def poisson_disc_nodes(radius, domain,
ntests=50,
rmax_factor=1.5,
build_rtree=False,
**kwargs):
'''
Generates nodes within a two or three dimensional domain. This first
generate nodes with Poisson disc sampling, and then the nodes are dispersed
to ensure a more even distribution. This function is considerably slower
than `min_energy_nodes` but it has the advantage of directly specifying the
node spacing.
Parameters
----------
radius : float or callable
The radius for each disc. This is the minimum allowable distance
between the nodes generated by Poisson disc sampling. This can be a
float or a function that takes a (n, d) array of locations and returns
an (n,) array of disc radii.
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices
build_rtree : bool, optional
If `True`, then an R-tree will be built to speed up computational
geometry operations. This should be set to `True` if there are many
(>10,000) simplices making up the domain
**kwargs
Additional arguments passed to `prepare_nodes`
Returns
-------
(n, d) float array
Nodes positions
dict
The indices of nodes belonging to each group. There will always be a
group called 'interior' containing the nodes that are not on the
boundary. By default there is a group containing all the boundary nodes
called 'boundary:all'. If `boundary_groups` was specified, then those
groups will be included in this dictionary and their names will be
given a 'boundary:' prefix. If `boundary_groups_with_ghosts` was
specified then those groups of ghost nodes will be included in this
dictionary and their names will be given a 'ghosts:' prefix.
(n, d) float array
Outward normal vectors for each node. If a node is not on the boundary
then its corresponding row will contain NaNs.
Notes
-----
It is assumed that `vert` and `smp` define a closed domain. If this is not
the case, then it is likely that an error message will be raised which says
"ValueError: No intersection found for segment
'''
domain = as_domain(domain)
if build_rtree:
logger.debug('Building R-tree...')
domain.build_rtree()
logger.debug('Done')
if np.isscalar(radius):
scalar_radius = radius
def radius(x):
return np.full(x.shape[0], scalar_radius)
def rho(x):
# the density function corresponding to the radius function
return 1.0/(radius(x)**x.shape[1])
nodes = poisson_discs(
radius, domain,
ntests=ntests,
rmax_factor=rmax_factor
)
out = prepare_nodes(nodes, domain, rho=rho, **kwargs)
return out
def disperse(nodes, domain,
iterations=20,
rho=None,
fixed_nodes=None,
neighbors=None,
delta=0.1):
'''
Disperses the nodes within the domain. The dispersion is analogous to
electrostatic repulsion, where neighboring nodes exert a repulsive force on
eachother. Each node steps in the direction of its net repulsive force with
a step size proportional to the distance to its nearest neighbor. If a node
is repelled into a boundary then it bounces back in.
Parameters
----------
nodes : (n, d) float array
Initial node positions
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices.
iterations : int, optional
Number of dispersion iterations.
rho : callable, optional
Takes an (n, d) array as input and returns the repulsion force for a
node at those position.
fixed_nodes : (k, d) float array, optional
Nodes which do not move and only provide a repulsion force.
neighbors : int, optional
The number of adjacent nodes used to determine repulsion forces for
each node.
delta : float, optional
The step size. Each node moves in the direction of the repulsion force
by a distance `delta` times the distance to the nearest neighbor.
Returns
-------
(n, d) float array
'''
domain = as_domain(domain)
nodes = np.asarray(nodes, dtype=float)
assert_shape(nodes, (None, domain.dim), 'nodes')
if rho is None:
def rho(x):
return np.ones(x.shape[0])
if fixed_nodes is None:
fixed_nodes = np.zeros((0, domain.dim), dtype=float)
else:
fixed_nodes = np.asarray(fixed_nodes)
assert_shape(fixed_nodes, (None, domain.dim), 'fixed_nodes')
if neighbors is None:
# the default number of neighboring nodes to use when computing the
# repulsion force is 3 for 2D and 4 for 3D
if domain.dim == 2:
neighbors = 3
elif domain.dim == 3:
neighbors = 4
# ensure that the number of neighboring nodes used for the repulsion force
# is less than or equal to the total number of nodes
neighbors = min(neighbors, nodes.shape[0] + fixed_nodes.shape[0] - 1)
for itr in range(iterations):
logger.debug(
'Starting node dispersion iterations %s of %s.'
