def __init__(self, ref_el, degree, variant, quad_deg):
sd = ref_el.get_spatial_dimension()
if sd != 3:
raise Exception("NedelecDual3D only works on tetrahedra")
nodes = []
t = ref_el.get_topology()
if variant == "integral":
# edge nodes are \int_F v\cdot t p ds where p \in P_{q-1}(edge)
# degree is q - 1
edge = ref_el.get_facet_element().get_facet_element()
Q = quadrature.make_quadrature(edge, quad_deg)
Pq = polynomial_set.ONPolynomialSet(edge, degree)
Pq_at_qpts = Pq.tabulate(Q.get_points())[tuple([0] * (1))]
for e in range(len(t[1])):
for i in range(Pq_at_qpts.shape[0]):
phi = Pq_at_qpts[i, :]
nodes.append(
functional.IntegralMomentOfEdgeTangentEvaluation(
ref_el, Q, phi, e))
# face nodes are \int_F v\cdot p dA where p \in P_{q-2}(f)^3 with p \cdot n = 0 (cmp. Monk)
# these are equivalent to dofs from Fenics book defined by
# \int_F v\times n \cdot p ds where p \in P_{q-2}(f)^2
if degree > 0:
facet = ref_el.get_facet_element()
Q = quadrature.make_quadrature(facet, quad_deg)
Pq = polynomial_set.ONPolynomialSet(facet, degree - 1, (sd, ))
Pq_at_qpts = Pq.tabulate(Q.get_points())[(0, 0)]
for f in range(len(t[2])):
# R is used to map [1,0,0] to tangent1 and [0,1,0] to tangent2
R = ref_el.compute_face_tangents(f)
# Skip last functionals because we only want p with p \cdot n = 0
for i in range(2 * Pq.get_num_members() // 3):
phi = Pq_at_qpts[i, ...]
phi = numpy.matmul(phi[:-1, ...].T, R)
nodes.append(
functional.MonkIntegralMoment(ref_el, Q, phi, f))
# internal nodes. These are \int_T v \cdot p dx where p \in P_{q-3}^3(T)
if degree > 1:
Q = quadrature.make_quadrature(ref_el, quad_deg)
qpts = Q.get_points()
Pkm2 = polynomial_set.ONPolynomialSet(ref_el, degree - 2)
zero_index = tuple([0 for i in range(sd)])
Pkm2_at_qpts = Pkm2.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm2_at_qpts.shape[0]):
phi_cur = Pkm2_at_qpts[i, :]
l_cur = functional.IntegralMoment(
ref_el, Q, phi_cur, (d, ), (sd, ))
nodes.append(l_cur)
elif variant == "point":
num_edges = len(t[1])
for i in range(num_edges):
# points to specify P_k on each edge
pts_cur = ref_el.make_points(1, i, degree + 2)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointEdgeTangentEvaluation(
ref_el, i, pt_cur)
nodes.append(f)
if degree > 0: # face tangents
num_faces = len(t[2])
for i in range(num_faces): # loop over faces
pts_cur = ref_el.make_points(2, i, degree + 2)
for j in range(len(pts_cur)): # loop over points
pt_cur = pts_cur[j]
for k in range(2): # loop over tangents
f = functional.PointFaceTangentEvaluation(
ref_el, i, k, pt_cur)
nodes.append(f)
if degree > 1: # internal moments
Q = quadrature.make_quadrature(ref_el, 2 * (degree + 1))
qpts = Q.get_points()
Pkm2 = polynomial_set.ONPolynomialSet(ref_el, degree - 2)
zero_index = tuple([0 for i in range(sd)])
Pkm2_at_qpts = Pkm2.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm2_at_qpts.shape[0]):
phi_cur = Pkm2_at_qpts[i, :]
f = functional.IntegralMoment(ref_el, Q, phi_cur,
(d, ), (sd, ))
nodes.append(f)
entity_ids = {}
# set to empty
for i in range(sd + 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# edge dof
num_pts_per_edge = len(ref_el.make_points(1, 0, degree + 2))
for i in range(len(t[1])):
entity_ids[1][i] = list(range(cur, cur + num_pts_per_edge))
cur += num_pts_per_edge
# face dof
if degree > 0:
num_pts_per_face = len(ref_el.make_points(2, 0, degree + 2))
for i in range(len(t[2])):
entity_ids[2][i] = list(range(cur, cur + 2 * num_pts_per_face))
cur += 2 * num_pts_per_face
if degree > 1:
num_internal_dof = Pkm2_at_qpts.shape[0] * sd
entity_ids[3][0] = list(range(cur, cur + num_internal_dof))
super(NedelecDual3D, self).__init__(nodes, ref_el, entity_ids)
def __init__(self, ref_el, degree):
sd = ref_el.get_spatial_dimension()
if sd != 3:
raise Exception("NedelecDual3D only works on tetrahedra")
nodes = []
t = ref_el.get_topology()
# how many edges
num_edges = len(t[1])
for i in range(num_edges):
# points to specify P_k on each edge
pts_cur = ref_el.make_points(1, i, degree + 2)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointEdgeTangentEvaluation(ref_el, i, pt_cur)
nodes.append(f)
if degree > 0: # face tangents
num_faces = len(t[2])
for i in range(num_faces): # loop over faces
pts_cur = ref_el.make_points(2, i, degree + 2)
for j in range(len(pts_cur)): # loop over points
pt_cur = pts_cur[j]
for k in range(2): # loop over tangents
f = functional.PointFaceTangentEvaluation(
ref_el, i, k, pt_cur)
nodes.append(f)
if degree > 1: # internal moments
Q = quadrature.make_quadrature(ref_el, 2 * (degree + 1))
qpts = Q.get_points()
Pkm2 = polynomial_set.ONPolynomialSet(ref_el, degree - 2)
zero_index = tuple([0 for i in range(sd)])
Pkm2_at_qpts = Pkm2.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm2_at_qpts.shape[0]):
phi_cur = Pkm2_at_qpts[i, :]
f = functional.IntegralMoment(ref_el, Q, phi_cur, (d, ))
nodes.append(f)
entity_ids = {}
# set to empty
for i in range(sd + 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# edge dof
num_pts_per_edge = len(ref_el.make_points(1, 0, degree + 2))
for i in range(len(t[1])):
entity_ids[1][i] = list(range(cur, cur + num_pts_per_edge))
cur += num_pts_per_edge
# face dof
if degree > 0:
num_pts_per_face = len(ref_el.make_points(2, 0, degree + 2))
for i in range(len(t[2])):
entity_ids[2][i] = list(range(cur, cur + 2 * num_pts_per_face))
cur += 2 * num_pts_per_face
if degree > 1:
num_internal_dof = Pkm2_at_qpts.shape[0] * sd
entity_ids[3][0] = list(range(cur, cur + num_internal_dof))
super(NedelecDual3D, self).__init__(nodes, ref_el, entity_ids)