def plot_element_values(n, nodes, values, resample_n=None,
node_tuples=None, show_nodes=False):
dims = len(nodes)
orig_nodes = nodes
orig_values = values
if resample_n is not None:
import modepy as mp
basis = mp.simplex_onb(dims, n)
fine_nodes = mp.equidistant_nodes(dims, resample_n)
values = np.dot(mp.resampling_matrix(basis, fine_nodes, nodes), values)
nodes = fine_nodes
n = resample_n
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
if dims == 1:
import matplotlib.pyplot as pt
pt.plot(nodes[0], values)
if show_nodes:
pt.plot(orig_nodes[0], orig_values, "x")
pt.show()
elif dims == 2:
import mayavi.mlab as mlab
mlab.triangular_mesh(
nodes[0], nodes[1], values, submesh(list(gnitstam(n, 2))))
if show_nodes:
mlab.points3d(orig_nodes[0], orig_nodes[1], orig_values,
scale_factor=0.05)
mlab.show()
else:
raise RuntimeError("unsupported dimensionality %d" % dims)
def _vis_connectivity(self):
"""
:return: an array of shape
``(vis_discr.nelements,nsubelements,primitive_element_size)``
"""
# Assume that we're using modepy's default node ordering.
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam, single_valued
vis_order = single_valued(group.order
for group in self.vis_discr.groups)
node_tuples = list(gnitstam(vis_order, self.vis_discr.dim))
from modepy.tools import submesh
el_connectivity = np.array(submesh(node_tuples), dtype=np.intp)
nelements = sum(group.nelements for group in self.vis_discr.groups)
vis_connectivity = np.empty((nelements, ) + el_connectivity.shape,
dtype=np.intp)
el_nr_base = 0
for group in self.vis_discr.groups:
assert len(node_tuples) == group.nunit_nodes
vis_connectivity[el_nr_base:el_nr_base + group.nelements] = (
np.arange(el_nr_base * group.nunit_nodes,
(el_nr_base + group.nelements) * group.nunit_nodes,
group.nunit_nodes)[:, np.newaxis, np.newaxis] +
el_connectivity)
el_nr_base += group.nelements
return vis_connectivity
def _vis_connectivity(self):
"""
:return: an array of shape
``(vis_discr.nelements,nsubelements,primitive_element_size)``
"""
# Assume that we're using modepy's default node ordering.
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam, single_valued
vis_order = single_valued(
group.order for group in self.vis_discr.groups)
node_tuples = list(gnitstam(vis_order, self.vis_discr.dim))
from modepy.tools import submesh
el_connectivity = np.array(
submesh(node_tuples),
dtype=np.intp)
nelements = sum(group.nelements for group in self.vis_discr.groups)
vis_connectivity = np.empty(
(nelements,) + el_connectivity.shape, dtype=np.intp)
el_nr_base = 0
for group in self.vis_discr.groups:
assert len(node_tuples) == group.nunit_nodes
vis_connectivity[el_nr_base:el_nr_base+group.nelements] = (
np.arange(
el_nr_base*group.nunit_nodes,
(el_nr_base+group.nelements)*group.nunit_nodes,
group.nunit_nodes
)[:, np.newaxis, np.newaxis]
+ el_connectivity)
el_nr_base += group.nelements
return vis_connectivity
def estimate_lebesgue_constant(n, nodes, visualize=False):
"""Estimate the
`Lebesgue constant
<https://en.wikipedia.org/wiki/Lebesgue_constant_(interpolation)>`_
of the *nodes* at polynomial order *n*.
:arg nodes: an array of shape *(dims, nnodes)* as returned by
:func:`modepy.warp_and_blend_nodes`.
