"""
`quadrature_tables` are organized as follows::
quadrature_tables = {
'<geometry1>' : {
order1 : QuadraturePoints(args1),
order2 : QuadraturePoints(args2),
...
},
'<geometry2>' : {
order1 : QuadraturePoints(args1),
order2 : QuadraturePoints(args2),
...
},
...
}
**Note** The order for quadratures on tensor product domains (`'2_4'`,
`'3_8'` geometries) in case of composite Gauss quadratures (products of
1D quadratures) holds for each component separately, so the actual
polynomial order may be much higher (up to `order * dimension`).
Naming conventions in problem description files::
`<family>_<order>_<dimension>`
Integral 'family' is just an arbitrary name given by user.
Low order quadrature coordinates and weights copied from The Finite Element
Method Displayed by Gouri Dhatt and Gilbert Touzat, Wiley-Interscience
Production, 1984.
The line integral (geometry '1_2') coordinates and weights are from Abramowitz,
M. and Stegun, I.A., Handbook of Mathematical Functions, Dover Publications,
New York, 1972. The triangle (geometry '2_3') coordinates and weights are from
Dunavant, D.A., High Degree Efficient Symmetrical Gaussian Quadrature Rules for
the Triangle, Int. J. Num. Meth. Eng., 21 (1985) pp 1129-1148 - only rules with
points inside the reference triangle are used. The actual values were copied
from PHAML (http://math.nist.gov/phaml/), see also Mitchell, W.F., PHAML User's
Guide, NISTIR 7374, 2006.
Quadrature rules for the quadrilateral (geometry '2_4') and hexahedron
(geometry '3_8') of order higher than 5 are computed as the tensor product of
the line (geometry '1_2') rules.
Quadrature rules for the triangle (geometry '2_3') and tetrahedron (geometry
'3_4') of order higher than 19 and 6, respectively follow A. Grundmann and
H.M. Moeller, Invariant integration formulas for the n-simplex by combinatorial
methods, SIAM J. Numer. Anal. 15 (1978), 282--290. The generating function was
adapted from pytools/hegde codes (http://mathema.tician.de/software/hedge) by
Andreas Kloeckner.
"""
from __future__ import absolute_import
import numpy as nm
from sfepy.base.base import output, assert_, Struct
from sfepy.discrete.simplex_cubature import get_simplex_cubature
simplex_geometries = ['1_2', '2_3', '3_4']
tp_geometries = ['2_4', '3_8']
_msg1 = 'WARNING: quadrature order %s is not available for geometry %s!'
_msg2 = 'WARNING: using %d instead!'
[docs]def get_actual_order(geometry, order):
"""
Return the actual integration order for given geometry.
Parameters
----------
geometry : str
The geometry key describing the integration domain,
see the keys of `quadrature_tables`.
Returns
-------
order : int
If `order` is in quadrature tables it is this
value. Otherwise it is the closest higher order. If no
higher order is available, a warning is printed and the
highest available order is used.
"""
table = quadrature_tables[geometry]
if order not in table:
orders = list(table.keys())
ii = nm.searchsorted(orders, order)
if ii >= len(orders):
omax = max(orders)
output(_msg1 % (order, geometry))
output(_msg2 % omax)
order = omax
else:
order = orders[ii]
return order
[docs]class QuadraturePoints(Struct):
"""
Representation of a set of quadrature points.
Parameters
----------
data : array_like
The array of shape `(n_point, dim + 1)` of quadrature point
coordinates (first `dim` columns) and weights (the last column).
coors : array_like, optional
Optionally, instead of using `data`, the coordinates and weights can
be provided separately - `data` are then ignored.
weights : array_like, optional
Optionally, instead of using `data`, the coordinates and weights can
be provided separately - `data` are then ignored.
bounds : (float, float), optional
The coordinates and weights should correspond to a reference
element in `[0, 1]` x `dim`. Provide the correct bounds if this is
not the case.
tp_fix : float, optional
The value that is used to multiply the tensor product element
volume (= 1.0) to get the correct volume.
weight_fix : float, optional
The value that is used to multiply the weights to get the correct
values.
symmetric : bool
If True, the integral is 1D and the given coordinates and weights are
symmetric w.r.t. the centre of bounds; only the non-negative
coordinates are given.
"""
[docs] @staticmethod
def from_table(geometry, order):
"""
Create a new :class:`QuadraturePoints` instance, given reference
element geometry name and polynomial order. For tensor product
geometries, the polynomial order is the 1D (line) order.
