# Source code for sfepy.discrete.fem.poly_spaces

```from __future__ import absolute_import
import numpy as nm
import numpy.linalg as nla

from sfepy.base.base import assert_, Struct
from sfepy.discrete import PolySpace
from sfepy.linalg import combine, insert_strided_axis
from six.moves import range
from functools import reduce

# Requires fixed vertex numbering!
vertex_maps = {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]],
2 : [[0, 0],
[1, 0],
[1, 1],
[0, 1]],
1 : [[0],
[1]],
0 : [[0]]}

[docs]class LagrangeNodes(Struct):
"""Helper class for defining nodes of Lagrange elements."""

[docs]    @staticmethod
def append_edges(nodes, nts, iseq, nt, edges, order):
delta = 1.0 / float(order)

for ii, edge in enumerate(edges):
n1 = nodes[edge[0],:].copy()
n2 = nodes[edge[1],:].copy()
for ie in range(order - 1):
c2 = ie + 1
c1 = order - c2
nts[iseq] = [nt, ii]
aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]    @staticmethod
def append_faces(nodes, nts, iseq, nt, faces, order):
delta = 1.0 / float(order)

for ii, face in enumerate(faces):
n1 = nodes[face[0],:].copy()
n2 = nodes[face[1],:].copy()
n3 = nodes[face[2],:].copy()
for i1 in range(order - 2):
for i2 in range(order - 2 - i1):
c3 = i1 + 1
c2 = i2 + 1
c1 = order - c3 - c2
nts[iseq] = [nt, ii]
aux = [int(round(tmp)) for tmp
in delta * (c1 * n1 + c2 * n2 + c3 * n3)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]    @staticmethod
def append_bubbles(nodes, nts, iseq, nt, order):
delta = 1.0 / float(order)

n1 = nodes[0,:].copy()
n2 = nodes[1,:].copy()
n3 = nodes[2,:].copy()
n4 = nodes[3,:].copy()
for i1 in range(order - 3):
for i2 in range(order - 3):
for i3 in range(order - 3 - i1 - i2):
c4 = i1 + 1
c3 = i2 + 1
c2 = i3 + 1
c1 = order - c4 - c3 - c2
nts[iseq] = [nt, 0]
aux = [int(round(tmp)) for tmp
in delta * (c1 * n1 + c2 * n2 + c3 * n3 + c4 * n4)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]    @staticmethod
def append_tp_edges(nodes, nts, iseq, nt, edges, ao):
delta = 1.0 / float(ao)
for ii, edge in enumerate(edges):
n1 = nodes[edge[0],:].copy()
n2 = nodes[edge[1],:].copy()
for ie in range(ao - 1):
c2 = ie + 1
c1 = ao - c2
nts[iseq] = [nt, ii]
aux = [int(round(tmp)) for tmp in delta * (c1 * n1 + c2 * n2)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]    @staticmethod
def append_tp_faces(nodes, nts, iseq, nt, faces, ao):
delta = 1.0 / (float(ao) ** 2)
for ii, face in enumerate(faces):
n1 = nodes[face[0],:].copy()
n2 = nodes[face[1],:].copy()
n3 = nodes[face[2],:].copy()
n4 = nodes[face[3],:].copy()
for i1 in range(ao - 1):
for i2 in range(ao - 1):
c4 = i1 + 1
c3 = i2 + 1
c2 = ao - c4
c1 = ao - c3
nts[iseq] = [nt, ii]
aux = [int(round(tmp)) for tmp
in delta * (c1 * c2 * n1 + c2 * c3 * n2
+ c3 * c4 * n3 + c4 * c1 * n4)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]    @staticmethod
def append_tp_bubbles(nodes, nts, iseq, nt, ao):
delta = 1.0 / (float(ao) ** 3)
n1 = nodes[0,:].copy()
n2 = nodes[1,:].copy()
n3 = nodes[2,:].copy()
n4 = nodes[3,:].copy()
n5 = nodes[4,:].copy()
n6 = nodes[5,:].copy()
n7 = nodes[6,:].copy()
n8 = nodes[7,:].copy()
for i1 in range(ao - 1):
for i2 in range(ao - 1):
for i3 in range(ao - 1):
c6 = i1 + 1
c5 = i2 + 1
c4 = i3 + 1
c3 = ao - c6
c2 = ao - c5
c1 = ao - c4
nts[iseq] = [nt, 0]
aux = [int(round(tmp)) for tmp
in delta * (c1 * c2 * c3 * n1 + c4 * c2 * c3 * n2
+ c5 * c4 * c3 * n3 + c1 * c3 * c5 * n4
+ c1 * c2 * c6 * n5 + c4 * c2 * c6 * n6
+ c5 * c4 * c6 * n7 + c1 * c6 * c5 * n8)]
nodes[iseq,:] = aux
iseq += 1
return iseq

[docs]class NodeDescription(Struct):
"""
Describe FE nodes defined on different parts of a reference element.
