/usr/share/pyshared/ffc/quadrature_schemes.py is in python-ffc 1.0.0-1.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 | """Quadrature schemes on cells
This module generates quadrature schemes on reference cells that integrate
exactly a polynomial of a given degree using a specified scheme. The
UFC definition of a reference cell is used.
Scheme options are:
scheme="default"
scheme="canonical" (collapsed Gauss scheme supplied by FIAT)
Background on the schemes:
Keast rules for tetrahedra:
Keast, P. Moderate-degree tetrahedral quadrature formulas, Computer
Methods in Applied Mechanics and Engineering 55(3):339-348, 1986.
http://dx.doi.org/10.1016/0045-7825(86)90059-9
"""
# Copyright (C) 2011 Garth N. Wells
#
# This file is part of FFC.
#
# FFC is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# FFC is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with FFC. If not, see <http://www.gnu.org/licenses/>.
#
# First added: 2011-04-19
# Last changed: 2011-04-19
# NumPy
from numpy import array, arange, float64
# UFL
import ufl
# FFC modules
from ffc.log import debug, error
from ffc.fiatinterface import reference_cell
from ffc.fiatinterface import create_quadrature as fiat_create_quadrature
# Dictionary mapping from domain (cell) to dimension
from ufl.geometry import domain2dim
def create_quadrature(shape, degree, scheme="default"):
"""
Generate quadrature rule (points, weights) for given shape
that will integrate an polynomial of order 'degree' exactly.
"""
# FIXME: KBO: Can this be handled more elegantly?
# Handle point case
if isinstance(shape, int) and shape == 0 or domain2dim[shape] == 0:
return ([()], array([1.0,]))
if scheme == "default":
if shape == "tetrahedron":
return _tetrahedron_scheme(degree)
elif shape == "triangle":
return _triangle_scheme(degree)
else:
return _fiat_scheme(shape, degree)
elif scheme == "canonical":
return _fiat_scheme(shape, degree)
else:
error("Unknown quadrature scheme: %s." % scheme)
def _fiat_scheme(shape, degree):
"""Get quadrature scheme from FIAT interface"""
# Number of points per axis for exact integration
num_points_per_axis = (degree + 1 + 1) / 2
# Create and return FIAT quadrature rulet
return fiat_create_quadrature(shape, num_points_per_axis)
def _triangle_scheme(degree):
"""Return a quadrature scheme on a triangle of specified order. Falls
back on canonical rule for higher orders."""
if degree == 0 or degree == 1:
# Scheme from Zienkiewicz and Taylor, 1 point, degree of precision 1
x = array([ [1.0/3.0, 1.0/3.0] ])
w = array([0.5])
elif degree == 2:
# Scheme from Strang and Fix, 3 points, degree of precision 2
x = array([ [1.0/6.0, 1.0/6.0],
[1.0/6.0, 2.0/3.0],
[2.0/3.0, 1.0/6.0] ])
w = arange(3, dtype=float64)
w[:] = 1.0/6.0
elif degree == 3:
# Scheme from Strang and Fix, 6 points, degree of precision 3
x = array([ [0.659027622374092, 0.231933368553031],
[0.659027622374092, 0.109039009072877],
[0.231933368553031, 0.659027622374092],
[0.231933368553031, 0.109039009072877],
[0.109039009072877, 0.659027622374092],
[0.109039009072877, 0.231933368553031] ])
w = arange(6, dtype=float64)
w[:] = 1.0/12.0
elif degree == 4:
# Scheme from Strang and Fix, 6 points, degree of precision 4
x = array([ [0.816847572980459, 0.091576213509771],
[0.091576213509771, 0.816847572980459],
[0.091576213509771, 0.091576213509771],
[0.108103018168070, 0.445948490915965],
[0.445948490915965, 0.108103018168070],
[0.445948490915965, 0.445948490915965] ])