"""
table = quadrature_tables[geometry]
if geometry in simplex_geometries:
if order > max_orders[geometry]:
oo = order // 2
dim = int(geometry[0])
tp_fix = 0.5 if dim == 2 else 1.0 / 6.0
coors, weights, exact = get_simplex_cubature(oo, dim)
qp = QuadraturePoints(None, coors=coors, weights=weights,
bounds=(-1.0, 1.0), tp_fix=tp_fix)
assert_(exact >= order)
else:
order = get_actual_order(geometry, order)
qp = table[order]
qp.order = order
else:
order1d = order
dim = int(geometry[0])
order = dim * order1d
if order <= max_orders[geometry]:
order = get_actual_order(geometry, order)
qp = table[order]
qp.order = order
else:
oo = get_actual_order('1_2', order1d)
qp1d = quadrature_tables['1_2'][oo]
weights = nm.outer(qp1d.weights, qp1d.weights)
nc = qp1d.coors.shape[0]
if dim == 3:
weights = nm.outer(qp1d.weights, weights)
iz, iy, ix = nm.mgrid[0:nc, 0:nc, 0:nc]
coors = nm.c_[qp1d.coors[ix.ravel()],
qp1d.coors[iy.ravel()],
qp1d.coors[iz.ravel()]].copy()
else:
iy, ix = nm.mgrid[0:nc, 0:nc]
coors = nm.c_[qp1d.coors[ix.ravel()],
qp1d.coors[iy.ravel()]].copy()
weights = weights.ravel()
qp = QuadraturePoints(None, coors=coors, weights=weights)
qp.order = dim * oo
return qp
def __init__(self, data, coors=None, weights=None, bounds=None, tp_fix=1.0,
weight_fix=1.0, symmetric=False):
if coors is None:
data = nm.array(data, dtype=nm.float64, ndmin=2)
self.coors = data[:,:-1].copy()
self.weights = data[:,-1].copy()
elif weights is not None:
self.coors = nm.array(coors, dtype=nm.float64, ndmin=2)
self.weights = nm.array(weights, dtype=nm.float64)
else:
raise ValueError('both "coors" and "weights" have to be provided!')
self.weights *= weight_fix
self.n_point, self.dim = self.coors.shape
self.bounds = (0, 1)
bbox = nm.array([self.bounds] * self.dim, dtype=nm.float64)
self.volume = nm.prod(bbox.sum(axis=1)) * tp_fix
if symmetric:
isym = 0 if data[0, 0] == 0 else None
if bounds is not None:
# Transform from given bounds to self.bounds.
bbox = nm.array([bounds] * self.dim, dtype=nm.float64)
volume = nm.prod(nm.diff(bbox, axis=1)) * tp_fix
a, b = bounds
c, d = self.bounds
c1 = (d - c) / (b - a)
c2 = ((b * c) - (a * d)) / (b - a)
self.coors = c1 * self.coors + c2
self.weights *= self.volume / volume
if symmetric:
if self.coors.shape[1] != 1:
msg = 'symmetric mode is allowed for 1D integrals only!'
raise ValueError(msg)
origin = 0.5 * (self.bounds[0] + self.bounds[1])
self.coors = nm.r_[2 * origin - self.coors[:isym:-1], self.coors]
self.weights = nm.r_[self.weights[:isym:-1], self.weights]
_QP = QuadraturePoints
quadrature_tables = {
'0_1' : {
1 : _QP([[0.0, 1.0]])
},
'1_2' : {
1 : _QP([[0.000000000000000e+00, 2.0]],
bounds=(-1.0, 1.0), symmetric=True),
3 : _QP([[0.577350269189626e+00, 1.0]],
bounds=(-1.0, 1.0), symmetric=True),
5 : _QP([[0.000000000000000e+00, 0.888888888888889e+00],
[0.774596669241483e+00, 0.555555555555556e+00]],
bounds=(-1.0, 1.0), symmetric=True),
7 : _QP([[0.339981043584856e+00, 0.652145154862546e+00],
[0.861136311594053e+00, 0.347854845137454e+00]],
bounds=(-1.0, 1.0), symmetric=True),
9 : _QP([[0.000000000000000e+00, 0.568888888888889e+00],
[0.538469310105683e+00, 0.478628670499366e+00],
[0.906179845938664e+00, 0.236926885056189e+00]],
bounds=(-1.0, 1.0), symmetric=True),
11 : _QP([[0.238619186083197e+00, 0.467913934572691e+00],
[0.661209386466265e+00, 0.360761573048139e+00],
[0.932469514203152e+00, 0.171324492379170e+00]],
bounds=(-1.0, 1.0), symmetric=True),
13 : _QP([[0.000000000000000e+00, 0.417959183673469e+00],
[0.405845151377397e+00, 0.381830050505119e+00],
[0.741531185599394e+00, 0.279705391489277e+00],
[0.949107912342759e+00, 0.129484966168870e+00]],
bounds=(-1.0, 1.0), symmetric=True),
15 : _QP([[0.183434642495650e+00, 0.362683783378362e+00],
