# -*- 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):
r"""A 3D tetrahedron element formulation with linear shape functions.
Notes
-----
The tetrahedron element is defined by four points (0-3). [1]
The shape functions :math:`\boldsymbol{h}` are given in terms of the coordinates
:math:`(r,s,t)`.
.. math::
\boldsymbol{h}(r,s,t) = \begin{bmatrix}
1-r-s-t \\
r \\
s \\
t
\end{bmatrix}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.Tetra()
>>> element.plot().show()
References
----------
.. [1] W. Schroeder, K. Martin and B. Lorensen. The Visualization
Toolkit, 4th ed. Kitware, 2006. ISBN: 978-1-930934-19-1.
"""
def __init__(self):
self.points = np.array(
[[0, 0, 0], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float
)
self.cells = np.arange(len(self.points)).reshape(1, -1)
self.cell_type = "tetra"
[docs]
def function(self, rst):
"Return the shape functions at given coordinates (r, s, t)."
r, s, t = rst
return np.array([1 - r - s - t, r, s, t])
[docs]
def gradient(self, rst):
"Return the gradient of shape functions at given coordinates (r, s, t)."
r, s, t = rst
return np.array([[-1, -1, -1], [1, 0, 0], [0, 1, 0], [0, 0, 1]], dtype=float)
[docs]
def hessian(self, rst):
"Return the hessian of shape functions at given coordinates (r, s, t)."
return np.zeros((4, 3, 3))
[docs]
class TetraMINI(Element):
r"""A 3D tetrahedron element formulation with bubble-enriched linear shape
functions.
Notes
-----
The MINI tetrahedron element is defined by five points (0-4). [1]
The shape functions :math:`\boldsymbol{h}` are given in terms of the coordinates
:math:`(r,s,t)`.
.. math::
\boldsymbol{h}(r,s,t) = \begin{bmatrix}
1-r-s-t \\
r \\
s \\
t \\
r s t (1-r-s-t)
\end{bmatrix}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.TetraMINI()
>>> element.plot().show()
References
----------
.. [1] W. Schroeder, K. Martin and B. Lorensen. The Visualization
Toolkit, 4th ed. Kitware, 2006. ISBN: 978-1-930934-19-1.
"""
def __init__(self, bubble_multiplier=1.0):
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.cells = np.arange(len(self.points) - 1).reshape(1, -1)
self.cell_type = "tetra"
self.bubble_multiplier = bubble_multiplier
[docs]
def function(self, rst):
"Return the shape functions at given coordinates (r, s, t)."
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):
"Return the gradient of shape functions at given coordinates (r, s, t)."
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]
def hessian(self, rst):
"Return the hessian of shape functions at given coordinates (r, s, t)."
r, s, t = rst
hess = np.zeros((5, 3, 3))
hess[4, 0] = self.bubble_multiplier * np.array(
[
-2 * s * t,
t * (1 - r - s - t) - t * (r + s),
s * (1 - r - s - t) - s * (r + t),
]
)
hess[4, 1] = self.bubble_multiplier * np.array(
[
t * (1 - r - s - t) - t * (r + s),
-2 * r * t,
r * (1 - r - s - t) - r * (t + s),
]
)
hess[4, 2] = self.bubble_multiplier * np.array(
[
s * (1 - r - s - t) - s * (r + t),
r * (1 - r - s - t) - r * (s + t),
-2 * r * s,
]
)
return hess
[docs]
class QuadraticTetra(Element):
r"""A 3D tetrahedron element formulation with quadratic shape functions.
Notes
-----
The quadratic tetrahedron element is defined by ten points (0-9). The element
includes a mid-edge point on each of the edges of the tetrahedron. The ordering of
the ten points defining the cell is point ids (0-3,4-9) where ids 0-3 are the four
tetra vertices; and point ids 4-9 are the mid-edge points between (0,1), (1,2),
(2,0), (0,3), (1,3), and (2,3). [1]
The shape functions :math:`\boldsymbol{h}` are given in terms of the coordinates
:math:`(r,s,t)`.
.. math::
\boldsymbol{h}(r,s,t) = \begin{bmatrix}
t_1 (2 t_1 - 1) \\
t_2 (2 t_2 - 1) \\
t_3 (2 t_3 - 1) \\
t_4 (2 t_4 - 1) \\
4 t_1 t_2 \\
4 t_2 t_3 \\
4 t_3 t_1 \\
4 t_1 t_4 \\
4 t_2 t_4 \\
4 t_3 t_4
\end{bmatrix}
with
.. math::
t_1 &= 1 - r - s - t
t_2 &= r
t_3 &= s
t_4 &= t
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.QuadraticTetra()
>>> element.plot().show()
References
----------
.. [1] W. Schroeder, K. Martin and B. Lorensen. The Visualization
Toolkit, 4th ed. Kitware, 2006. ISBN: 978-1-930934-19-1.
"""
def __init__(self):
self.points = np.zeros((10, 3))
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)
self.cells = np.arange(len(self.points)).reshape(1, -1)
self.cell_type = "tetra10"
[docs]
def function(self, rst):
"Return the shape functions at given coordinates (r, s, t)."
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):
"Return the gradient of shape functions at given coordinates (r, s, t)."
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)