% (itr + 1, iterations)
)
new_nodes = _disperse_step(nodes, rho, fixed_nodes, neighbors, delta)
# If the line segment connecting the new and old node crosses the
# boundary, then the node should bounce off the boundary.
crossed, = domain.intersection_count(nodes, new_nodes).nonzero()
# points where nodes intersected the boundary and the simplex they
# intersected at
intr_pnt, intr_idx = domain.intersection_point(
nodes[crossed],
new_nodes[crossed]
)
# normal vector to the intersection points
intr_norms = domain.normals[intr_idx]
# residual distance that the nodes wanted to travel beyond the boundary
res = new_nodes[crossed] - intr_pnt
# normal component of the residuals
res_perp = np.sum(res*intr_norms, axis=1)
# bounce nodes off the boundary
new_nodes[crossed] -= 2*intr_norms*res_perp[:, None]
# check to see if the bounced nodes are still crossing the boundary. If
# they are, then set them back to their original position. Do not
# bother with multiple bounces.
still_crossed, = domain.intersection_count(
nodes[crossed],
new_nodes[crossed]
).nonzero()
new_nodes[crossed[still_crossed]] = nodes[crossed[still_crossed]]
nodes = new_nodes
return nodes
def min_energy_nodes(n, domain,
rho=None,
build_rtree=False,
start=0,
**kwargs):
'''
Generates nodes within a two or three dimensional. This first generates
nodes with a rejection sampling algorithm, and then the nodes are dispersed
to ensure a more even distribution.
Parameters
----------
n : int
The number of nodes generated during rejection sampling. This is not
necessarily equal to the number of nodes returned.
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices
rho : function, optional
Node density function. Takes a (n, d) array of coordinates and returns
an (n,) array of desired node densities at those coordinates. This
function should be normalized to be between 0 and 1.
build_rtree : bool, optional
If `True`, then an R-tree will be built to speed up computational
geometry operations. This should be set to `True` if there are many
(>10,000) simplices making up the domain.
start : int, optional
The starting index for the Halton sequence, which is used to propose
new points. Setting this value is akin to setting the seed for a random
number generator.
**kwargs
Additional arguments passed to `prepare_nodes`
Returns
-------
(n, d) float array
Nodes positions
dict
The indices of nodes belonging to each group. There will always be a
group called 'interior' containing the nodes that are not on the
boundary. By default there is a group containing all the boundary nodes
called 'boundary:all'. If `boundary_groups` was specified, then those
groups will be included in this dictionary and their names will be
given a 'boundary:' prefix. If `boundary_groups_with_ghosts` was
specified then those groups of ghost nodes will be included in this
dictionary and their names will be given a 'ghosts:' prefix.
(n, d) float array
Outward normal vectors for each node. If a node is not on the boundary
then its corresponding row will contain NaNs.
