from itertools import chain, repeat
import gem
import numpy
import sympy
try:
import symengine
symbolics = symengine
except ImportError:
symbolics = sympy
from FIAT.polynomial_set import mis
from FIAT.reference_element import UFCQuadrilateral
from gem.utils import cached_property
from finat.finiteelementbase import FiniteElementBase
from finat.physically_mapped import Citations, DirectlyDefinedElement
from finat.sympy2gem import sympy2gem
[docs]class DirectSerendipity(DirectlyDefinedElement, FiniteElementBase):
def __init__(self, cell, degree):
if Citations is not None:
Citations().register("Arbogast2017")
# These elements only known currently on quads
assert isinstance(cell, UFCQuadrilateral)
self._cell = cell
self._degree = degree
self._deriv_cache = {}
@property
def cell(self):
return self._cell
@property
def complex(self):
return self._cell
@property
def degree(self):
return self._degree
@property
def formdegree(self):
return 0
[docs] def entity_dofs(self):
if self.degree == 1:
return {0: {i: [i] for i in range(4)},
1: {i: [] for i in range(4)},
2: {0: []}}
elif self.degree == 2:
return {0: {i: [i] for i in range(4)},
1: {i: [i+4] for i in range(4)},
2: {0: []}}
else:
return {0: {i: [i] for i in range(4)},
1: {i: list(range(4 + i * (self.degree-1),
4 + (i + 1) * (self.degree-1)))
for i in range(4)},
2: {0: list(range(4 + 4 * (self.degree - 1),
self.space_dimension()))}}
[docs] def space_dimension(self):
return 4 if self.degree == 1 else (self.degree+1)*(self.degree+2)//2 + 2
@property
def index_shape(self):
return (self.space_dimension(),)
@property
def value_shape(self):
return ()
@cached_property
def _basis(self):
return ds_sym(self.cell.topology, self.degree, sp=symbolics)
def _basis_deriv(self, xx, alpha):
key = (tuple(xx), alpha)
_, _, phis = self._basis
try:
return self._deriv_cache[key]
except KeyError:
dphi = tuple(diff(phi, xx, alpha) for phi in phis)
return self._deriv_cache.setdefault(key, dphi)
[docs] def basis_evaluation(self, order, ps, entity=None, coordinate_mapping=None):
'''Return code for evaluating the element at known points on the
reference element.
:param order: return derivatives up to this order.
:param ps: the point set.
:param entity: the cell entity on which to tabulate.
'''
# Build everything in sympy
vs, xx, _ = self._basis
# and convert -- all this can be used for each derivative!
phys_verts = coordinate_mapping.physical_vertices()
phys_points = gem.partial_indexed(
coordinate_mapping.physical_points(ps, entity=entity),
ps.indices)
repl = dict((vs[idx], phys_verts[idx])
for idx in numpy.ndindex(vs.shape))
repl.update(zip(xx, phys_points))
mapper = gem.node.Memoizer(sympy2gem)
mapper.bindings = repl
result = {}
for i in range(order+1):
alphas = mis(2, i)
for alpha in alphas:
dphis = self._basis_deriv(xx, alpha)
result[alpha] = gem.ListTensor(list(map(mapper, dphis)))
return result
[docs] def point_evaluation(self, order, refcoords, entity=None):
raise NotImplementedError("Not done yet, sorry!")
[docs] def mapping(self):
return "physical"
[docs]def xysub(x, y):
return {x[0]: y[0], x[1]: y[1]}
[docs]def ds1_sym(ct, *, vs=None, sp=symbolics):
"""Constructs lowest-order case of Arbogast's directly defined C^0 serendipity
elements, which are a special case.
:param ct: The cell topology of the reference quadrilateral.
:param vs: (Optional) coordinates of cell on which to construct the basis.
If it is None, this function constructs symbols for the vertices.
:returns: a 3-tuple containing symbols for the physical cell coordinates and the
physical cell independent variables (e.g. "x" and "y") and a list
of the four basis functions.
