def test_deprecated_nodal_face_mass_matrix(dims, order=3):
# FIXME DEPRECATED remove along with nodal_face_mass_matrix (>=2022)
vol_shape = mp.Simplex(dims)
vol_space = mp.space_for_shape(vol_shape, order)
vertices = mp.unit_vertices_for_shape(vol_shape)
volume_nodes = mp.edge_clustered_nodes_for_space(vol_space, vol_shape)
volume_basis = mp.basis_for_space(vol_space, vol_shape)
from modepy.matrices import nodal_face_mass_matrix
for face in mp.faces_for_shape(vol_shape):
face_space = mp.space_for_shape(face, order)
face_nodes = mp.edge_clustered_nodes_for_space(face_space, face)
face_vertices = vertices[:, face.volume_vertex_indices]
fmm = nodal_face_mass_matrix(
volume_basis.functions, volume_nodes,
face_nodes, order, face_vertices)
fmm2 = nodal_face_mass_matrix(
volume_basis.functions,
volume_nodes, face_nodes, order+1, face_vertices)
error = la.norm(fmm - fmm2, np.inf) / la.norm(fmm2, np.inf)
logger.info("fmm error: %.5e", error)
assert error < 5e-11, f"error {error:.5e} on face {face.face_index}"
fmm[np.abs(fmm) < 1e-13] = 0
nnz = np.sum(fmm > 0)
logger.info("fmm: nnz %d\n%s", nnz, fmm)
logger.info("mass matrix:\n%s", mp.mass_matrix(
mp.basis_for_space(face_space, face).functions,
mp.edge_clustered_nodes_for_space(face_space, face)))
def test_modal_mass_matrix_for_face(dims, shape_cls, order=3):
vol_shape = shape_cls(dims)
vol_space = mp.space_for_shape(vol_shape, order)
vol_basis = mp.basis_for_space(vol_space, vol_shape)
from modepy.matrices import modal_mass_matrix_for_face
for face in mp.faces_for_shape(vol_shape):
face_space = mp.space_for_shape(face, order)
face_basis = mp.basis_for_space(face_space, face)
face_quad = mp.quadrature_for_space(mp.space_for_shape(face, 2*order), face)
face_quad2 = mp.quadrature_for_space(
mp.space_for_shape(face, 2*order+2), face)
fmm = modal_mass_matrix_for_face(
face, face_quad, face_basis.functions, vol_basis.functions)
fmm2 = modal_mass_matrix_for_face(
face, face_quad2, face_basis.functions, vol_basis.functions)
error = la.norm(fmm - fmm2, np.inf) / la.norm(fmm2, np.inf)
logger.info("fmm error: %.5e", error)
assert error < 1e-11, f"error {error:.5e} on face {face.face_index}"
fmm[np.abs(fmm) < 1e-13] = 0
nnz = np.sum(fmm > 0)
logger.info("fmm: nnz %d\n%s", nnz, fmm)
def test_resampling_matrix(dims, shape_cls, ncoarse=5, nfine=10):
shape = shape_cls(dims)
coarse_space = mp.space_for_shape(shape, ncoarse)
fine_space = mp.space_for_shape(shape, nfine)
coarse_nodes = mp.edge_clustered_nodes_for_space(coarse_space, shape)
coarse_basis = mp.basis_for_space(coarse_space, shape)
fine_nodes = mp.edge_clustered_nodes_for_space(fine_space, shape)
fine_basis = mp.basis_for_space(fine_space, shape)
my_eye = np.dot(
mp.resampling_matrix(fine_basis.functions, coarse_nodes, fine_nodes),
mp.resampling_matrix(coarse_basis.functions, fine_nodes, coarse_nodes))
assert la.norm(my_eye - np.eye(len(my_eye))) < 3e-13
my_eye_least_squares = np.dot(
mp.resampling_matrix(coarse_basis.functions, coarse_nodes, fine_nodes,
least_squares_ok=True),
mp.resampling_matrix(coarse_basis.functions, fine_nodes, coarse_nodes),
)
assert la.norm(my_eye_least_squares - np.eye(len(my_eye_least_squares))) < 4e-13
def from_mesh_interp_matrix(grp: NodalElementGroupBase) -> np.ndarray:
meg = grp.mesh_el_group
meg_space = type(grp.space)(meg.dim, meg.order)
return mp.resampling_matrix(
mp.basis_for_space(meg_space, grp.shape).functions, grp.unit_nodes,
meg.unit_nodes)
def is_affine_simplex_group(group, abs_tol=None):
if abs_tol is None:
abs_tol = 1.0e-13
if not isinstance(group, SimplexElementGroup):
raise TypeError("expected a 'SimplexElementGroup' not '%s'" %