Notes
-----
It is assumed that `vert` and `smp` define a closed domain. If this is not
the case, then it is likely that an error message will be raised which says
"ValueError: No intersection found for segment
Examples
--------
make 9 nodes within the unit square
>>> vert = np.array([[0, 0], [1, 0], [1, 1], [0, 1]])
>>> smp = np.array([[0, 1], [1, 2], [2, 3], [3, 0]])
>>> out = min_energy_nodes(9, (vert, smp))
view the nodes
>>> out[0]
array([[ 0.50325675, 0. ],
[ 0.00605261, 1. ],
[ 1. , 0.51585247],
[ 0. , 0.00956821],
[ 1. , 0.99597894],
[ 0. , 0.5026365 ],
[ 1. , 0.00951112],
[ 0.48867638, 1. ],
[ 0.54063894, 0.47960892]])
view the indices of nodes making each group
>>> out[1]
{'boundary:all': array([7, 6, 5, 4, 3, 2, 1, 0]),
'interior': array([8])}
view the outward normal vectors for each node, note that the normal vector
for the interior node is `nan`
>>> out[2]
array([[ 0., -1.],
[ 0., 1.],
[ 1., -0.],
[ -1., -0.],
[ 1., -0.],
[ -1., -0.],
[ 1., -0.],
[ 0., 1.],
[ nan, nan]])
'''
domain = as_domain(domain)
if build_rtree:
logger.debug('Building R-tree...')
domain.build_rtree()
logger.debug('Done')
if rho is None:
def rho(x):
return np.ones(x.shape[0])
nodes = rejection_sampling(n, rho, domain, start=start)
out = prepare_nodes(nodes, domain, rho=rho, **kwargs)
return out
def prepare_nodes(nodes, domain,
rho=None,
iterations=20,
neighbors=None,
dispersion_delta=0.1,
pinned_nodes=None,
snap_delta=0.5,
boundary_groups=None,
boundary_groups_with_ghosts=None,
ghost_delta=0.5,
include_vertices=False,
orient_simplices=True):
'''
Prepares a set of nodes for solving PDEs with the RBF and RBF-FD method.
This includes: dispersing the nodes away from eachother to ensure a more
even spacing, snapping nodes to the boundary, determining the normal
vectors for each node, determining the group that each node belongs to,
creating ghost nodes, sorting the nodes so that adjacent nodes are close in
memory, and verifying that no two nodes are anomalously close to eachother.
The function returns a set of nodes, the normal vectors for each node, and
a dictionary identifying which group each node belongs to.
Parameters
----------
nodes : (n, d) float arrary
An initial sampling of nodes within the domain
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices
rho : function, optional
Node density function. Takes a (n, d) array of coordinates and returns
an (n,) array of desired node densities at those coordinates. This is
used during the node dispersion step.
iterations : int, optional
Number of dispersion iterations.
neighbors : int, optional
Number of neighboring nodes to use when calculating the repulsion
force. This defaults to 3 for 2D nodes and 4 for 3D nodes.
dispersion_delta : float, optional
Scaling factor for the node step size in each iteration. The step size
is equal to `dispersion_delta` times the distance to the nearest
neighbor.
pinned_nodes : (k, d) array, optional
Nodes which do not move and only provide a repulsion force. These nodes
are included in the set of nodes returned by this function and they are
in the group named "pinned".
snap_delta : float, optional
Controls the maximum snapping distance. The maximum snapping distance
for each node is `snap_delta` times the distance to the nearest
neighbor. This defaults to 0.5.
boundary_groups: dict, optional
Dictionary defining the boundary groups. The keys are the names of the
groups and the values are lists of simplex indices making up each
group. This function will return a dictionary identifying which nodes
belong to each boundary group. By default, there is a single group
named 'all' for the entire boundary. Specifically, The default value is
`{'all':range(len(smp))}`.
boundary_groups_with_ghosts: list of strs, optional
List of boundary groups that will be given ghost nodes. By default, no
boundary groups are given ghost nodes. The groups specified here must
exist in `boundary_groups`.
ghost_delta : float, optional
How far the ghost nodes should be from their corresponding boundary
node. The distance is `ghost_delta` times the distance to the nearest
neighbor.
include_vertices : bool, optional
If `True`, then the vertices will be included in the output nodes. Each
vertex will be assigned to the boundary group that its adjoining
simplices are part of. If the simplices are in multiple groups, then
the vertex will be assigned to the group containing the simplex that
comes first in `smp`.
orient_simplices : bool, optional
If `False` then it is assumed that the simplices are already oriented
such that their normal vectors point outward.
