# Copyright (C) 2021-2025 Université Gustave Eiffel.
# This file is part of the EasyFEA project.
# EasyFEA is distributed under the terms of the GNU General Public License v3, see LICENSE.txt and CREDITS.md for more information.
"""Quad element module."""
import numpy as np
from .._group_elems import _GroupElem
from ...utilities import _types
[docs]
class QUAD4(_GroupElem):
# v
# ^
# |
# 3-----------2
# | | |
# | | |
# | +---- | --> u
# | |
# | |
# 0-----------1
def __init__(
self, gmshId: int, connect: _types.IntArray, coordGlob: _types.FloatArray
):
super().__init__(gmshId, connect, coordGlob)
@property
def origin(self) -> list[int]:
return [-1, -1]
@property
def triangles(self) -> list[int]:
# fmt: off
return [0,1,3,
1,2,3]
# fmt: on
@property
def surfaces(self) -> _types.IntArray:
return np.array([[0, 1, 2, 3, 0]], dtype=int)
@property
def faces(self) -> _types.IntArray:
return np.arange(self.nPe, dtype=int)
[docs]
def Get_Local_Coords(self):
list_x = [-1, 1, 1, -1]
list_y = [-1, -1, 1, 1]
local_coords = np.array([list_x, list_y]).T
return local_coords
[docs]
def _N(self) -> _types.FloatArray:
N1 = lambda r, s: (r - 1) * (s - 1) / 4
N2 = lambda r, s: -(r + 1) * (s - 1) / 4
N3 = lambda r, s: (r + 1) * (s + 1) / 4
N4 = lambda r, s: -(r - 1) * (s + 1) / 4
N = np.array([N1, N2, N3, N4]).reshape(-1, 1)
return N
[docs]
def _dN(self) -> _types.FloatArray:
dN1 = [lambda r, s: s / 4 - 1 / 4, lambda r, s: r / 4 - 1 / 4]
dN2 = [lambda r, s: 1 / 4 - s / 4, lambda r, s: -r / 4 - 1 / 4]
dN3 = [lambda r, s: s / 4 + 1 / 4, lambda r, s: r / 4 + 1 / 4]
dN4 = [lambda r, s: -s / 4 - 1 / 4, lambda r, s: 1 / 4 - r / 4]
dN = np.array([dN1, dN2, dN3, dN4])
return dN
[docs]
def _ddN(self) -> _types.FloatArray:
return super()._ddN()
[docs]
def _dddN(self) -> _types.FloatArray:
return super()._dddN()
[docs]
def _ddddN(self) -> _types.FloatArray:
return super()._ddddN()
[docs]
class QUAD8(_GroupElem):
# v
# ^
# |
# 3-----6-----2
# | | |
# | | |
# 7 +---- 5 --> u
# | |
# | |
# 0-----4-----1
def __init__(
self, gmshId: int, connect: _types.IntArray, coordGlob: _types.FloatArray
):
super().__init__(gmshId, connect, coordGlob)
@property
def origin(self) -> list[int]:
return [-1, -1]
@property
def triangles(self) -> list[int]:
# fmt: off
return [4,5,7,
5,6,7,
0,4,7,
4,1,5,
5,2,6,
6,3,7]
# fmt: on
@property
def surfaces(self) -> _types.IntArray:
return np.array([[0, 4, 1, 5, 2, 6, 3, 7, 0]], dtype=int)
@property
def faces(self) -> _types.IntArray:
return np.arange(self.nPe, dtype=int)
[docs]
def Get_Local_Coords(self):
list_x = [-1, 1, 1, -1, 0, 1, 0, -1]
list_y = [-1, -1, 1, 1, -1, 0, 1, 0]
local_coords = np.array([list_x, list_y]).T
return local_coords
[docs]
def _N(self) -> _types.FloatArray:
N1 = lambda r, s: -(r - 1) * (s - 1) * (r + s + 1) / 4
N2 = lambda r, s: -(r + 1) * (s - 1) * (r - s - 1) / 4
N3 = lambda r, s: (r + 1) * (s + 1) * (r + s - 1) / 4