type(group).__name__)
# get matrices
basis = mp.basis_for_space(group._modepy_space, group._modepy_shape)
vinv = la.inv(mp.vandermonde(basis.functions, group.unit_nodes))
diff = mp.differentiation_matrices(
basis.functions, basis.gradients, group.unit_nodes)
if not isinstance(diff, tuple):
diff = (diff,)
# construct all second derivative matrices (including cross terms)
from itertools import product
mats = []
for n in product(range(group.dim), repeat=2):
if n[0] > n[1]:
continue
mats.append(vinv.dot(diff[n[0]].dot(diff[n[1]])))
# check just the first element for a non-affine local-to-global mapping
ddx_coeffs = np.einsum("aij,bj->abi", mats, group.nodes[:, 0, :])
norm_inf = np.max(np.abs(ddx_coeffs))
if norm_inf > abs_tol:
return False
# check all elements for a non-affine local-to-global mapping
ddx_coeffs = np.einsum("aij,bcj->abci", mats, group.nodes)
norm_inf = np.max(np.abs(ddx_coeffs))
return norm_inf < abs_tol
def test_deprecated_modal_face_mass_matrix(dims, order=3):
# FIXME DEPRECATED remove along with modal_face_mass_matrix (>=2022)
shape = mp.Simplex(dims)
space = mp.space_for_shape(shape, order)
vertices = mp.unit_vertices_for_shape(shape)
basis = mp.basis_for_space(space, shape)
from modepy.matrices import modal_face_mass_matrix
for face in mp.faces_for_shape(shape):
face_vertices = vertices[:, face.volume_vertex_indices]
fmm = modal_face_mass_matrix(
basis.functions, order, face_vertices)
fmm2 = modal_face_mass_matrix(
basis.functions, order+1, face_vertices)
error = la.norm(fmm - fmm2, np.inf) / la.norm(fmm2, np.inf)
logger.info("fmm error: %.5e", error)
assert error < 1e-11, f"error {error:.5e} on face {face.face_index}"
fmm[np.abs(fmm) < 1e-13] = 0
nnz = np.sum(fmm > 0)
logger.info("fmm: nnz %d\n%s", nnz, fmm)
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 test_nodal_mass_matrix_for_face(dims, shape_cls, order=3):
vol_shape = shape_cls(dims)
vol_space = mp.space_for_shape(vol_shape, order)
volume_nodes = mp.edge_clustered_nodes_for_space(vol_space, vol_shape)
volume_basis = mp.basis_for_space(vol_space, vol_shape)
from modepy.matrices import (nodal_mass_matrix_for_face,
nodal_quad_mass_matrix_for_face)
for face in mp.faces_for_shape(vol_shape):
face_space = mp.space_for_shape(face, order)
face_basis = mp.basis_for_space(face_space, face)
face_nodes = mp.edge_clustered_nodes_for_space(face_space, face)
face_quad = mp.quadrature_for_space(mp.space_for_shape(face, 2*order), face)
face_quad2 = mp.quadrature_for_space(
mp.space_for_shape(face, 2*order+2), face)
fmm = nodal_mass_matrix_for_face(
face, face_quad, face_basis.functions, volume_basis.functions,
volume_nodes, face_nodes)
fmm2 = nodal_quad_mass_matrix_for_face(
face, face_quad2, volume_basis.functions, volume_nodes)
for f_face in face_basis.functions:
fval_nodal = f_face(face_nodes)
fval_quad = f_face(face_quad2.nodes)
assert (
la.norm(fmm@fval_nodal - fmm2@fval_quad, np.inf)
/ la.norm(fval_quad)) < 3e-15
fmm[np.abs(fmm) < 1e-13] = 0
nnz = np.sum(fmm > 0)
logger.info("fmm: nnz %d\n%s", nnz, fmm)
logger.info("mass matrix:\n%s",
mp.mass_matrix(face_basis.functions, face_nodes))
def _(meg: _ModepyElementGroup, el_tess_info, elements):
shape = meg._modepy_shape
space = meg._modepy_space
# get midpoints in reference coordinates
midpoints = -1 + np.array(
_get_ref_midpoints(shape, el_tess_info.ref_vertices))
# resample midpoints to ambient coordinates
resampling_mat = mp.resampling_matrix(
mp.basis_for_space(space, shape).functions, midpoints.T,
meg.unit_nodes)
resampled_midpoints = np.einsum("mu,deu->edm", resampling_mat,
meg.nodes[:, elements])
return dict(zip(elements, resampled_midpoints))
