# -*- 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
[docs]class Tetra(Element):
def __init__(self):
super().__init__(shape=(4, 3))
self.points = np.array(
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float
)
[docs] def function(self, rst):
"linear tetrahedral shape functions"
r, s, t = rst
return np.array([1 - r - s - t, r, s, t])
[docs] def gradient(self, rst):
"linear tetrahedral gradient of shape functions"
r, s, t = rst
return np.array([[-1, -1, -1], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float)
[docs]class TetraMINI(Element):
def __init__(self, bubble_multiplier=1.0):
super().__init__(shape=(5, 3))
self.points = np.array(
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1], [1 / 3, 1 / 3, 1 / 3]],
dtype=float,
)
self.bubble_multiplier = bubble_multiplier
[docs] def function(self, rst):
"linear bubble-enriched tetrahedral basis functions"
r, s, t = rst
a = self.bubble_multiplier
return np.array([1 - r - s - t, r, s, t, a * r * s * t * (1 - r - s - t)])
[docs] def gradient(self, rst):
"linear bubble-enriched tetrahedral derivative of basis functions"
r, s, t = rst
a = self.bubble_multiplier
return np.array(
[
[-1, -1, -1],
[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
[
a * (s * t * (1 - r - s - t) - r * s * t),
a * (r * t * (1 - r - s - t) - r * s * t),
a * (r * s * (1 - r - s - t) - r * s * t),
],
],
dtype=float,
)
[docs]class QuadraticTetra(Element):
def __init__(self):
super().__init__(shape=(10, 3))
self.points = np.zeros(self.shape)
self.points[:4] = np.array(
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float
)
self.points[4] = np.mean(self.points[[0, 1]], axis=0)
self.points[5] = np.mean(self.points[[1, 2]], axis=0)
self.points[6] = np.mean(self.points[[2, 0]], axis=0)
self.points[7] = np.mean(self.points[[0, 3]], axis=0)
self.points[8] = np.mean(self.points[[1, 3]], axis=0)
self.points[9] = np.mean(self.points[[2, 3]], axis=0)
[docs] def function(self, rst):
"quadratic tetrahedral shape functions"
r, s, t = rst
t1 = 1 - r - s - t
t2 = r
t3 = s
t4 = t
h = np.array(
[
t1 * (2 * t1 - 1),
t2 * (2 * t2 - 1),
t3 * (2 * t3 - 1),
t4 * (2 * t4 - 1),
4 * t1 * t2,
4 * t2 * t3,
4 * t3 * t1,
4 * t1 * t4,
4 * t2 * t4,
4 * t3 * t4,
]
)
return h
[docs] def gradient(self, rst):
"quadratic tetrahedral gradient of shape functions"
r, s, t = rst
t1 = 1 - r - s - t
t2 = r
t3 = s
t4 = t
dhdt = np.array(
[
[4 * t1 - 1, 0, 0, 0],
[0, 4 * t2 - 1, 0, 0],
[0, 0, 4 * t3 - 1, 0],
[0, 0, 0, 4 * t4 - 1],
[4 * t2, 4 * t1, 0, 0],
[0, 4 * t3, 4 * t2, 0],
[4 * t3, 0, 4 * t1, 0],
[4 * t4, 0, 0, 4 * t1],
[0, 4 * t4, 0, 4 * t2],
[0, 0, 4 * t4, 4 * t3],
]
)
dtdr = np.array([[-1, -1, -1], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float)
return np.dot(dhdt, dtdr)