[0.525532409916329e+00, 0.313706645877887e+00],
[0.796666477413627e+00, 0.222381034453374e+00],
[0.960289856497536e+00, 0.101228536290376e+00]],
bounds=(-1.0, 1.0), symmetric=True),
17 : _QP([[0.000000000000000e+00, 0.330239355001260e+00],
[0.324253423403809e+00, 0.312347077040003e+00],
[0.613371432700590e+00, 0.260610696402935e+00],
[0.836031107326636e+00, 0.180648160694857e+00],
[0.968160239507626e+00, 0.081274388361574e+00]],
bounds=(-1.0, 1.0), symmetric=True),
19 : _QP([[0.148874338981631e+00, 0.295524224714753e+00],
[0.433395394129247e+00, 0.269266719309996e+00],
[0.679409568299024e+00, 0.219086362515982e+00],
[0.865063366688985e+00, 0.149451349150581e+00],
[0.973906528517172e+00, 0.066671344308688e+00]],
bounds=(-1.0, 1.0), symmetric=True),
23 : _QP([[0.125233408511469e+00, 0.249147045813403e+00],
[0.367831498998180e+00, 0.233492536538355e+00],
[0.587317954286617e+00, 0.203167426723066e+00],
[0.769902674194305e+00, 0.160078328543346e+00],
[0.904117256370475e+00, 0.106939325995318e+00],
[0.981560634246719e+00, 0.047175336386512e+00]],
bounds=(-1.0, 1.0), symmetric=True),
31 : _QP([[0.095012509837637440185e+00, 0.189450610455068496285e+00],
[0.281603550779258913230e+00, 0.182603415044923588867e+00],
[0.458016777657227386342e+00, 0.169156519395002538189e+00],
[0.617876244402643748447e+00, 0.149595988816576732081e+00],
[0.755404408355003033895e+00, 0.124628971255533872052e+00],
[0.865631202387831743880e+00, 0.095158511682492784810e+00],
[0.944575023073232576078e+00, 0.062253523938647892863e+00],
[0.989400934991649932596e+00, 0.027152459411754094852e+00]],
bounds=(-1.0, 1.0), symmetric=True),
39 : _QP([[0.076526521133497333755e+00, 0.152753387130725850698e+00],
[0.227785851141645078080e+00, 0.149172986472603746788e+00],
[0.373706088715419560673e+00, 0.142096109318382051329e+00],
[0.510867001950827098004e+00, 0.131688638449176626898e+00],
[0.636053680726515025453e+00, 0.118194531961518417312e+00],
[0.746331906460150792614e+00, 0.101930119817240435037e+00],
[0.839116971822218823395e+00, 0.083276741576704748725e+00],
[0.912234428251325905868e+00, 0.062672048334109063570e+00],
[0.963971927277913791268e+00, 0.040601429800386941331e+00],
[0.993128599185094924786e+00, 0.017614007139152118312e+00]],
bounds=(-1.0, 1.0), symmetric=True),
47 : _QP([[0.064056892862605626085e+00, 0.127938195346752156974e+00],
[0.191118867473616309159e+00, 0.125837456346828296121e+00],
[0.315042679696163374387e+00, 0.121670472927803391204e+00],
[0.433793507626045138487e+00, 0.115505668053725601353e+00],
[0.545421471388839535658e+00, 0.107444270115965634783e+00],
[0.648093651936975569252e+00, 0.097618652104113888270e+00],
[0.740124191578554364244e+00, 0.086190161531953275917e+00],
[0.820001985973902921954e+00, 0.073346481411080305734e+00],
[0.886415527004401034213e+00, 0.059298584915436780746e+00],
[0.938274552002732758524e+00, 0.044277438817419806169e+00],
[0.974728555971309498198e+00, 0.028531388628933663181e+00],
[0.995187219997021360180e+00, 0.012341229799987199547e+00]],
bounds=(-1.0, 1.0), symmetric=True),
},
'2_3' : {
1 : _QP([[1.0/3.0, 1.0/3.0, 0.5]],
tp_fix=0.5),
2 : _QP([[1.0/6.0, 1.0/6.0, 1.0/6.0],
[2.0/3.0, 1.0/6.0, 1.0/6.0],
[1.0/6.0, 2.0/3.0, 1.0/6.0]],
tp_fix=0.5),
3 : _QP([[1.0/3.0, 1.0/3.0,-27.0/96.0],
[1.0/5.0, 1.0/5.0, 25.0/96.0],
[3.0/5.0, 1.0/5.0, 25.0/96.0],
[1.0/5.0, 3.0/5.0, 25.0/96.0]],
tp_fix=0.5),
4 : _QP([[0.445948490915965e+00, 0.445948490915965e+00, 0.223381589678011e+00],
[0.108103018168070e+00, 0.445948490915965e+00, 0.223381589678011e+00],
[0.445948490915965e+00, 0.108103018168070e+00, 0.223381589678011e+00],
[0.091576213509771e+00, 0.091576213509771e+00, 0.109951743655322e+00],