Returns
-------
(m, d) float array
Nodes positions
dict
The indices of nodes belonging to each group. There will always be a
group called 'interior' containing the nodes that are not on the
boundary. By default there is a group containing all the boundary nodes
called 'boundary:all'. If `boundary_groups` was specified, then those
groups will be included in this dictionary and their names will be
given a 'boundary:' prefix. If `boundary_groups_with_ghosts` was
specified then those groups of ghost nodes will be included in this
dictionary and their names will be given a 'ghosts:' prefix.
(n, d) float array
Outward normal vectors for each node. If a node is not on the boundary
then its corresponding row will contain NaNs.
'''
domain = as_domain(domain)
if orient_simplices:
logger.debug('Orienting simplices...')
domain.orient_simplices()
logger.debug('Done')
nodes = np.asarray(nodes, dtype=float)
assert_shape(nodes, (None, domain.dim), 'nodes')
# the `fixed_nodes` are used to provide a repulsion force during
# dispersion, but they do not move.
fixed_nodes = np.zeros((0, domain.dim), dtype=float)
if pinned_nodes is not None:
pinned_nodes = np.asarray(pinned_nodes, dtype=float)
assert_shape(pinned_nodes, (None, domain.dim), 'pinned_nodes')
fixed_nodes = np.vstack((fixed_nodes, pinned_nodes))
if include_vertices:
fixed_nodes = np.vstack((fixed_nodes, domain.vertices))
logger.debug('Dispersing nodes...')
nodes = disperse(
nodes, domain,
iterations=iterations,
rho=rho,
fixed_nodes=fixed_nodes,
neighbors=neighbors,
delta=dispersion_delta
)
logger.debug('Done')
# append the domain vertices to the collection of nodes if requested
if include_vertices:
nodes = np.vstack((nodes, domain.vertices))
# snap nodes to the boundary, identifying which simplex each node
# was snapped to
logger.debug('Snapping nodes to boundary...')
nodes, smpid = domain.snap(nodes, delta=snap_delta)
logger.debug('Done')
normals = np.full_like(nodes, np.nan)
normals[smpid >= 0] = domain.normals[smpid[smpid >= 0]]
# create a dictionary identifying which nodes belong to which group
groups = {}
groups['interior'], = (smpid == -1).nonzero()
# append the user specified pinned nodes
if pinned_nodes is not None:
pinned_idx = np.arange(pinned_nodes.shape[0]) + nodes.shape[0]
pinned_normals = np.full_like(pinned_nodes, np.nan)
nodes = np.vstack((nodes, pinned_nodes))
normals = np.vstack((normals, pinned_normals))
groups['pinned'] = pinned_idx
logger.debug('Grouping boundary nodes...')
if boundary_groups is None:
boundary_groups = {'all': np.arange(len(domain.simplices))}
else:
boundary_groups = {
str(k): np.array(v, dtype=int) for k, v in boundary_groups.items()
}
# Validate the user-specified boundary groups
simplex_counts = Counter(chain(*boundary_groups.values()))
for idx in range(len(domain.simplices)):
if simplex_counts[idx] != 1:
logger.warning(
'Simplex %s is specified %s times in the boundary groups.'
% (idx, simplex_counts[idx])
)
extra = set(simplex_counts).difference(range(len(domain.simplices)))
if extra:
raise ValueError(
'The simplex indices %s were specified in the boundary groups '
'but do not exist.' % extra
)
if boundary_groups_with_ghosts is None:
boundary_groups_with_ghosts = []
# find the mapping from simplex indices to node indices, then use
# `boundary_groups` to find which nodes belong to each boundary group
smp_to_nodes = [[] for _ in range(len(domain.simplices))]
for i, j in enumerate(smpid):
if j != -1:
smp_to_nodes[j].append(i)
for bnd_name, bnd_smp in boundary_groups.items():
bnd_idx = list(chain.from_iterable(smp_to_nodes[i] for i in bnd_smp))
groups['boundary:%s' % bnd_name] = np.array(bnd_idx, dtype=int)
logger.debug('Done')
logger.debug('Creating ghost nodes...')