"""

def _describe_facets(self, ii):
nts = self.node_types[ii]
ik = nm.where(nts[1:,1] > nts[:-1,1])[0]

if len(ik) == 0:
ifacets = None
n_dof = 0

else:
ii = ii.astype(nm.int32)

ik = nm.r_[0, ik + 1, nts.shape[0]]
ifacets = [ii[ik[ir] : ik[ir+1]] for ir in range(len(ik) - 1)]
n_dof = len(ifacets[0])

return ifacets, n_dof

def _describe_other(self, ii):
if len(ii):
return ii, len(ii)

else:
return None, 0

def _get_facet_nodes(self, ifacets, nodes):
if ifacets is None:
return None

else:
return [nodes[ii] for ii in ifacets]

def _get_nodes(self, ii, nodes):
if ii is None:
return None

else:
return nodes[ii]

def __init__(self, node_types, nodes):
self.node_types = node_types

# Vertex nodes.
ii = nm.where(node_types[:,0] == 0)[0]
self.vertex, self.n_vertex_nod = self._describe_other(ii)
self.vertex_nodes = self._get_nodes(self.vertex, nodes)

# Edge nodes.
ii = nm.where(node_types[:,0] == 1)[0]
self.edge, self.n_edge_nod = self._describe_facets(ii)
self.edge_nodes = self._get_facet_nodes(self.edge, nodes)

# Face nodes.
ii = nm.where(node_types[:,0] == 2)[0]
self.face, self.n_face_nod = self._describe_facets(ii)
self.face_nodes = self._get_facet_nodes(self.face, nodes)

# Bubble nodes.
ii = nm.where(node_types[:,0] == 3)[0]
self.bubble, self.n_bubble_nod = self._describe_other(ii)
self.bubble_nodes = self._get_nodes(self.bubble, nodes)

[docs]    def has_extra_nodes(self):
"""
Return True if the element has some edge, face or bubble nodes.
"""
return (self.n_edge_nod + self.n_face_nod + self.n_bubble_nod) > 0

[docs]class FEPolySpace(PolySpace):
"""
Base for FE polynomial space classes.
"""
[docs]    def get_mtx_i(self):
return self.mtx_i

[docs]    def describe_nodes(self):
return NodeDescription(self.nts, self.nodes)

[docs]class LagrangePolySpace(FEPolySpace):

[docs]    def create_context(self, cmesh, eps, check_errors, i_max, newton_eps,
tdim=None):
from sfepy.discrete.fem.extmods.bases import CLagrangeContext

ref_coors = self.geometry.coors

if cmesh is not None:
mesh_coors = cmesh.coors

conn = cmesh.get_conn(cmesh.tdim, 0)
mesh_conn = conn.indices.reshape(cmesh.n_el, -1).astype(nm.int32)

if tdim is None:
tdim = cmesh.tdim

else:
mesh_coors = mesh_conn = None

if tdim is None:
raise ValueError('supply either cmesh or tdim!')

ctx = CLagrangeContext(order=self.order,
tdim=tdim,
nodes=self.nodes,
ref_coors=ref_coors,
mesh_coors=mesh_coors,
mesh_conn=mesh_conn,
mtx_i=self.get_mtx_i(),
eps=eps,
check_errors=check_errors,
i_max=i_max,
newton_eps=newton_eps)

return ctx

def _eval_base(self, coors, diff=0, ori=None,
suppress_errors=False, eps=1e-15):
"""
See :func:`PolySpace.eval_base()`.
"""
if diff == 2:
base = self._eval_hessian(coors)

else:
base = self.eval_ctx.evaluate(coors, diff=diff,
eps=eps,
check_errors=not suppress_errors)
return base

[docs]class LagrangeSimplexPolySpace(LagrangePolySpace):
"""Lagrange polynomial space on a simplex domain."""
name = 'lagrange_simplex'

def __init__(self, name, geometry, order, init_context=True):
PolySpace.__init__(self, name, geometry, order)

n_v = geometry.n_vertex

mtx = nm.ones((n_v, n_v), nm.float64)
mtx[0:n_v-1,:] = nm.transpose(geometry.coors)
self.mtx_i = nm.ascontiguousarray(nla.inv(mtx))
self.rhs = nm.ones((n_v,), nm.float64)

self.nodes, self.nts, node_coors = self._define_nodes()
self.node_coors = nm.ascontiguousarray(node_coors)
self.n_nod = self.nodes.shape[0]

if init_context:
self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8,
tdim=n_v - 1)

else:
self.eval_ctx = None

def _define_nodes(self):
# Factorial.
fac = lambda n : reduce(lambda a, b : a * (b + 1), range(n), 1)

geometry = self.geometry
n_v, dim = geometry.n_vertex, geometry.dim
order = self.order

n_nod = fac(order + dim) // (fac(order) * fac(dim))
## print n_nod, gd
nodes = nm.zeros((n_nod, n_v), nm.int32)
nts = nm.zeros((n_nod, 2), nm.int32)

if order == 0:
nts[0,:] = [3, 0]
nodes[0,:] = nm.zeros((n_v,), nm.int32)

else:
iseq = 0

# Vertex nodes.
nts[0:n_v,0] = 0
nts[0:n_v,1] = nm.arange(n_v, dtype = nm.int32)
aux = order * nm.identity(n_v, dtype = nm.int32)
nodes[iseq:iseq+n_v,:] = aux
iseq += n_v

if dim == 0:
pass

elif dim == 1:
iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 3,
[[0, 1]], order)
elif dim == 2:
iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 1,
geometry.edges, order)
iseq = LagrangeNodes.append_faces(nodes, nts, iseq, 3,
[[0, 1, 2]], order)
elif dim == 3:
iseq = LagrangeNodes.append_edges(nodes, nts, iseq, 1,
geometry.edges, order)
iseq = LagrangeNodes.append_faces(nodes, nts, iseq, 2,
geometry.faces, order)
iseq = LagrangeNodes.append_bubbles(nodes, nts, iseq, 3,
order)
else:
raise NotImplementedError

## print nm.concatenate((nts, nodes), 1)

