5690355937288492),
(0.5690355937288492, 0.21548220313557542,
0.21548220313557542),
(0.0, 0.5690355937288492, 0.21548220313557542),
(0.0, 0.21548220313557542, 0.5690355937288492),
(0.0, 0.21548220313557542, 0.21548220313557542),
(0.5690355937288492, 0.0, 0.21548220313557542),
(0.21548220313557542, 0.0, 0.5690355937288492),
(0.21548220313557542, 0.0, 0.21548220313557542),
(0.5690355937288492, 0.21548220313557542, 0.0),
(0.21548220313557542, 0.5690355937288492, 0.0),
(0.21548220313557542, 0.21548220313557542, 0.0),
]
) # 12 points (3 points on each facet, 1st two parallel to edge 0)
alpha = 1 / 6
x.extend([
(alpha, alpha, 0.5),
(0.5, alpha, alpha),
(alpha, 0.5, alpha),
(alpha, alpha, alpha),
]) # 4 points inside the cell
w = arange(32, dtype=float64)
w[0:4] = 0.00068688236002531922325120561367839
w[4:16] = 0.0015107814913526136472998739890272
w[16:28] = 0.0050062894680040258624242888174649
w[28:32] = 0.021428571428571428571428571428571
else:
raise ValueError("Dimension not supported")
elif degree == 4:
if sd == 2:
alpha = 0.2113248654051871 # 0.2113248654051871
beta1 = 0.4247639617258106 # 0.4247639617258106
beta2 = 0.130791593829745 # 0.130791593829745
x = list(ref_el.vertices)
for e in range(3):
x.extend(ref_el.make_points(1, e, 2)) # edge midpoints
x.extend([
(1 - alpha, alpha),
(alpha, 1 - alpha),
(0.0, 1 - alpha),
(0.0, alpha),
(alpha, 0.0),
(1 - alpha, 0.0),
] # edge points
)
x.extend([(beta1, beta1), (1 - 2 * beta1, beta1),
(beta1, 1 - 2 * beta1)]) # points in center of cell
x.extend([(beta2, beta2), (1 - 2 * beta2, beta2),
(beta2, 1 - 2 * beta2)]) # points in center of cell
w = arange(18, dtype=float64)
w[0:3] = 0.003174603174603175 # chk
w[3:6] = 0.0126984126984127 # chk 0.0126984126984127
w[6:12] = 0.01071428571428571 # chk 0.01071428571428571
w[12:15] = 0.07878121446939182 # chk 0.07878121446939182
w[15:18] = 0.05058386489568756 # chk 0.05058386489568756
else:
raise ValueError("Dimension not supported")
elif degree == 5:
if sd == 2:
alpha1 = 0.3632980741536860e-00
alpha2 = 0.1322645816327140e-00
beta1 = 0.4578368380791611e-00
beta2 = 0.2568591072619591e-00
beta3 = 0.5752768441141011e-01
gamma1 = 0.7819258362551702e-01
delta1 = 0.2210012187598900e-00
x = list(ref_el.vertices)
x.extend([
(1 - alpha1, alpha1),
(alpha1, 1 - alpha1),
(0.0, 1 - alpha1),
(0.0, alpha1),
(alpha1, 0.0),
(1 - alpha1, 0.0),
] # edge points
)
x.extend([
(1 - alpha2, alpha2),
(alpha2, 1 - alpha2),
(0.0, 1 - alpha2),
(0.0, alpha2),
(alpha2, 0.0),
(1 - alpha2, 0.0),
] # edge points
)
x.extend([(beta1, beta1), (1 - 2 * beta1, beta1),
(beta1, 1 - 2 * beta1)]) # points in center of cell
x.extend([(beta2, beta2), (1 - 2 * beta2, beta2),
(beta2, 1 - 2 * beta2)]) # points in center of cell
x.extend([(beta3, beta3), (1 - 2 * beta3, beta3),
(beta3, 1 - 2 * beta3)]) # points in center of cell
x.extend([
(gamma1, delta1),
(1 - gamma1 - delta1, delta1),
(gamma1, 1 - gamma1 - delta1),
(delta1, gamma1),
(1 - gamma1 - delta1, gamma1),
(delta1, 1 - gamma1 - delta1),
] # edge points
)
w = arange(30, dtype=float64)
w[0:3] = 0.7094239706792450e-03
w[3:9] = 0.6190565003676629e-02
w[9:15] = 0.3480578640489211e-02
w[15:18] = 0.3453043037728279e-01
w[18:21] = 0.4590123763076286e-01
w[21:24] = 0.1162613545961757e-01
w[24:30] = 0.2727857596999626e-01
else:
raise ValueError("Dimension not supported")
# Return scheme
return QuadratureRule(T, x, w)