def warp_and_blend_nodes_2d(n, node_tuples=None):
try:
alpha = _alpha_opt_2d[n-1]
except IndexError:
alpha = 5/3
if node_tuples is None:
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
node_tuples = list(gnitstam(n, 2))
else:
if len(node_tuples) != (n+1)*(n+2)//2:
raise ValueError("node_tuples list does not have the correct length")
# shape: (2, nnodes)
unit_nodes = (np.array(node_tuples, dtype=np.float64)/n*2 - 1).T
from modepy.tools import (
unit_to_barycentric,
barycentric_to_equilateral,
equilateral_to_unit)
bary = unit_to_barycentric(unit_nodes)
return equilateral_to_unit(
barycentric_to_equilateral(bary)
+ _2d_equilateral_shift(n, bary, alpha))
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 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
def warp_and_blend_nodes_2d(n, node_tuples=None):
try:
alpha = _alpha_opt_2d[n - 1]
except IndexError:
alpha = 5 / 3
space = PN(2, n)
if node_tuples is None:
node_tuples = node_tuples_for_space(space)
else:
if len(node_tuples) != space.space_dim:
raise ValueError(
"'node_tuples' list does not have the correct length")
# shape: (2, nnodes)
unit_nodes = (np.array(node_tuples, dtype=np.float64) / n * 2 - 1).T
from modepy.tools import (unit_to_barycentric, barycentric_to_equilateral,
equilateral_to_unit)
bary = unit_to_barycentric(unit_nodes)
return equilateral_to_unit(
barycentric_to_equilateral(bary) +
_2d_equilateral_shift(n, bary, alpha))
def warp_and_blend_nodes_3d(n, node_tuples=None):
try:
alpha = _alpha_opt_3d[n-1]
except IndexError:
alpha = 1.
if node_tuples is None:
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
node_tuples = list(gnitstam(n, 3))
else:
if len(node_tuples) != (n+1)*(n+2)*(n+3)//6:
raise ValueError("node_tuples list does not have the correct length")
# shape: (3, nnodes)
unit_nodes = (np.array(node_tuples, dtype=np.float64)/n*2 - 1).T
from modepy.tools import (
unit_to_barycentric,
barycentric_to_equilateral,
equilateral_to_unit,
EQUILATERAL_VERTICES)
bary = unit_to_barycentric(unit_nodes)
equi = barycentric_to_equilateral(bary)
equi_vertices = EQUILATERAL_VERTICES[3]
# total number of nodes and tolerance
tol = 1e-8
shift = np.zeros_like(equi)
for i1, i2, i3, i4, vertex_step in [
(0, 1, 2, 3, -1),
(1, 2, 3, 0, -1),
(2, 3, 0, 1, -1),
(3, 0, 1, 2, -1),
]:
vi2, vi3, vi4 = [(i1 + vertex_step*i) % 4 for i in range(1, 4)]
# all vertices have the same distance from the origin
tangent1 = equi_vertices[vi3] - equi_vertices[vi4]
tangent1 /= la.norm(tangent1)
tangent2 = equi_vertices[vi2] - equi_vertices[vi3]
tangent2 -= np.dot(tangent1, tangent2)*tangent1
tangent2 /= la.norm(tangent2)
sub_bary = bary[[i2, i3, i4]]
warp1, warp2 = _2d_equilateral_shift(n, sub_bary, alpha)
l1 = bary[i1]
l2, l3, l4 = sub_bary
blend = l2*l3*l4
denom = (l2+0.5*l1)*(l3+0.5*l1)*(l4+0.5*l1)
denom_ok = denom > tol
blend[denom_ok] = (
(1+(alpha*l1[denom_ok])**2)
* blend[denom_ok]
/ denom[denom_ok])
shift = shift + (blend*warp1)[np.newaxis, :] * tangent1[:, np.newaxis]
shift = shift + (blend*warp2)[np.newaxis, :] * tangent2[:, np.newaxis]
is_face = (l1 < tol) & ((l2 > tol) | (l3 > tol) | (l4 > tol))
# assign face warp separately
shift[:, is_face] = (
+ warp1[is_face][np.newaxis, :] * tangent1[:, np.newaxis]
+ warp2[is_face][np.newaxis, :] * tangent2[:, np.newaxis]
)
return equilateral_to_unit(equi + shift)
def warp_and_blend_nodes_3d(n, node_tuples=None):
try:
alpha = _alpha_opt_3d[n - 1]
except IndexError:
alpha = 1.
if node_tuples is None:
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
node_tuples = list(gnitstam(n, 3))
else:
if len(node_tuples) != (n + 1) * (n + 2) * (n + 3) // 6:
raise ValueError(
"node_tuples list does not have the correct length")
# shape: (3, nnodes)
unit_nodes = (np.array(node_tuples, dtype=np.float64) / n * 2 - 1).T
from modepy.tools import (unit_to_barycentric, barycentric_to_equilateral,
equilateral_to_unit, EQUILATERAL_VERTICES)
bary = unit_to_barycentric(unit_nodes)
equi = barycentric_to_equilateral(bary)
equi_vertices = EQUILATERAL_VERTICES[3]
# total number of nodes and tolerance
tol = 1e-8
shift = np.zeros_like(equi)
for i1, i2, i3, i4, vertex_step in [
(0, 1, 2, 3, -1),
(1, 2, 3, 0, -1),
(2, 3, 0, 1, -1),
(3, 0, 1, 2, -1),
]:
vi2, vi3, vi4 = [(i1 + vertex_step * i) % 4 for i in range(1, 4)]
# all vertices have the same distance from the origin
tangent1 = equi_vertices[vi3] - equi_vertices[vi4]
tangent1 /= la.norm(tangent1)
tangent2 = equi_vertices[vi2] - equi_vertices[vi3]
tangent2 -= np.dot(tangent1, tangent2) * tangent1
tangent2 /= la.norm(tangent2)
sub_bary = bary[[i2, i3, i4]]
warp1, warp2 = _2d_equilateral_shift(n, sub_bary, alpha)
l1 = bary[i1]
l2, l3, l4 = sub_bary
blend = l2 * l3 * l4
denom = (l2 + 0.5 * l1) * (l3 + 0.5 * l1) * (l4 + 0.5 * l1)
denom_ok = denom > tol
blend[denom_ok] = ((1 + (alpha * l1[denom_ok])**2) * blend[denom_ok] /
denom[denom_ok])
shift = shift + (blend * warp1)[np.newaxis, :] * tangent1[:,
np.newaxis]
shift = shift + (blend * warp2)[np.newaxis, :] * tangent2[:,
np.newaxis]
is_face = (l1 < tol) & ((l2 > tol) | (l3 > tol) | (l4 > tol))
# assign face warp separately
shift[:, is_face] = (
+warp1[is_face][np.newaxis, :] * tangent1[:, np.newaxis] +
warp2[is_face][np.newaxis, :] * tangent2[:, np.newaxis])
return equilateral_to_unit(equi + shift)