[0.816847572980459e+00, 0.091576213509771e+00, 0.109951743655322e+00],
[0.091576213509771e+00, 0.816847572980459e+00, 0.109951743655322e+00]],
tp_fix=0.5, weight_fix=0.5),
5 : _QP([[0.333333333333333e+00, 0.333333333333333e+00, 0.225000000000000e+00],
[0.470142064105115e+00, 0.470142064105115e+00, 0.132394152788506e+00],
[0.059715871789770e+00, 0.470142064105115e+00, 0.132394152788506e+00],
[0.470142064105115e+00, 0.059715871789770e+00, 0.132394152788506e+00],
[0.101286507323456e+00, 0.101286507323456e+00, 0.125939180544827e+00],
[0.797426985353087e+00, 0.101286507323456e+00, 0.125939180544827e+00],
[0.101286507323456e+00, 0.797426985353087e+00, 0.125939180544827e+00]],
tp_fix=0.5, weight_fix=0.5),
6 : _QP([[0.249286745170910e+00, 0.249286745170910e+00, 0.116786275726379e+00],
[0.501426509658179e+00, 0.249286745170910e+00, 0.116786275726379e+00],
[0.249286745170910e+00, 0.501426509658179e+00, 0.116786275726379e+00],
[0.063089014491502e+00, 0.063089014491502e+00, 0.050844906370207e+00],
[0.873821971016996e+00, 0.063089014491502e+00, 0.050844906370207e+00],
[0.063089014491502e+00, 0.873821971016996e+00, 0.050844906370207e+00],
[0.310352451033784e+00, 0.636502499121399e+00, 0.082851075618374e+00],
[0.636502499121399e+00, 0.310352451033784e+00, 0.082851075618374e+00],
[0.053145049844817e+00, 0.636502499121399e+00, 0.082851075618374e+00],
[0.636502499121399e+00, 0.053145049844817e+00, 0.082851075618374e+00],
[0.310352451033784e+00, 0.053145049844817e+00, 0.082851075618374e+00],
[0.053145049844817e+00, 0.310352451033784e+00, 0.082851075618374e+00]],
tp_fix=0.5, weight_fix=0.5),
7 : _QP([[0.333333333333333e+00, 0.333333333333333e+00,-0.149570044467682e+00],
[0.260345966079040e+00, 0.260345966079040e+00, 0.175615257433208e+00],
[0.479308067841920e+00, 0.260345966079040e+00, 0.175615257433208e+00],
[0.260345966079040e+00, 0.479308067841920e+00, 0.175615257433208e+00],
[0.065130102902216e+00, 0.065130102902216e+00, 0.053347235608838e+00],
[0.869739794195568e+00, 0.065130102902216e+00, 0.053347235608838e+00],
[0.065130102902216e+00, 0.869739794195568e+00, 0.053347235608838e+00],
[0.312865496004874e+00, 0.638444188569810e+00, 0.077113760890257e+00],
[0.638444188569810e+00, 0.312865496004874e+00, 0.077113760890257e+00],
[0.048690315425316e+00, 0.638444188569810e+00, 0.077113760890257e+00],
[0.638444188569810e+00, 0.048690315425316e+00, 0.077113760890257e+00],
[0.312865496004874e+00, 0.048690315425316e+00, 0.077113760890257e+00],
[0.048690315425316e+00, 0.312865496004874e+00, 0.077113760890257e+00]],
tp_fix=0.5, weight_fix=0.5),
8 : _QP([[0.333333333333333e+00, 0.333333333333333e+00, 0.144315607677787e+00],
[0.459292588292723e+00, 0.459292588292723e+00, 0.095091634267285e+00],
[0.081414823414554e+00, 0.459292588292723e+00, 0.095091634267285e+00],
[0.459292588292723e+00, 0.081414823414554e+00, 0.095091634267285e+00],
[0.170569307751760e+00, 0.170569307751760e+00, 0.103217370534718e+00],
[0.658861384496480e+00, 0.170569307751760e+00, 0.103217370534718e+00],
[0.170569307751760e+00, 0.658861384496480e+00, 0.103217370534718e+00],
[0.050547228317031e+00, 0.050547228317031e+00, 0.032458497623198e+00],
[0.898905543365938e+00, 0.050547228317031e+00, 0.032458497623198e+00],
[0.050547228317031e+00, 0.898905543365938e+00, 0.032458497623198e+00],
[0.263112829634638e+00, 0.728492392955404e+00, 0.027230314174435e+00],
[0.728492392955404e+00, 0.263112829634638e+00, 0.027230314174435e+00],
[0.008394777409958e+00, 0.728492392955404e+00, 0.027230314174435e+00],
[0.728492392955404e+00, 0.008394777409958e+00, 0.027230314174435e+00],
[0.263112829634638e+00, 0.008394777409958e+00, 0.027230314174435e+00],
[0.008394777409958e+00, 0.263112829634638e+00, 0.027230314174435e+00]],
