# Copyright (C) 2008 Robert C. Kirby (Texas Tech University)
# Modified by Andrew T. T. McRae (Imperial College London)
#
# This file is part of FIAT (https://www.fenicsproject.org)
#
# SPDX-License-Identifier: LGPL-3.0-or-later
from FIAT import (polynomial_set, expansions, quadrature, dual_set,
finite_element, functional)
from itertools import chain
import numpy
from FIAT.check_format_variant import check_format_variant
[docs]
def NedelecSpace2D(ref_el, k):
"""Constructs a basis for the 2d H(curl) space of the first kind
which is (P_k)^2 + P_k rot( x )"""
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("NedelecSpace2D requires 2d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk_from_Pkp1 = vec_Pkp1.take(vec_Pk_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * k + 2)
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkH_at_Qpts = PkH.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
PkH_crossx_coeffs = numpy.zeros((PkH.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
def rot_x_foo(a):
if a == 0:
return 1, 1.0
elif a == 1:
return 0, -1.0
for i in range(PkH.get_num_members()):
for j in range(sd):
(ind, sign) = rot_x_foo(j)
for k in range(Pkp1.get_num_members()):
PkH_crossx_coeffs[i, j, k] = sign * sum(Qwts * PkH_at_Qpts[i, :] * Qpts[:, ind] * Pkp1_at_Qpts[k, :])
# for l in range( len( Qpts ) ):
# PkH_crossx_coeffs[i,j,k] += Qwts[ l ] \
# * PkH_at_Qpts[i,l] \
# * Qpts[l][ind] \
# * Pkp1_at_Qpts[k,l] \
# * sign
PkHcrossx = polynomial_set.PolynomialSet(ref_el,
k + 1,
k + 1,
vec_Pkp1.get_expansion_set(),
PkH_crossx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk_from_Pkp1,
PkHcrossx)
[docs]
def NedelecSpace3D(ref_el, k):
"""Constructs a nodal basis for the 3d first-kind Nedelec space"""
sd = ref_el.get_spatial_dimension()
if sd != 3:
raise Exception("NedelecSpace3D requires 3d reference element")
vec_Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1, (sd,))
dimPkp1 = expansions.polynomial_dimension(ref_el, k + 1)
dimPk = expansions.polynomial_dimension(ref_el, k)
if k > 0:
dimPkm1 = expansions.polynomial_dimension(ref_el, k - 1)
else:
dimPkm1 = 0
vec_Pk_indices = list(chain(*(range(i * dimPkp1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pk = vec_Pkp1.take(vec_Pk_indices)
vec_Pke_indices = list(chain(*(range(i * dimPkp1 + dimPkm1, i * dimPkp1 + dimPk)
for i in range(sd))))
vec_Pke = vec_Pkp1.take(vec_Pke_indices)
Pkp1 = polynomial_set.ONPolynomialSet(ref_el, k + 1)
Q = quadrature.make_quadrature(ref_el, 2 * (k + 1))
Qpts = numpy.array(Q.get_points())
Qwts = numpy.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkCrossXcoeffs = numpy.zeros((vec_Pke.get_num_members(),
sd,
Pkp1.get_num_members()), "d")
Pke_qpts = vec_Pke.tabulate(Qpts)[zero_index]
Pkp1_at_Qpts = Pkp1.tabulate(Qpts)[zero_index]
for i in range(vec_Pke.get_num_members()):
for j in range(sd): # vector components
qwts_cur_bf_val = (
Qpts[:, (j + 2) % 3] * Pke_qpts[i, (j + 1) % 3, :] -
Qpts[:, (j + 1) % 3] * Pke_qpts[i, (j + 2) % 3, :]) * Qwts
PkCrossXcoeffs[i, j, :] = numpy.dot(Pkp1_at_Qpts, qwts_cur_bf_val)
# for k in range( Pkp1.get_num_members() ):
# PkCrossXcoeffs[i,j,k] = sum( Qwts * cur_bf_val * Pkp1_at_Qpts[k,:] )
# for l in range( len( Qpts ) ):
# cur_bf_val = Qpts[l][(j+2)%3] \
# * Pke_qpts[i,(j+1)%3,l] \
# - Qpts[l][(j+1)%3] \
# * Pke_qpts[i,(j+2)%3,l]
# PkCrossXcoeffs[i,j,k] += Qwts[l] \
# * cur_bf_val \
# * Pkp1_at_Qpts[k,l]
PkCrossX = polynomial_set.PolynomialSet(ref_el,
k + 1,
k + 1,
vec_Pkp1.get_expansion_set(),
PkCrossXcoeffs,
vec_Pkp1.get_dmats())
return polynomial_set.polynomial_set_union_normalized(vec_Pk, PkCrossX)
[docs]
class NedelecDual2D(dual_set.DualSet):
"""Dual basis for first-kind Nedelec in 2D."""