tree = KDTree(nodes)
for bnd_name in boundary_groups_with_ghosts:
bnd_idx = groups['boundary:%s' % bnd_name]
spacing = ghost_delta*tree.query(nodes[bnd_idx], 2)[0][:, 1]
ghost_idx = np.arange(bnd_idx.shape[0]) + nodes.shape[0]
ghost_nodes = nodes[bnd_idx] + spacing[:, None]*normals[bnd_idx]
ghost_normals = np.full_like(ghost_nodes, np.nan)
nodes = np.vstack((nodes, ghost_nodes))
normals = np.vstack((normals, ghost_normals))
groups['ghosts:%s' % bnd_name] = ghost_idx
logger.debug('Done')
logger.debug('Sorting nodes...')
sort_idx = neighbor_argsort(nodes)
nodes = nodes[sort_idx]
normals = normals[sort_idx]
reverse_sort_idx = np.argsort(sort_idx)
groups = {k: reverse_sort_idx[v] for k, v in groups.items()}
logger.debug('Done')
logger.debug('Checking the quality of the generated nodes...')
_check_spacing(nodes, rho)
logger.debug('Done')
return nodes, groups, normals
def disperse(nodes,
domain,
rho=None,
fixed_nodes=None,
neighbors=None,
delta=0.1):
'''
Slightly disperses the nodes within the domain. The disperson is
analogous to electrostatic repulsion, where neighboring node exert a
repulsive force on eachother. If a node is repelled into a boundary
then it bounces back in.
Parameters
----------
nodes : (n, d) float array
Initial node positions
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices.
rho : callable, optional
Takes an (n, d) array as input and returns the repulsion force for
a node at those position.
fixed_nodes : (k, d) float array, optional
Nodes which do not move and only provide a repulsion force
neighbors : int, optional
The number of adjacent nodes used to determine repulsion forces
for each node
delta : float, optional
The step size. Each node moves in the direction of the repulsion
force by a distance `delta` times the distance to the nearest
neighbor.
Returns
-------
(n, d) float array
'''
domain = as_domain(domain)
nodes = np.asarray(nodes, dtype=float)
assert_shape(nodes, (None, domain.dim), 'nodes')
if rho is None:
def rho(x):
return np.ones(x.shape[0])
if fixed_nodes is None:
fixed_nodes = np.zeros((0, domain.dim), dtype=float)
else:
fixed_nodes = np.asarray(fixed_nodes)
assert_shape(fixed_nodes, (None, domain.dim), 'fixed_nodes')
if neighbors is None:
# the default number of neighboring nodes to use when computing
# the repulsion force is 4 for 2D and 5 for 3D
if domain.dim == 2:
neighbors = 4
elif domain.dim == 3:
neighbors = 5
# ensure that the number of neighboring nodes used for the repulsion
# force is less than or equal to the total number of nodes
neighbors = min(neighbors, nodes.shape[0] + fixed_nodes.shape[0])
# if m is 0 or 1 then the nodes remain stationary
if neighbors <= 1:
return np.array(nodes, copy=True)
# node positions after repulsion
out = _disperse(nodes, rho, fixed_nodes, neighbors, delta)
# indices of nodes which are now outside the domain
crossed = domain.intersection_count(nodes, out) > 0
crossed, = crossed.nonzero()
# points where nodes intersected the boundary and the simplex they
# intersected at
intr_pnt, intr_idx = domain.intersection_point(nodes[crossed],
out[crossed])
# normal vector to intersection points
intr_norms = domain.normals[intr_idx]
# distance that the node wanted to travel beyond the boundary
res = out[crossed] - intr_pnt
# bounce node off the boundary
out[crossed] -= 2*intr_norms*np.sum(res*intr_norms, 1)[:, None]
# check to see if the bounced nodes still intersect the boundary. If
# they do, then set them back to their original position
still_crossed = domain.intersection_count(nodes[crossed],
out[crossed]) > 0
out[crossed[still_crossed]] = nodes[crossed[still_crossed]]
return out
def prepare_nodes(nodes, domain,
rho=None,
iterations=20,
neighbors=None,
dispersion_delta=0.1,
pinned_nodes=None,
snap_delta=0.5,
boundary_groups=None,
boundary_groups_with_ghosts=None,
include_vertices=False,
orient_simplices=True):
'''
Prepares a set of nodes for solving PDEs with the RBF and RBF-FD
method. This includes: dispersing the nodes away from eachother to
ensure a more even spacing, snapping nodes to the boundary,
determining the normal vectors for each node, determining the group
that each node belongs to, creating ghost nodes, sorting the nodes
so that adjacent nodes are close in memory, and verifying that no
two nodes are anomalously close to eachother.