def test_diff_matrix(dims, shape_cls, order=5):
shape = shape_cls(dims)
space = mp.space_for_shape(shape, order)
nodes = mp.edge_clustered_nodes_for_space(space, shape)
basis = mp.basis_for_space(space, shape)
diff_mat = mp.differentiation_matrices(basis.functions, basis.gradients, nodes)
if isinstance(diff_mat, tuple):
diff_mat = diff_mat[0]
f = np.sin(nodes[0])
df_dx = np.cos(nodes[0])
df_dx_num = np.dot(diff_mat, f)
error = la.norm(df_dx - df_dx_num) / la.norm(df_dx)
logger.info("error: %.5e", error)
assert error < 2.0e-4, error
def _test_node_vertex_consistency_resampling(mesh, mgrp, tol):
if mesh.vertices is None:
return True
if mgrp.nelements == 0:
return True
if isinstance(mgrp, _ModepyElementGroup):
basis = mp.basis_for_space(mgrp._modepy_space, mgrp._modepy_shape).functions
else:
raise TypeError(f"unsupported group type: {type(mgrp).__name__}")
resampling_mat = mp.resampling_matrix(
basis,
mgrp.vertex_unit_coordinates().T,
mgrp.unit_nodes)
# dim, nelments, nnvertices
map_vertices = np.einsum(
"ij,dej->dei", resampling_mat, mgrp.nodes)
grp_vertices = mesh.vertices[:, mgrp.vertex_indices]
per_element_vertex_errors = np.sqrt(np.sum(
np.sum((map_vertices - grp_vertices)**2, axis=0),
axis=-1))
if tol is None:
tol = 1e3 * np.finfo(per_element_vertex_errors.dtype).eps
from meshmode.mesh.processing import find_bounding_box
bbox_min, bbox_max = find_bounding_box(mesh)
size = la.norm(bbox_max-bbox_min)
max_el_vertex_error = np.max(per_element_vertex_errors)
assert max_el_vertex_error < tol*size, max_el_vertex_error
return True
def _(meg: _ModepyElementGroup, el_tess_info, elements):
shape = meg._modepy_shape
space = meg._modepy_space
# get child unit node coordinates.
from meshmode.mesh.refinement.utils import map_unit_nodes_to_children
child_unit_nodes = np.hstack(
list(map_unit_nodes_to_children(meg, meg.unit_nodes, el_tess_info)))
# resample child nodes to ambient coordinates
resampling_mat = mp.resampling_matrix(
mp.basis_for_space(space, shape).functions, child_unit_nodes,
meg.unit_nodes)
resampled_unit_nodes = np.einsum("cu,deu->edc", resampling_mat,
meg.nodes[:, elements])
ambient_dim = len(meg.nodes)
nunit_nodes = len(meg.unit_nodes[0])
return {
el: resampled_unit_nodes[iel].reshape((ambient_dim, -1, nunit_nodes))
for iel, el in enumerate(elements)
}
def basis_obj(self):
return mp.basis_for_space(self.space, self.shape)
def make_face_restriction(actx, discr, group_factory, boundary_tag,
per_face_groups=False):
"""Create a mesh, a discretization and a connection to restrict
a function on *discr* to its values on the edges of element faces
denoted by *boundary_tag*.
:arg boundary_tag: The boundary tag for which to create a face
restriction. May be
:class:`FACE_RESTR_INTERIOR`
to indicate interior faces, or
:class:`FACE_RESTR_ALL`
to make a discretization consisting of all (interior and
boundary) faces.
:arg per_face_groups: If *True*, the resulting discretization is
guaranteed to have groups organized as::
(grp0, face0), (grp0, face1), ... (grp0, faceN),
(grp1, face0), (grp1, face1), ... (grp1, faceN), ...
each with the elements in the same order as the originating
group. If *False*, volume and boundary groups correspond with
each other one-to-one, and an interpolation batch is created
per face.
:return: a
:class:`meshmode.discretization.connection.DirectDiscretizationConnection`
representing the new connection. The new boundary discretization can be
obtained from the
:attr:`meshmode.discretization.connection.DirectDiscretizationConnection.to_discr`
attribute of the return value, and the corresponding new boundary mesh
from that.
