class BAR2(ContinuumElement):
"""2-node, isoparametric element
Notes
-----
Node and element face numbering
0-------1
[0]
"""
ndi = 1
nshr = 0
name = "BAR2"
num_node = 2
num_coord = 1
num_dof_per_node = 1
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord, self.num_node)
args = (self.num_coord-1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
class TRIE3(ContinuumElement):
"""Element function for linear CST element defined in [1]
Notes
-----
Node and element face numbering
2
|\
[2] | \ [1]
| \
0-----1
[0]
References
----------
1. Taylor & Hughes (1981), p.49
"""
ndi = 3
nshr = 1
name = "TRIE3"
num_node = 3
num_coord = 2
num_dof_per_node = 2
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
@staticmethod
def b_matrix(shg, shgbar=None, **kwargs):
"""Assemble and return the B matrix"""
B = zeros((4, 6))
B[0, 0::2] = shg[0, :]
B[1, 1::2] = shg[1, :]
B[3, 0::2] = shg[1, :]
B[3, 1::2] = shg[0, :]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords,
cls.thickness)
class QE4FI(ContinuumElement):
"""4-node, isoparametric, plane strain element. Full integration
Notes
-----
Node and element face numbering
[2]
3-------2
| |
[3] | | [1]
| |
0-------1
[0]
"""
ndi = 3
nshr = 1
name = "QE4FI"
num_node = 4
num_coord = 2
num_dof_per_node = 2
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
def b_matrix(self, shg, shgbar=None, **kwargs):
"""Assemble and return the B matrix"""
B = zeros(
(self.ndi + self.nshr, self.num_node * self.num_dof_per_node))
B[0, 0::2] = shg[0, :]
B[1, 1::2] = shg[1, :]
B[3, 0::2] = shg[1, :]
B[3, 1::2] = shg[0, :]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords,
cls.thickness)
class TET4(ContinuumElement):
"""Trilinear tet element"""
ndi = 3
nshr = 3
name = "TET4"
num_node = 4
num_coord = 3
num_dof_per_node = 3
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
@staticmethod
def b_matrix(shg, shgbar=None, **kwargs):
B = zeros((6, 12))
B[0, 0::3] = shg[0, :]
B[1, 1::3] = shg[1, :]
B[2, 2::3] = shg[2, :]
B[3, 0::3] = shg[1, :]
B[3, 1::3] = shg[0, :]
B[4, 1::3] = shg[2, :]
B[4, 2::3] = shg[1, :]
B[5, 0::3] = shg[2, :]
B[5, 2::3] = shg[0, :]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords)
class HEX8RI(ContinuumElement):
"""Trilinear hex element
Notes
-----
Integration points in parentheses
7---------------6
/|(7) (6)/|
/ | / |
/ | / |
/ | / |
/ (4)| (5)/ |
4---------------5 |
| | | |
| 3---------|-----2
| / (3) | (2)/
| / | /
| / | /
| / | /
|/ (0) (1)|/
0---------------1
"""
ndi = 3
nshr = 3
name = "HEX8RI"
num_node = 8
num_coord = 3
num_dof_per_node = 3
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord,
self.num_node,
reduced=True)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
def b_matrix(self, shg, shgbar=None, **kwargs):
B = zeros((6, 24))
B[0, 0::3] = shg[0, :]
B[1, 1::3] = shg[1, :]
B[2, 2::3] = shg[2, :]
B[3, 0::3] = shg[1, :]
B[3, 1::3] = shg[0, :]
B[4, 1::3] = shg[2, :]
B[4, 2::3] = shg[1, :]
B[5, 0::3] = shg[2, :]
B[5, 2::3] = shg[0, :]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords)
class QE8FI(ContinuumElement):
"""8-node, isoparametric, plane strain element
Notes
-----
Node and element face numbering
[2]
3---6---2
| |
[3] 7 5 [1]
| |
0---4---1
[0]
"""
ndi = 3
nshr = 1
name = "QE8FI"
num_node = 8
num_coord = 2
num_dof_per_node = 2
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord, self.num_node)
args = (self.num_coord-1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
def b_matrix(self, shg, shgbar=None, **kwargs):
"""Assemble and return the B matrix"""
B = zeros((self.ndi+self.nshr, self.num_node * self.num_dof_per_node))
B[0, 0::2] = shg[0, :]
B[1, 1::2] = shg[1, :]
B[3, 0::2] = shg[1, :]
B[3, 1::2] = shg[0, :]
if shgbar is None:
return B
# mean dilatational formulation
for a in range(self.num_node):
i = 2 * a
j = i + 1
bb1 = (shgbar[0, a] - shg[0, a]) / 2.
bb2 = (shgbar[1, a] - shg[1, a]) / 2.
B[0, i:i+2] += [bb1, bb2]
B[1, i:i+2] += [bb1, bb2]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node,
coords, thickness=cls.thickness)
class QE4SRI(ContinuumElementSRI):
"""4-node, isoparametric, plane strain element
Notes
-----
Node and element face numbering
[2]
3-------2
| |
[3] | | [1]
| |
0-------1
[0]
"""
ndi = 3
nshr = 1
name = "QE4SRI"
num_node = 4
num_coord = 2
num_dof_per_node = 2
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
self.sri_e = QE4RI(**{"hourglass stiffness": 0.})
@staticmethod
def iso_contrib(c, num_coord, nshr):
# modify base function to have zeros in rows/columns associated with
# plane strain
I = array([1 for i in range(2)] + [0 for i in range(2)])
a = dot(I, c) / num_coord
return outer(a, I)
def b_matrix(self, shg, shgbar=None, **kwargs):
"""Assemble and return the B matrix"""
B = zeros(
(self.ndi + self.nshr, self.num_node * self.num_dof_per_node))
B[0, 0::2] = shg[0, :]
B[1, 1::2] = shg[1, :]
B[3, 0::2] = shg[1, :]
B[3, 1::2] = shg[0, :]
if shgbar is None:
return B
# mean dilatational formulation
for a in range(self.num_node):
i = 2 * a
j = i + 1
bb1 = (shgbar[0, a] - shg[0, a]) / 2.
bb2 = (shgbar[1, a] - shg[1, a]) / 2.
