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quadrature.py
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# -*- coding: utf-8 -*-
"""
Created on Sat Nov 14 22:44:18 2015
"""
__author__ = 'Konstantinos Tatsis'
import numpy as np
import abc
class Quadrature:
def __init__(self, points, weights):
self.info = np.hstack((points, weights))
class Gauss(Quadrature):
@classmethod
def inLine(cls, rule):
"""
Gauss quadrature rule in a one-dimensional linear domain.
Parameters
----------
rule: {1, 2, 3, 4, 5}
The integration rule, as described in the table below.
----------------------
Rule Points Degree
----------------------
1 1 1
2 2 3
3 3 5
4 4 7
5 5 9
----------------------
Returns
-------
quadrature: Gauss
The quadrature points and weights.
Raises
------
TypeError
If an invalid rule is specified.
"""
if rule == 1:
points = np.array([0])
weights = np.array([2])
elif rule == 2:
p = np.sqrt(3)/3
points = np.array([[+p], [-p]])
weights = np.array([[1], [1]])
elif rule == 3:
p = np.sqrt(3/5)
points = np.array([[-p], [0], [+p]])
weights = np.array([[5/9], [8/9], [5/9]])
elif rule == 4:
p1 = np.sqrt(525+70*np.sqrt(30))/35
p2 = np.sqrt(525-70*np.sqrt(30))/35
points = np.array([[-p1], [-p2], [+p2], [+p1]])
w1 = (18-np.sqrt(30))/36
w2 = (18+np.sqrt(30))/36
weights = np.array([[w1], [w2], [w2], [w1]])
elif rule == 5:
p1 = np.sqrt(5+2*np.sqrt(10/7))/3
p2 = np.sqrt(5-2*np.sqrt(10/7))/3
points = np.array([[-p1], [-p2], [0], [+p1], [+p2]])
w1 = (322-13*np.sqrt(70))/900
w2 = (322+13*np.sqrt(70))/900
weights = np.array([[w1], [w2], [128/225], [w1], [w2]])
else:
raise TypeError('Invalid integration rule.')
return cls(points, weights)
@classmethod
def inTriangle(cls, rule):
"""
Parameters
----------
rule: {1, 3, -3, 6, -6, 7, 12}
The integration rule, as described in the table below.
------------------------------------------------------------------
Rule Points Degree Comments
------------------------------------------------------------------
1 1 1 Centroid rule, useful for Tri3 stiffness.
3 3 2 Useful for Tri6 stiffness and Tri3 mass.
-3 3 2 Midpoint rule, less accurate than 3.
6 6 4 Useful for Tri10 stiffness and Tri6 mass.
-6 6 3 A linear combination of 3 and -3 rules.
7 7 5 Radon's formula, useful for Tri10 stiffness.
12 12 6 Useful for Tri10 mass.
------------------------------------------------------------------
Returns
-------
quadrature: Gauss
The quadrature points and weights.
Raises
------
TypeError
If an invalid rule is specified.
"""
if rule == 1:
points = np.array([[1/3, 1/3, 1/3]])
weights = np.array([[1]])
elif rule == 3:
p1, p2 = 1/6, 2/3
points = np.array([[p2, p1, p1], [p1, p2, p1], [p1, p1, p2]])
weights = np.array([[1/3], [1/3], [1/3]])
elif rule == -3:
points = np.array([[0, 0.5, 0.5], [0.5, 0, 0.5], [0.5, 0.5, 0]])
weights = np.array([[1/3], [1/3], [1/3]])
elif rule == 6:
p1 = 0
p2 = 0
elif rule == -6:
pass
elif rule == 7:
p0 = 1/3
p1, p2 = (6+np.sqrt(15))/21, (6-np.sqrt(15))/21
p3, p4 = (9+2*np.sqrt(15))/21, (9-2*np.sqrt(15))/21
points = np.array([
[p0, p0], # 1
[p1, p4], # 2
[p1, p1], # 3
[p4, p1], # 4
[p3, p2], # 5
[p2, p3], # 6
[p2, p2]]) # 7
w0, w1, w2 = 9/40, (155+np.sqrt(15))/1200, (155-np.sqrt(15))/1200
weights = np.array([[w0, w1, w1, w1, w2, w2, w2]]).T
elif rule == 12:
pass
else:
raise TypeError('Invalid integration rule.')
return cls(points, weights)
@classmethod
def inQuadrilateral(cls, rule):
"""
Parameters
----------
rule: {1, 3, -3, 6, -6, 7, 12}
The integration rule, as described in the table below.
