# -*- 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 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
)
class QuadraticQuad(Element):
r"""Quadratic serendipity quadrilateral element.
.. code-block::
^ s
3 (-1/ 1) |6 ( 0/ 1) 2 ( 1/ 1)
o-----------o-----------o
| | |
| | |
| | |
|7 (-1/ 0) | |5 ( 1/ 0)
o -----|-----------o-----> r
| | |
| | |
| |
| |
o-----------o-----------o
0 (-1/-1) 4 ( 0/-1) 1 ( 1/-1)
Attributes
----------
points : ndarray
Array with point locations in natural coordinate system
"""
def __init__(self):
super().__init__(shape=(8, 2))
self.points = np.array(
[
[-1, -1],
[1, -1],
[1, 1],
[-1, 1],
[0, -1],
[1, 0],
[0, 1],
[-1, 0],
],
dtype=float,
)
def function(self, rs):
r"""Quadratic serendipity quadrilateral - shape functions."""
r, s = rs
ra, sa = self.points.T
h = (1 + ra * r) * (1 + sa * s) * (ra * r + sa * s - 1) / 4
h[ra == 0] = (1 - r**2) * (1 + sa[ra == 0] * s) / 2
h[sa == 0] = (1 + ra[sa == 0] * r) * (1 - s**2) / 2
return h
def gradient(self, rs):
r"""Quadratic serendipity quadrilateral - gradient of shape functions.
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
ra, sa = self.points.T
dhdr = (
ra * (1 + sa * s) * (ra * r + sa * s - 1) / 4
+ (1 + ra * r) * (1 + sa * s) * ra / 4
)
dhdr[ra == 0] = -2 * r * (1 + sa[ra == 0] * s) / 2
dhdr[sa == 0] = ra[sa == 0] * (1 - s**2) / 2
dhds = (1 + ra * r) * sa * (ra * r + sa * s - 1) / 4 + (1 + ra * r) * (
1 + sa * s
) * sa / 4
dhds[ra == 0] = (1 - r**2) * sa[ra == 0] / 2
dhds[sa == 0] = (1 + ra[sa == 0] * r) * -2 * s / 2
return np.vstack([dhdr, dhds]).T
class BiQuadraticQuad(Element):
r"""Bi-Quadratic Lagrange quadrilateral element.
.. code-block::
^ s
3 (-1/ 1) |6 ( 0/ 1) 2 ( 1/ 1)
o-----------o-----------o
| | |
| | |
| | |
|7 (-1/ 0) |8 ( 0/ 0) |5 ( 1/ 0)
o -----o-----------o-----> r
| | |
| | |
| |
| |
o-----------o-----------o
0 (-1/-1) 4 ( 0/-1) 1 ( 1/-1)
Attributes
----------
points : ndarray
Array with point locations in natural coordinate system
"""
def __init__(self):
super().__init__(shape=(9, 2))
self._lagrange = ArbitraryOrderLagrange(order=2, dim=2)
self._vertices = np.array([0, 2, 8, 6])
self._edges = np.array([1, 5, 7, 3])
self._faces = np.array([4])
self._volume = np.array([], dtype=int)
self._permute = np.concatenate(
(self._vertices, self._edges, self._faces, self._volume)
)
self.points = self._lagrange.points[self._permute]
def function(self, rs):
r"""Bi-Quadratic Lagrange quadrilateral - shape functions."""
return self._lagrange.function(rs)[self._permute]
def gradient(self, rs):
r"""Bi-Quadratic Lagrange quadrilateral - gradient of shape functions.
Arguments
---------
rs : ndarray
Point as coordinate vector for gradient of shape function evaluation
Returns
-------
ndarray
Gradient of shape functions evaluated at given location
"""
return self._lagrange.gradient(rs)[self._permute, :]