:arg visualize: visualize the function that gives rise to the
returned Lebesgue constant. (2D only for now)
:return: the Lebesgue constant, a scalar
.. versionadded:: 2013.2
"""
from modepy.matrices import vandermonde
from modepy.modes import simplex_onb
dims = len(nodes)
basis = simplex_onb(dims, n)
vdm = vandermonde(basis, nodes)
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
huge_n = 30*n
equi_node_tuples = list(gnitstam(huge_n, dims))
tons_of_equi_nodes = (
np.array(equi_node_tuples, dtype=np.float64)
/ huge_n * 2 - 1).T
eq_vdm = vandermonde(basis, tons_of_equi_nodes)
eq_to_out = la.solve(vdm.T, eq_vdm.T).T
lebesgue_worst = np.sum(np.abs(eq_to_out), axis=1)
lebesgue_constant = np.max(lebesgue_worst)
if visualize:
print("Lebesgue constant: %g" % lebesgue_constant)
from modepy.tools import submesh
import mayavi.mlab as mlab
mlab.figure(bgcolor=(1, 1, 1))
mlab.triangular_mesh(
tons_of_equi_nodes[0],
tons_of_equi_nodes[1],
lebesgue_worst / lebesgue_constant,
submesh(equi_node_tuples))
x, y = np.mgrid[-1:1:20j, -1:1:20j]
mlab.mesh(x, y, 0*x, representation="wireframe", color=(0.4, 0.4, 0.4),
line_width=0.6)
mlab.show()
return lebesgue_constant
def _vis_connectivity(self):
"""
:return: a list of :class:`_VisConnectivityGroup` instances.
"""
# Assume that we're using modepy's default node ordering.
from pytools import (
generate_nonnegative_integer_tuples_summing_to_at_most as gnitstam,
generate_nonnegative_integer_tuples_below as gnitb)
from meshmode.mesh import TensorProductElementGroup, SimplexElementGroup
result = []
from pyvisfile.vtk import (
VTK_LINE, VTK_TRIANGLE, VTK_TETRA,
VTK_QUAD, VTK_HEXAHEDRON)
subel_nr_base = 0
for group in self.vis_discr.groups:
if isinstance(group.mesh_el_group, SimplexElementGroup):
node_tuples = list(gnitstam(group.order, group.dim))
from modepy.tools import submesh
el_connectivity = np.array(
submesh(node_tuples),
dtype=np.intp)
vtk_cell_type = {
1: VTK_LINE,
2: VTK_TRIANGLE,
3: VTK_TETRA,
}[group.dim]
elif isinstance(group.mesh_el_group, TensorProductElementGroup):
node_tuples = list(gnitb(group.order+1, group.dim))
node_tuple_to_index = dict(
(nt, i) for i, nt in enumerate(node_tuples))
def add_tuple(a, b):
return tuple(ai+bi for ai, bi in zip(a, b))
el_offsets = {
1: [(0,), (1,)],
2: [(0, 0), (1, 0), (1, 1), (0, 1)],
3: [
(0, 0, 0),
(1, 0, 0),
(1, 1, 0),
(0, 1, 0),
(0, 0, 1),
(1, 0, 1),
(1, 1, 1),
(0, 1, 1),
]
}[group.dim]
el_connectivity = np.array([
[
node_tuple_to_index[add_tuple(origin, offset)]
for offset in el_offsets]
for origin in gnitb(group.order, group.dim)])
vtk_cell_type = {
1: VTK_LINE,
2: VTK_QUAD,
3: VTK_HEXAHEDRON,
}[group.dim]
else:
raise NotImplementedError("visualization for element groups "
"of type '%s'" % type(group.mesh_el_group).__name__)
assert len(node_tuples) == group.nunit_nodes
vis_connectivity = (
group.node_nr_base + np.arange(
0, group.nelements*group.nunit_nodes, group.nunit_nodes
)[:, np.newaxis, np.newaxis]
+ el_connectivity).astype(np.intp)
vgrp = _VisConnectivityGroup(
vis_connectivity=vis_connectivity,
vtk_cell_type=vtk_cell_type,
subelement_nr_base=subel_nr_base)
result.append(vgrp)
subel_nr_base += vgrp.nsubelements
return result
from __future__ import absolute_import
import numpy as np
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