tp_fix=0.5, weight_fix=0.5),
9 : _QP([[0.333333333333333e+00, 0.333333333333333e+00, 0.097135796282799e+00],
[0.489682519198738e+00, 0.489682519198738e+00, 0.031334700227139e+00],
[0.020634961602525e+00, 0.489682519198738e+00, 0.031334700227139e+00],
[0.489682519198738e+00, 0.020634961602525e+00, 0.031334700227139e+00],
[0.437089591492937e+00, 0.437089591492937e+00, 0.077827541004774e+00],
[0.125820817014127e+00, 0.437089591492937e+00, 0.077827541004774e+00],
[0.437089591492937e+00, 0.125820817014127e+00, 0.077827541004774e+00],
[0.188203535619033e+00, 0.188203535619033e+00, 0.079647738927210e+00],
[0.623592928761935e+00, 0.188203535619033e+00, 0.079647738927210e+00],
[0.188203535619033e+00, 0.623592928761935e+00, 0.079647738927210e+00],
[0.044729513394453e+00, 0.044729513394453e+00, 0.025577675658698e+00],
[0.910540973211095e+00, 0.044729513394453e+00, 0.025577675658698e+00],
[0.044729513394453e+00, 0.910540973211095e+00, 0.025577675658698e+00],
[0.221962989160766e+00, 0.741198598784498e+00, 0.043283539377289e+00],
[0.741198598784498e+00, 0.221962989160766e+00, 0.043283539377289e+00],
[0.036838412054736e+00, 0.741198598784498e+00, 0.043283539377289e+00],
[0.741198598784498e+00, 0.036838412054736e+00, 0.043283539377289e+00],
[0.221962989160766e+00, 0.036838412054736e+00, 0.043283539377289e+00],
[0.036838412054736e+00, 0.221962989160766e+00, 0.043283539377289e+00]],
tp_fix=0.5, weight_fix=0.5),
10 : _QP([[0.333333333333333e+00, 0.333333333333333e+00, 0.908179903827540e-01],
[0.485577633383657e+00, 0.485577633383657e+00, 0.367259577564670e-01],
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19 : _QP([[0.333333333333333e+00, 0.333333333333333e+00, 0.329063313889190e-01],
[0.489609987073006e+00, 0.489609987073006e+00, 0.103307318912720e-01],
[0.207800258539870e-01, 0.489609987073006e+00, 0.103307318912720e-01],
[0.489609987073006e+00, 0.207800258539870e-01, 0.103307318912720e-01],
[0.454536892697893e+00, 0.454536892697893e+00, 0.223872472630160e-01],
[0.909262146042150e-01, 0.454536892697893e+00, 0.223872472630160e-01],
[0.454536892697893e+00, 0.909262146042150e-01, 0.223872472630160e-01],
[0.401416680649431e+00, 0.401416680649431e+00, 0.302661258694680e-01],
[0.197166638701138e+00, 0.401416680649431e+00, 0.302661258694680e-01],
[0.401416680649431e+00, 0.197166638701138e+00, 0.302661258694680e-01],
[0.255551654403098e+00, 0.255551654403098e+00, 0.304909678021980e-01],
[0.488896691193805e+00, 0.255551654403098e+00, 0.304909678021980e-01],
[0.255551654403098e+00, 0.488896691193805e+00, 0.304909678021980e-01],
[0.177077942152130e+00, 0.177077942152130e+00, 0.241592127416410e-01],
[0.645844115695741e+00, 0.177077942152130e+00, 0.241592127416410e-01],
[0.177077942152130e+00, 0.645844115695741e+00, 0.241592127416410e-01],
[0.110061053227952e+00, 0.110061053227952e+00, 0.160508035868010e-01],
[0.779877893544096e+00, 0.110061053227952e+00, 0.160508035868010e-01],
[0.110061053227952e+00, 0.779877893544096e+00, 0.160508035868010e-01],
[0.555286242518400e-01, 0.555286242518400e-01, 0.808458026178400e-02],
[0.888942751496321e+00, 0.555286242518400e-01, 0.808458026178400e-02],
[0.555286242518400e-01, 0.888942751496321e+00, 0.808458026178400e-02],
[0.126218637772290e-01, 0.126218637772290e-01, 0.207936202748500e-02],
[0.974756272445543e+00, 0.126218637772290e-01, 0.207936202748500e-02],
[0.126218637772290e-01, 0.974756272445543e+00, 0.207936202748500e-02],
[0.395754787356943e+00, 0.600633794794645e+00, 0.388487690498100e-02],
[0.600633794794645e+00, 0.395754787356943e+00, 0.388487690498100e-02],
[0.361141784841200e-02, 0.600633794794645e+00, 0.388487690498100e-02],
[0.600633794794645e+00, 0.361141784841200e-02, 0.388487690498100e-02],
