def basis_transformation(self, coordinate_mapping):
"""The extra 6 dofs removed here correspond to the constraints."""
V = numpy.zeros((30, 24), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
W = _evaluation_transform(coordinate_mapping)
# Put into the right rows and columns.
V[0:3, 0:3] = V[3:6, 3:6] = V[6:9, 6:9] = W
V[9:21, 9:21] = _edge_transform(self.cell, coordinate_mapping)
# internal DOFs
detJ = coordinate_mapping.detJ_at([1/3, 1/3])
V[21:24, 21:24] = W / detJ
# RESCALING FOR CONDITIONING
h = coordinate_mapping.cell_size()
for e in range(3):
eff_h = h[e]
for i in range(24):
V[i, 3*e] = V[i, 3*e]/(eff_h*eff_h)
V[i, 1+3*e] = V[i, 1+3*e]/(eff_h*eff_h)
V[i, 2+3*e] = V[i, 2+3*e]/(eff_h*eff_h)
# Note: that the edge DOFs are scaled by edge lengths in FIAT implies
# that they are already have the necessary rescaling to improve
# conditioning.
return ListTensor(V.T)
def basis_transformation(self, coordinate_mapping):
Js = [
coordinate_mapping.jacobian_at(vertex)
for vertex in self.cell.get_vertices()
]
h = coordinate_mapping.cell_size()
d = self.cell.get_dimension()
numbf = self.space_dimension()
M = numpy.eye(numbf, dtype=object)
for multiindex in numpy.ndindex(M.shape):
M[multiindex] = Literal(M[multiindex])
cur = 0
for i in range(d + 1):
cur += 1 # skip the vertex
J = Js[i]
for j in range(d):
for k in range(d):
M[cur + j, cur + k] = J[j, k] / h[i]
cur += d
return ListTensor(M)
def basis_transformation(self, coordinate_mapping):
"""The extra 6 dofs removed here correspond to the constraints."""
V = numpy.zeros((30, 24), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
J = coordinate_mapping.jacobian_at([1 / 3, 1 / 3])
W = numpy.zeros((3, 3), dtype=object)
W[0, 0] = J[1, 1] * J[1, 1]
W[0, 1] = -2 * J[1, 1] * J[0, 1]
W[0, 2] = J[0, 1] * J[0, 1]
W[1, 0] = -1 * J[1, 1] * J[1, 0]
W[1, 1] = J[1, 1] * J[0, 0] + J[0, 1] * J[1, 0]
W[1, 2] = -1 * J[0, 1] * J[0, 0]
W[2, 0] = J[1, 0] * J[1, 0]
W[2, 1] = -2 * J[1, 0] * J[0, 0]
W[2, 2] = J[0, 0] * J[0, 0]
# Put into the right rows and columns.
V[0:3, 0:3] = V[3:6, 3:6] = V[6:9, 6:9] = W
V[9:21, 9:21] = _edge_transform(self.cell, coordinate_mapping)
for i in range(21, 24):
V[i, i] = Literal(1)
return ListTensor(V.T)
def basis_transformation(self, coordinate_mapping):
V = numpy.zeros((20, 9), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
for i in range(0, 9, 3):
V[i, i] = Literal(1)
V[i+2, i+2] = Literal(1)
T = self.cell
# This bypasses the GEM wrapper.
that = numpy.array([T.compute_normalized_edge_tangent(i) for i in range(3)])
nhat = numpy.array([T.compute_normal(i) for i in range(3)])
detJ = coordinate_mapping.detJ_at([1/3, 1/3])
J = coordinate_mapping.jacobian_at([1/3, 1/3])
J_np = numpy.array([[J[0, 0], J[0, 1]],
[J[1, 0], J[1, 1]]])
JTJ = J_np.T @ J_np
for e in range(3):
# Compute alpha and beta for the edge.
Ghat_T = numpy.array([nhat[e, :], that[e, :]])
(alpha, beta) = Ghat_T @ JTJ @ that[e, :] / detJ
# Stuff into the right rows and columns.
idx = 3*e + 1
V[idx, idx-1] = Literal(-1) * alpha / beta
V[idx, idx] = Literal(1) / beta
return ListTensor(V.T)
def basis_transformation(self, coordinate_mapping):
# Jacobians at edge midpoints
J = coordinate_mapping.jacobian_at([1/3, 1/3])
rns = coordinate_mapping.reference_normals()
pns = coordinate_mapping.physical_normals()
pts = coordinate_mapping.physical_tangents()
pel = coordinate_mapping.physical_edge_lengths()
V = numpy.eye(6, dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
for i in range(3):
V[i+3, i+3] = (rns[i, 0]*(pns[i, 0]*J[0, 0] + pns[i, 1]*J[1, 0])
+ rns[i, 1]*(pns[i, 0]*J[0, 1] + pns[i, 1]*J[1, 1]))
for i, c in enumerate([(1, 2), (0, 2), (0, 1)]):
B12 = (rns[i, 0]*(pts[i, 0]*J[0, 0] + pts[i, 1]*J[1, 0])
+ rns[i, 1]*(pts[i, 0]*J[0, 1] + pts[i, 1]*J[1, 1]))
V[3+i, c[0]] = -1*B12 / pel[i]
V[3+i, c[1]] = B12 / pel[i]
# diagonal post-scaling to patch up conditioning
h = coordinate_mapping.cell_size()
for e in range(3):
v0id, v1id = [i for i in range(3) if i != e]
for i in range(6):
V[i, 3+e] = 2*V[i, 3+e] / (h[v0id] + h[v1id])
return ListTensor(V.T)
def basis_transformation(self, coordinate_mapping):
"""Note, the extra 3 dofs which are removed here
correspond to the constraints."""