------------------------------------------------------------------
Rule Points Degree Comments
------------------------------------------------------------------
1 1 1 Used in reduced and selective integration.
2 4 3 Used for Quad4 stiffness and mass.
3 9 5 Used for Quad8 and Quad9 stiffness and mass.
4 16 7 Used for Quad16 stiffness and mass.
5 25 9 -
------------------------------------------------------------------
"""
# The order of Gauss points follows the definition of Shape functions:
# Order 1: V
# Order 2: V
# Order 3: V
# Order 4: X
# Order 5: X
if rule == 1:
points = np.array([[0, 0]])
weights = np.array([[2, 2]])
elif rule == 2:
p = np.sqrt(3)/3
points = np.array([
[+p, +p], # 1
[-p, +p], # 2
[-p, -p], # 3
[+p, -p]]) # 4
weights = np.ones((4, 2))
elif rule == 3.:
p = np.sqrt(3/5)
points = np.array([
[+p, +p], # 1
[-p, -p], # 2
[-p, +p], # 3
[+p, -p], # 4
[ 0, +p], # 5
[-p, 0], # 6
[ 0, -p], # 7
[+p, 0], # 8
[ 0, 0]]) # 9
w0 = 8/9
w1 = 5/9
weights = np.array([
[w1, w1], # 1
[w1, w1], # 2
[w1, w1], # 3
[w1, w1], # 4
[w0, w1], # 5
[w1, w0], # 6
[w0, w1], # 7
[w1, w0], # 8
[w0, w0]]) # 9
elif rule == 4:
p1 = np.sqrt((3+2*np.sqrt(6/5))/7)
p2 = np.sqrt((3-2*np.sqrt(6/5))/7)
points = np.array([
[-p1, -p1], # 1
[-p2, -p1], # 2
[+p2, -p1], # 3
[+p1, -p1], # 4
[+p1, -p2], # 5
[+p1, +p2], # 6
[+p1, +p1], # 7
[+p2, +p1], # 8
[-p2, +p1], # 9
[-p1, +p1], # 10
[-p1, +p2], # 11
[-p1, -p2], # 12
[-p2, -p2], # 13
[+p2, -p2], # 14
[+p2, +p2], # 15
[-p2, +p2]]) # 16
w1 = (18-np.sqrt(30))/36
w2 = (18+np.sqrt(30))/36
weights = np.array([
[w1, w1], # 1
[w2, w1], # 2
[w2, w1], # 3
[w1, w1], # 4
[w1, w2], # 5
[w1, w2], # 6
[w1, w1], # 7
[w2, w1], # 8
[w2, w1], # 9
[w1, w1], # 10
[w1, w2], # 11
[w1, w2], # 12
[w2, w2], # 13
[w2, w2], # 14
[w2, w2], # 15
[w2, w2]]) # 16
elif rule==5:
p1 = np.sqrt(5+2*np.sqrt(10/7))/3
p2 = np.sqrt(5-2*np.sqrt(10/7))/3
points = np.array([
[-p1, -p1], # 1
[-p2, -p1], # 2
[ 0, -p1], # 3
[+p2, -p1], # 4
[+p1, -p1], # 5
[-p1, -p2], # 6
[-p2, -p2], # 7
[ 0, -p2], # 8
[+p2, -p2], # 9
[+p1, -p2], # 10
[-p1, 0], # 11
[-p2, 0], # 12
[ 0, 0], # 13
[+p2, 0], # 14
[+p1, 0], # 15
[-p1, +p2], # 16
[-p2, +p2], # 17
[ 0, +p2], # 18
[+p2, +p2], # 19
[+p1, +p2], # 20
[-p1, +p1], # 21
[-p2, +p1], # 22
[ 0, +p1], # 23
[+p2, +p1], # 24
[+p1, +p1]]) # 25
w0 = 128/225
w1 = (322-13*np.sqrt(70))/900
w2 = (322+13*np.sqrt(70))/900
weights = np.array([
[w1, w1], # 1
[w2, w1], # 2
[w0, w1], # 3
[w2, w1], # 4
[w1, w1], # 5
[w1, w2], # 6
[w2, w2], # 7
[w0, w2], # 8
[w2, w2], # 9
[w1, w2], # 10
[w1, w0], # 11
[w2, w0], # 12
[w0, w0], # 13
[w2, w0], # 14
[w1, w0], # 15
[w1, w2], # 16
[w2, w2], # 17
[w0, w2], # 18
[w2, w2], # 19
[w1, w2], # 20
[w1 ,w1], # 21
[w2, w1], # 22
[w0, w1], # 23
[w2, w1], # 24
[w1, w1]]) # 25
else:
raise TypeError('Invalid integration rule.')
return cls(points, weights)
@classmethod
def inTetrahedron(cls, rule):
"""
Parameters
----------
rule: {1, 4, 8, -8, 14, -14, 15, -15, 24}
The integration rule, as described in the table below.