"""
if boundary_tag is None:
boundary_tag = FACE_RESTR_INTERIOR
from warnings import warn
warn("passing *None* for boundary_tag is deprecated--pass "
"FACE_RESTR_INTERIOR instead",
DeprecationWarning, stacklevel=2)
logger.info("building face restriction: start")
# {{{ gather boundary vertices
bdry_vertex_vol_nrs = _get_face_vertices(discr.mesh, boundary_tag)
vol_to_bdry_vertices = np.empty(
discr.mesh.vertices.shape[-1],
discr.mesh.vertices.dtype)
vol_to_bdry_vertices.fill(-1)
vol_to_bdry_vertices[bdry_vertex_vol_nrs] = np.arange(
len(bdry_vertex_vol_nrs), dtype=np.intp)
bdry_vertices = discr.mesh.vertices[:, bdry_vertex_vol_nrs]
# }}}
from meshmode.mesh import Mesh, _ModepyElementGroup
bdry_mesh_groups = []
connection_data = {}
if boundary_tag not in [FACE_RESTR_ALL, FACE_RESTR_INTERIOR]:
btag_bit = discr.mesh.boundary_tag_bit(boundary_tag)
for igrp, (grp, fagrp_map) in enumerate(
zip(discr.groups, discr.mesh.facial_adjacency_groups)):
mgrp = grp.mesh_el_group
if not isinstance(mgrp, _ModepyElementGroup):
raise NotImplementedError("can only take boundary of "
"meshes based on SimplexElementGroup and "
"TensorProductElementGroup")
# {{{ pull together per-group face lists
group_boundary_faces = []
if boundary_tag is FACE_RESTR_INTERIOR:
for fagrp in fagrp_map.values():
if fagrp.ineighbor_group is None:
# boundary faces -> not looking for those
continue
group_boundary_faces.extend(
zip(fagrp.elements, fagrp.element_faces))
elif boundary_tag is FACE_RESTR_ALL:
group_boundary_faces.extend(
(iel, iface)
for iface in range(grp.mesh_el_group.nfaces)
for iel in range(grp.nelements)
)
else:
bdry_grp = fagrp_map.get(None)
if bdry_grp is not None:
nb_el_bits = -bdry_grp.neighbors
face_relevant_flags = (nb_el_bits & btag_bit) != 0
group_boundary_faces.extend(
zip(
bdry_grp.elements[face_relevant_flags],
bdry_grp.element_faces[face_relevant_flags]))
# }}}
batch_base = 0
# group by face_index
for face in mgrp._modepy_faces:
batch_boundary_el_numbers_in_grp = np.array([
ibface_el
for ibface_el, ibface_face in group_boundary_faces
if ibface_face == face.face_index
], dtype=np.intp)
# {{{ preallocate arrays for mesh group
nbatch_elements = len(batch_boundary_el_numbers_in_grp)
if per_face_groups or face.face_index == 0:
if per_face_groups:
ngroup_bdry_elements = nbatch_elements
else:
ngroup_bdry_elements = len(group_boundary_faces)
# make up some not-terrible nodes for the boundary Mesh
space = mp.space_for_shape(face, mgrp.order)
bdry_unit_nodes = mp.edge_clustered_nodes_for_space(space, face)
vol_basis = mp.basis_for_space(
mgrp._modepy_space, mgrp._modepy_shape).functions
vertex_indices = np.empty(
(ngroup_bdry_elements, face.nvertices),
mgrp.vertex_indices.dtype)
nbdry_unit_nodes = bdry_unit_nodes.shape[-1]
nodes = np.empty(
(discr.ambient_dim, ngroup_bdry_elements, nbdry_unit_nodes),
dtype=np.float64)
# }}}
new_el_numbers = batch_base + np.arange(nbatch_elements)
if not per_face_groups:
batch_base += nbatch_elements
# {{{ no per-element axes in these computations
face_unit_nodes = face.map_to_volume(bdry_unit_nodes)
resampling_mat = mp.resampling_matrix(
vol_basis,
face_unit_nodes, mgrp.unit_nodes)
# }}}
# {{{ build information for mesh element group
# Find vertex_indices
glob_face_vertices = mgrp.vertex_indices[
batch_boundary_el_numbers_in_grp][:, face.volume_vertex_indices]
vertex_indices[new_el_numbers] = \
vol_to_bdry_vertices[glob_face_vertices]
# Find nodes
nodes[:, new_el_numbers, :] = np.einsum(
"ij,dej->dei",
resampling_mat,
mgrp.nodes[:, batch_boundary_el_numbers_in_grp, :])
# }}}
connection_data[igrp, face.face_index] = _ConnectionBatchData(
group_source_element_indices=batch_boundary_el_numbers_in_grp,
group_target_element_indices=new_el_numbers,
face=face,
)
is_last_face = face.face_index + 1 == mgrp.nfaces
if per_face_groups or is_last_face:
bdry_mesh_group = type(mgrp)(
mgrp.order, vertex_indices, nodes,
unit_nodes=bdry_unit_nodes)
bdry_mesh_groups.append(bdry_mesh_group)
bdry_mesh = Mesh(bdry_vertices, bdry_mesh_groups)
from meshmode.discretization import Discretization
bdry_discr = Discretization(
actx, bdry_mesh, group_factory)
connection = _build_boundary_connection(
actx, discr, bdry_discr, connection_data,
per_face_groups)
logger.info("building face restriction: done")
return connection
def from_mesh_interp_matrix(self):
meg = self.mesh_el_group
meg_space = mp.QN(meg.dim, meg.order)
return mp.resampling_matrix(
mp.basis_for_space(meg_space, self._shape).functions,
self.unit_nodes, meg.unit_nodes)
def _basis(self):
return mp.basis_for_space(self._space, self._shape)