# -*- 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 ConstantQuad(Element):
r"""Quadrilateral element with constant shape functions.
.. code-block::
^ s
3 (-1/ 1) | 2 ( 1/ 1)
o-----------|-----------o
| | |
| | |
| | |
| | |
| -----|-----------|-----> r
| | |
| | |
| |
| |
o-----------------------o
0 (-1,-1) 1 ( 1/-1)
Attributes
----------
points : ndarray
Array with point locations in natural coordinate system
"""
def __init__(self):
super().__init__(shape=(1, 2))
self.points = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]], dtype=float)
[docs] def function(self, rst):
r"""Constant quadrilateral - shape functions.
.. math::
\boldsymbol{h}(\boldsymbol{r}) = \begin{bmatrix}
1
\end{bmatrix}
Arguments
---------
rs : ndarray
Point as coordinate vector for shape function evaluation
Returns
-------
ndarray
Shape functions evaluated at given location
"""
return np.array([1])
[docs] def gradient(self, rst):
r"""Constant quadrilateral - gradient of shape functions.
.. math::
\frac{\partial \boldsymbol{h}}{\partial \boldsymbol{r}} =
\begin{bmatrix}
0
\end{bmatrix}
Arguments
---------
rs : ndarray
Point as coordinate vector for gradient of shape function evaluation
Returns
-------
ndarray
Gradient of shape functions evaluated at given location
"""
return np.array([[0, 0]])
[docs]class Quad(Element):
r"""Quadrilateral element with linear shape functions.
.. code-block::
^ s
3 (-1/ 1) | 2 ( 1/ 1)
o-----------|-----------o
| | |
| | |
| | |
| | |
| -----|-----------|-----> r
| | |
| | |
| |
| |
o-----------------------o
0 (-1,-1) 1 ( 1/-1)
Attributes
----------
points : ndarray
Array with point locations in natural coordinate system
"""
def __init__(self):
super().__init__(shape=(4, 2))
self.points = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]], dtype=float)
[docs] def function(self, rs):
r"""Linear quadrilateral - shape functions.
.. math::
\boldsymbol{h}(\boldsymbol{r}) = \frac{1}{4} \begin{bmatrix}
(1-r)(1-s) \\ (1+r)(1-s) \\ (1+r)(1+s) \\ (1-r)(1+s)
\end{bmatrix}
Arguments
---------
rs : ndarray
Point as coordinate vector for shape function evaluation
Returns
-------
ndarray
Shape functions evaluated at given location
"""
r, s = rs
return (
np.array(
[
(1 - r) * (1 - s),
(1 + r) * (1 - s),
(1 + r) * (1 + s),
(1 - r) * (1 + s),
]
)
* 0.25
)
[docs] def gradient(self, rs):
r"""Linear quadrilateral - gradient of shape functions.
.. math::
\frac{\partial \boldsymbol{h}}{\partial \boldsymbol{r}} =
\frac{1}{4} \begin{bmatrix}
-(1-s) & -(1-r) \\
(1-s) & -(1+r) \\
(1+s) & (1+r) \\
-(1+s) & (1-r)
\end{bmatrix}
Arguments
---------
rs : ndarray
Point as coordinate vector for gradient of shape function evaluation
Returns
-------
ndarray
Gradient of shape functions evaluated at given location
"""
r, s = rs
return (
np.array(
[
[-(1 - s), -(1 - r)],
[(1 - s), -(1 + r)],
[(1 + s), (1 + r)],
[-(1 + s), (1 - r)],
]
)
* 0.25
)