def elast_hex8(coord, params):
"""Hexaedral element with 8 nodes for classic elasticity
Parameters
----------
coord : coord
Coordinates of the element.
params : list
List with material parameters in the following order:
[Young modulus, Poisson coefficient, density].
Returns
-------
stiff_mat : ndarray (float)
Local stifness matrix.
mass_mat : ndarray (float)
Local mass matrix.
"""
E, nu, rho = params
C = fem.elast_mat((E, nu))
stiff_mat = np.zeros((24, 24))
mass_mat = np.zeros((24, 24))
gpts, gwts = gau.gauss_nd(2, ndim=3)
for cont in range(gpts.shape[0]):
r = gpts[cont, 0]
s = gpts[cont, 1]
t = gpts[cont, 2]
H, B, det = fem.elast_diff_3d(r, s, t, coord)
factor = det * gwts[cont]
stiff_mat += factor * (B.T @ C @ B)
mass_mat += rho*factor * (H.T @ H)
return stiff_mat, mass_mat
def elast_axi_quad9(coord, params):
"""
Quadrilateral element with 9 nodes for classic elasticity
under plane-strain
Parameters
----------
coord : coord
Coordinates of the element.
params : list
List with material parameters in the following order:
[Young modulus, Poisson coefficient, density].
Returns
-------
stiff_mat : ndarray (float)
Local stifness matrix.
mass_mat : ndarray (float)
Local mass matrix.
"""
E, nu, rho = params
C = fem.elast_mat_axi((E, nu))
stiff_mat = np.zeros((18, 18))
mass_mat = np.zeros((18, 18))
gpts, gwts = gau.gauss_nd(3, ndim=2)
for cont in range(gpts.shape[0]):
r = gpts[cont, 0]
s = gpts[cont, 1]
H, B, det = fem.elast_diff_axi(r, s, coord, fem.shape_quad9)
factor = det * gwts[cont]
stiff_mat += factor * (B.T @ C @ B)
mass_mat += rho*factor * (H.T @ H)
return stiff_mat, mass_mat
def elast_quad9(coord, params):
"""
Quadrilateral element with 9 nodes for classic elasticity
under plane-strain
Parameters
----------
coord : coord
Coordinates of the element.
params : list
List with material parameters in the following order:
[Young modulus, Poisson coefficient, density].
Returns
-------
stiff_mat : ndarray (float)
Local stifness matrix.
mass_mat : ndarray (float)
Local mass matrix.
"""
stiff_mat = np.zeros((18, 18))
mass_mat = np.zeros((18, 18))
C = fem.umat(params[:2])
if len(params) == 2:
dens = 1
else:
dens = params[-1]
gpts, gwts = gau.gauss_nd(3, ndim=2)
for cont in range(gpts.shape[0]): # pylint: disable=E1136 # pylint/issues/3139
r, s = gpts[cont, :]
H, B, det = fem.elast_diff_2d(r, s, coord, fem.shape_quad9)
factor = det * gwts[cont]
stiff_mat += factor * (B.T @ C @ B)
mass_mat += dens * factor * (H.T @ H)
return stiff_mat, mass_mat
def test_gauss_nd():
"""Test for ND Gauss integration"""
npts = 4
ndim = 2
pts, wts = gauss.gauss_nd(npts, ndim=ndim)
fun = lambda x, y: 3*x*y**2 - x**3
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += wts[cont] * fun(x, y)
assert np.isclose(inte, 0)
npts = 10
ndim = 2
pts, wts = gauss.gauss_nd(npts, ndim=ndim)
fun = lambda x, y: np.exp(-x**2 - y**2)
inte = 0.0
for cont, pt in enumerate(pts):
x, y = pt
inte += wts[cont] * fun(x, y)
assert np.isclose(inte, 2.23098514140413)
def elast_quad4(coord, params):
"""Quadrilateral element with 4 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (4, 2).
params : tuple
Material parameters in the following order:
young : float
Young modulus (>0).
poisson : float
Poisson coefficient (-1, 0.5).
dens : float, optional
Density (>0).
Returns
-------
stiff_mat : ndarray
Local stiffness matrix for the element (8, 8).
mass_mat : ndarray
Local mass matrix for the element (8, 8).