"""
if vs is None:
vs = numpy.asarray(list(zip(sp.symbols('x:4'),
sp.symbols('y:4'))))
else:
vs = numpy.asarray(vs)
xx = numpy.asarray(sp.symbols("x,y"))
ts = numpy.zeros((4, 2), dtype=object)
for e in range(4):
v0id, v1id = ct[1][e][:]
for j in range(2):
ts[e, :] = vs[v1id, :] - vs[v0id, :]
ns = numpy.zeros((4, 2), dtype=object)
for e in (0, 3):
ns[e, 0] = -ts[e, 1]
ns[e, 1] = ts[e, 0]
for e in (1, 2):
ns[e, 0] = ts[e, 1]
ns[e, 1] = -ts[e, 0]
xstars = numpy.zeros((4, 2), dtype=object)
for e in range(4):
v0id, v1id = ct[1][e][:]
xstars[e, :] = (vs[v0id, :] + vs[v1id])/2
lams = [(xx-xstars[i, :]) @ ns[i, :] for i in range(4)]
RV = (lams[0] - lams[1]) / (lams[0] + lams[1])
RH = (lams[2] - lams[3]) / (lams[2] + lams[3])
Rs = [RV, RH]
xis = []
for e in range(4):
dct = xysub(xx, xstars[e, :])
i = 2*((3-e)//2)
j = i + 1
xi = lams[i] * lams[j] * (1 + (-1)**(e+1) * Rs[e//2]) / lams[i].subs(dct) / lams[j].subs(dct) / 2
xis.append(xi)
d = xysub(xx, vs[0, :])
r = lams[1] * lams[3] / lams[1].subs(d) / lams[3].subs(d)
d = xysub(xx, vs[2, :])
r -= lams[0] * lams[3] / lams[0].subs(d) / lams[3].subs(d)
d = xysub(xx, vs[3, :])
r += lams[0] * lams[2] / lams[0].subs(d) / lams[2].subs(d)
d = xysub(xx, vs[1, :])
r -= lams[1] * lams[2] / lams[1].subs(d) / lams[2].subs(d)
R = r - sum([r.subs(xysub(xx, xstars[i, :])) * xis[i] for i in range(4)])
n03 = numpy.array([[0, -1], [1, 0]]) @ (vs[3, :] - vs[0, :])
lam03 = (xx - vs[0, :]) @ n03
n12 = numpy.array([[0, -1], [1, 0]]) @ (vs[2, :] - vs[1, :])
lam12 = (xx - vs[2, :]) @ n12
phi0tilde = lam12 - lam12.subs({xx[0]: vs[3, 0], xx[1]: vs[3, 1]}) * (1 + R) / 2
phi1tilde = lam03 - lam03.subs({xx[0]: vs[2, 0], xx[1]: vs[2, 1]}) * (1 - R) / 2
phi2tilde = lam03 - lam03.subs({xx[0]: vs[1, 0], xx[1]: vs[1, 1]}) * (1 - R) / 2
phi3tilde = lam12 - lam12.subs({xx[0]: vs[0, 0], xx[1]: vs[0, 1]}) * (1 + R) / 2
phis = []
for i, phitilde in enumerate([phi0tilde, phi1tilde, phi2tilde, phi3tilde]):
phi = phitilde / phitilde.subs({xx[0]: vs[i, 0], xx[1]: vs[i, 1]})
phis.append(phi)
return vs, xx, numpy.asarray(phis)
[docs]def newton_dd(nds, fs):
"""Constructs Newton's divided differences for the input arrays,
which may include symbolic values."""
n = len(nds)
mat = numpy.zeros((n, n), dtype=object)
mat[:, 0] = fs[:]
for j in range(1, n):
for i in range(n-j):
mat[i, j] = (mat[i+1, j-1] - mat[i, j-1]) / (nds[i+j] - nds[i])
return mat[0, :]
[docs]def newton_poly(nds, fs, xsym):
"""Constructs Lagrange interpolating polynomial passing through
x values nds and y values fs. Returns a a symbolic object in terms
of independent variable xsym."""
coeffs = newton_dd(nds, fs)
result = coeffs[-1]
n = len(coeffs)
for i in range(n-2, -1, -1):
result = result * (xsym - nds[i]) + coeffs[i]
return result
[docs]def diff(expr, xx, alpha):
"""Differentiate expr with respect to xx.
:arg expr: symengine/symengine Expression to differentiate.
:arg xx: iterable of coordinates to differentiate with respect to.
:arg alpha: derivative multiindex, one entry for each entry of xx
indicating how many derivatives in that direction.
:returns: New symengine/symengine expression."""
if isinstance(expr, sympy.Expr):
return expr.diff(*(zip(xx, alpha)))
else:
return symengine.diff(expr, *(chain(*(repeat(x, a) for x, a in zip(xx, alpha)))))
[docs]def dsr_sym(ct, r, *, vs=None, sp=symbolics):
"""Constructs higher-order (>= 2) case of Arbogast's directly defined C^0 serendipity
elements, which include all polynomials of degree r plus a couple of rational
functions.
:param ct: The cell topology of the reference quadrilateral.
:param vs: (Optional) coordinates of cell on which to construct the basis.
If it is None, this function constructs symbols for the vertices.