from six.moves import zip
# prepare plot and eval nodes on triangle
dims = 2
node_n = 40
node_tuples = list(gnitstam(node_n, dims))
plot_nodes = np.array(node_tuples, dtype=np.float64) / node_n
eval_nodes = 2*(plot_nodes - 0.5).T
# get triangle submesh
from modepy.tools import submesh
tri_subtriangles = np.array(submesh(node_tuples))
# evaluate each basis function, build global tri mesh
node_count = 0
all_nodes = []
all_triangles = []
all_values = []
from modepy.modes import simplex_onb
p = 3
stretch_factor = 1.5
for (i, j), basis_func in zip(
gnitstam(p, dims),
simplex_onb(dims, p),
import numpy as np
from pytools import generate_nonnegative_integer_tuples_summing_to_at_most \
as gnitstam
# prepare plot and eval nodes on triangle
dims = 2
node_n = 40
node_tuples = list(gnitstam(node_n, dims))
plot_nodes = np.array(node_tuples, dtype=np.float64) / node_n
eval_nodes = 2 * (plot_nodes - 0.5).T
# get triangle submesh
from modepy.tools import submesh
tri_subtriangles = np.array(submesh(node_tuples))
# evaluate each basis function, build global tri mesh
node_count = 0
all_nodes = []
all_triangles = []
all_values = []
from modepy.modes import simplex_onb
p = 3
stretch_factor = 1.5
for (i, j), basis_func in zip(
gnitstam(p, dims),
simplex_onb(dims, p),
):
def _vis_connectivity(self):
"""
:return: a list of :class:`_VisConnectivityGroup` instances.
"""
# Assume that we're using modepy's default node ordering.
from pytools import (
generate_nonnegative_integer_tuples_summing_to_at_most as gnitstam,
generate_nonnegative_integer_tuples_below as gnitb)
from meshmode.mesh import TensorProductElementGroup, SimplexElementGroup
result = []
from pyvisfile.vtk import (VTK_LINE, VTK_TRIANGLE, VTK_TETRA, VTK_QUAD,
VTK_HEXAHEDRON)
subel_nr_base = 0
for group in self.vis_discr.groups:
if isinstance(group.mesh_el_group, SimplexElementGroup):
node_tuples = list(gnitstam(group.order, group.dim))
from modepy.tools import submesh
el_connectivity = np.array(submesh(node_tuples), dtype=np.intp)
vtk_cell_type = {
1: VTK_LINE,
2: VTK_TRIANGLE,
3: VTK_TETRA,
}[group.dim]
elif isinstance(group.mesh_el_group, TensorProductElementGroup):
node_tuples = list(gnitb(group.order + 1, group.dim))
node_tuple_to_index = dict(
(nt, i) for i, nt in enumerate(node_tuples))
def add_tuple(a, b):
return tuple(ai + bi for ai, bi in zip(a, b))
el_offsets = {
1: [(0, ), (1, )],
2: [(0, 0), (1, 0), (1, 1), (0, 1)],
3: [
(0, 0, 0),
(1, 0, 0),
(1, 1, 0),
(0, 1, 0),
(0, 0, 1),
(1, 0, 1),
(1, 1, 1),
(0, 1, 1),
]
}[group.dim]
el_connectivity = np.array([[
node_tuple_to_index[add_tuple(origin, offset)]
for offset in el_offsets
] for origin in gnitb(group.order, group.dim)])
vtk_cell_type = {
1: VTK_LINE,
2: VTK_QUAD,
3: VTK_HEXAHEDRON,
}[group.dim]
else:
raise NotImplementedError("visualization for element groups "
"of type '%s'" %
type(group.mesh_el_group).__name__)
assert len(node_tuples) == group.nunit_nodes
vis_connectivity = (
group.node_nr_base +
np.arange(0, group.nelements * group.nunit_nodes,
group.nunit_nodes)[:, np.newaxis, np.newaxis] +
el_connectivity).astype(np.intp)
vgrp = _VisConnectivityGroup(vis_connectivity=vis_connectivity,
vtk_cell_type=vtk_cell_type,
subelement_nr_base=subel_nr_base)
result.append(vgrp)
subel_nr_base += vgrp.nsubelements
return result
