# -*- 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 Triangle(Element):
def __init__(self):
super().__init__(shape=(3, 2))
self.points = np.array([[0, 0], [1, 0], [0, 1]], dtype=float)
[docs] def function(self, rs):
"linear triangle shape functions"
r, s = rs
return np.array([1 - r - s, r, s])
[docs] def gradient(self, rs):
"linear triangle gradient of shape functions"
r, s = rs
return np.array([[-1, -1], [1, 0], [0, 1]], dtype=float)
[docs]class TriangleMINI(Element):
def __init__(self, bubble_multiplier=1.0):
super().__init__(shape=(4, 2))
self.points = np.array([[0, 0], [1, 0], [0, 1], [1 / 3, 1 / 3]], dtype=float)
self.bubble_multiplier = bubble_multiplier
[docs] def function(self, rs):
"linear bubble-enriched triangle shape functions"
r, s = rs
a = self.bubble_multiplier
return np.array([1 - r - s, r, s, a * r * s * (1 - r - s)])
[docs] def gradient(self, rs):
"linear bubble-enriched triangle gradient of shape functions"
r, s = rs
a = self.bubble_multiplier
return np.array(
[
[-1, -1],
[1, 0],
[0, 1],
[a * (s * (1 - r - s) - r * s), a * (r * (1 - r - s) - r * s)],
],
dtype=float,
)
[docs]class QuadraticTriangle(Element):
def __init__(self):
super().__init__(shape=(6, 2))
self.points = np.zeros(self.shape)
self.points[:3] = np.array([[0, 0], [1, 0], [0, 1]], dtype=float)
self.points[3] = np.mean(self.points[[0, 1]], axis=0)
self.points[4] = np.mean(self.points[[1, 2]], axis=0)
self.points[5] = np.mean(self.points[[2, 0]], axis=0)
[docs] def function(self, rs):
"quadratic triangle shape functions"
r, s = rs
h = np.array(
[1 - r - s, r, s, 4 * r * (1 - r - s), 4 * r * s, 4 * s * (1 - r - s)]
)
h[0] += -h[3] / 2 - h[5] / 2
h[1] += -h[3] / 2 - h[4] / 2
h[2] += -h[4] / 2 - h[5] / 2
return h
[docs] def gradient(self, rs):
"quadratic triangle gradient of shape functions"
r, s = rs
t1 = 1 - r - s
t2 = r
t3 = s
dhdr_a = np.array([[-1, -1], [1, 0], [0, 1]], dtype=float)
dhdr_b = np.array(
[
[4 * (t1 - t2), -4 * t2],
[4 * t3, 4 * t2],
[-4 * t3, 4 * (t1 - t2)],
]
)
dhdr = np.vstack((dhdr_a, dhdr_b))
dhdr[0] += -dhdr[3] / 2 - dhdr[5] / 2
dhdr[1] += -dhdr[3] / 2 - dhdr[4] / 2
dhdr[2] += -dhdr[4] / 2 - dhdr[5] / 2
return dhdr