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 = 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 = ONPolynomialSet(ref_el, k + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * k + 2)
Qpts = np.array(Q.get_points())
Qwts = np.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 = np.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 = PolynomialSet(ref_el, k + 1, k + 1,
vec_Pkp1.get_expansion_set(), PkH_crossx_coeffs,
vec_Pkp1.get_dmats())
return polynomial_set_union_normalized(vec_Pk_from_Pkp1, PkHcrossx)
def __init__(self, ref_el, degree):
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("Nedelec2D only works on triangles")
nodes = []
t = ref_el.get_topology()
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 = 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, ))
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().__init__(nodes, ref_el, entity_ids)
def RTSpace(ref_el, deg):
"""Constructs a basis for the the Raviart-Thomas space
(P_k)^d + P_k x"""
sd = ref_el.get_spatial_dimension()
vec_Pkp1 = ONPolynomialSet(ref_el, deg + 1, (sd, ))
dimPkp1 = expansions.polynomial_dimension(ref_el, deg + 1)
dimPk = expansions.polynomial_dimension(ref_el, deg)
dimPkm1 = expansions.polynomial_dimension(ref_el, deg - 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 = ONPolynomialSet(ref_el, deg + 1)
PkH = Pkp1.take(list(range(dimPkm1, dimPk)))
Q = quadrature.make_quadrature(ref_el, 2 * deg + 2)
# have to work on this through "tabulate" interface
# first, tabulate PkH at quadrature points
Qpts = np.array(Q.get_points())
Qwts = np.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]
PkHx_coeffs = np.zeros((PkH.get_num_members(), sd, Pkp1.get_num_members()),
"d")
for i in range(PkH.get_num_members()):
for j in range(sd):
fooij = PkH_at_Qpts[i, :] * Qpts[:, j] * Qwts
PkHx_coeffs[i, j, :] = np.dot(Pkp1_at_Qpts, fooij)
PkHx = PolynomialSet(ref_el, deg, deg + 1, vec_Pkp1.get_expansion_set(),
PkHx_coeffs, vec_Pkp1.get_dmats())
return polynomial_set_union_normalized(vec_Pk_from_Pkp1, PkHx)
def to_riesz(self, poly_set):
r"""This method gives the action of the entire dual set
on each member of the expansion set underlying poly_set.
Then, applying the linear functionals of the dual set to an
arbitrary polynomial in poly_set is accomplished by (generalized)
matrix multiplication.
For scalar-valued spaces, this produces a matrix
:\math:`R_{i, j}` such that
:\math:`\ell_i(f) = \sum_{j} a_j \ell_i(\phi_j)`
for :\math:`f=\sum_{j} a_j \phi_j`.
More generally, it will have shape concatenating
the number of functionals in the dual set, the value shape
of functions it takes, and the number of members of the
expansion set.
"""
# This rather technical code queries the low-level information
# in pt_dict and deriv_dict
# for each functional to find out where it evaluates its
# inputs and/or their derivatives. Then, it tabulates the
# expansion set one time for all the function values and
# another for all of the derivatives. This circumvents
# needing to call the to_riesz method of each functional and
# also limits the number of different calls to tabulate.
tshape = self.nodes[0].target_shape
num_nodes = len(self.nodes)
es = poly_set.get_expansion_set()
ed = poly_set.get_embedded_degree()
num_exp = es.get_num_members(poly_set.get_embedded_degree())
riesz_shape = tuple([num_nodes] + list(tshape) + [num_exp])
self.mat = np.zeros(riesz_shape, "d")
# Dictionaries mapping pts to which functionals they come from
pts_to_ells = collections.OrderedDict()
dpts_to_ells = collections.OrderedDict()
for i, ell in enumerate(self.nodes):
for pt in ell.pt_dict:
if pt in pts_to_ells:
pts_to_ells[pt].append(i)
else:
pts_to_ells[pt] = [i]
for pt in ell.deriv_dict:
if pt in dpts_to_ells:
dpts_to_ells[pt].append(i)
else:
dpts_to_ells[pt] = [i]
# Now tabulate the function values
pts = list(pts_to_ells.keys())
expansion_values = es.tabulate(ed, pts)
for j, pt in enumerate(pts):
which_ells = pts_to_ells[pt]
for k in which_ells:
pt_dict = self.nodes[k].pt_dict
wc_list = pt_dict[pt]
for i in range(num_exp):
for (w, c) in wc_list:
self.mat[k][c][i] += w * expansion_values[i, j]
# Tabulate the derivative values that are needed
max_deriv_order = max([ell.max_deriv_order for ell in self.nodes])
if max_deriv_order > 0:
dpts = list(dpts_to_ells.keys())
# It's easiest/most efficient to get derivatives of the
# expansion set through the polynomial set interface.
# This is creating a short-lived set to do just this.
expansion = ONPolynomialSet(self.ref_el, ed)
dexpansion_values = expansion.tabulate(dpts, max_deriv_order)
for j, pt in enumerate(dpts):
which_ells = dpts_to_ells[pt]
for k in which_ells:
dpt_dict = self.nodes[k].deriv_dict
wac_list = dpt_dict[pt]
for i in range(num_exp):
for (w, alpha, c) in wac_list:
self.mat[k][c][i] += w * dexpansion_values[alpha][
i, j]
return self.mat
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 = 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 = ONPolynomialSet(ref_el, k + 1)
Q = quadrature.make_quadrature(ref_el, 2 * (k + 1))
Qpts = np.array(Q.get_points())
Qwts = np.array(Q.get_weights())
zero_index = tuple([0 for i in range(sd)])
PkCrossXcoeffs = np.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, :] = np.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 = PolynomialSet(ref_el, k + 1, k + 1,
vec_Pkp1.get_expansion_set(), PkCrossXcoeffs,
vec_Pkp1.get_dmats())
return polynomial_set_union_normalized(vec_Pk, PkCrossX)
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 = 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().__init__(nodes, ref_el, entity_ids)