# -*- coding: utf-8 -*-
"""
_______ _______ ___ __ __ _______ _______
| || || | | | | || || |
| ___|| ___|| | | | | || _ || ___|
| |___ | |___ | | | |_| || |_| || |___
| ___|| ___|| |___ | || ___|| ___|
| | | |___ | || || | | |___
|___| |_______||_______||_______||___| |_______|
This file is part of felupe.
Felupe is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
Felupe 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 General Public License for more details.
You should have received a copy of the GNU General Public License
along with Felupe. If not, see <http://www.gnu.org/licenses/>.
"""
import numpy as np
from ._base import Element
from ._lagrange import ArbitraryOrderLagrange
[docs]class ConstantHexahedron(Element):
def __init__(self):
self.points = np.array(
[
[-1, -1, -1],
[1, -1, -1],
[1, 1, -1],
[-1, 1, -1],
[-1, -1, 1],
[1, -1, 1],
[1, 1, 1],
[-1, 1, 1],
],
dtype=float,
)
super().__init__(shape=(1, 3))
[docs] def function(self, rst):
"constant hexahedron shape functions"
return np.array([1])
[docs] def gradient(self, rst):
"constant hexahedron gradient of shape functions"
return np.array([[0, 0, 0]])
[docs]class Hexahedron(Element):
def __init__(self):
self.points = np.array(
[
[-1, -1, -1],
[1, -1, -1],
[1, 1, -1],
[-1, 1, -1],
[-1, -1, 1],
[1, -1, 1],
[1, 1, 1],
[-1, 1, 1],
],
dtype=float,
)
super().__init__(shape=(8, 3))
[docs] def function(self, rst):
"linear hexahedron shape functions"
r, s, t = rst
return (
np.array(
[
(1 - r) * (1 - s) * (1 - t),
(1 + r) * (1 - s) * (1 - t),
(1 + r) * (1 + s) * (1 - t),
(1 - r) * (1 + s) * (1 - t),
(1 - r) * (1 - s) * (1 + t),
(1 + r) * (1 - s) * (1 + t),
(1 + r) * (1 + s) * (1 + t),
(1 - r) * (1 + s) * (1 + t),
]
)
* 0.125
)
[docs] def gradient(self, rst):
"linear hexahedron gradient of shape functions"
r, s, t = rst
return (
np.array(
[
[-(1 - s) * (1 - t), -(1 - r) * (1 - t), -(1 - r) * (1 - s)],
[(1 - s) * (1 - t), -(1 + r) * (1 - t), -(1 + r) * (1 - s)],
[(1 + s) * (1 - t), (1 + r) * (1 - t), -(1 + r) * (1 + s)],
[-(1 + s) * (1 - t), (1 - r) * (1 - t), -(1 - r) * (1 + s)],
[-(1 - s) * (1 + t), -(1 - r) * (1 + t), (1 - r) * (1 - s)],
[(1 - s) * (1 + t), -(1 + r) * (1 + t), (1 + r) * (1 - s)],
[(1 + s) * (1 + t), (1 + r) * (1 + t), (1 + r) * (1 + s)],
[-(1 + s) * (1 + t), (1 - r) * (1 + t), (1 - r) * (1 + s)],
]
)
* 0.125
)
[docs]class QuadraticHexahedron(Element):
def __init__(self):
self.points = np.array(
[
[-1, -1, -1],
[1, -1, -1],
[1, 1, -1],
[-1, 1, -1],
#
[-1, -1, 1],
[1, -1, 1],
[1, 1, 1],
[-1, 1, 1],
#
[0, -1, -1],
[1, 0, -1],
[0, 1, -1],
[-1, 0, -1],
#
[0, -1, 1],
[1, 0, 1],
[0, 1, 1],
[-1, 0, 1],
#
[-1, -1, 0],
[1, -1, 0],
[1, 1, 0],
[-1, 1, 0],
],
dtype=float,
)
super().__init__(shape=(20, 3))
[docs] def function(self, rst):
"quadratic serendipity hexahedron shape functions"
r, s, t = rst
return np.array(