-----------------------------------------------------------------
Rule Points Degree Comments
-----------------------------------------------------------------
1 1 1 Centroid rule, useful for Tet4 stiffness.
4 4 2 Useful for Tet4 mass and Tet10 stiffness.
8 8 3
-8 8 3 Has corners and face centers as points.
14 14 4 Useful for Tet10 mass.
-14 14 3 Has edge midpoints as sample points.
15 15 5 Useful for Tet21 stiffness.
-15 15 4 Less accurate than above.
24 24 6 Useful for Tet21 stiffness.
-----------------------------------------------------------------
"""
if rule == 1:
p, w = 1/4, 1
points = np.array([[p, p, p, p]])
weights = np.array([[w]])
elif rule == 4:
p1 = (5-np.sqrt(5))/20
p2 = (5+3*np.sqrt(5))/20
points = np.array([
[p2, p1, p1, p1],
[p1, p2, p1, p1],
[p1, p1, p2, p1],
[p1, p1, p1, p2]])
weights = np.array([[1/4], [1/4], [1/4], [1/4]])
elif rule == 8:
p1 = (55-3*np.sqrt(17)+np.sqrt(1022-134*np.sqrt(17)))/196
p2 = (55-3*np.sqrt(17)-np.sqrt(1022-134*np.sqrt(17)))/196
w1 = 1/8+np.sqrt((1715161837-406006699*np.sqrt(17))/23101)/3120
w2 = 1/8-np.sqrt((1715161837-406006699*np.sqrt(17))/23101)/3120
points = np.array([
[1-3*p1, p1, p1, p1], # 1
[p1, 1-3*p1, p1, p1], # 2
[p1, p1, 1-3*p1, p1], # 3
[p1, p1, p1, 1-3*p1], # 4
[1-3*p2, p2, p2, p2], # 5
[p2, 1-3*p2, p2, p2], # 6
[p2, p2, 1-3*p2, p2], # 7
[p2, p2, p2, 1-3*p2]]) # 8
weights = np.array([
[w1], [w1], [w1], [w1], [w2], [w2], [w2], [w2]])
elif rule == -8:
w1, w2 = 1/40, 9/40
points = np.array([
[1, 0, 0, 0], # 1
[0, 1, 0, 0], # 2
[0, 0, 1, 0], # 3
[0, 0, 0, 1], # 4
[0, 1, 1, 1], # 5
[1, 0, 1, 1], # 6
[1, 1, 0, 1], # 7
[1, 1, 1, 0]]) # 8
weights = np.array([
[w1], [w1], [w1], [w1], [w2], [w2], [w2], [w2]])
elif rule == 14:
p1 = 0
p2 = 0
p3 = 0
w1 = 1
w2 = 1
w3 = 1
points = np.array([
[1-3*p1, p1, p1, p1], # 1
[p1, 1-3*p1, p1, p1], # 2
[p1, p1, 1-3*p1, p1], # 3
[p1, p1, p1, 1-3*p1], # 4
[1-3*p2, p2, p2, p2], # 5
[p2, 1-3*p2, p2, p2], # 6
[p2, p2, 1-3*p2, p2], # 7
[p2, p2, p2, 1-3*p2], # 8
[0.5-p3, 0.5-p3, p3, p3], # 9
[0.5-p3, p3, 0.5-p3, p3], # 10
[0.5-p3, p3, p3, 0.5-p3], # 11
[p3, 0.5-p3, 0.5-p3, p3], # 12
[p3, 0.5-p3, p3, 0.5-p3], # 13
[p3, p3, 0.5-p3, 0.5-p3]]) # 14
weights = np.array([
[w1], [w1], [w1], [w1],
[w2], [w2], [w2], [w2],
[w3], [w3], [w3], [w3], [w3], [w3]])
elif rule == -14:
p1 = (243-51*np.sqrt(11)+2*np.sqrt(16486-9723*np.sqrt(11)/2))/356