# Check orders.
orders = nm.sum(nodes, 1)
if not nm.all(orders == order):
raise AssertionError('wrong orders! (%d == all of %s)'
% (order, orders))

# Coordinates of the nodes.
if order == 0:
tmp = nm.ones((n_nod, n_v), nm.int32)
node_coors = nm.dot(tmp, geometry.coors) / n_v

else:
node_coors = nm.dot(nodes, geometry.coors) / order

return nodes, nts, node_coors

def _eval_hessian(self, coors):
"""
Evaluate the second derivatives of the basis.
"""
def get_bc(coor):
rhs = nm.concatenate((coor, [1]))
bc = nm.dot(self.mtx_i, rhs)

return bc

def get_val(bc, node, omit=[]):
val = nm.ones(1, nm.float64)
for i1 in range(bc.shape[0]):
if i1 in omit: continue

for i2 in range(node[i1]):
val *= (self.order * bc[i1] - i2) / (i2 + 1.0)

return val

def get_der(bc1, node1, omit=[]):
val = nm.zeros(1, nm.float64)
for i1 in range(node1):
if i1 in omit: continue

aux = nm.ones(1, nm.float64)
for i2 in range(node1):
if (i1 == i2) or (i2 in omit): continue
aux *= (self.order * bc1 - i2) / (i2 + 1.0)

val += aux * self.order / (i1 + 1.0)

return val

n_v = self.mtx_i.shape[0]
dim = n_v - 1

mi = self.mtx_i[:, :dim]
bfgg = nm.zeros((coors.shape[0], dim, dim, self.n_nod),
dtype=nm.float64)

for ic, coor in enumerate(coors):
bc = get_bc(coor)

for ii, node in enumerate(self.nodes):
for ig1, bc1 in enumerate(bc): # 1. derivative w.r.t. bc1.
for ig2, bc2 in enumerate(bc): # 2. derivative w.r.t. bc2.
if ig1 == ig2:
val = get_val(bc, node, omit=[ig1])

vv = 0.0
for i1 in range(node[ig1]):
aux = get_der(bc2, node[ig2], omit=[i1])
vv += aux * self.order / (i1 + 1.0)

val *= vv

else:
val = get_val(bc, node, omit=[ig1, ig2])
val *= get_der(bc1, node[ig1])
val *= get_der(bc2, node[ig2])

bfgg[ic, :, :, ii] += val * mi[ig1] * mi[ig2][:, None]

return bfgg

[docs]class LagrangeSimplexBPolySpace(LagrangeSimplexPolySpace):
"""Lagrange polynomial space with forced bubble function on a simplex
domain."""
name = 'lagrange_simplex_bubble'

def __init__(self, name, geometry, order, init_context=True):
LagrangeSimplexPolySpace.__init__(self, name, geometry, order,
init_context=False)

nodes, nts, node_coors = self.nodes, self.nts, self.node_coors

shape = [nts.shape[0] + 1, 2]
nts = nm.resize(nts, shape)
nts[-1,:] = [3, 0]

shape = [nodes.shape[0] + 1, nodes.shape[1]]
nodes = nm.resize(nodes, shape)
# Make a 'hypercubic' (cubic in 2D) node.
nodes[-1,:] = 1

n_v = self.geometry.n_vertex
tmp = nm.ones((n_v,), nm.int32)

node_coors = nm.vstack((node_coors,
nm.dot(tmp, self.geometry.coors) / n_v))

self.nodes, self.nts = nodes, nts
self.node_coors = nm.ascontiguousarray(node_coors)

self.bnode = nodes[-1:,:]

self.n_nod = self.nodes.shape[0]

if init_context:
self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8,
tdim=n_v - 1)

else:
self.eval_ctx = None

[docs]    def create_context(self, *args, **kwargs):
ctx = LagrangePolySpace.create_context(self, *args, **kwargs)
ctx.is_bubble = 1

return ctx

[docs]class LagrangeTensorProductPolySpace(LagrangePolySpace):
"""Lagrange polynomial space on a tensor product domain."""
name = 'lagrange_tensor_product'

def __init__(self, name, geometry, order, init_context=True):
PolySpace.__init__(self, name, geometry, order)

g1d = Struct(n_vertex = 2,
dim = 1,
coors = self.bbox[:,0:1].copy())
self.ps1d = LagrangeSimplexPolySpace('P_aux', g1d, order,
init_context=False)

self.nodes, self.nts, node_coors = self._define_nodes()
self.node_coors = nm.ascontiguousarray(node_coors)
self.n_nod = self.nodes.shape[0]

if init_context:
tdim = int(nm.sqrt(geometry.n_vertex))
self.eval_ctx = self.create_context(None, 0, 1e-15, 100, 1e-8,
tdim=tdim)

else:
self.eval_ctx = None

def _define_nodes(self):
geometry = self.geometry
order = self.order

n_v, dim = geometry.n_vertex, geometry.dim

vertex_map = order * nm.array(vertex_maps[dim], dtype=nm.int32)

n_nod = (order + 1) ** dim
nodes = nm.zeros((n_nod, 2 * dim), nm.int32)
nts = nm.zeros((n_nod, 2), nm.int32)

if order == 0:
nts[0,:] = [3, 0]
nodes[0,:] = nm.zeros((n_nod,), nm.int32)

else:
iseq = 0