[0.395754787356943e+00, 0.361141784841200e-02, 0.388487690498100e-02],
[0.361141784841200e-02, 0.395754787356943e+00, 0.388487690498100e-02],
[0.307929983880436e+00, 0.557603261588784e+00, 0.255741606120220e-01],
[0.557603261588784e+00, 0.307929983880436e+00, 0.255741606120220e-01],
[0.134466754530780e+00, 0.557603261588784e+00, 0.255741606120220e-01],
[0.557603261588784e+00, 0.134466754530780e+00, 0.255741606120220e-01],
[0.307929983880436e+00, 0.134466754530780e+00, 0.255741606120220e-01],
[0.134466754530780e+00, 0.307929983880436e+00, 0.255741606120220e-01],
[0.264566948406520e+00, 0.720987025817365e+00, 0.888090357333800e-02],
[0.720987025817365e+00, 0.264566948406520e+00, 0.888090357333800e-02],
[0.144460257761150e-01, 0.720987025817365e+00, 0.888090357333800e-02],
[0.720987025817365e+00, 0.144460257761150e-01, 0.888090357333800e-02],
[0.264566948406520e+00, 0.144460257761150e-01, 0.888090357333800e-02],
[0.144460257761150e-01, 0.264566948406520e+00, 0.888090357333800e-02],
[0.358539352205951e+00, 0.594527068955871e+00, 0.161245467617310e-01],
[0.594527068955871e+00, 0.358539352205951e+00, 0.161245467617310e-01],
[0.469335788381780e-01, 0.594527068955871e+00, 0.161245467617310e-01],
[0.594527068955871e+00, 0.469335788381780e-01, 0.161245467617310e-01],
[0.358539352205951e+00, 0.469335788381780e-01, 0.161245467617310e-01],
[0.469335788381780e-01, 0.358539352205951e+00, 0.161245467617310e-01],
[0.157807405968595e+00, 0.839331473680839e+00, 0.249194181749100e-02],
[0.839331473680839e+00, 0.157807405968595e+00, 0.249194181749100e-02],
[0.286112035056700e-02, 0.839331473680839e+00, 0.249194181749100e-02],
[0.839331473680839e+00, 0.286112035056700e-02, 0.249194181749100e-02],
[0.157807405968595e+00, 0.286112035056700e-02, 0.249194181749100e-02],
[0.286112035056700e-02, 0.157807405968595e+00, 0.249194181749100e-02],
[0.750505969759110e-01, 0.701087978926173e+00, 0.182428401189510e-01],
[0.701087978926173e+00, 0.750505969759110e-01, 0.182428401189510e-01],
[0.223861424097916e+00, 0.701087978926173e+00, 0.182428401189510e-01],
[0.701087978926173e+00, 0.223861424097916e+00, 0.182428401189510e-01],
[0.750505969759110e-01, 0.223861424097916e+00, 0.182428401189510e-01],
[0.223861424097916e+00, 0.750505969759110e-01, 0.182428401189510e-01],
[0.142421601113383e+00, 0.822931324069857e+00, 0.102585637361990e-01],
[0.822931324069857e+00, 0.142421601113383e+00, 0.102585637361990e-01],
[0.346470748167600e-01, 0.822931324069857e+00, 0.102585637361990e-01],
[0.822931324069857e+00, 0.346470748167600e-01, 0.102585637361990e-01],
[0.142421601113383e+00, 0.346470748167600e-01, 0.102585637361990e-01],
[0.346470748167600e-01, 0.142421601113383e+00, 0.102585637361990e-01],
[0.654946280829380e-01, 0.924344252620784e+00, 0.379992885530200e-02],
[0.924344252620784e+00, 0.654946280829380e-01, 0.379992885530200e-02],
[0.101611192962780e-01, 0.924344252620784e+00, 0.379992885530200e-02],
[0.924344252620784e+00, 0.101611192962780e-01, 0.379992885530200e-02],
[0.654946280829380e-01, 0.101611192962780e-01, 0.379992885530200e-02],
[0.101611192962780e-01, 0.654946280829380e-01, 0.379992885530200e-02]],
tp_fix=0.5, weight_fix=0.5),
},
'2_4' : {
2 : _QP([[ nm.sqrt(2.0/3.0), 0.0 , 4.0/3.0],
[-1/nm.sqrt(6) , 1/nm.sqrt(2), 4.0/3.0],
[-1/nm.sqrt(6) ,-1/nm.sqrt(2), 4.0/3.0]], bounds=(-1.0, 1.0)),
3 : _QP([[-1/nm.sqrt(3),-1/nm.sqrt(3), 1.0],
[ 1/nm.sqrt(3),-1/nm.sqrt(3), 1.0],
[ 1/nm.sqrt(3), 1/nm.sqrt(3), 1.0],
[-1/nm.sqrt(3), 1/nm.sqrt(3), 1.0]], bounds=(-1.0, 1.0)),
5 : _QP([[ nm.sqrt(7.0/15.0), 0.0 , 0.816326530612245],
[-nm.sqrt(7.0/15.0), 0.0 , 0.816326530612245],
[ 0.0 , nm.sqrt(7.0/15.0), 0.816326530612245],
[ 0.0 ,-nm.sqrt(7.0/15.0), 0.816326530612245],