:returns: a 3-tuple containing symbols for the physical cell coordinates and the
physical cell independent variables (e.g. "x" and "y") and a list
of the four basis functions.
"""
if vs is None: # do vertices symbolically
vs = numpy.asarray(list(zip(sp.symbols('x:4'),
sp.symbols('y:4'))))
else:
vs = numpy.asarray(vs)
xx = numpy.asarray(sp.symbols("x,y"))
ts = numpy.zeros((4, 2), dtype=object)
for e in range(4):
v0id, v1id = ct[1][e][:]
ts[e, :] = vs[v1id, :] - vs[v0id, :]
ns = numpy.zeros((4, 2), dtype=object)
for e in (0, 3):
ns[e, 0] = -ts[e, 1]
ns[e, 1] = ts[e, 0]
for e in (1, 2):
ns[e, 0] = ts[e, 1]
ns[e, 1] = -ts[e, 0]
# midpoints of each edge
xstars = numpy.zeros((4, 2), dtype=object)
for e in range(4):
v0id, v1id = ct[1][e][:]
xstars[e, :] = (vs[v0id, :] + vs[v1id])/2
lams = [(xx-xstars[i, :]) @ ns[i, :] for i in range(4)]
# # internal functions
bubble = numpy.prod(lams)
if r < 4:
internal_bfs = []
internal_nodes = []
elif r == 4: # Just one point
xbar = sum(vs[i, 0] for i in range(4)) / 4
ybar = sum(vs[i, 1] for i in range(4)) / 4
internal_bfs = [bubble / bubble.subs(xysub(xx, (xbar, ybar)))]
internal_nodes = [(xbar, ybar)]
else: # build a triangular lattice inside the quad
dx0 = (vs[1, :] - vs[0, :]) / (r-2)
dx1 = (vs[2, :] - vs[0, :]) / (r-2)
# Vertices of the triangle
v0 = vs[0, :] + dx0 + dx1
v1 = vs[0, :] + (r-3) * dx0 + dx1
v2 = vs[0, :] + dx0 + (r-3) * dx1
# Pardon the fortran, but these are barycentric coordinates...
bary = numpy.zeros((3,), dtype="object")
y12 = v1[1] - v2[1]
x21 = v2[0] - v1[0]
x02 = v0[0] - v2[0]
y02 = v0[1] - v2[1]
det = y12 * x02 + x21 * y02
delx = xx[0] - v2[0]
dely = xx[1] - v2[1]
bary[0] = (y12 * delx + x21 * dely) / det
bary[1] = (-y02 * delx + x02 * dely) / det
bary[2] = 1 - bary[0] - bary[1]
# And this bit directly constructs the Lagrange polynomials
# of degree r-4 on the triangular lattice inside the triangle.
# This trick is restricted to equispaced points, but we're on a much
# lower degree than r. This bypasses symbolic inversion/etc otherwise
# required to build the Lagrange polynomials.
rm4 = r - 4
lags = []
internal_nodes = []
for i in range(rm4, -1, -1):
for j in range(rm4-i, -1, -1):
k = rm4 - i - j
internal_nodes.append((v0 * i + v1 * j + v2 * k)/rm4)
ii = (i, j, k)
lag_cur = sp.Integer(1)
for q in range(3):
for p in range(ii[q]):
lag_cur *= (rm4 * bary[q] - p) / (ii[q] - p)
lags.append(lag_cur.simplify())
internal_bfs = []
for lag, nd in zip(lags, internal_nodes):
foo = lag * bubble
internal_bfs.append(foo / foo.subs(xysub(xx, nd)))
RV = (lams[0] - lams[1]) / (lams[0] + lams[1])
RH = (lams[2] - lams[3]) / (lams[2] + lams[3])
# R for each edge (1 on edge, zero on opposite
Rs = [(1 - RV) / 2, (1 + RV) / 2, (1 - RH) / 2, (1 + RH) / 2]
nodes1d = [sp.Rational(i, r) for i in range(1, r)]
s = sp.Symbol('s')