# Vertex nodes.
nts[0:n_v,0] = 0
nts[0:n_v,1] = nm.arange(n_v, dtype=nm.int32)
if dim == 3:
for ii in range(n_v):
i1, i2, i3 = vertex_map[ii]
nodes[iseq,:] = [order - i1, i1,
order - i2, i2,
order - i3, i3]
iseq += 1
elif dim == 2:
for ii in range(n_v):
i1, i2 = vertex_map[ii]
nodes[iseq,:] = [order - i1, i1, order - i2, i2]
iseq += 1
else:
for ii in range(n_v):
i1 = vertex_map[ii][0]
nodes[iseq,:] = [order - i1, i1]
iseq += 1

if dim == 1:
iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 3,
[[0, 1]], order)
elif dim == 2:
iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1,
geometry.edges, order)
iseq = LagrangeNodes.append_tp_faces(nodes, nts, iseq, 3,
[[0, 1, 2, 3]], order)
elif dim == 3:
iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1,
geometry.edges, order)
iseq = LagrangeNodes.append_tp_faces(nodes, nts, iseq, 2,
geometry.faces, order)
iseq = LagrangeNodes.append_tp_bubbles(nodes, nts, iseq, 3,
order)
else:
raise NotImplementedError

# Check orders.
orders = nm.sum(nodes, 1)
if not nm.all(orders == order * dim):
raise AssertionError('wrong orders! (%d == all of %s)'
% (order * dim, orders))

# Coordinates of the nodes.
if order == 0:
tmp = nm.ones((n_nod, n_v), nm.int32)
node_coors = nm.dot(tmp, geometry.coors) / n_v

else:
c_min, c_max = self.bbox[:,0]

cr = nm.arange(2 * dim)
node_coors = (nodes[:,cr[::2]] * c_min
+ nodes[:,cr[1::2]] * c_max) / order

return nodes, nts, node_coors

def _eval_base_debug(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):
"""Python version of eval_base()."""
dim = self.geometry.dim

ev = self.ps1d.eval_base

if diff:
base = nm.ones((coors.shape[0], dim, self.n_nod), dtype=nm.float64)

for ii in range(dim):
self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy()
self.ps1d.n_nod = self.n_nod

for iv in range(dim):
if ii == iv:
base[:,iv:iv+1,:] *= ev(coors[:,ii:ii+1].copy(),
diff=True,
suppress_errors=suppress_errors,
eps=eps)

else:
base[:,iv:iv+1,:] *= ev(coors[:,ii:ii+1].copy(),
diff=False,
suppress_errors=suppress_errors,
eps=eps)

else:
base = nm.ones((coors.shape[0], 1, self.n_nod), dtype=nm.float64)

for ii in range(dim):
self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy()
self.ps1d.n_nod = self.n_nod

base *= ev(coors[:,ii:ii+1].copy(),
diff=diff,
suppress_errors=suppress_errors,
eps=eps)

return base

def _eval_hessian(self, coors):
"""
Evaluate the second derivatives of the basis.
"""
evh = self.ps1d.eval_base

dim = self.geometry.dim
bfgg = nm.zeros((coors.shape[0], dim, dim, self.n_nod),
dtype=nm.float64)

v0s = []
v1s = []
v2s = []
for ii in range(dim):
self.ps1d.nodes = self.nodes[:,2*ii:2*ii+2].copy()
self.ps1d.n_nod = self.n_nod
ev = self.ps1d.create_context(None, 0, 1e-15, 100, 1e-8,
tdim=1).evaluate

v0s.append(ev(coors[:, ii:ii+1].copy())[:, 0, :])
v1s.append(ev(coors[:, ii:ii+1].copy(), diff=1)[:, 0, :])
v2s.append(evh(coors[:, ii:ii+1], diff=2)[:, 0, 0, :])

for ir in range(dim):
vv = v2s[ir] # Destroys v2s!