p2 = (243-51*np.sqrt(11)-2*np.sqrt(16486-9723*np.sqrt(11)/2))/356
w1 = 31/280+np.sqrt((13686301-3809646*np.sqrt(11))/5965)/600
w2 = 31/280-np.sqrt((13686301-3809646*np.sqrt(11))/5965)/600
points = np.array([
[1-3*p1, p1, p1, p1], # 1
[p1, 1-3*p1, p1, p1], # 2
[p1, p1, 1-3*p1, p1], # 3
[p1, p1, p1, 1-3*p1], # 4
[1-3*p2, p2, p2, p2], # 5
[p2, 1-3*p2, p2, p2], # 6
[p2, p2, 1-3*p2, p2], # 7
[p2, p2, p2, 1-3*p2], # 8
[0.5, 0.5, 0, 0], # 9
[0.5, 0, 0.5, 0], # 10
[0.5, 0, 0, 0.5], # 11
[0, 0.5, 0.5, 0], # 12
[0, 0.5, 0, 0.5], # 13
[0, 0, 0.5, 0.5]]) # 14
weights = np.array([
[w1], [w1], [w1], [w1],
[w2], [w2], [w2], [w2],
[w3], [w3], [w3], [w3], [w3], [w3]])
elif rule == 15:
p1 = (7-np.sqrt(15))/34
p2 = 7/17-p1
p3 = (10-2*np.sqrt(15))/40
w1 = (2665+14*np.sqrt(15))/37800
w2 = (2665-14*np.sqrt(15))/37800
points = np.array([
[1-3*p1, p1, p1, p1], # 1
[p1, 1-3*p1, p1, p1], # 2
[p1, p1, 1-3*p1, p1], # 3
[p1, p1, p1, 1-3*p1], # 4
[1-3*p2, p2, p2, p2], # 5
[p2, 1-3*p2, p2, p2], # 6
[p2, p2, 1-3*p2, p2], # 7
[p2, p2, p2, 1-3*p2], # 8
[0.5-p3, 0.5-p3, p3, p3], # 9
[0.5-p3, p3, 0.5-p3, p3], # 10
[0.5-p3, p3, p3, 0.5-p3], # 11
[p3, 0.5-p3, 0.5-p3, p3], # 12
[p3, 0.5-p3, p3, 0.5-p3], # 13
[p3, p3, 0.5-p3, 0.5-p3], # 14
[0.25, 0.25, 0.25, 0.25]]) # 15
weights = np.array([
[w1], [w1], [w1], [w1],
[w2], [w2], [w2], [w2],
[w3], [w3], [w3], [w3], [w3], [w3], 16/135])
elif rule == -15:
p1 = (13-np.sqrt(91))/52
p2, p3 = 1/3, 1/11
w1 = 81/2240
w2 = 161051/2304960
w3 = 338/5145
points = np.array([
[0, p2, p2, p2], # 1
[p2, 0, p2, p2], # 2
[p2, p2, 0, p2], # 3
[p2, p2, p2, 0], # 4
[8/11, p3, p3, p3], # 5
[p3, 8/11, p3, p3], # 6
[p3, p3, 8/11, p3], # 7
[p3, p3, p3, 8/11], # 8
[0.5-p1, 0.5-p1, p1, p1], # 9
[0.5-p1, p1, 0.5-p1, p1], # 10
[0.5-p1, p1, p1, 0.5-p1], # 11
[p1, 0.5-p1, 0.5-p1, p1], # 12
[p1, 0.5-p1, p1, 0.5-p1], # 13
[p1, p1, 0.5-p1, 0.5-p1], # 14
[0.25, 0.25, 0.25, 0.25]]) # 15
weights = np.array([
[w1], [w1], [w1], [w1],
[w2], [w2], [w2], [w2],
[w3], [w3], [w3], [w3], [w3], [w3], 6544/36015])
elif rule == 24:
p1 = 0
p2 = 0
p3 = 0
p4=(3-np.sqrt(5))/12
p4j=(5+np.sqrt(5))/12
p4k=(1+np.sqrt(5))/12
w1 = 1
w2 = 1
w3 = 1
w4 = 27/560
points = np.array([
[1-3*p1, p1, p1, p1], # 1
[p1, 1-3*p1, p1, p1], # 2
[p1, p1, 1-3*p1, p1], # 3
[p1, p1, p1, 1-3*p1], # 4
[1-3*p2, p2, p2, p2], # 5
[p2, 1-3*p2, p2, p2], # 6
[p2, p2, 1-3*p2, p2], # 7
[p2, p2, p2, 1-3*p2], # 8
[1-3*p3, p3, p3, p3], # 9