The function returns a set of nodes, the normal vectors for each
node, and a dictionary identifying which group each node belongs to.
Parameters
----------
nodes : (n, d) float arrary
An initial sampling of nodes within the domain
domain : (p, d) float array and (q, d) int array
Vertices of the domain and connectivity of the vertices
rho : function, optional
Node density function. Takes a (n, d) array of coordinates and
returns an (n,) array of desired node densities at those
coordinates. This is used during the node dispersion step.
iterations : int, optional
Number of dispersion iterations.
neighbors : int, optional
Number of neighboring nodes to use when calculating the repulsion
force. This defaults to 4 for 2D nodes and 5 for 3D nodes.
dispersion_delta : float, optional
Scaling factor for the node step size in each iteration. The step
size is equal to `dispersion_delta` times the distance to the
nearest neighbor.
pinned_nodes : (k, d) array, optional
Nodes which do not move and only provide a repulsion force. These
nodes are included in the set of nodes returned by this function
and they are in the group named "pinned".
snap_delta : float, optional
Controls the maximum snapping distance. The maximum snapping
distance for each node is `snap_delta` times the distance to the
nearest neighbor. This defaults to 0.5.
boundary_groups: dict, optional
Dictionary defining the boundary groups. The keys are the names of
the groups and the values are lists of simplex indices making up
each group. This function will return a dictionary identifying
which nodes belong to each boundary group. By default, there is a
single group named 'all' for the entire boundary. Specifically,
The default value is `{'all':range(len(smp))}`.
boundary_groups_with_ghosts: list of strs, optional
List of boundary groups that will be given ghost nodes. By
default, no boundary groups are given ghost nodes. The groups
specified here must exist in `boundary_groups`.
include_vertices : bool, optional
If `True`, then the vertices will be included in the output nodes.
Each vertex will be assigned to the boundary group that its
adjoining simplices are part of. If the simplices are in multiple
groups, then the vertex will be assigned to the group containing
the simplex that comes first in `smp`.
orient_simplices : bool, optional
If `False` then it is assumed that the simplices are already
oriented such that their normal vectors point outward.
Returns
-------
(m, d) float array
Nodes positions
dict
The indices of nodes belonging to each group. There will always be
a group called 'interior' containing the nodes that are not on the
boundary. By default there is a group containing all the boundary
nodes called 'boundary:all'. If `boundary_groups` was specified,
then those groups will be included in this dictionary and their
names will be given a 'boundary:' prefix. If
`boundary_groups_with_ghosts` was specified then those groups of
ghost nodes will be included in this dictionary and their names
will be given a 'ghosts:' prefix.
(n, d) float array
Outward normal vectors for each node. If a node is not on the
boundary then its corresponding row will contain NaNs.