for ik in range(dim):
if ik == ir: continue
vv *= v0s[ik]

bfgg[:, ir, ir, :] = vv

for ic in range(dim):
if ic == ir: continue
val = v1s[ir] * v1s[ic]
for ik in range(dim):
if (ik == ir) or (ik == ic): continue
val *= v0s[ik]

bfgg[:, ir, ic, :] += val

return bfgg

[docs]    def get_mtx_i(self):
return self.ps1d.mtx_i

[docs]class LagrangeWedgePolySpace(FEPolySpace):
"""
"""
name = 'lagrange_wedge'

def __init__(self, name, geometry, order, init_context=True):
from sfepy.discrete.fem.geometry_element import GeometryElement

PolySpace.__init__(self, name, geometry, order)

geom_1_2 = GeometryElement('1_2')
geom_2_3 = GeometryElement('2_3')

ps1 = LagrangeTensorProductPolySpace(f'{name}_1', geom_1_2, order,
#  init_context=False)
init_context=init_context)
ps2 = LagrangeSimplexPolySpace(f'{name}_2', geom_2_3, order,
init_context=init_context)
#    init_context=False)

geom_3_8 = GeometryElement('3_8')
ps0 = LagrangeTensorProductPolySpace(f'{name}_0', geom_3_8, order,
init_context=False)
geom_3_4 = GeometryElement('3_4')
ps0b = LagrangeSimplexPolySpace(f'{name}_0b', geom_3_4, order,
init_context=False)

n_nod = ps2.n_nod * ps1.n_nod
nd2 = ps2.nodes.shape[1]
nd = nd2 + ps1.nodes.shape[1]
self.nodes = nm.empty((n_nod, nd), nm.int32)
self.nodes[:, :nd2] = nm.tile(ps2.nodes, (ps1.n_nod, 1))
self.nodes[:, nd2:] = nm.repeat(ps1.nodes, ps2.n_nod, axis=0)
self.nts = nm.vstack([ps2.nts, ps1.nts])
self.nts[ps2.n_nod:, 1] += ps2.n_nod

self.node_coors = nm.empty((n_nod, 3), nm.int32)
self.node_coors[:, :2] = nm.tile(ps2.node_coors, (ps1.n_nod, 1))
self.node_coors[:, 2] = nm.repeat(ps1.node_coors, ps2.n_nod,
axis=0)[:, 0]

self.ps = [ps2, ps1]
self.n_nod = n_nod
self.eval_ctx = None

def _eval_base(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):

nd = self.geometry.dim if diff else 1
base = nm.empty((coors.shape[0], nd, self.n_nod), nm.float64)

for qp1 in nm.unique(coors[:, 2]):
idxs = coors[:, 2] == qp1
coors2 = coors[idxs, :2]
base2 = self.ps[0]._eval_base(coors2, False, ori,
suppress_errors, eps)
coors1 = coors[idxs, 2][:, None]
base1 = self.ps[1]._eval_base(coors1, False, ori,
suppress_errors, eps)

if diff:
base2d = self.ps[0]._eval_base(coors2, True, ori,
suppress_errors, eps)
base1d = self.ps[1]._eval_base(coors1, True, ori,
suppress_errors, eps)
base_r = base2d[:, 0, None, :] * base1[:, 0, :, None]
base_s = base2d[:, 1, None, :] * base1[:, 0, :, None]
base_t = base2[:, 0, None, :] * base1d[:, 0, :, None]

base[idxs, 0] = base_r.reshape((-1, self.n_nod))
base[idxs, 1] = base_s.reshape((-1, self.n_nod))
base[idxs, 2] = base_t.reshape((-1, self.n_nod))

else:
base_ = base2[:, 0, None, :] * base1[:, 0, :, None]
base[idxs] = base_.reshape(-1, 1, self.n_nod)

return base

[docs]class SerendipityTensorProductPolySpace(FEPolySpace):
"""
Serendipity polynomial space using Lagrange functions.

Notes
-----
- Orders >= 4 (with bubble functions) are not supported.
- Does not use CLagrangeContext, basis functions are hardcoded.
- `self.nodes`, `self.node_coors` are not used for basis evaluation and
assembling.
"""
name = 'serendipity_tensor_product'
supported_orders = {1, 2, 3}

from sfepy.discrete.fem._serendipity import all_bfs

def __init__(self, name, geometry, order):
import sympy as sm

if geometry.dim < 2:
raise ValueError('serendipity elements need dimension 2 or 3! (%d)'
% geometry.dim)
if order not in self.supported_orders:
raise ValueError('serendipity elements support only orders %s! (%d)'
% (self.supported_orders, order))
PolySpace.__init__(self, name, geometry, order)

self.nodes, self.nts, self.node_coors = self._define_nodes()
self.n_nod = self.nodes.shape[0]

bfs = self.all_bfs[geometry.dim][order]
self.bfs = bfs[0]
self.bfgs = bfs[1]

x, y, z = sm.symbols('x y z')
vs = [x, y, z][:geometry.dim]

self._bfs = [sm.lambdify(vs, bf) for bf in self.bfs]
self._bfgs = [[sm.lambdify(vs, bfg) for bfg in bfgs]
for bfgs in self.bfgs]

[docs]    def create_context(self, cmesh, eps, check_errors, i_max, newton_eps,
tdim=None):
pass

def _define_nodes(self):
geometry = self.geometry
order = self.order

n_v, dim = geometry.n_vertex, geometry.dim

vertex_map = order * nm.array(vertex_maps[dim], dtype=nm.int32)

# Only for orders 1, 2, 3!
if dim == 2:
n_nod = 4 * self.order

else:
n_nod = 8 + 12 * (self.order - 1)

nodes = nm.zeros((n_nod, 2 * dim), nm.int32)
nts = nm.zeros((n_nod, 2), nm.int32)

if order == 0:
nts[0, :] = [3, 0]
nodes[0, :] = nm.zeros((n_nod,), nm.int32)

else:
iseq = 0

# Vertex nodes.
nts[0:n_v, 0] = 0
nts[0:n_v, 1] = nm.arange(n_v, dtype=nm.int32)

if dim == 3:
for ii in range(n_v):
i1, i2, i3 = vertex_map[ii]
nodes[iseq, :] = [order - i1, i1,
order - i2, i2,
order - i3, i3]
iseq += 1

else: # dim == 2:
for ii in range(n_v):
i1, i2 = vertex_map[ii]
nodes[iseq, :] = [order - i1, i1, order - i2, i2]
iseq += 1

if dim == 2:
iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1,
geometry.edges, order)

elif dim == 3:
iseq = LagrangeNodes.append_tp_edges(nodes, nts, iseq, 1,
geometry.edges, order)

else:
raise NotImplementedError

# Coordinates of the nodes.
c_min, c_max = self.bbox[:, 0]

cr = nm.arange(2 * dim)
node_coors = (nodes[:, cr[::2]] * c_min
+ nodes[:, cr[1::2]] * c_max) / order

return nodes, nts, nm.ascontiguousarray(node_coors)

def _eval_base(self, coors, diff=0, ori=None,
suppress_errors=False, eps=1e-15):
"""
See :func:`PolySpace.eval_base()`.