[p3, 1-3*p3, p3, p3], # 10
[p3, p3, 1-3*p3, p3], # 11
[p3, p3, p3, 1-3*p3], # 12
[p4j, p4k, p4, p4], # 13
[p4j, p4, p4k, p4], # 14
[p4j, p4, p4, p4k], # 15
[p4, p4j, p4k, p4], # 16
[p4, p4j, p4, p4k], # 17
[p4, p4, p4j, p4k], # 18
[p4k, p4j, p4, p4], # 19
[p4k, p4, p4j, p4], # 20
[p4k, p4, p4, p4j], # 21
[p4, p4k, p4j, p4], # 22
[p4, p4k, p4, p4j], # 23
[p4, p4, p4k, p4j]]) # 24
weights = np.array([
[w1], [w1], [w1], [w1],
[w2], [w2], [w2], [w2],
[w3], [w3], [w3], [w3],
[w4], [w4], [w4], [w4],
[w4], [w4], [w4], [w4],
[w4], [w4], [w4], [w4]])
else:
raise TypeError('Invalid integration rule.')
return cls(points, weights)
@classmethod
def inWedge(cls, rule):
pass
@classmethod
def inPyramid(cls, rule):
pass
@classmethod
def inHexahedron(cls, rule):
"""
Parameters
----------
rule: {1, 2, 3, 4, 5}
The integration rule, as described in the table below.
-----------------------------------------------------------------
Rule Points Degree Comments
-----------------------------------------------------------------
1 1 1 Used in reduced and selective integration.
2 8 3 Useful for Hex8 stiffness and mass.
3 27 5 Useful for Hex20 & 27 stiffness and mass.
4 64 7 Rarely used.
5 125 9 rarely used.
-----------------------------------------------------------------
"""
if rule == 1:
points = np.array([[0, 0, 0]])
weights = np.array([[2, 2, 2]])
elif rule == 2:
p = np.sqrt(3)/3
points = np.array([
[-p, -p, -p], # 1
[+p, -p, -p], # 2
[+p, +p, -p], # 3
[-p, +p, -p], # 4
[-p, -p, +p], # 5
[+p, -p, +p], # 6
[+p, +p, +p], # 7
[-p, +p, +p]]) # 8
weights = np.ones((8, 3))
elif rule == 3:
p = np.sqrt(3/5)
points = np.array([
[ 0, 0, -p], # 1
[-p, -p, -p], # 2
[ 0, -p, -p], # 3
[+p, -p, -p], # 4
[+p, 0, -p], # 5
[+p, +p, -p], # 6
[ 0, +p, -p], # 7
[-p, +p, -p], # 8
[-p, 0, -p], # 9
[ 0, 0, 0], # 10
[-p, -p, 0], # 11
[ 0, -p, 0], # 12
[+p, -p, 0], # 13
[+p, 0, 0], # 14
[+p, +p, 0], # 15
[ 0, +p, 0], # 16
[-p, +p, 0], # 17
[-p, 0, 0], # 18
[ 0, 0, +p], # 19
[-p, -p, +p], # 20
[ 0, -p, +p], # 21
[+p, -p, +p], # 22
[+p, 0, +p], # 23
[+p, +p, +p], # 24
[ 0, +p, +p], # 25
[-p, +p, +p], # 26
[-p, 0, +p]]) # 27
w0 = 8/9
w1 = 5/9
weights = np.array([
[w0, w0, w1], # 1
[w1, w1, w1], # 2
[w0, w1, w1], # 3
[w1, w1, w1], # 4
[w1, w0, w1], # 5
[w1, w1, w1], # 6
[w0, w1, w1], # 7
[w1, w1, w1], # 8
[w1, w0, w1], # 9
[w0, w0, w0], # 10
[w1, w1, w0], # 11
[w0, w1, w0], # 12
[w1, w1, w0], # 13