[ 0.881917103688197, 0.881917103688197, 0.183673469387755],
[ 0.881917103688197,-0.881917103688197, 0.183673469387755],
[-0.881917103688197, 0.881917103688197, 0.183673469387755],
[-0.881917103688197,-0.881917103688197, 0.183673469387755]], bounds=(-1.0, 1.0)),
},
'3_4' : {
1 : _QP([[ 1.0/4.0, 1.0/4.0, 1.0/4.0, 1.0/6.0]], tp_fix=1.0/6.0),
2 : _QP([[ (5-nm.sqrt(5))/20 , (5-nm.sqrt(5))/20 , (5-nm.sqrt(5))/20 , 1.0/24.0],
[ (5-nm.sqrt(5))/20 , (5-nm.sqrt(5))/20 , (5+3*nm.sqrt(5))/20, 1.0/24.0],
[ (5-nm.sqrt(5))/20 , (5+3*nm.sqrt(5))/20, (5-nm.sqrt(5))/20 , 1.0/24.0],
[ (5+3*nm.sqrt(5))/20, (5-nm.sqrt(5))/20 , (5-nm.sqrt(5))/20 , 1.0/24.0]], tp_fix=1.0/6.0),
3 : _QP([[ 1.0/4.0, 1.0/4.0, 1.0/4.0,-2.0/15.0],
[ 1.0/6.0, 1.0/6.0, 1.0/6.0, 3.0/40.0],
[ 1.0/6.0, 1.0/6.0, 1.0/2.0, 3.0/40.0],
[ 1.0/6.0, 1.0/2.0, 1.0/6.0, 3.0/40.0],
[ 1.0/2.0, 1.0/6.0, 1.0/6.0, 3.0/40.0]], tp_fix=1.0/6.0),
4 : _QP([[-0.5000000000000000, -0.5000000000000000, -0.5000000000000000, -0.1052444444444440],
[-0.8571428571428570, -0.8571428571428570, -0.8571428571428570, 0.0609777777777780],
[-0.8571428571428570, -0.8571428571428570, 0.5714285714285710, 0.0609777777777780],
[-0.8571428571428570, 0.5714285714285710, -0.8571428571428570, 0.0609777777777780],
[ 0.5714285714285710, -0.8571428571428570, -0.8571428571428570, 0.0609777777777780],
[-0.2011928476664020, -0.2011928476664020, -0.7988071523335980, 0.1991111111111110],
[-0.2011928476664020, -0.7988071523335980, -0.2011928476664020, 0.1991111111111110],
[-0.7988071523335980, -0.2011928476664020, -0.2011928476664020, 0.1991111111111110],
[-0.2011928476664020, -0.7988071523335980, -0.7988071523335980, 0.1991111111111110],
[-0.7988071523335980, -0.2011928476664020, -0.7988071523335980, 0.1991111111111110],
[-0.7988071523335980, -0.7988071523335980, -0.2011928476664020, 0.1991111111111110]],
bounds=(-1.0, 1.0), tp_fix=1.0/6.0),
6 : _QP([[-0.5707942574816960, -0.5707942574816960, -0.5707942574816960, 0.0532303336775570],
[-0.2876172275549120, -0.5707942574816960, -0.5707942574816960, 0.0532303336775570],
[-0.5707942574816960, -0.2876172275549120, -0.5707942574816960, 0.0532303336775570],
[-0.5707942574816960, -0.5707942574816960, -0.2876172275549120, 0.0532303336775570],
[-0.9186520829307770, -0.9186520829307770, -0.9186520829307770, 0.0134362814070940],
[0.7559562487923320, -0.9186520829307770, -0.9186520829307770, 0.0134362814070940],
[-0.9186520829307770, 0.7559562487923320, -0.9186520829307770, 0.0134362814070940],
[-0.9186520829307770, -0.9186520829307770, 0.7559562487923320, 0.0134362814070940],
[-0.3553242197154490, -0.3553242197154490, -0.3553242197154490, 0.0738095753915400],
[-0.9340273408536530, -0.3553242197154490, -0.3553242197154490, 0.0738095753915400],
[-0.3553242197154490, -0.9340273408536530, -0.3553242197154490, 0.0738095753915400],
[-0.3553242197154490, -0.3553242197154490, -0.9340273408536530, 0.0738095753915400],
[-0.8726779962499650, -0.8726779962499650, -0.4606553370833680, 0.0642857142857140],
[-0.8726779962499650, -0.4606553370833680, -0.8726779962499650, 0.0642857142857140],
[-0.8726779962499650, -0.8726779962499650, 0.2060113295832980, 0.0642857142857140],
[-0.8726779962499650, 0.2060113295832980, -0.8726779962499650, 0.0642857142857140],
[-0.8726779962499650, -0.4606553370833680, 0.2060113295832980, 0.0642857142857140],
[-0.8726779962499650, 0.2060113295832980, -0.4606553370833680, 0.0642857142857140],
[-0.4606553370833680, -0.8726779962499650, -0.8726779962499650, 0.0642857142857140],
[-0.4606553370833680, -0.8726779962499650, 0.2060113295832980, 0.0642857142857140],
[-0.4606553370833680, 0.2060113295832980, -0.8726779962499650, 0.0642857142857140],
[0.2060113295832980, -0.8726779962499650, -0.4606553370833680, 0.0642857142857140],
