Source code for felupe.element._quad

# -*- coding: utf-8 -*-
"""
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|   |___ |   |___ |   |    |  |_|  ||   |_| ||   |___ 
|    ___||    ___||   |___ |       ||    ___||    ___|
|   |    |   |___ |       ||       ||   |    |   |___ 
|___|    |_______||_______||_______||___|    |_______|

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): 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, rst): "quadratic quad shape functions" return self._lagrange.function(rst)[self._permute] def gradient(self, rst): "quadratic quad gradient of shape functions" return self._lagrange.gradient(rst)[self._permute, :]