[
-(1 - r) * (1 - s) * (1 - t) * (2 + r + s + t) * 0.125,
-(1 + r) * (1 - s) * (1 - t) * (2 - r + s + t) * 0.125,
-(1 + r) * (1 + s) * (1 - t) * (2 - r - s + t) * 0.125,
-(1 - r) * (1 + s) * (1 - t) * (2 + r - s + t) * 0.125,
#
-(1 - r) * (1 - s) * (1 + t) * (2 + r + s - t) * 0.125,
-(1 + r) * (1 - s) * (1 + t) * (2 - r + s - t) * 0.125,
-(1 + r) * (1 + s) * (1 + t) * (2 - r - s - t) * 0.125,
-(1 - r) * (1 + s) * (1 + t) * (2 + r - s - t) * 0.125,
#
(1 - r) * (1 + r) * (1 - s) * (1 - t) * 0.25,
(1 - s) * (1 + s) * (1 + r) * (1 - t) * 0.25,
(1 - r) * (1 + r) * (1 + s) * (1 - t) * 0.25,
(1 - s) * (1 + s) * (1 - r) * (1 - t) * 0.25,
#
(1 - r) * (1 + r) * (1 - s) * (1 + t) * 0.25,
(1 - s) * (1 + s) * (1 + r) * (1 + t) * 0.25,
(1 - r) * (1 + r) * (1 + s) * (1 + t) * 0.25,
(1 - s) * (1 + s) * (1 - r) * (1 + t) * 0.25,
#
(1 - t) * (1 + t) * (1 - r) * (1 - s) * 0.25,
(1 - t) * (1 + t) * (1 + r) * (1 - s) * 0.25,
(1 - t) * (1 + t) * (1 + r) * (1 + s) * 0.25,
(1 - t) * (1 + t) * (1 - r) * (1 + s) * 0.25,
]
)
[docs] def gradient(self, rst):
"quadratic serendipity hexahedron gradient of shape functions"
r, s, t = rst
return np.array(
[
[
(1 - s) * (1 - t) * (2 + r + s + t) * 0.125
- (1 - r) * (1 - s) * (1 - t) * 0.125,
(1 - r) * (1 - t) * (2 + r + s + t) * 0.125
- (1 - r) * (1 - s) * (1 - t) * 0.125,
(1 - r) * (1 - s) * (2 + r + s + t) * 0.125
- (1 - r) * (1 - s) * (1 - t) * 0.125,
],
[
-(1 - s) * (1 - t) * (2 - r + s + t) * 0.125
+ (1 + r) * (1 - s) * (1 - t) * 0.125,
(1 + r) * (1 - t) * (2 - r + s + t) * 0.125
- (1 + r) * (1 - s) * (1 - t) * 0.125,
(1 + r) * (1 - s) * (2 - r + s + t) * 0.125
- (1 + r) * (1 - s) * (1 - t) * 0.125,
],
[
-(1 + s) * (1 - t) * (2 - r - s + t) * 0.125
+ (1 + r) * (1 + s) * (1 - t) * 0.125,
-(1 + r) * (1 - t) * (2 - r - s + t) * 0.125
+ (1 + r) * (1 + s) * (1 - t) * 0.125,
(1 + r) * (1 + s) * (2 - r - s + t) * 0.125
- (1 + r) * (1 + s) * (1 - t) * 0.125,
],
[
(1 + s) * (1 - t) * (2 + r - s + t) * 0.125
- (1 - r) * (1 + s) * (1 - t) * 0.125,
-(1 - r) * (1 - t) * (2 + r - s + t) * 0.125
+ (1 - r) * (1 + s) * (1 - t) * 0.125,
(1 - r) * (1 + s) * (2 + r - s + t) * 0.125
- (1 - r) * (1 + s) * (1 - t) * 0.125,
],
#
[
(1 - s) * (1 + t) * (2 + r + s - t) * 0.125
- (1 - r) * (1 - s) * (1 + t) * 0.125,
(1 - r) * (1 + t) * (2 + r + s - t) * 0.125
- (1 - r) * (1 - s) * (1 + t) * 0.125,
-(1 - r) * (1 - s) * (2 + r + s - t) * 0.125
+ (1 - r) * (1 - s) * (1 + t) * 0.125,
],
[
-(1 - s) * (1 + t) * (2 - r + s - t) * 0.125
+ (1 + r) * (1 - s) * (1 + t) * 0.125,
(1 + r) * (1 + t) * (2 - r + s - t) * 0.125
- (1 + r) * (1 - s) * (1 + t) * 0.125,
-(1 + r) * (1 - s) * (2 - r + s - t) * 0.125
+ (1 + r) * (1 - s) * (1 + t) * 0.125,
],
[
-(1 + s) * (1 + t) * (2 - r - s - t) * 0.125
+ (1 + r) * (1 + s) * (1 + t) * 0.125,
-(1 + r) * (1 + t) * (2 - r - s - t) * 0.125
+ (1 + r) * (1 + s) * (1 + t) * 0.125,
