# -*- 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):
r"""A 2D triangle element formulation with linear shape functions.
Notes
-----
The triangle element is defined by three points (0-2). [1]
The shape functions :math:`\boldsymbol{h}` are given in terms of the coordinates
:math:`(r,s)` [2]_.
.. math::
\boldsymbol{h}(r,s) = \begin{bmatrix}
1-r-s \\
r \\
s
\end{bmatrix}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.Triangle()
>>> 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.
.. [2] K. J. Bathe, Finite element procedures, 2nd ed. K. J. Bathe, Watertown, MA,
2014.
"""
def __init__(self):
self.points = np.array([[0, 0], [1, 0], [0, 1]], dtype=float)
self.cells = np.arange(len(self.points)).reshape(1, -1)
self.cell_type = "triangle"
[docs]
def function(self, rs):
"Return the shape functions at given coordinates (r, s)."
r, s = rs
return np.array([1 - r - s, r, s])
[docs]
def gradient(self, rs):
"Return the gradient of shape functions at given coordinates (r, s)."
r, s = rs
return np.array([[-1, -1], [1, 0], [0, 1]], dtype=float)
[docs]
def hessian(self, rs):
"Return the hessian of shape functions at given coordinates (r, s)."
r, s = rs
return np.zeros((3, 2, 2))
[docs]
class TriangleMINI(Element):
r"""A 2D triangle element formulation with bubble-enriched linear shape functions.
Notes
-----
The MINI triangle 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)`.
.. math::
\boldsymbol{h}(r,s) = \begin{bmatrix}
1-r-s \\
r \\
s \\
r s (1-r-s)
\end{bmatrix}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.TriangleMINI()
>>> 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], [1, 0], [0, 1], [1 / 3, 1 / 3]], dtype=float)
self.cells = np.arange(len(self.points) - 1).reshape(1, -1)
self.cell_type = "triangle"
self.bubble_multiplier = bubble_multiplier
[docs]
def function(self, rs):
"Return the shape functions at given coordinates (r, s)."
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):
"Return the gradient of shape functions at given coordinates (r, s)."
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]
def hessian(self, rs):
"Return the hessian of shape functions at given coordinates (r, s)."
r, s = rs
hess = np.zeros((4, 2, 2))
hess[3] = self.bubble_multiplier * np.array(
[
[-2 * s, (1 - r - s) - (s + r)],
[(1 - r - s) - (r + s), -2 * r],
]
)
return hess
[docs]
class QuadraticTriangle(Element):
r"""A 2D triangle element formulation with quadratic shape functions.
Notes
-----
The quadratic triangle element is defined by six points (0-5). The element includes
three mid-edge points besides the three triangle vertices. The ordering of the three
points defining the element is point ids (0-2,3-5) where id #3 is the mid-edge point
between points (0,1); id #4 is the mid-edge point between points (1,2); and id #5 is
the mid-edge point between points (2,0). [1]
The shape functions :math:`\boldsymbol{h}` are given in terms of the coordinates
:math:`(r,s)` [2]_.
.. math::
\boldsymbol{h}(r,s) = \begin{bmatrix}
1-r-s \\
r \\
s \\
4 r (1-r-s) \\
4 r s \\
4 s (1-r-s)
\end{bmatrix}
Examples
--------
.. pyvista-plot::
:force_static:
>>> import felupe as fem
>>>
>>> element = fem.QuadraticTriangle()
>>> 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.
.. [2] K. J. Bathe, Finite element procedures, 2nd ed. K. J. Bathe, Watertown, MA,
2014.
"""
def __init__(self):
self.points = np.zeros((6, 2))
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)
self.cells = np.arange(len(self.points)).reshape(1, -1)
self.cell_type = "triangle6"
[docs]
def function(self, rs):
"Return the shape functions at given coordinates (r, s)."
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):
"Return the gradient of shape functions at given coordinates (r, s)."
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
