from FIAT import (finite_element, functional, dual_set,
polynomial_set, lagrange)
import numpy
[docs]
class BDFMDualSet(dual_set.DualSet):
def __init__(self, ref_el, degree):
# Initialize containers for map: mesh_entity -> dof number and
# dual basis
entity_ids = {}
nodes = []
sd = ref_el.get_spatial_dimension()
t = ref_el.get_topology()
# Define each functional for the dual set
# codimension 1 facet normals.
# note this will die for degree greater than 1.
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointScaledNormalEvaluation(ref_el, i, pt_cur)
nodes.append(f)
# codimension 1 facet tangents.
# because the tangent component is discontinuous, these actually
# count as internal nodes.
tangent_count = 0
for i in range(len(t[sd - 1])):
pts_cur = ref_el.make_points(sd - 1, i, sd + degree - 1)
tangent_count += len(pts_cur)
for j in range(len(pts_cur)):
pt_cur = pts_cur[j]
f = functional.PointEdgeTangentEvaluation(ref_el, i, pt_cur)
nodes.append(f)
# sets vertices (and in 3d, edges) to have no nodes
for i in range(sd - 1):
entity_ids[i] = {}
for j in range(len(t[i])):
entity_ids[i][j] = []
cur = 0
# set codimension 1 (edges 2d, faces 3d) dof
pts_facet_0 = ref_el.make_points(sd - 1, 0, sd + degree)
pts_per_facet = len(pts_facet_0)
entity_ids[sd - 1] = {}
for i in range(len(t[sd - 1])):
entity_ids[sd - 1][i] = list(range(cur, cur + pts_per_facet))
cur += pts_per_facet
# internal nodes
entity_ids[sd] = {0: list(range(cur, cur + tangent_count))}
cur += tangent_count
super(BDFMDualSet, self).__init__(nodes, ref_el, entity_ids)
[docs]
def BDFMSpace(ref_el, order):
sd = ref_el.get_spatial_dimension()
if sd != 2:
raise Exception("BDFM_k elements only valid for dim 2")
# Note that order will be 2.
# Linear vector valued space. Since the embedding degree of this element
# is 2, this is implemented by taking the quadratic space and selecting
# the linear polynomials.
vec_poly_set = polynomial_set.ONPolynomialSet(ref_el, order, (sd,))
# Linears are the first three polynomials in each dimension.
vec_poly_set = vec_poly_set.take([0, 1, 2, 6, 7, 8])
# Scalar quadratic Lagrange element.
lagrange_ele = lagrange.Lagrange(ref_el, order)
# Select the dofs associated with the edges.
edge_dofs_dict = lagrange_ele.dual.get_entity_ids()[sd - 1]
edge_dofs = numpy.array([(edge, dof)
for edge, dofs in list(edge_dofs_dict.items())
for dof in dofs])
tangent_polys = lagrange_ele.poly_set.take(edge_dofs[:, 1])
new_coeffs = numpy.zeros((tangent_polys.get_num_members(), sd, tangent_polys.coeffs.shape[-1]))
# Outer product of the tangent vectors with the quadratic edge polynomials.
for i, (edge, dof) in enumerate(edge_dofs):
tangent = ref_el.compute_edge_tangent(edge)
new_coeffs[i, :, :] = numpy.outer(tangent, tangent_polys.coeffs[i, :])
bubble_set = polynomial_set.PolynomialSet(ref_el,
order,
order,
vec_poly_set.get_expansion_set(),
new_coeffs,
vec_poly_set.get_dmats())
element_set = polynomial_set.polynomial_set_union_normalized(bubble_set, vec_poly_set)
return element_set
[docs]
class BrezziDouglasFortinMarini(finite_element.CiarletElement):
"""The BDFM element"""
def __init__(self, ref_el, degree):
if degree != 2:
raise Exception("BDFM_k elements only valid for k == 2")
poly_set = BDFMSpace(ref_el, degree)
dual = BDFMDualSet(ref_el, degree - 1)
formdegree = ref_el.get_spatial_dimension() - 1
super(BrezziDouglasFortinMarini, self).__init__(poly_set, dual, degree, formdegree,
mapping="contravariant piola")