w = arange(6, dtype=float64)
w[0:3] = 0.109951743655322
w[3:6] = 0.223381589678011
w = w/2.0
elif degree == 5:
# Scheme from Strang and Fix, 7 points, degree of precision 5
x = array([ [0.33333333333333333, 0.33333333333333333],
[0.79742698535308720, 0.10128650732345633],
[0.10128650732345633, 0.79742698535308720],
[0.10128650732345633, 0.10128650732345633],
[0.05971587178976981, 0.47014206410511505],
[0.47014206410511505, 0.05971587178976981],
[0.47014206410511505, 0.47014206410511505] ])
w = arange(7, dtype=float64)
w[0] = 0.22500000000000000
w[1:4] = 0.12593918054482717
w[4:7] = 0.13239415278850616
w = w/2.0
elif degree == 6:
# Scheme from Strang and Fix, 12 points, degree of precision 6
x = array([ [0.873821971016996, 0.063089014491502],
[0.063089014491502, 0.873821971016996],
[0.063089014491502, 0.063089014491502],
[0.501426509658179, 0.249286745170910],
[0.249286745170910, 0.501426509658179],
[0.249286745170910, 0.249286745170910],
[0.636502499121399, 0.310352451033785],
[0.636502499121399, 0.053145049844816],
[0.310352451033785, 0.636502499121399],
[0.310352451033785, 0.053145049844816],
[0.053145049844816, 0.636502499121399],
[0.053145049844816, 0.310352451033785] ])
w = arange(12, dtype=float64)
w[0:3] = 0.050844906370207
w[3:6] = 0.116786275726379
w[6:12] = 0.082851075618374
w = w/2.0
else:
# Get canonical scheme
x, w = _fiat_scheme("triangle", degree)
# Return scheme
return x, w
def _tetrahedron_scheme(degree):
"""Return a quadrature scheme on a tetrahedron of specified
degree. Falls back on canonical rule for higher orders"""
if degree == 0 or degree == 1:
# Scheme from Zienkiewicz and Taylor, 1 point, degree of precision 1
x = array([ [1.0/4.0, 1.0/4.0, 1.0/4.0] ])
w = array([1.0/6.0])
elif degree == 2:
# Scheme from Zienkiewicz and Taylor, 4 points, degree of precision 2
a, b = 0.585410196624969, 0.138196601125011
x = array([ [a, b, b],
[b, a, b],
[b, b, a],
[b, b, b] ])
w = arange(4, dtype=float64)
w[:] = 1.0/24.0
elif degree == 3:
# Scheme from Zienkiewicz and Taylor, 5 points, degree of precision 3
# Note: this scheme has a negative weight
x = array([ [0.2500000000000000, 0.2500000000000000, 0.2500000000000000],
[0.5000000000000000, 0.1666666666666666, 0.1666666666666666],
[0.1666666666666666, 0.5000000000000000, 0.1666666666666666],
[0.1666666666666666, 0.1666666666666666, 0.5000000000000000],
[0.1666666666666666, 0.1666666666666666, 0.1666666666666666] ])
w = arange(5, dtype=float64)
w[0] = -0.8
w[1:5] = 0.45
w = w/6.0
elif degree == 4:
# Keast rule, 14 points, degree of precision 4
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST5)
x = array([ [0.0000000000000000, 0.5000000000000000, 0.5000000000000000],
[0.5000000000000000, 0.0000000000000000, 0.5000000000000000],
[0.5000000000000000, 0.5000000000000000, 0.0000000000000000],
[0.5000000000000000, 0.0000000000000000, 0.0000000000000000],
[0.0000000000000000, 0.5000000000000000, 0.0000000000000000],
[0.0000000000000000, 0.0000000000000000, 0.5000000000000000],
[0.6984197043243866, 0.1005267652252045, 0.1005267652252045],
[0.1005267652252045, 0.1005267652252045, 0.1005267652252045],
[0.1005267652252045, 0.1005267652252045, 0.6984197043243866],
[0.1005267652252045, 0.6984197043243866, 0.1005267652252045],
[0.0568813795204234, 0.3143728734931922, 0.3143728734931922],
[0.3143728734931922, 0.3143728734931922, 0.3143728734931922],
[0.3143728734931922, 0.3143728734931922, 0.0568813795204234],