def __init__(self, ref_el, degree, variant, quad_deg):
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("Nedelec2D only works on triangles")
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()
Q = quadrature.make_quadrature(edge, quad_deg)
Pq = polynomial_set.ONPolynomialSet(edge, degree)
Pq_at_qpts = Pq.tabulate(Q.get_points())[tuple([0]*(sd - 1))]
for e in range(len(t[sd - 1])):
for i in range(Pq_at_qpts.shape[0]):
phi = Pq_at_qpts[i, :]
nodes.append(functional.IntegralMomentOfEdgeTangentEvaluation(ref_el, Q, phi, e))
# internal nodes. These are \int_T v \cdot p dx where p \in P_{q-2}^2
if degree > 0:
Q = quadrature.make_quadrature(ref_el, quad_deg)
qpts = Q.get_points()
Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
zero_index = tuple([0 for i in range(sd)])
Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm1_at_qpts.shape[0]):
phi_cur = Pkm1_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])
# edge tangents
for i in range(num_edges):
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)
# internal moments
if degree > 0:
Q = quadrature.make_quadrature(ref_el, 2 * (degree + 1))
qpts = Q.get_points()
Pkm1 = polynomial_set.ONPolynomialSet(ref_el, degree - 1)
zero_index = tuple([0 for i in range(sd)])
Pkm1_at_qpts = Pkm1.tabulate(qpts)[zero_index]
for d in range(sd):
for i in range(Pkm1_at_qpts.shape[0]):
phi_cur = Pkm1_at_qpts[i, :]
l_cur = functional.IntegralMoment(ref_el, Q, phi_cur, (d,), (sd,))
nodes.append(l_cur)
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
# edges
num_edge_pts = 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_edge_pts))
cur += num_edge_pts
# moments against P_{degree-1} internally, if degree > 0
if degree > 0:
num_internal_dof = sd * Pkm1_at_qpts.shape[0]
entity_ids[2][0] = list(range(cur, cur + num_internal_dof))
super(NedelecDual2D, self).__init__(nodes, ref_el, entity_ids)
[docs]
class NedelecDual3D(dual_set.DualSet):
"""Dual basis for first-kind Nedelec in 3D."""
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)
[docs]
class Nedelec(finite_element.CiarletElement):
"""
Nedelec finite element
:arg ref_el: The reference element.
:arg k: The degree.
:arg variant: optional variant specifying the types of nodes.
variant can be chosen from ["point", "integral", "integral(quadrature_degree)"]
"point" -> dofs are evaluated by point evaluation. Note that this variant has suboptimal
convergence order in the H(curl)-norm
"integral" -> dofs are evaluated by quadrature rule. The quadrature degree is chosen to integrate
polynomials of degree 5*k so that most expressions will be interpolated exactly. This is important
when you want to have (nearly) curl-preserving interpolation.
"integral(quadrature_degree)" -> dofs are evaluated by quadrature rule of degree quadrature_degree
"""
def __init__(self, ref_el, k, variant=None):
degree = k - 1
(variant, quad_deg) = check_format_variant(variant, degree, "Nedelec")
if ref_el.get_spatial_dimension() == 3:
poly_set = NedelecSpace3D(ref_el, degree)
dual = NedelecDual3D(ref_el, degree, variant, quad_deg)
elif ref_el.get_spatial_dimension() == 2:
poly_set = NedelecSpace2D(ref_el, degree)
dual = NedelecDual2D(ref_el, degree, variant, quad_deg)
else:
raise Exception("Not implemented")
formdegree = 1 # 1-form
super(Nedelec, self).__init__(poly_set, dual, degree, formdegree,
mapping="covariant piola")