B[0, i:i + 2] += [bb1, bb2]
B[1, i:i + 2] += [bb1, bb2]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords,
cls.thickness)
class QE4IM(ContinuumElementI):
"""4-node, isoparametric, plane strain element, incompatible modes
Notes
-----
Node and element face numbering
[2]
3-------2
| |
[3] | | [1]
| |
0-------1
[0]
"""
ndi = 3
nshr = 1
name = "QE4IM"
num_node = 4
num_coord = 2
num_dof_per_node = 2
num_incompat_modes = 2
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord, self.num_node)
args = (self.num_coord-1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.thickness = 1.
def b_matrix(self, shg, shgbar=None, lcoords=None, coords=None, det=None,
disp=0, **kwargs):
"""Assemble and return the B matrix"""
B = zeros((self.ndi+self.nshr, self.num_node * self.num_dof_per_node))
B[0, 0::2] = shg[0, :]
B[1, 1::2] = shg[1, :]
B[3, 0::2] = shg[1, :]
B[3, 1::2] = shg[0, :]
if not disp:
return B
# Algorithm in
# The Finite Element Method: Its Basis and Fundamentals
# By Olek C Zienkiewicz, Robert L Taylor, J.Z. Zhu
# Jacobian at element centroid
dNdxi = self.shape.grad([0., 0.])
dxdxi = dot(dNdxi, coords)
J0 = inv(dxdxi)
dt0 = determinant(dxdxi)
xi, eta = lcoords
dNdxi = array([[-2. * xi, 0.], [0., -2. * eta]])
dNdx = dt0 / det * dot(J0, dNdxi)
G1 = array([[dNdx[0, 0], 0],
[0, dNdx[0, 1]],
[0, 0],
[dNdx[0, 1], dNdx[0, 0]]])
G2 = array([[dNdx[1, 0], 0],
[0, dNdx[1, 1]],
[0, 0],
[dNdx[1, 1], dNdx[1, 0]]])
G = concatenate((G1, G2), axis=1)
return B, G
# Algorithm in Taylor's original paper and
# The Finite Element Method: Linear Static and Dynamic
# Finite Element Analysis
# By Thomas J. R. Hughe
xi = self.gauss_coords
n = self.num_gauss
dxdxi = asum([coords[i, 0] * xi[i, 0] for i in range(n)])
dxdeta = asum([coords[i, 0] * xi[i, 1] for i in range(n)])
dydxi = asum([coords[i, 1] * xi[i, 0] for i in range(n)])
dydeta = asum([coords[i, 1] * xi[i, 1] for i in range(n)])
xi, eta = lcoords
G1 = array([[-xi * dydeta, 0],
[0, xi * dxdeta],
[0, 0],
[xi * dxdeta, -xi * dydeta]])
G2 = array([[-eta * dydxi, 0.],
[0, eta * dxdxi],
[0, 0],
[eta * dxdxi, -eta * dydxi]])
G = 2. / det * concatenate((G1, G2), axis=1)
return B, G
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords, cls.thickness)
class HEX8SRI(ContinuumElementSRI):
"""Trilinear hex element
Notes
-----
Integration points in parentheses
7---------------6
/|(7) (6)/|
/ | / |
/ | / |
/ | / |
/ (4)| (5)/ |
4---------------5 |
| | | |
| 3---------|-----2
| / (3) | (2)/
| / | /
| / | /
| / | /
|/ (0) (1)|/
0---------------1
"""
ndi = 3
nshr = 3
name = "HEX8SRI"
num_node = 8
num_coord = 3
num_dof_per_node = 3
type = geom.get_elem_type(num_coord, num_node)
cp = geom.elem_center_coord(num_coord, num_node)
xp = geom.elem_corner_coord(num_coord, num_node)
def __init__(self):
# integration scheme for this element
self.integration = IntegrationProperties(self.num_coord, self.num_node)
self.shape = ShapefunctionPrototype(self.num_coord, self.num_node)
# element boundary properties
self.num_face_nodes = geom.num_face_nodes(self.num_coord,
self.num_node)
args = (self.num_coord - 1, self.num_face_nodes)
self.bndry = ShapefunctionPrototype(*args)
self.bndry.integration = IntegrationProperties(*args)
self.sri_e = HEX8RI(**{"hourglass stiffness": 0.})
@staticmethod
def iso_contrib(c):
I = array([1 for i in range(3)] + [0 for i in range(3)])
a = dot(I, c) / num_coord
return outer(a, I)
def b_matrix(self, shg, shgbar=None, **kwargs):
B = zeros((6, 24))
B[0, 0::3] = shg[0, :]
B[1, 1::3] = shg[1, :]
B[2, 2::3] = shg[2, :]
B[3, 0::3] = shg[1, :]
B[3, 1::3] = shg[0, :]
B[4, 1::3] = shg[2, :]
B[4, 2::3] = shg[1, :]
B[5, 0::3] = shg[2, :]
B[5, 2::3] = shg[0, :]
return B
@classmethod
def volume(cls, coords):
return geom.elem_volume(cls.num_coord, cls.num_node, coords)