[0.3143728734931922, 0.0568813795204234, 0.3143728734931922] ])
w = arange(14, dtype=float64)
w[0:6] = 0.0190476190476190
w[6:10] = 0.0885898247429807
w[10:14] = 0.1328387466855907
w = w/6.0
elif degree == 5:
# Keast rule, 15 points, degree of precision 5
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST6)
x = array([ [0.2500000000000000, 0.2500000000000000, 0.2500000000000000],
[0.0000000000000000, 0.3333333333333333, 0.3333333333333333],
[0.3333333333333333, 0.3333333333333333, 0.3333333333333333],
[0.3333333333333333, 0.3333333333333333, 0.0000000000000000],
[0.3333333333333333, 0.0000000000000000, 0.3333333333333333],
[0.7272727272727273, 0.0909090909090909, 0.0909090909090909],
[0.0909090909090909, 0.0909090909090909, 0.0909090909090909],
[0.0909090909090909, 0.0909090909090909, 0.7272727272727273],
[0.0909090909090909, 0.7272727272727273, 0.0909090909090909],
[0.4334498464263357, 0.0665501535736643, 0.0665501535736643],
[0.0665501535736643, 0.4334498464263357, 0.0665501535736643],
[0.0665501535736643, 0.0665501535736643, 0.4334498464263357],
[0.0665501535736643, 0.4334498464263357, 0.4334498464263357],
[0.4334498464263357, 0.0665501535736643, 0.4334498464263357],
[0.4334498464263357, 0.4334498464263357, 0.0665501535736643] ])
w = arange(15, dtype=float64)
w[0] = 0.1817020685825351
w[1:5] = 0.0361607142857143
w[5:9] = 0.0698714945161738
w[9:15] = 0.0656948493683187
w = w/6.0
elif degree == 6:
# Keast rule, 24 points, degree of precision 6
# Values taken from http://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tet/quadrature_rules_tet.html
# (KEAST7)
x = array([ [0.3561913862225449, 0.2146028712591517, 0.2146028712591517],
[0.2146028712591517, 0.2146028712591517, 0.2146028712591517],
[0.2146028712591517, 0.2146028712591517, 0.3561913862225449],
[0.2146028712591517, 0.3561913862225449, 0.2146028712591517],
[0.8779781243961660, 0.0406739585346113, 0.0406739585346113],
[0.0406739585346113, 0.0406739585346113, 0.0406739585346113],
[0.0406739585346113, 0.0406739585346113, 0.8779781243961660],
[0.0406739585346113, 0.8779781243961660, 0.0406739585346113],
[0.0329863295731731, 0.3223378901422757, 0.3223378901422757],
[0.3223378901422757, 0.3223378901422757, 0.3223378901422757],
[0.3223378901422757, 0.3223378901422757, 0.0329863295731731],
[0.3223378901422757, 0.0329863295731731, 0.3223378901422757],
[0.2696723314583159, 0.0636610018750175, 0.0636610018750175],
[0.0636610018750175, 0.2696723314583159, 0.0636610018750175],
[0.0636610018750175, 0.0636610018750175, 0.2696723314583159],
[0.6030056647916491, 0.0636610018750175, 0.0636610018750175],
[0.0636610018750175, 0.6030056647916491, 0.0636610018750175],
[0.0636610018750175, 0.0636610018750175, 0.6030056647916491],
[0.0636610018750175, 0.2696723314583159, 0.6030056647916491],
[0.2696723314583159, 0.6030056647916491, 0.0636610018750175],
[0.6030056647916491, 0.0636610018750175, 0.2696723314583159],
[0.0636610018750175, 0.6030056647916491, 0.2696723314583159],
[0.2696723314583159, 0.0636610018750175, 0.6030056647916491],
[0.6030056647916491, 0.2696723314583159, 0.0636610018750175] ])
w = arange(24, dtype=float64)
w[0:4] = 0.0399227502581679
w[4:8] = 0.0100772110553207
w[8:12] = 0.0553571815436544
w[12:24] = 0.0482142857142857
w = w/6.0
else:
# Get canonical scheme
x, w = _fiat_scheme("tetrahedron", degree)
# Return scheme
return x, w
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