N4 = lambda r, s: (r - 1) * (s + 1) * (r - s + 1) / 4
N5 = lambda r, s: (r - 1) * (r + 1) * (s - 1) / 2
N6 = lambda r, s: -(r + 1) * (s - 1) * (s + 1) / 2
N7 = lambda r, s: -(r - 1) * (r + 1) * (s + 1) / 2
N8 = lambda r, s: (r - 1) * (s - 1) * (s + 1) / 2
N = np.array([N1, N2, N3, N4, N5, N6, N7, N8]).reshape(-1, 1)
return N
[docs]
def _dN(self) -> _types.FloatArray:
dN1 = [
lambda r, s: -(r - 1) * (s - 1) / 4 - (s - 1) * (r + s + 1) / 4,
lambda r, s: -(r - 1) * (s - 1) / 4 - (r - 1) * (r + s + 1) / 4,
]
dN2 = [
lambda r, s: -(r + 1) * (s - 1) / 4 - (s - 1) * (r - s - 1) / 4,
lambda r, s: (r + 1) * (s - 1) / 4 - (r + 1) * (r - s - 1) / 4,
]
dN3 = [
lambda r, s: (r + 1) * (s + 1) / 4 + (s + 1) * (r + s - 1) / 4,
lambda r, s: (r + 1) * (s + 1) / 4 + (r + 1) * (r + s - 1) / 4,
]
dN4 = [
lambda r, s: (r - 1) * (s + 1) / 4 + (s + 1) * (r - s + 1) / 4,
lambda r, s: -(r - 1) * (s + 1) / 4 + (r - 1) * (r - s + 1) / 4,
]
dN5 = [
lambda r, s: (r - 1) * (s - 1) / 2 + (r + 1) * (s - 1) / 2,
lambda r, s: (r - 1) * (r + 1) / 2,
]
dN6 = [
lambda r, s: -(s - 1) * (s + 1) / 2,
lambda r, s: -(r + 1) * (s - 1) / 2 - (r + 1) * (s + 1) / 2,
]
dN7 = [
lambda r, s: -(r - 1) * (s + 1) / 2 - (r + 1) * (s + 1) / 2,
lambda r, s: -(r - 1) * (r + 1) / 2,
]
dN8 = [
lambda r, s: (s - 1) * (s + 1) / 2,
lambda r, s: (r - 1) * (s - 1) / 2 + (r - 1) * (s + 1) / 2,
]
dN = np.array([dN1, dN2, dN3, dN4, dN5, dN6, dN7, dN8])
return dN
[docs]
def _ddN(self) -> _types.FloatArray:
ddN1 = [lambda r, s: 1 / 2 - s / 2, lambda r, s: 1 / 2 - r / 2]
ddN2 = [lambda r, s: 1 / 2 - s / 2, lambda r, s: r / 2 + 1 / 2]
ddN3 = [lambda r, s: s / 2 + 1 / 2, lambda r, s: r / 2 + 1 / 2]
ddN4 = [lambda r, s: s / 2 + 1 / 2, lambda r, s: 1 / 2 - r / 2]
ddN5 = [lambda r, s: s - 1, lambda r, s: 0]
ddN6 = [lambda r, s: 0, lambda r, s: -r - 1]
ddN7 = [lambda r, s: -s - 1, lambda r, s: 0]
ddN8 = [lambda r, s: 0, lambda r, s: r - 1]
ddN = np.array([ddN1, ddN2, ddN3, ddN4, ddN5, ddN6, ddN7, ddN8])
return ddN
[docs]
def _dddN(self) -> _types.FloatArray:
return super()._dddN()
[docs]
def _ddddN(self) -> _types.FloatArray:
return super()._ddddN()
[docs]
class QUAD9(_GroupElem):
# v
# ^
# |
# 3-----6-----2
# | | |
# | | |
# 7 8---- 5 --> u
# | |
# | |
# 0-----4-----1
def __init__(
self, gmshId: int, connect: _types.IntArray, coordGlob: _types.FloatArray
):
super().__init__(gmshId, connect, coordGlob)
@property
def origin(self) -> list[int]:
return [-1, -1]
@property
def triangles(self) -> list[int]:
# fmt: off
return [0, 4, 8,
4, 1, 5,
4, 5, 8,
0, 8, 7,
7, 8, 6,
8, 5, 2,
8, 2, 6,
7, 6, 3]
# fmt: on
@property
def surfaces(self) -> _types.IntArray:
return np.array([[0, 4, 1, 5, 2, 6, 3, 7, 0]], dtype=int)
@property
def faces(self) -> _types.IntArray:
return np.arange(self.nPe, dtype=int)
[docs]
def Get_Local_Coords(self):
list_x = [-1, 1, 1, -1, 0, 1, 0, -1, 0]
list_y = [-1, -1, 1, 1, -1, 0, 1, 0, 0]
local_coords = np.array([list_x, list_y]).T