"""
dim = self.geometry.dim
if diff:
bdim = dim

else:
bdim = 1

base = nm.empty((coors.shape[0], bdim, self.n_nod), dtype=nm.float64)

if diff == 0:
for ib, bf in enumerate(self._bfs):
base[:, 0, ib] = bf(*coors.T)

elif diff == 1:
for ib, bfg in enumerate(self._bfgs):
for ig in range(dim):
base[:, ig, ib] = bfg[ig](*coors.T)

else:
raise NotImplementedError

return base

[docs]class LobattoTensorProductPolySpace(FEPolySpace):
"""
Hierarchical polynomial space using Lobatto functions.

Each row of the `nodes` attribute defines indices of Lobatto functions that
need to be multiplied together to evaluate the corresponding shape
function. This defines the ordering of basis functions on the reference
element.
"""
name = 'lobatto_tensor_product'

def __init__(self, name, geometry, order):
PolySpace.__init__(self, name, geometry, order)

aux = self._define_nodes()
self.nodes, self.nts, node_coors, self.face_axes, self.sfnodes = aux
self.node_coors = nm.ascontiguousarray(node_coors)
self.n_nod = self.nodes.shape[0]

aux = nm.where(self.nodes > 0, self.nodes, 1)
self.node_orders = nm.prod(aux, axis=1)
self.edge_indx = nm.where(self.nts[:, 0] == 1)[0]
self.face_indx = nm.where(self.nts[:, 0] == 2)[0]

self.face_axes_nodes = self._get_face_axes_nodes(self.face_axes)

def _get_counts(self):
order = self.order
dim = self.geometry.dim

n_nod = (order + 1) ** dim
n_per_edge = (order - 1)
n_per_face = (order - 1) ** (dim - 1)
n_bubble = (order - 1) ** dim

return n_nod, n_per_edge, n_per_face, n_bubble

def _define_nodes(self):
geometry = self.geometry
order = self.order

n_v, dim = geometry.n_vertex, geometry.dim

n_nod, n_per_edge, n_per_face, n_bubble = self._get_counts()

nodes = nm.zeros((n_nod, dim), nm.int32)
nts = nm.zeros((n_nod, 2), nm.int32)

# Vertex nodes.
nts[0:n_v, 0] = 0
nts[0:n_v, 1] = nm.arange(n_v, dtype=nm.int32)
nodes[0:n_v] = nm.array(vertex_maps[dim], dtype=nm.int32)
ii = n_v

# Edge nodes.
if (dim > 1) and (n_per_edge > 0):
ik = nm.arange(2, order + 1, dtype=nm.int32)
zo = nm.zeros((n_per_edge, 2), dtype=nm.int32)
zo[:, 1] = 1
for ie, edge in enumerate(geometry.edges):
n1, n2 = nodes[edge]
ifix = nm.where(n1 == n2)[0]
irun = nm.where(n1 != n2)[0][0]
ic = n1[ifix]

nodes[ii:ii + n_per_edge, ifix] = zo[:, ic]
nodes[ii:ii + n_per_edge, irun] = ik
nts[ii:ii + n_per_edge] = [[1, ie]]
ii += n_per_edge

# 3D face nodes.
face_axes = []
sfnodes = None
if (dim == 3) and (n_per_face > 0):
n_face = len(geometry.faces)
sfnodes = nm.zeros((n_per_face * n_face, dim), nm.int32)
ii0 = ii

ik = nm.arange(2, order + 1, dtype=nm.int32)
zo = nm.zeros((n_per_face, 2), dtype=nm.int32)
zo[:, 1] = 1

for ifa, face in enumerate(geometry.faces):
ns = nodes[face]

diff = nm.diff(ns, axis=0)
asum = nm.abs(diff).sum(axis=0)
ifix = nm.where(asum == 0)[0][0]
ic = ns[0, ifix]
irun1 = nm.where(asum == 2)[0][0]
irun2 = nm.where(asum == 1)[0][0]

iy, ix = nm.meshgrid(ik, ik)

nodes[ii:ii + n_per_face, ifix] = zo[:, ic]
nodes[ii:ii + n_per_face, irun1] = ix.ravel()
nodes[ii:ii + n_per_face, irun2] = iy.ravel()
nts[ii:ii + n_per_face] = [[2, ifa]]

ij = ii - ii0
sfnodes[ij:ij + n_per_face, ifix] = zo[:, ic]
sfnodes[ij:ij + n_per_face, irun1] = iy.ravel()
sfnodes[ij:ij + n_per_face, irun2] = ix.ravel()

face_axes.append([irun1, irun2])

ii += n_per_face

face_axes = nm.array(face_axes)