'''
domain = as_domain(domain)
nodes = np.asarray(nodes, dtype=float)
assert_shape(nodes, (None, domain.dim), 'nodes')
# the `fixed_nodes` are used to provide a repulsion force during
# dispersion, but they do not move. TODO There is chance that one of
# the points in `fixed_nodes` is equal to a point in `nodes`. This
# situation should be handled
fixed_nodes = np.zeros((0, domain.dim), dtype=float)
if pinned_nodes is not None:
pinned_nodes = np.asarray(pinned_nodes, dtype=float)
assert_shape(pinned_nodes, (None, domain.dim), 'pinned_nodes')
fixed_nodes = np.vstack((fixed_nodes, pinned_nodes))
if include_vertices:
fixed_nodes = np.vstack((fixed_nodes, domain.vertices))
for i in range(iterations):
logger.debug('starting node dispersion iterations %s of %s'
% (i + 1, iterations))
nodes = disperse(nodes, domain,
rho=rho,
fixed_nodes=fixed_nodes,
neighbors=neighbors,
delta=dispersion_delta)
# append the domain vertices to the collection of nodes if requested
if include_vertices:
nodes = np.vstack((nodes, domain.vertices))
# snap nodes to the boundary, identifying which simplex each node
# was snapped to
logger.debug('snapping nodes to boundary ...')
nodes, smpid = domain.snap(nodes, delta=snap_delta)
logger.debug('done')
# get the normal vectors for the boundary nodes
if orient_simplices:
logger.debug('orienting simplices ...')
domain.orient_simplices()
logger.debug('done')
normals = np.full_like(nodes, np.nan)
normals[smpid >= 0] = domain.normals[smpid[smpid >= 0]]
# create a dictionary identifying which nodes belong to which group
groups = {}
groups['interior'], = (smpid == -1).nonzero()
# append the user specified pinned nodes
if pinned_nodes is not None:
pinned_idx = np.arange(pinned_nodes.shape[0]) + nodes.shape[0]
pinned_normals = np.full_like(pinned_nodes, np.nan)
nodes = np.vstack((nodes, pinned_nodes))
normals = np.vstack((normals, pinned_normals))
groups['pinned'] = pinned_idx
if boundary_groups is None:
boundary_groups = {'all': range(len(domain.simplices))}
# TODO: There should be a test to make sure each simplex belongs to
# at most one group.
if boundary_groups_with_ghosts is None:
boundary_groups_with_ghosts = []
# create groups for the boundary nodes
logger.debug('grouping boundary nodes and generating ghosts ...')
for k, v in boundary_groups.items():
# convert the list of simplices in the boundary group to a set,
# because it is much faster to determine membership of a set
v = set(v)
bnd_idx = np.array([i for i, j in enumerate(smpid) if j in v])
groups['boundary:' + k] = bnd_idx
if k in boundary_groups_with_ghosts:
# append ghost nodes if requested
dist = KDTree(nodes).query(nodes[bnd_idx], 2)[0][:, [1]]
ghost_idx = np.arange(bnd_idx.shape[0]) + nodes.shape[0]
ghost_nodes = nodes[bnd_idx] + 0.5*dist*normals[bnd_idx]
ghost_normals = np.full_like(ghost_nodes, np.nan)
nodes = np.vstack((nodes, ghost_nodes))
normals = np.vstack((normals, ghost_normals))
groups['ghosts:' + k] = ghost_idx
logger.debug('done')
# sort `nodes` so that spatially adjacent nodes are close together
logger.debug('sorting nodes ...')
sort_idx = neighbor_argsort(nodes)
logger.debug('done')
nodes = nodes[sort_idx]
normals = normals[sort_idx]
reverse_sort_idx = np.argsort(sort_idx)
groups = {}
for k, v in groups.items():
if len(v) > 0: groups[k] = reverse_sort_idx[v]
#groups = {k: reverse_sort_idx[v] for k, v in groups.items()}
logger.debug('checking the quality of the generated nodes ...')
_check_spacing(nodes, rho)
logger.debug('done')
return nodes, groups, normals
