def test_barycentric_coordinate_map(dims):
from random import Random
rng = Random(17)
n = 5
unit = np.empty((dims, n))
from modepy.tools import (
pick_random_simplex_unit_coordinate,
unit_to_barycentric,
barycentric_to_unit,
barycentric_to_equilateral,
equilateral_to_unit,)
for i in range(n):
unit[:, i] = pick_random_simplex_unit_coordinate(rng, dims)
bary = unit_to_barycentric(unit)
assert (np.abs(np.sum(bary, axis=0) - 1) < 1e-15).all()
assert (bary >= 0).all()
unit2 = barycentric_to_unit(bary)
assert la.norm(unit-unit2) < 1e-14
equi = barycentric_to_equilateral(bary)
unit3 = equilateral_to_unit(equi)
assert la.norm(unit-unit3) < 1e-14
def test_barycentric_coordinate_map(dims):
from random import Random
rng = Random(17)
n = 5
unit = np.empty((dims, n))
from modepy.tools import (
pick_random_simplex_unit_coordinate,
unit_to_barycentric,
barycentric_to_unit,
barycentric_to_equilateral,
equilateral_to_unit,)
for i in range(n):
unit[:, i] = pick_random_simplex_unit_coordinate(rng, dims)
bary = unit_to_barycentric(unit)
assert (np.abs(np.sum(bary, axis=0) - 1) < 1e-15).all()
assert (bary >= 0).all()
unit2 = barycentric_to_unit(bary)
assert la.norm(unit-unit2) < 1e-14
equi = barycentric_to_equilateral(bary)
unit3 = equilateral_to_unit(equi)
assert la.norm(unit-unit3) < 1e-14
def get_simplex_element_flip_matrix(order, unit_nodes, permutation=None):
"""
Generate a resampling matrix that corresponds to a
permutation of the barycentric coordinates being applied.
The default permutation is to swap the
first two barycentric coordinates.
:param order: The order of the function space on the simplex,
(see second argument in
:fun:`modepy.simplex_best_available_basis`)
:param unit_nodes: A np array of unit nodes with shape
*(dim, nunit_nodes)*
:param permutation: Either *None*, or a tuple of shape
storing a permutation:
the *i*th barycentric coordinate gets mapped to
the *permutation[i]*th coordinate.
:return: A numpy array of shape *(nunit_nodes, nunit_nodes)*
which, when its transpose is right-applied
to the matrix of nodes (shaped *(dim, nunit_nodes)*),
corresponds to the permutation being applied
"""
from modepy.tools import barycentric_to_unit, unit_to_barycentric
bary_unit_nodes = unit_to_barycentric(unit_nodes)
flipped_bary_unit_nodes = bary_unit_nodes.copy()
if permutation is None:
flipped_bary_unit_nodes[0, :] = bary_unit_nodes[1, :]
flipped_bary_unit_nodes[1, :] = bary_unit_nodes[0, :]
else:
flipped_bary_unit_nodes[permutation, :] = bary_unit_nodes
flipped_unit_nodes = barycentric_to_unit(flipped_bary_unit_nodes)
dim = unit_nodes.shape[0]
shape = mp.Simplex(dim)
space = mp.PN(dim, order)
basis = mp.basis_for_space(space, shape)
flip_matrix = mp.resampling_matrix(basis.functions, flipped_unit_nodes,
unit_nodes)
flip_matrix[np.abs(flip_matrix) < 1e-15] = 0
# Flipping twice should be the identity
if permutation is None:
assert la.norm(
np.dot(flip_matrix, flip_matrix) -
np.eye(len(flip_matrix))) < 1e-13
return flip_matrix
def flip_simplex_element_group(vertices, grp, grp_flip_flags):
from modepy.tools import barycentric_to_unit, unit_to_barycentric
from meshmode.mesh import SimplexElementGroup
if not isinstance(grp, SimplexElementGroup):
raise NotImplementedError(
"flips only supported on "
"exclusively SimplexElementGroup-based meshes")
# Swap the first two vertices on elements to be flipped.
new_vertex_indices = grp.vertex_indices.copy()
new_vertex_indices[grp_flip_flags, 0] \
= grp.vertex_indices[grp_flip_flags, 1]
new_vertex_indices[grp_flip_flags, 1] \
= grp.vertex_indices[grp_flip_flags, 0]
# Generate a resampling matrix that corresponds to the
# first two barycentric coordinates being swapped.
bary_unit_nodes = unit_to_barycentric(grp.unit_nodes)
flipped_bary_unit_nodes = bary_unit_nodes.copy()
flipped_bary_unit_nodes[0, :] = bary_unit_nodes[1, :]
flipped_bary_unit_nodes[1, :] = bary_unit_nodes[0, :]
flipped_unit_nodes = barycentric_to_unit(flipped_bary_unit_nodes)
flip_matrix = mp.resampling_matrix(
mp.simplex_best_available_basis(grp.dim, grp.order),
flipped_unit_nodes, grp.unit_nodes)