# Bubble nodes.
if n_bubble > 0:
ik = nm.arange(2, order + 1, dtype=nm.int32)
nodes[ii:] = nm.array([aux for aux in combine([ik] * dim)])
nts[ii:ii + n_bubble] = [[3, 0]]
ii += n_bubble

assert_(ii == n_nod)

# Coordinates of the "nodes". All nodes on a facet have the same
# coordinates - the centre of the facet.
c_min, c_max = self.bbox[:, 0]

node_coors = nm.zeros(nodes.shape, dtype=nm.float64)
node_coors[:n_v] = nodes[:n_v]

if (dim > 1) and (n_per_edge > 0):
ie = nm.where(nts[:, 0] == 1)[0]
node_coors[ie] = node_coors[geometry.edges[nts[ie, 1]]].mean(1)

if (dim == 3) and (n_per_face > 0):
ifa = nm.where(nts[:, 0] == 2)[0]
node_coors[ifa] = node_coors[geometry.faces[nts[ifa, 1]]].mean(1)

if n_bubble > 0:
ib = nm.where(nts[:, 0] == 3)[0]
node_coors[ib] = node_coors[geometry.conn].mean(0)

return nodes, nts, node_coors, face_axes, sfnodes

def _get_face_axes_nodes(self, face_axes):
if not len(face_axes): return None

nodes = self.nodes[self.face_indx]
n_per_face = self._get_counts()[2]
anodes = nm.tile(nodes[:n_per_face, face_axes[0]], (6, 1))

return anodes

def _eval_base(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):
"""
See PolySpace.eval_base().
"""
from .extmods.lobatto_bases import eval_lobatto_tensor_product as ev
c_min, c_max = self.bbox[:, 0]

base = ev(coors, self.nodes, c_min, c_max, self.order, diff)

if ori is not None:
ebase = nm.tile(base, (ori.shape[0], 1, 1, 1))

if self.edge_indx.shape[0]:
# Orient edge functions.
ie, ii = nm.where(ori[:, self.edge_indx] == 1)
ii = self.edge_indx[ii]
ebase[ie, :, :, ii] *= -1.0

if self.face_indx.shape[0]:
# Orient face functions.
fori = ori[:, self.face_indx]

# ... normal axis order
ie, ii = nm.where((fori == 1) | (fori == 2))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0

# ... swapped axis order
sbase = ev(coors, self.sfnodes, c_min, c_max, self.order, diff)
sbase = insert_strided_axis(sbase, 0, ori.shape[0])

# ...overwrite with swapped axes basis.
ie, ii = nm.where(fori >= 4)
ii2 = self.face_indx[ii]
ebase[ie, :, :, ii2] = sbase[ie, :, :, ii]

# ...deal with orientation.
ie, ii = nm.where((fori == 5) | (fori == 6))
ii = self.face_indx[ii]
ebase[ie, :, :, ii] *= -1.0

base = ebase

return base

[docs]class BernsteinSimplexPolySpace(FEPolySpace):
"""
Bernstein polynomial space on simplex domains.

Notes
-----
Naive proof-of-concept implementation, does not use recurrent formulas or
Duffy transformation to obtain tensor product structure.
"""
name = 'bernstein_simplex'

def __init__(self, name, geometry, order):
PolySpace.__init__(self, name, geometry, order)

self.nodes, self.nts, self.node_coors = self._define_nodes()
self.n_nod = self.nodes.shape[0]
self.eval_ctx = None

def _define_nodes(self):
nodes, nts, node_coors = LagrangeSimplexPolySpace._define_nodes(self)
return nodes, nts, node_coors

@staticmethod
def _get_barycentric(coors):
dim = coors.shape[1]

bcoors = nm.empty((coors.shape[0], dim + 1))
bcoors[:, 0] = 1.0 - coors.sum(axis=1)
bcoors[:, 1:] = coors
return bcoors

def _eval_base(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):
"""
See PolySpace.eval_base().
"""
from scipy.special import factorial

dim = self.geometry.dim
if diff:
bdim = dim

bgrad = nm.zeros((dim + 1, dim), dtype=nm.float64)

else:
bdim = 1

base = nm.ones((coors.shape[0], bdim, self.n_nod), dtype=nm.float64)
if dim == 0:
return base

bcoors = self._get_barycentric(coors)

fs = factorial(nm.arange(0, self.order + 1))
of = fs[-1]
if not diff:
for ii, node in enumerate(self.nodes):
coef = of / nm.prod(fs[node])
val = coef * nm.prod(nm.power(bcoors, node), axis=1)
base[:, 0, ii] = val

else:
for ii, node in enumerate(self.nodes):
coef = of / nm.prod(fs[node])
for ider in range(dim):
dval = 0.0
for ib in range(dim + 1):
ex = node[ib]
val = coef
for im in range(dim + 1):
if ib == im:
val *= (ex *
nm.power(bcoors[:, im], ex - 1) *

else:
val *= nm.power(bcoors[:, im], node[im])

dval += val

base[:, ider, ii] = dval

return base

[docs]class BernsteinTensorProductPolySpace(FEPolySpace):
"""
Bernstein polynomial space.

Each row of the `nodes` attribute defines indices of 1D Bernstein basis
functions that need to be multiplied together to evaluate the corresponding
shape function. This defines the ordering of basis functions on the
reference element.