[w1, w0, w0], # 14
[w1, w1, w0], # 15
[w0, w1, w0], # 16
[w1, w1, w0], # 17
[w1, w0, w0], # 18
[w0, w0, w1], # 19
[w1, w1, w1], # 20
[w0, w1, w1], # 21
[w1, w1, w1], # 22
[w1, w0, w1], # 23
[w1, w1, w1], # 24
[w0, w1, w1], # 25
[w1, w1, w1], # 26
[w1, w0, w1]]) # 27
elif rule == 4:
p1 = np.sqrt((3+2*np.sqrt(6/5))/7)
p2 = np.sqrt((3-2*np.sqrt(6/5))/7)
points = np.array([
[-p1, -p1, -p1], # 1
[-p2, -p1, -p1], # 2
[+p2, -p1, -p1], # 3
[+p1, -p1, -p1], # 4
[+p1, -p2, -p1], # 5
[+p1, +p2, -p1], # 6
[+p1, +p1, -p1], # 7
[+p2, +p1, -p1], # 8
[-p2, +p1, -p1], # 9
[-p1, +p1, -p1], # 10
[-p1, +p2, -p1], # 11
[-p1, -p2, -p1], # 12
[-p2, -p2, -p1], # 13
[+p2, -p2, -p1], # 14
[+p2, +p2, -p1], # 15
[-p2, +p2, -p1], # 16
[-p1, -p1, -p2], # 17
[-p2, -p1, -p2], # 18
[+p2, -p1, -p2], # 19
[+p1, -p1, -p2], # 20
[+p1, -p2, -p2], # 21
[+p1, +p2, -p2], # 22
[+p1, +p1, -p2], # 23
[+p2, +p1, -p2], # 24
[-p2, +p1, -p2], # 25
[-p1, +p1, -p2], # 26
[-p1, +p2, -p2], # 27
[-p1, -p2, -p2], # 28
[-p2, -p2, -p2], # 29
[+p2, -p2, -p2], # 30
[+p2, +p2, -p2], # 31
[-p2, +p2, -p2], # 32
[-p1, -p1, +p2], # 33
[-p2, -p1, +p2], # 34
[+p2, -p1, +p2], # 35
[+p1, -p1, +p2], # 36
[+p1, -p2, +p2], # 37
[+p1, +p2, +p2], # 38
[+p1, +p1, +p2], # 39
[+p2, +p1, +p2], # 40
[-p2, +p1, +p2], # 41
[-p1, +p1, +p2], # 42
[-p1, +p2, +p2], # 43
[-p1, -p2, +p2], # 44
[-p2, -p2, +p2], # 45
[+p2, -p2, +p2], # 46
[+p2, +p2, +p2], # 47
[-p2, +p2, +p2], # 48
[-p1, -p1, +p1], # 49
[-p2, -p1, +p1], # 50
[+p2, -p1, +p1], # 51
[+p1, -p1, +p1], # 52
[+p1, -p2, +p1], # 53
[+p1, +p2, +p1], # 54
[+p1, +p1, +p1], # 55
[+p2, +p1, +p1], # 56
[-p2, +p1, +p1], # 57
[-p1, +p1, +p1], # 58
[-p1, +p2, +p1], # 59
[-p1, -p2, +p1], # 60
[-p2, -p2, +p1], # 61
[+p2, -p2, +p1], # 62
[+p2, +p2, +p1], # 63
[-p2, +p2, +p1]]) # 64
w1 = (18-np.sqrt(30))/36
w2 = (18+np.sqrt(30))/36
weights = np.array([
[w1, w1, w1], # 1
[w2, w1, w1], # 2
[w2, w1, w1], # 3
[w1, w1, w1], # 4
[w1, w2, w1], # 5
[w1, w2, w1], # 6
[w1, w1, w1], # 7
[w2, w1, w1], # 8
[w2, w1, w1], # 9
[w1, w1, w1], # 10
[w1, w2, w1], # 11
[w1, w2, w1], # 12
[w2, w2, w1], # 13
[w2, w2, w1], # 14
[w2, w2, w1], # 15
[w2, w2, w1], # 16
[w1, w1, w2], # 17
[w2, w1, w2], # 18
[w2, w1, w2], # 19
[w1, w1, w2], # 20
[w1, w2, w2], # 21
[w1, w2, w2], # 22
[w1, w1, w2], # 23
[w2, w1, w2], # 24
[w2, w1, w2], # 25
[w1, w1, w2], # 26
[w1, w2, w2], # 27
[w1, w2, w2], # 28