Examples
--------
>>> coord = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]])
>>> params = [8/3, 1/3, 1]
>>> stiff, mass = uel4nquad(coord, params)
>>> stiff_ex = 1/6 * np.array([
... [ 8, 3, -5, 0, -4, -3, 1, 0],
... [ 3, 8, 0, 1, -3, -4, 0, -5],
... [-5, 0, 8, -3, 1, 0, -4, 3],
... [ 0, 1, -3, 8, 0, -5, 3, -4],
... [-4, -3, 1, 0, 8, 3, -5, 0],
... [-3, -4, 0, -5, 3, 8, 0, 1],
... [ 1, 0, -4, 3, -5, 0, 8, -3],
... [ 0, -5, 3, -4, 0, 1, -3, 8]])
>>> mass_ex = 1/9 * np.array([
... [4, 0, 2, 0, 1, 0, 2, 0],
... [0, 4, 0, 2, 0, 1, 0, 2],
... [2, 0, 4, 0, 2, 0, 1, 0],
... [0, 2, 0, 4, 0, 2, 0, 1],
... [1, 0, 2, 0, 4, 0, 2, 0],
... [0, 1, 0, 2, 0, 4, 0, 2],
... [2, 0, 1, 0, 2, 0, 4, 0],
... [0, 2, 0, 1, 0, 2, 0, 4]])
>>> np.allclose(stiff, stiff_ex)
True
>>> np.allclose(mass, mass_ex)
True
"""
stiff_mat = np.zeros([8, 8])
mass_mat = np.zeros([8, 8])
C = fem.umat(params[:2])
if len(params) == 2:
dens = 1
else:
dens = params[-1]
gpts, gwts = gau.gauss_nd(2)
for cont in range(gpts.shape[0]): # pylint: disable=E1136 # pylint/issues/3139
r, s = gpts[cont, :]
H, B, det = fem.elast_diff_2d(r, s, coord, fem.shape_quad4)
factor = det * gwts[cont]
stiff_mat += factor * (B.T @ C @ B)
mass_mat += dens*factor* (H.T @ H)
return stiff_mat, mass_mat
def uel4nquad(coord, params):
"""Quadrilateral element with 4 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (4, 2).
params : tuple
Material parameters in the following order:
young : float
Young modulus (>0).
poisson : float
Poisson coefficient (-1, 0.5).
dens : float, optional
Density (>0).
Returns
-------
stiff_mat : ndarray
Local stiffness matrix for the element (8, 8).
mass_mat : ndarray
Local mass matrix for the element (8, 8).
Examples
--------
>>> coord = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]])
>>> params = [8/3, 1/3, 1]
>>> stiff, mass = uel4nquad(coord, params)
>>> stiff_ex = 1/6 * np.array([
... [ 8, 3, -5, 0, -4, -3, 1, 0],
... [ 3, 8, 0, 1, -3, -4, 0, -5],
... [-5, 0, 8, -3, 1, 0, -4, 3],
... [ 0, 1, -3, 8, 0, -5, 3, -4],
... [-4, -3, 1, 0, 8, 3, -5, 0],
... [-3, -4, 0, -5, 3, 8, 0, 1],
... [ 1, 0, -4, 3, -5, 0, 8, -3],
... [ 0, -5, 3, -4, 0, 1, -3, 8]])
>>> mass_ex = 1/9 * np.array([
... [4, 0, 2, 0, 1, 0, 2, 0],
... [0, 4, 0, 2, 0, 1, 0, 2],
... [2, 0, 4, 0, 2, 0, 1, 0],
... [0, 2, 0, 4, 0, 2, 0, 1],
... [1, 0, 2, 0, 4, 0, 2, 0],
... [0, 1, 0, 2, 0, 4, 0, 2],
... [2, 0, 1, 0, 2, 0, 4, 0],
... [0, 2, 0, 1, 0, 2, 0, 4]])
>>> np.allclose(stiff, stiff_ex)
True
>>> np.allclose(mass, mass_ex)
True
"""
stiff_mat = np.zeros([8, 8])
mass_mat = np.zeros([8, 8])
C = fem.umat(params[:2])
if len(params) == 2:
dens = 1
else:
dens = params[-1]
gpts, gwts = gau.gauss_nd(2)
for i in range(gpts.shape[0]):
ri = gpts[i, 0]
si = gpts[i, 1]
ddet, B = fem.stdm4NQ(ri, si, coord)
N = fem.sha4(ri, si)
factor = ddet * gwts[i]
stiff_mat += factor * (B.T @ C @ B)
mass_mat += dens * factor * (N.T @ N)
return stiff_mat, mass_mat