"""
name = 'bernstein_tensor_product'
def __init__(self, name, geometry, order):
PolySpace.__init__(self, name, geometry, order)

self.nodes, self.nts, self.node_coors = self._define_nodes()
self.n_nod = self.nodes.shape[0]
self.eval_ctx = None

def _define_nodes(self):
nn, nts, node_coors = LagrangeTensorProductPolySpace._define_nodes(self)
nodes = nn[:, 1::2]

return nodes, nts, node_coors

def _eval_base(self, coors, diff=False, ori=None,
suppress_errors=False, eps=1e-15):
"""
See PolySpace.eval_base().
"""
from sfepy.discrete.iga.extmods.igac import eval_bernstein_basis as ev

dim = self.geometry.dim
if diff:
bdim = dim

else:
bdim = 1

base = nm.ones((coors.shape[0], bdim, self.n_nod), dtype=nm.float64)
degree = self.order
n_efuns_max = degree + 1

for iq, qp in enumerate(coors):
B = nm.empty((dim, n_efuns_max), dtype=nm.float64)
dB_dxi = nm.empty((dim, n_efuns_max), dtype=nm.float64)
for ii in range(dim):
ev(B[ii, :], dB_dxi[ii, :], qp[ii], degree)

if not diff:
for ii, ni in enumerate(self.nodes.T):
base[iq, 0, :] *= B[ii, ni]

else:
for ii, ni in enumerate(self.nodes.T):
for iv in range(bdim):
if ii == iv:
base[iq, iv, :] *= dB_dxi[ii, ni]

else:
base[iq, iv, :] *= B[ii, ni]

return base

[docs]def get_lgl_nodes(p):
"""
Compute the Legendre-Gauss-Lobatto nodes and weights.
"""
from numpy.polynomial.legendre import legvander

# Use the Chebyshev-Gauss-Lobatto nodes as the first guess.
xs = nm.cos(nm.pi * nm.arange(p + 1) / p)

eps = nm.finfo(nm.float64).eps
xs0 = 2.0
while nm.linalg.norm(xs - xs0, ord=nm.inf) > eps:
xs0 = xs
V = legvander(xs, p)
xs = xs0 - (xs * V[:, p] - V[:,p-1]) / ((p+1) * V[:,p])

ws = 2.0 / (p * (p+1) * V[:,p]**2)

return xs, ws

[docs]def eval_lagrange1d_basis(coors, ncoors):
n_nod = len(ncoors)
n_coors = len(coors)

val = nm.ones((n_coors, n_nod), dtype=nm.float64)
dval = nm.zeros((n_coors, n_nod), dtype=nm.float64)
for ib in range(n_nod):
for ic in range(n_nod):
if ib != ic:
val[:, ib] *= ((coors - ncoors[ic])
/ (ncoors[ib] - ncoors[ic]))

for ik in range(n_nod):
if ib == ik: continue
aux = 1.0 / (ncoors[ib] - ncoors[ik])
for ic in range(n_nod):
if (ib != ic) and (ik != ic):
aux *= ((coors - ncoors[ic])
/ (ncoors[ib] - ncoors[ic]))

dval[:, ib] += aux

return val, dval

[docs]class SEMTensorProductPolySpace(FEPolySpace):
"""
Spectral element method polynomial space = Lagrange polynomial space with
Legendre-Gauss-Lobatto nodes. The same nodes and corresponding weights
should be used for numerical quadrature to obtain a diagonal mass matrix.
"""
name = 'sem_tensor_product'

def __init__(self, name, geometry, order, init_context=True):
PolySpace.__init__(self, name, geometry, order)

(self.nodes, self.nts,
node_coors, self.node_weights,
self.node_coors1d, self.weights1d) = self._define_nodes()
self.node_coors = nm.ascontiguousarray(node_coors)

self.n_nod = self.nodes.shape[0]
self.eval_ctx = None

def _define_nodes(self):
nn, nts, node_coors = LagrangeTensorProductPolySpace._define_nodes(self)
nodes = nn[:, 1::2]

node_coors1d, weights1d = get_lgl_nodes(self.order)
# Transform node_coors1d from [1, -1] to [0, 1].
node_coors1d = 0.5 * (1 - node_coors1d)
weights1d *= 0.5

node_weights = nm.ones_like(node_coors[:, 0])
for ii, ni in enumerate(nodes.T):
node_coors[:, ii] = node_coors1d[ni]
node_weights[:] *= weights1d[ni]

return nodes, nts, node_coors, node_weights, node_coors1d, weights1d

def _eval_base(self, coors, diff=0, ori=None,
suppress_errors=False, eps=1e-15):
"""
See :func:`PolySpace.eval_base()`.
"""
dim = self.geometry.dim
bdim = dim if diff else 1

assert diff in (0, 1)

out = nm.ones((coors.shape[0], bdim, self.n_nod), dtype=nm.float64)
vals = []
dvals = []
for ii in range(dim):
b1d, db1d = eval_lagrange1d_basis(coors[:, ii], self.node_coors1d)
vals.append(b1d)
dvals.append(db1d)

if diff == 0:
for ii, ni in enumerate(self.nodes.T):
out[:, 0, :] *= vals[ii][:, ni]

else:
for ii, ni in enumerate(self.nodes.T):
for iv in range(bdim):
if ii == iv:
out[:, iv, :] *= dvals[ii][:, ni]

else:
out[:, iv, :] *= vals[ii][:, ni]

return out
```