[w2, w2, w2], # 29
[w2, w2, w2], # 30
[w2, w2, w2], # 31
[w2, w2, w2], # 32
[w1, w1, w2], # 33
[w2, w1, w2], # 34
[w2, w1, w2], # 35
[w1, w1, w2], # 36
[w1, w2, w2], # 37
[w1, w2, w2], # 38
[w1, w1, w2], # 39
[w2, w1, w2], # 40
[w2, w1, w2], # 41
[w1, w1, w2], # 42
[w1, w2, w2], # 43
[w1, w2, w2], # 44
[w2, w2, w2], # 45
[w2, w2, w2], # 46
[w2, w2, w2], # 47
[w2, w2, w2], # 48
[w1, w1, w1], # 49
[w2, w1, w1], # 50
[w2, w1, w1], # 51
[w1, w1, w1], # 52
[w1, w2, w1], # 53
[w1, w2, w1], # 54
[w1, w1, w1], # 55
[w2, w1, w1], # 56
[w2, w1, w1], # 57
[w1, w1, w1], # 58
[w1, w2, w1], # 59
[w1, w2, w1], # 60
[w2, w2, w1], # 61
[w2, w2, w1], # 62
[w2, w2, w1], # 63
[w2, w2, w1]]) # 64
elif rule == 5:
p1 = np.sqrt(5+2*np.sqrt(10/7))/3
p2 = np.sqrt(5-2*np.sqrt(10/7))/3
points = np.array([
[-p1, -p1, -p1], # 1
[-p2, -p1, -p1], # 2
[ 0, -p1, -p1], # 3
[+p2, -p1, -p1], # 4
[+p1, -p1, -p1], # 5
[-p1, -p2, -p1], # 6
[-p2, -p2, -p1], # 7
[ 0, -p2, -p1], # 8
[+p2, -p2, -p1], # 9
[+p1, -p2, -p1], # 10
[-p1, 0, -p1], # 11
[-p2, 0, -p1], # 12
[ 0, 0, -p1], # 13
[+p2, 0, -p1], # 14
[+p1, 0, -p1], # 15
[-p1, +p2, -p1], # 16
[-p2, +p2, -p1], # 17
[ 0, +p2, -p1], # 18
[+p2, +p2, -p1], # 19
[+p1, +p2, -p1], # 20
[-p1, +p1, -p1], # 21
[-p2, +p1, -p1], # 22
[ 0, +p1, -p1], # 23
[+p2, +p1, -p1], # 24
[+p1, +p1, -p1], # 25
[-p1, -p1, -p2], # 26
[-p2, -p1, -p2], # 27
[ 0, -p1, -p2], # 28
[+p2, -p1, -p2], # 29
[+p1, -p1, -p2], # 30
[-p1, -p2, -p2], # 31
[-p2, -p2, -p2], # 32
[ 0, -p2, -p2], # 33
[+p2, -p2, -p2], # 34
[+p1, -p2, -p2], # 35
[-p1, 0, -p2], # 36
[-p2, 0, -p2], # 37
[ 0, 0, -p2], # 38
[+p2, 0, -p2], # 39
[+p1, 0, -p2], # 40
[-p1, p2, -p2], # 41
[-p2, p2, -p2], # 42
[ 0, p2, -p2], # 43
[+p2, p2, -p2], # 44
[+p1, p2, -p2], # 45
[-p1, p1, -p2], # 46
[-p2, p1, -p2], # 47
[ 0, p1, -p2], # 48
[+p2, p1, -p2], # 49
[+p1, p1, -p2], # 50
[-p1, -p1, 0], # 51
[-p2, -p1, 0], # 52
[ 0, -p1, 0], # 53
[+p2, -p1, 0], # 54
[+p1, -p1, 0], # 55
[-p1, -p2, 0], # 56
[-p2, -p2, 0], # 57
[ 0, -p2, 0], # 58
[+p2, -p2, 0], # 59
[+p1, -p2, 0], # 60
[-p1, 0, 0], # 61
[-p2, 0, 0], # 62
[ 0, 0, 0], # 63
[+p2, 0, 0], # 64
[+p1, 0, 0], # 65
[-p1, +p2, 0], # 66
[-p2, +p2, 0], # 67
[ 0, +p2, 0], # 68
[+p2, +p2, 0], # 69
[+p1, +p2, 0], # 70
[-p1, +p1, 0], # 71
[-p2, +p1, 0], # 72
[ 0, +p1, 0], # 73
[+p2, +p1, 0], # 74
[+p1, +p1, 0], # 75
[-p1, -p1, +p2], # 76
[-p2, -p1, +p2], # 77
[ 0, -p1, +p2], # 78