return local_coords
[docs]
def _N(self) -> _types.FloatArray:
N1 = lambda r, s: r * s * (r - 1) * (s - 1) / 4
N2 = lambda r, s: r * s * (r + 1) * (s - 1) / 4
N3 = lambda r, s: r * s * (r + 1) * (s + 1) / 4
N4 = lambda r, s: r * s * (r - 1) * (s + 1) / 4
N5 = lambda r, s: -s * (r - 1) * (r + 1) * (s - 1) / 2
N6 = lambda r, s: -r * (r + 1) * (s - 1) * (s + 1) / 2
N7 = lambda r, s: -s * (r - 1) * (r + 1) * (s + 1) / 2
N8 = lambda r, s: -r * (r - 1) * (s - 1) * (s + 1) / 2
N9 = lambda r, s: (r - 1) * (r + 1) * (s - 1) * (s + 1)
N = np.array([N1, N2, N3, N4, N5, N6, N7, N8, N9]).reshape(-1, 1)
return N
[docs]
def _dN(self) -> _types.FloatArray:
dN1 = [
lambda r, s: r * s * (s - 1) / 4 + s * (r - 1) * (s - 1) / 4,
lambda r, s: r * s * (r - 1) / 4 + r * (r - 1) * (s - 1) / 4,
]
dN2 = [
lambda r, s: r * s * (s - 1) / 4 + s * (r + 1) * (s - 1) / 4,
lambda r, s: r * s * (r + 1) / 4 + r * (r + 1) * (s - 1) / 4,
]
dN3 = [
lambda r, s: r * s * (s + 1) / 4 + s * (r + 1) * (s + 1) / 4,
lambda r, s: r * s * (r + 1) / 4 + r * (r + 1) * (s + 1) / 4,
]
dN4 = [
lambda r, s: r * s * (s + 1) / 4 + s * (r - 1) * (s + 1) / 4,
lambda r, s: r * s * (r - 1) / 4 + r * (r - 1) * (s + 1) / 4,
]
dN5 = [
lambda r, s: -s * (r - 1) * (s - 1) / 2 - s * (r + 1) * (s - 1) / 2,
lambda r, s: -s * (r - 1) * (r + 1) / 2 - (r - 1) * (r + 1) * (s - 1) / 2,
]
dN6 = [
lambda r, s: -r * (s - 1) * (s + 1) / 2 - (r + 1) * (s - 1) * (s + 1) / 2,
lambda r, s: -r * (r + 1) * (s - 1) / 2 - r * (r + 1) * (s + 1) / 2,
]
dN7 = [
lambda r, s: -s * (r - 1) * (s + 1) / 2 - s * (r + 1) * (s + 1) / 2,
lambda r, s: -s * (r - 1) * (r + 1) / 2 - (r - 1) * (r + 1) * (s + 1) / 2,
]
dN8 = [
lambda r, s: -r * (s - 1) * (s + 1) / 2 - (r - 1) * (s - 1) * (s + 1) / 2,
lambda r, s: -r * (r - 1) * (s - 1) / 2 - r * (r - 1) * (s + 1) / 2,
]
dN9 = [
lambda r, s: (r - 1) * (s - 1) * (s + 1) + (r + 1) * (s - 1) * (s + 1),
lambda r, s: (r - 1) * (r + 1) * (s - 1) + (r - 1) * (r + 1) * (s + 1),
]
dN = np.array([dN1, dN2, dN3, dN4, dN5, dN6, dN7, dN8, dN9])
return dN
[docs]
def _ddN(self) -> _types.FloatArray:
ddN1 = [lambda r, s: s * (s - 1) / 2, lambda r, s: r * (r - 1) / 2]
ddN2 = [lambda r, s: s * (s - 1) / 2, lambda r, s: r * (r + 1) / 2]
ddN3 = [lambda r, s: s * (s + 1) / 2, lambda r, s: r * (r + 1) / 2]
ddN4 = [lambda r, s: s * (s + 1) / 2, lambda r, s: r * (r - 1) / 2]
ddN5 = [lambda r, s: -s * (s - 1), lambda r, s: -(r - 1) * (r + 1)]
ddN6 = [lambda r, s: -(s - 1) * (s + 1), lambda r, s: -r * (r + 1)]
ddN7 = [lambda r, s: -s * (s + 1), lambda r, s: -(r - 1) * (r + 1)]
ddN8 = [lambda r, s: -(s - 1) * (s + 1), lambda r, s: -r * (r - 1)]
ddN9 = [lambda r, s: 2 * (s - 1) * (s + 1), lambda r, s: 2 * (r - 1) * (r + 1)]
ddN = np.array([ddN1, ddN2, ddN3, ddN4, ddN5, ddN6, ddN7, ddN8, ddN9])
return ddN
[docs]
def _dddN(self) -> _types.FloatArray:
return super()._dddN()
[docs]
def _ddddN(self) -> _types.FloatArray:
return super()._ddddN()