T = self.cell
V = numpy.zeros((18, 15), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
V[:12, :12] = _edge_transform(T, coordinate_mapping)
# internal dofs
for i in range(12, 15):
V[i, i] = Literal(1)
return ListTensor(V.T)
def basis_transformation(self, coordinate_mapping):
"""Note, the extra 3 dofs which are removed here
correspond to the constraints."""
T = self.cell
V = numpy.zeros((18, 15), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
V[:12, :12] = _edge_transform(T, coordinate_mapping)
# internal dofs
W = _evaluation_transform(coordinate_mapping)
detJ = coordinate_mapping.detJ_at([1/3, 1/3])
V[12:15, 12:15] = W / detJ
# Note: that the edge DOFs are scaled by edge lengths in FIAT implies
# that they are already have the necessary rescaling to improve
# conditioning.
return ListTensor(V.T)
def list_tensor(self, o, *ops):
return ListTensor(ops)
def basis_transformation(self, coordinate_mapping):
# Jacobians at edge midpoints
J = coordinate_mapping.jacobian_at([1 / 3, 1 / 3])
rns = coordinate_mapping.reference_normals()
pns = coordinate_mapping.physical_normals()
pts = coordinate_mapping.physical_tangents()
pel = coordinate_mapping.physical_edge_lengths()
V = numpy.zeros((21, 21), dtype=object)
for multiindex in numpy.ndindex(V.shape):
V[multiindex] = Literal(V[multiindex])
for v in range(3):
s = 6 * v
V[s, s] = Literal(1)
for i in range(2):
for j in range(2):
V[s + 1 + i, s + 1 + j] = J[j, i]
V[s + 3, s + 3] = J[0, 0] * J[0, 0]
V[s + 3, s + 4] = 2 * J[0, 0] * J[1, 0]
V[s + 3, s + 5] = J[1, 0] * J[1, 0]
V[s + 4, s + 3] = J[0, 0] * J[0, 1]
V[s + 4, s + 4] = J[0, 0] * J[1, 1] + J[1, 0] * J[0, 1]
V[s + 4, s + 5] = J[1, 0] * J[1, 1]
V[s + 5, s + 3] = J[0, 1] * J[0, 1]
V[s + 5, s + 4] = 2 * J[0, 1] * J[1, 1]
V[s + 5, s + 5] = J[1, 1] * J[1, 1]
for e in range(3):
v0id, v1id = [i for i in range(3) if i != e]
# nhat . J^{-T} . t
foo = (rns[e, 0] * (J[0, 0] * pts[e, 0] + J[1, 0] * pts[e, 1]) +
rns[e, 1] * (J[0, 1] * pts[e, 0] + J[1, 1] * pts[e, 1]))
# vertex points
V[18 + e, 6 * v0id] = -15 / 8 * (foo / pel[e])
V[18 + e, 6 * v1id] = 15 / 8 * (foo / pel[e])
# vertex derivatives
for i in (0, 1):
V[18 + e, 6 * v0id + 1 + i] = -7 / 16 * foo * pts[e, i]
V[18 + e, 6 * v1id + 1 + i] = V[18 + e, 6 * v0id + 1 + i]
# second derivatives
tau = [
pts[e, 0] * pts[e, 0], 2 * pts[e, 0] * pts[e, 1],
pts[e, 1] * pts[e, 1]
]
for i in (0, 1, 2):
V[18 + e, 6 * v0id + 3 + i] = -1 / 32 * (pel[e] * foo * tau[i])
V[18 + e, 6 * v1id + 3 + i] = 1 / 32 * (pel[e] * foo * tau[i])
V[18 + e, 18 +
e] = (rns[e, 0] * (J[0, 0] * pns[e, 0] + J[1, 0] * pns[e, 1]) +
rns[e, 1] * (J[0, 1] * pns[e, 0] + J[1, 1] * pns[e, 1]))
# Patch up conditioning
h = coordinate_mapping.cell_size()
for v in range(3):
for k in range(2):
for i in range(21):
V[i, 6 * v + 1 + k] = V[i, 6 * v + 1 + k] / h[v]
for k in range(3):
for i in range(21):
V[i, 6 * v + 3 + k] = V[i, 6 * v + 3 + k] / (h[v] * h[v])
for e in range(3):
v0id, v1id = [i for i in range(3) if i != e]
for i in range(21):
V[i, 18 + e] = 2 * V[i, 18 + e] / (h[v0id] + h[v1id])
return ListTensor(V.T)