-(1 + r) * (1 + s) * (2 - r - s - t) * 0.125
+ (1 + r) * (1 + s) * (1 + t) * 0.125,
],
[
(1 + s) * (1 + t) * (2 + r - s - t) * 0.125
- (1 - r) * (1 + s) * (1 + t) * 0.125,
-(1 - r) * (1 + t) * (2 + r - s - t) * 0.125
+ (1 - r) * (1 + s) * (1 + t) * 0.125,
-(1 - r) * (1 + s) * (2 + r - s - t) * 0.125
+ (1 - r) * (1 + s) * (1 + t) * 0.125,
],
#
[
-2 * r * (1 - s) * (1 - t) * 0.25,
-(1 - r) * (1 + r) * (1 - t) * 0.25,
-(1 - r) * (1 + r) * (1 - s) * 0.25,
],
[
(1 - s) * (1 + s) * (1 - t) * 0.25,
-2 * s * (1 + r) * (1 - t) * 0.25,
-(1 - s) * (1 + s) * (1 + r) * 0.25,
],
[
-2 * r * (1 + s) * (1 - t) * 0.25,
(1 - r) * (1 + r) * (1 - t) * 0.25,
-(1 - r) * (1 + r) * (1 + s) * 0.25,
],
[
-(1 - s) * (1 + s) * (1 - t) * 0.25,
-2 * s * (1 - r) * (1 - t) * 0.25,
-(1 - s) * (1 + s) * (1 - r) * 0.25,
],
#
[
-2 * r * (1 - s) * (1 + t) * 0.25,
-(1 - r) * (1 + r) * (1 + t) * 0.25,
(1 - r) * (1 + r) * (1 - s) * 0.25,
],
[
(1 - s) * (1 + s) * (1 + t) * 0.25,
-2 * s * (1 + r) * (1 + t) * 0.25,
(1 - s) * (1 + s) * (1 + r) * 0.25,
],
[
-2 * r * (1 + s) * (1 + t) * 0.25,
(1 - r) * (1 + r) * (1 + t) * 0.25,
(1 - r) * (1 + r) * (1 + s) * 0.25,
],
[
-(1 - s) * (1 + s) * (1 + t) * 0.25,
-2 * s * (1 - r) * (1 + t) * 0.25,
(1 - s) * (1 + s) * (1 - r) * 0.25,
],
#
[
-(1 - t) * (1 + t) * (1 - s) * 0.25,
-(1 - t) * (1 + t) * (1 - r) * 0.25,
-2 * t * (1 - r) * (1 - s) * 0.25,
],
[
(1 - t) * (1 + t) * (1 - s) * 0.25,
-(1 - t) * (1 + t) * (1 + r) * 0.25,
-2 * t * (1 + r) * (1 - s) * 0.25,
],
[
(1 - t) * (1 + t) * (1 + s) * 0.25,
(1 - t) * (1 + t) * (1 + r) * 0.25,
-2 * t * (1 + r) * (1 + s) * 0.25,
],
[
-(1 - t) * (1 + t) * (1 + s) * 0.25,
(1 - t) * (1 + t) * (1 - r) * 0.25,
-2 * t * (1 - r) * (1 + s) * 0.25,
],
]
)
[docs]class TriQuadraticHexahedron(Element):
def __init__(self):
super().__init__(shape=(27, 3))
self.points = np.array(
[
[-1, -1, -1],
[1, -1, -1],
[1, 1, -1],
[-1, 1, -1],
#
[-1, -1, 1],
[1, -1, 1],
[1, 1, 1],
[-1, 1, 1],
#
[0, -1, -1],
[1, 0, -1],
[0, 1, -1],
[-1, 0, -1],
#
[0, -1, 1],
[1, 0, 1],
[0, 1, 1],
[-1, 0, 1],
#
[-1, -1, 0],
[1, 1, 0],
[1, 1, 0],
[-1, 1, 0],
#
[0, -1, 0],
[1, 0, 0],
[0, 1, 0],
[-1, 0, 0],
#
[-1, 0, 0],
[1, 0, 0],
[0, -1, 0],
[0, 1, 0],
#
[0, 0, 0],
],
dtype=float,
)
self._lagrange = ArbitraryOrderLagrange(order=2, dim=3)
self._vertices = np.array([0, 2, 8, 6, 18, 20, 26, 24])
self._edges = np.array([1, 5, 7, 3, 19, 23, 25, 21, 9, 11, 17, 15])
self._faces = np.array([12, 14, 10, 16, 4, 22])
self._volume = np.array([13])
self._permute = np.concatenate(
(self._vertices, self._edges, self._faces, self._volume)
)
[docs] def function(self, rst):
"quadratic hexahedron shape functions"
return self._lagrange.function(rst)[self._permute]
[docs] def gradient(self, rst):
"quadratic hexahedron gradient of shape functions"
return self._lagrange.gradient(rst)[self._permute, :]