[0.2060113295832980, -0.8726779962499650, -0.8726779962499650, 0.0642857142857140],
[0.2060113295832980, -0.4606553370833680, -0.8726779962499650, 0.0642857142857140]],
bounds=(-1.0, 1.0), tp_fix=1.0/6.0),
},
'3_6': {
3: _QP([[1.0/3.0, 1.0/3.0, 0.5, 0.5]],
tp_fix=1/2),
6: _QP([[1.0/6.0, 1.0/6.0, 0.2113248654051871, 1.0/12.0],
[2.0/3.0, 1.0/6.0, 0.2113248654051871, 1.0/12.0],
[1.0/6.0, 2.0/3.0, 0.2113248654051871, 1.0/12.0],
[1.0/6.0, 1.0/6.0, 0.7886751345948129, 1.0/12.0],
[2.0/3.0, 1.0/6.0, 0.7886751345948129, 1.0/12.0],
[1.0/6.0, 2.0/3.0, 0.7886751345948129, 1.0/12.0]],
tp_fix=1/2),
9: _QP([[0.1081030181680702, 0.4459484909159649, 0.1127016653792583, 0.0310252207886127],
[0.4459484909159649, 0.1081030181680702, 0.1127016653792583, 0.0310252207886127],
[0.4459484909159649, 0.4459484909159649, 0.1127016653792583, 0.0310252207886127],
[0.8168475729804585, 0.0915762135097707, 0.1127016653792583, 0.0152710755076836],
[0.0915762135097707, 0.8168475729804585, 0.1127016653792583, 0.0152710755076836],
[0.0915762135097707, 0.0915762135097707, 0.1127016653792583, 0.0152710755076836],
[0.1081030181680702, 0.4459484909159649, 0.5000000000000000, 0.0496403532617803],
[0.4459484909159649, 0.1081030181680702, 0.5000000000000000, 0.0496403532617803],
[0.4459484909159649, 0.4459484909159649, 0.5000000000000000, 0.0496403532617803],
[0.8168475729804585, 0.0915762135097707, 0.5000000000000000, 0.0244337208122937],
[0.0915762135097707, 0.8168475729804585, 0.5000000000000000, 0.0244337208122937],
[0.0915762135097707, 0.0915762135097707, 0.5000000000000000, 0.0244337208122937],
[0.1081030181680702, 0.4459484909159649, 0.8872983346207417, 0.0310252207886127],
[0.4459484909159649, 0.1081030181680702, 0.8872983346207417, 0.0310252207886127],
[0.4459484909159649, 0.4459484909159649, 0.8872983346207417, 0.0310252207886127],
[0.8168475729804585, 0.0915762135097707, 0.8872983346207417, 0.0152710755076836],
[0.0915762135097707, 0.8168475729804585, 0.8872983346207417, 0.0152710755076836],
[0.0915762135097707, 0.0915762135097707, 0.8872983346207417, 0.0152710755076836]],
tp_fix=1/2),
},
'3_8' : {
2 : _QP([[ 0.0 , nm.sqrt(2.0/3.0),-1/nm.sqrt(3), 2.0],
[ 0.0 ,-nm.sqrt(2.0/3.0),-1/nm.sqrt(3), 2.0],
[ nm.sqrt(2.0/3.0), 0.0 , 1/nm.sqrt(3), 2.0],
[-nm.sqrt(2.0/3.0), 0.0 , 1/nm.sqrt(3), 2.0]], bounds=(-1.0, 1.0)),
3 : _QP([[-1.0, 0.0, 0.0, 4.0/3.0],
[ 1.0, 0.0, 0.0, 4.0/3.0],
[ 0.0,-1.0, 0.0, 4.0/3.0],
[ 0.0, 1.0, 0.0, 4.0/3.0],
[ 0.0, 0.0,-1.0, 4.0/3.0],
[ 0.0, 0.0, 1.0, 4.0/3.0]], bounds=(-1.0, 1.0)),
5 : _QP([[-nm.sqrt(19.0/30.0), 0.0 , 0.0 , 320.0/361.0],
[ nm.sqrt(19.0/30.0), 0.0 , 0.0 , 320.0/361.0],
[ 0.0 ,-nm.sqrt(19.0/30.0), 0.0 , 320.0/361.0],
[ 0.0 , nm.sqrt(19.0/30.0), 0.0 , 320.0/361.0],
[ 0.0 , 0.0 ,-nm.sqrt(19.0/30.0), 320.0/361.0],
[ 0.0 , 0.0 , nm.sqrt(19.0/30.0), 320.0/361.0],
[ nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), 121.0/361.0],
[ nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), 121.0/361.0],
[ nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), 121.0/361.0],
[ nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), 121.0/361.0],
[-nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), 121.0/361.0],
[-nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), 121.0/361.0],
[-nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), nm.sqrt(19.0/33.0), 121.0/361.0],
[-nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0),-nm.sqrt(19.0/33.0), 121.0/361.0]], bounds=(-1.0, 1.0)),
},
}
del _QP
def _get_max_orders():
max_orders = {}
for key, table in quadrature_tables.items():
orders = list(table.keys())
max_orders[key] = max(orders)
return max_orders
max_orders = _get_max_orders()