# for each edge:
# I need its adjacent two edges and its opposite edge
# and its "tunnel R" RH or RV
# This is very 2d specific.
opposite_edges = {e: [eother for eother in ct[1]
if set(ct[1][e]).intersection(ct[1][eother]) == set()][0]
for e in ct[1]}
adjacent_edges = {e: tuple(sorted([eother for eother in ct[1]
if eother != e
and set(ct[1][e]).intersection(ct[1][eother])
!= set()]))
for e in ct[1]}
ae = adjacent_edges
tunnel_R_edges = {e: ((lams[ae[e][0]] - lams[ae[e][1]])
/ (lams[ae[e][0]] + lams[ae[e][1]]))
for e in range(4)}
edge_nodes = []
for ed in range(4):
((v0x, v0y), (v1x, v1y)) = vs[ct[1][ed], :]
delx = v1x - v0x
dely = v1y - v0y
edge_nodes.append([(v0x+nd*delx, v0y+nd*dely) for nd in nodes1d])
# subtracts off the value of function at internal nodes times those
# internal basis functions
def nodalize(f):
return f - sum(f.subs(xysub(xx, nd)) * bf
for bf, nd in zip(internal_bfs, internal_nodes))
edge_bfs = []
if r == 2:
for ed in range(4):
lamadj0 = lams[adjacent_edges[ed][0]]
lamadj1 = lams[adjacent_edges[ed][1]]
ephi = lamadj0 * lamadj1 * Rs[ed]
phi = nodalize(ephi) / ephi.subs(xysub(xx, xstars[ed]))
edge_bfs.append([phi])
else:
for ed in range(4):
((v0x, v0y), (v1x, v1y)) = vs[ct[1][ed], :]
Rcur = tunnel_R_edges[ed]
lam_op = lams[opposite_edges[ed]]
edge_bfs_cur = []
for i in range(len(nodes1d)):
# strike out i:th node
idcs = [j for j in range(len(nodes1d)) if i != j]
nodes1d_cur = [nodes1d[j] for j in idcs]
edge_nodes_cur = [edge_nodes[ed][j]
for j in idcs]
# construct the 1d interpolation with remaining nodes
pvals = []
for nd in edge_nodes_cur:
sub = xysub(xx, nd)
pval_cur = (-1 * Rcur.subs(sub)**(r-2)
/ lam_op.subs(sub))
pvals.append(pval_cur)
ptilde = newton_poly(nodes1d_cur, pvals, s)
xt = xx @ ts[ed]
vt0 = numpy.asarray((v0x, v0y)) @ ts[ed]
vt1 = numpy.asarray((v1x, v1y)) @ ts[ed]
p = ptilde.subs({s: (xt-vt0) / (vt1-vt0)})
prebf = (lams[adjacent_edges[ed][0]]
* lams[adjacent_edges[ed][1]]
* (lams[opposite_edges[ed]] * p
+ Rcur**(r-2) * Rs[ed]))
prebf = nodalize(prebf)
bfcur = prebf / prebf.subs(xysub(xx, edge_nodes[ed][i]))
edge_bfs_cur.append(bfcur)
edge_bfs.append(edge_bfs_cur)
# vertex basis functions
vertex_to_adj_edges = {i: tuple([e for e in ct[1] if i in ct[1][e]])
for i in ct[0]}
vertex_to_off_edges = {i: tuple([e for e in ct[1] if i not in ct[1][e]])
for i in ct[0]}
vertex_bfs = []
for v in range(4):
ed0, ed1 = vertex_to_off_edges[v]
lam0 = lams[ed0]
lam1 = lams[ed1]
prebf = lam0 * lam1
# subtract off edge values
for adj_ed in vertex_to_adj_edges[v]:
edge_nodes_cur = edge_nodes[adj_ed]
edge_bfs_cur = edge_bfs[adj_ed]
for k, (nd, edbf) in enumerate(zip(edge_nodes_cur, edge_bfs_cur)):
sb = xysub(xx, nd)
prebf -= lam0.subs(sb) * lam1.subs(sb) * edbf
bf = nodalize(prebf) / prebf.subs(xysub(xx, vs[v, :]))
vertex_bfs.append(bf)
bfs = vertex_bfs
for edbfs in edge_bfs:
bfs.extend(edbfs)
bfs.extend(internal_bfs)
nds = [tuple(vs[i, :]) for i in range(4)]
for ends in edge_nodes:
nds.extend(ends)
nds.extend(internal_nodes)
return vs, xx, numpy.asarray(bfs)
[docs]def ds_sym(ct, r, *, vs=None, sp=symbolics):
"""Symbolically Constructs Arbogast's directly defined C^0 serendipity elements,
which include all polynomials of degree r plus a couple of rational functions.
:param ct: The cell topology of the reference quadrilateral.
:param vs: (Optional) coordinates of cell on which to construct the basis.
If it is None, this function constructs symbols for the vertices.
:returns: a 3-tuple containing symbols for the physical cell coordinates and the
physical cell independent variables (e.g. "x" and "y") and a list
of the four basis functions.
"""
if r == 1:
return ds1_sym(ct, vs=vs, sp=sp)
else:
return dsr_sym(ct, r, vs=vs, sp=sp)