flip_matrix[np.abs(flip_matrix) < 1e-15] = 0
# Flipping twice should be the identity
assert la.norm(
np.dot(flip_matrix, flip_matrix) - np.eye(len(flip_matrix))) < 1e-13
# Apply the flip matrix to the nodes.
new_nodes = grp.nodes.copy()
new_nodes[:, grp_flip_flags] = np.einsum("ij,dej->dei", flip_matrix,
grp.nodes[:, grp_flip_flags])
return SimplexElementGroup(grp.order,
new_vertex_indices,
new_nodes,
unit_nodes=grp.unit_nodes)
def flip_simplex_element_group(vertices, grp, grp_flip_flags):
from modepy.tools import barycentric_to_unit, unit_to_barycentric
from meshmode.mesh import SimplexElementGroup
if not isinstance(grp, SimplexElementGroup):
raise NotImplementedError("flips only supported on "
"exclusively SimplexElementGroup-based meshes")
# Swap the first two vertices on elements to be flipped.
new_vertex_indices = grp.vertex_indices.copy()
new_vertex_indices[grp_flip_flags, 0] \
= grp.vertex_indices[grp_flip_flags, 1]
new_vertex_indices[grp_flip_flags, 1] \
= grp.vertex_indices[grp_flip_flags, 0]
# Generate a resampling matrix that corresponds to the
# first two barycentric coordinates being swapped.
bary_unit_nodes = unit_to_barycentric(grp.unit_nodes)
flipped_bary_unit_nodes = bary_unit_nodes.copy()
flipped_bary_unit_nodes[0, :] = bary_unit_nodes[1, :]
flipped_bary_unit_nodes[1, :] = bary_unit_nodes[0, :]
flipped_unit_nodes = barycentric_to_unit(flipped_bary_unit_nodes)
flip_matrix = mp.resampling_matrix(
mp.simplex_best_available_basis(grp.dim, grp.order),
flipped_unit_nodes, grp.unit_nodes)
flip_matrix[np.abs(flip_matrix) < 1e-15] = 0
# Flipping twice should be the identity
assert la.norm(
np.dot(flip_matrix, flip_matrix)
- np.eye(len(flip_matrix))) < 1e-13
# Apply the flip matrix to the nodes.
new_nodes = grp.nodes.copy()
new_nodes[:, grp_flip_flags] = np.einsum(
"ij,dej->dei",
flip_matrix, grp.nodes[:, grp_flip_flags])
return SimplexElementGroup(
grp.order, new_vertex_indices, new_nodes,
unit_nodes=grp.unit_nodes)
def flip_matrix(self):
"""
:return: The matrix which should be applied to the
*(dim, nunitnodes)*-shaped array of nodes corresponding to
an element in order to change orientation - <-> +.
The matrix will be *(dim, dim)* and orthogonal with
*np.float64* type entries.
"""
if self._flip_matrix is None:
# This is very similar to :mod:`meshmode` in processing.py
# the function :function:`from_simplex_element_group`, but
# we needed to use firedrake nodes
from modepy.tools import barycentric_to_unit, unit_to_barycentric
# Generate a resampling matrix that corresponds to the
# first two barycentric coordinates being swapped.
bary_unit_nodes = unit_to_barycentric(self.unit_nodes())
flipped_bary_unit_nodes = bary_unit_nodes.copy()
flipped_bary_unit_nodes[0, :] = bary_unit_nodes[1, :]
flipped_bary_unit_nodes[1, :] = bary_unit_nodes[0, :]
flipped_unit_nodes = barycentric_to_unit(flipped_bary_unit_nodes)
from modepy import resampling_matrix, simplex_best_available_basis
flip_matrix = resampling_matrix(
simplex_best_available_basis(self.dim(),
self.analog().degree),
flipped_unit_nodes, self.unit_nodes())
flip_matrix[np.abs(flip_matrix) < 1e-15] = 0
# Flipping twice should be the identity
assert la.norm(
np.dot(flip_matrix, flip_matrix) -
np.eye(len(flip_matrix))) < 1e-13
self._flip_matrix = flip_matrix
return self._flip_matrix
def test_barycentric_coordinate_map(dims):
n = 100
from modepy.tools import (
unit_to_barycentric,
barycentric_to_unit,
barycentric_to_equilateral,
equilateral_to_unit,
)
rng = np.random.Generator(np.random.PCG64(17))
unit = nd.random_nodes_for_shape(shp.Simplex(dims), n, rng=rng)
bary = unit_to_barycentric(unit)
assert (np.abs(np.sum(bary, axis=0) - 1) < 1e-15).all()
assert (bary >= 0).all()
unit2 = barycentric_to_unit(bary)
assert la.norm(unit - unit2) < 1e-14
equi = barycentric_to_equilateral(bary)
unit3 = equilateral_to_unit(equi)
assert la.norm(unit - unit3) < 1e-14
