def str_el6(coord, ul):
"""Compute the strains at each element integration point
This one is used for 6-noded triangular elements.
Parameters
----------
coord : ndarray
Coordinates of the nodes of the element (6, 2).
ul : ndarray
Array with displacements for the element.
Returns
-------
epsGT : ndarray
Strain components for the Gauss points.
xl : ndarray
Configuration of the Gauss points after deformation.
"""
epsl = np.zeros([3])
epsG = np.zeros([3, 7])
xl = np.zeros([7, 2])
XW, XP = gau.gpoints7()
for i in range(7):
ri = XP[i, 0]
si = XP[i, 1]
ddet, B = stdm6NT(ri, si, coord)
epsl = np.dot(B, ul)
epsG[:, i] = epsl[:]
N = sha6(ri, si)
xl[i, 0] = sum(N[0, 2*i]*coord[i, 0] for i in range(6))
xl[i, 1] = sum(N[0, 2*i]*coord[i, 1] for i in range(6))
return epsG.T, xl
def uel6ntrian(coord, enu, Emod):
"""Triangular element with 6 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (6, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (12, 12).
Examples
--------
>>> coord = Matrix([
... [0, 0],
... [1, 0],
... [0, 1],
... [0.5, 0],
... [0.5, 0.5],
... [0, 0.5]])
>>> stiff = uel6ntrian(coord, S(1)/3, S(8)/3)
>>> stiff_ex = 1/6 * Matrix([
... [12, 6, 3, 1, 1, 1, -12, -4, 0, 0, -4, -4],
... [6, 12, 1, 1, 1, 3, -4, -4, 0, 0, -4, -12],
... [3, 1, 9, 0, 0, -1, -12, -4, 0, 4, 0, 0],
... [1, 1, 0, 3, -1, 0, -4, -4, 4, 0, 0, 0],
... [1, 1, 0, -1, 3, 0, 0, 0, 0, 4, -4, -4],
... [1, 3, -1, 0, 0, 9, 0, 0, 4, 0, -4, -12],
... [-12, -4, -12, -4, 0, 0, 32, 8, -8, -8, 0, 8],
... [-4, -4, -4, -4, 0, 0, 8, 32, -8, -24, 8, 0],
... [0, 0, 0, 4, 0, 4, -8, -8, 32, 8, -24, -8],
... [0, 0, 4, 0, 4, 0, -8, -24, 8, 32, -8, -8],
... [-4, -4, 0, 0, -4, -4, 0, 8, -24, -8, 32, 8],
... [-4, -12, 0, 0, -4, -12, 8, 0, -8, -8, 8, 32]])
>>> (stiff - stiff_ex).norm()/stiff_ex.norm() < 1e-6
True
"""
kl = np.zeros([12, 12])
C = fem.umat(enu, Emod)
XW, XP = gau.gpoints7()
ngpts = 7
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm6NT(ri, si, coord)
kl = kl + 0.5*B.T*C*B*alf*ddet
return kl
def str_el6(coord, ul):
"""Compute the strains at each element integration point
This one is used for 6-noded triangular elements.
Parameters
----------
coord : ndarray
Coordinates of the nodes of the element (6, 2).
ul : ndarray
Array with displacements for the element.
Returns
-------
epsGT : ndarray
Strain components for the Gauss points.
xl : ndarray
Configuration of the Gauss points after deformation.
"""
epsl = np.zeros([3])
epsG = np.zeros([3, 7])
epsGT = np.zeros([7, 3])
xl = np.zeros([7, 2])
x, y = symbols('x y')
XW, XP = gau.gpoints7()
for i in range(7):
ri = XP[i, 0]
si = XP[i, 1]
ddet, B = stdm6NT(ri, si, coord)
epsl = B*ul
epsG[:, i] = epsl[:]
N = sha6(ri, si)
NN = N.subs([(x, ri), (y, si)])
xl[i, 0] = sum(NN[0, 2*i]*coord[i, 0] for i in range(6))
xl[i, 1] = sum(NN[0, 2*i]*coord[i, 1] for i in range(6))
epsGT = epsG.T
return epsGT, xl
def uel6ntrian(coord, par):
"""Triangular element with 6 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (6, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (12, 12).
Examples
--------
>>> coord = np.array([
... [0, 0],
... [1, 0],
... [0, 1],
... [0.5, 0],
... [0.5, 0.5],
... [0, 0.5]])
>>> stiff = uel6ntrian(coord,1/3, 8/3)
>>> stiff_ex = 1/6 * np.array([
... [12, 6, 3, 1, 1, 1, -12, -4, 0, 0, -4, -4],
... [6, 12, 1, 1, 1, 3, -4, -4, 0, 0, -4, -12],
... [3, 1, 9, 0, 0, -1, -12, -4, 0, 4, 0, 0],
... [1, 1, 0, 3, -1, 0, -4, -4, 4, 0, 0, 0],
... [1, 1, 0, -1, 3, 0, 0, 0, 0, 4, -4, -4],
... [1, 3, -1, 0, 0, 9, 0, 0, 4, 0, -4, -12],
... [-12, -4, -12, -4, 0, 0, 32, 8, -8, -8, 0, 8],
... [-4, -4, -4, -4, 0, 0, 8, 32, -8, -24, 8, 0],
... [0, 0, 0, 4, 0, 4, -8, -8, 32, 8, -24, -8],
... [0, 0, 4, 0, 4, 0, -8, -24, 8, 32, -8, -8],
... [-4, -4, 0, 0, -4, -4, 0, 8, -24, -8, 32, 8],
... [-4, -12, 0, 0, -4, -12, 8, 0, -8, -8, 8, 32]])
>>> np.allclose(stiff, stiff_ex)
True
"""
Emod = par[0]
enu = par[1]
rho = par[2]
calpha = par[3]
cbeta = par[4]
kl = np.zeros([12, 12])
ml = np.zeros([12, 12])
cl = np.zeros([12, 12])
C = fem.umat(enu, Emod, rho)
XW, XP = gau.gpoints7()
ngpts = 7
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm6NT(ri, si, coord)
N = fem.sha6(ri, si)
kl = kl + 0.5 * np.dot(np.dot(B.T, C), B) * alf * ddet
ml = ml + 0.5 * rho * np.dot(N.T, N) * alf * ddet
cl = calpha * kl + cbeta * ml
return kl, ml, cl