[+p2, -p1, +p2], # 79
[+p1, -p1, +p2], # 80
[-p1, -p2, +p2], # 81
[-p2, -p2, +p2], # 82
[ 0, -p2, +p2], # 83
[+p2, -p2, +p2], # 84
[+p1, -p2, +p2], # 85
[-p1, 0, +p2], # 86
[-p2, 0, +p2], # 87
[ 0, 0, +p2], # 88
[+p2, 0, +p2], # 89
[+p1, 0, +p2], # 90
[-p1, +p2, +p2], # 91
[-p2, +p2, +p2], # 92
[ 0, +p2, +p2], # 93
[+p2, +p2, +p2], # 94
[+p1, +p2, +p2], # 95
[-p1, +p1, +p2], # 96
[-p2, +p1, +p2], # 97
[ 0, +p1, +p2], # 98
[+p2, +p1, +p2], # 99
[+p1, +p1, +p2], # 100
[-p1, -p1, +p1], # 101
[-p2, -p1, +p1], # 102
[ 0, -p1, +p1], # 103
[+p2, -p1, +p1], # 104
[+p1, -p1, +p1], # 105
[-p1, -p2, +p1], # 106
[-p2, -p2, +p1], # 107
[ 0, -p2, +p1], # 108
[+p2, -p2, +p1], # 109
[+p1, -p2, +p1], # 110
[-p1, 0, +p1], # 111
[-p2, 0, +p1], # 112
[ 0, 0, +p1], # 113
[+p2, 0, +p1], # 114
[+p1, 0, +p1], # 115
[-p1, +p2, +p1], # 116
[-p2, +p2, +p1], # 117
[ 0, +p2, +p1], # 118
[+p2, +p2, +p1], # 119
[+p1, +p2, +p1], # 120
[-p1, +p1, +p1], # 121
[-p2, +p1, +p1], # 122
[ 0, +p1, +p1], # 123
[+p2, +p1, +p1], # 124
[+p1, +p1, +p1]]) # 125
w0 = 128/225
w1 = (322-13*np.sqrt(70))/900
w2 = (322+13*np.sqrt(70))/900
weights = np.array([
[w1, w1, w1], # 1
[w2, w1, w1], # 2
[w0, w1, w1], # 3
[w2, w1, w1], # 4
[w1, w1, w1], # 5
[w1, w2, w1], # 6
[w2, w2, w1], # 7
[w0, w2, w1], # 8
[w2, w2, w1], # 9
[w1, w2, w1], # 10
[w1, w0, w1], # 11
[w2, w0, w1], # 12
[w0, w0, w1], # 13
[w2, w0, w1], # 14
[w1, w0, w1], # 15
[w1, w2, w1], # 16
[w2, w2, w1], # 17
[w0, w2, w1], # 18
[w2, w2, w1], # 19
[w1, w2, w1], # 20
[w1, w1, w1], # 21
[w2, w1, w1], # 22
[w0, w1, w1], # 23
[w2, w1, w1], # 24
[w1, w1, w1], # 25
[w1, w1, w2], # 26
[w2, w1, w2], # 27
[w0, w1, w2], # 28
[w2, w1, w2], # 29
[w1, w1, w2], # 30
[w1, w2, w2], # 31
[w2, w2, w2], # 32
[w0, w2, w2], # 33
[w2, w2, w2], # 34
[w1, w2, w2], # 35
[w1, w0, w2], # 36
[w2, w0, w2], # 37
[w0, w0, w2], # 38
[w2, w0, w2], # 39
[w1, w0, w2], # 40
[w1, w2, w2], # 41
[w2, w2, w2], # 42
[w0, w2, w2], # 43
[w2, w2, w2], # 44
[w1, w2, w2], # 45
[w1, w1, w2], # 46
[w2, w1, w2], # 47
[w0, w1, w2], # 48
[w2, w1, w2], # 49
[w1, w1, w2], # 50
[w1, w1, w0], # 51
[w2, w1, w0], # 52
[w0, w1, w0], # 53
[w2, w1, w0], # 54
[w1, w1, w0], # 55
[w1, w2, w0], # 56
[w2, w2, w0], # 57
[w0, w2, w0], # 58
[w2, w2, w0], # 59
[w1, w2, w0], # 60
[w1, w0, w0], # 61
[w2, w0, w0], # 62
[w0, w0, w0], # 63
[w2, w0, w0], # 64
[w1, w0, w0], # 65
[w1, w2, w0], # 66