def uel3ntrian(coord, par):
"""Triangular element with 3 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (3, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (6, 6).
Examples
--------
>>> coord = np.array([
... [0, 0],
... [1, 0],
... [0, 1]])
>>> stiff = uel3ntrian(coord, 1/3, 8/3)
>>> stiff_ex = 1/2 * np.array([
... [4, 2, -3, -1, -1, -1],
... [2, 4, -1, -1, -1, -3],
... [-3, -1, 3, 0, 0, 1],
... [-1, -1, 0, 1, 1, 0],
... [-1, -1, 0, 1, 1, 0],
... [-1, -3, 1, 0, 0, 3]])
>>> np.allclose(stiff, stiff_ex)
True
"""
Emod = par[0]
enu = par[1]
rho = par[2]
calpha = par[3]
cbeta = par[4]
kl = np.zeros([6, 6])
ml = np.zeros([6, 6])
cl = np.zeros([6, 6])
C = fem.umat(enu, Emod, rho)
XW, XP = gau.gpoints3()
ngpts = 3
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm3NT(ri, si, coord)
N = fem.sha3(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
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 uel9nquad(coord, par):
"""Quadrilateral element with 4 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (4, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (8, 8).
Examples
--------
>>> coord = np.array([[-1, -1], [1, -1], [1, 1], [-1, 1]])
>>> stiff = uel4nquad(coord, 1/3, 8/3)
>>> 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]])
>>> np.allclose(stiff, stiff_ex)
True
"""
Emod = par[0]
enu = par[1]
rho = par[2]
calpha = par[3]
cbeta = par[4]
kl = np.zeros([18, 18])
ml = np.zeros([18, 18])
cl = np.zeros([18, 18])
C = fem.umat(enu, Emod, rho)
XW, XP = gau.gpoints3x3()
ngpts = 9
for i in range(0, ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm9NQ(ri, si, coord)
N = fem.sha9(ri, si)
kl = kl + np.dot(np.dot(B.T, C), B) * alf * ddet
ml = ml + rho * np.dot(N.T, N) * alf * ddet
cl = calpha * kl + cbeta * ml
return kl, ml, cl
def uel3ntrian(coord, enu, Emod):
"""Triangular element with 3 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (3, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (6, 6).
Examples
--------
>>> coord = Matrix([
... [0, 0],
... [1, 0],
... [0, 1]])
>>> stiff = uel3ntrian(coord, S(1)/3, S(8)/3)
>>> stiff_ex = 1/2 * Matrix([
... [4, 2, -3, -1, -1, -1],
... [2, 4, -1, -1, -1, -3],
... [-3, -1, 3, 0, 0, 1],
... [-1, -1, 0, 1, 1, 0],
... [-1, -1, 0, 1, 1, 0],
... [-1, -3, 1, 0, 0, 3]])
>>> (stiff - stiff_ex).norm()/stiff_ex.norm() < 1e-6
True
"""
kl = np.zeros([6, 6])
C = fem.umat(enu, Emod)
XW, XP = gau.gpoints3()
ngpts = 3
for i in range(ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm3NT(ri, si, coord)
kl = kl + 0.5*B.T*C*B*alf*ddet
return kl
def uel4nquad(coord, enu, Emod):
"""Quadrilateral element with 4 nodes
Parameters
----------
coord : ndarray
Coordinates for the nodes of the element (4, 2).
enu : float
Poisson coefficient (-1, 0.5).
Emod : float
Young modulus (>0).
Returns
-------
kl : ndarray
Local stiffness matrix for the element (8, 8).
Examples
--------
>>> coord = Matrix([[-1, -1], [1, -1], [1, 1], [-1, 1]])
>>> stiff = uel4nquad(coord, S(1)/3, S(8)/3)
>>> stiff_ex = 1/6 * Matrix([
... [ 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]])
>>> (stiff - stiff_ex).norm()/stiff_ex.norm() < 1e-6
True
"""
kl = np.zeros([8, 8])
C = fem.umat(enu, Emod)
XW, XP = gau.gpoints2x2()
ngpts = 4
for i in range(0, ngpts):
ri = XP[i, 0]
si = XP[i, 1]
alf = XW[i]
ddet, B = fem.stdm4NQ(ri, si, coord)
kl = kl + B.T*C*B*alf*ddet
return kl
def locstrain3nT(ul, coord, enu, Emod):
"""
Plot strain and stress for a linear 3-noded element in the
natural space
Parameters
----------
ul : ndarray (float)
Array with the displacements for the element.
coord : ndarray (float)
Coordinates for the nodes of the element.
enu : float
Poisson coefficient in the range (-1, 0.5).
Emod : float
Young modulus, should be greater than zero.
"""
r = np.zeros([3, 2])
eG = np.zeros([3, 3])
sG = np.zeros([3, 3])
eps = np.zeros([3, 3])
e = np.zeros([3])
s = np.zeros([3])
C = np.zeros([3, 3])
r[0, 0] = 0.0
r[0, 1] = 0.0
r[1, 0] = 1.0
r[1, 1] = 0.0
r[2, 0] = 0.0
r[2, 1] = 1.0
C = fe.umat(enu, Emod)
for i in range(3):
ri = r[i, 0]
si = r[i, 1]
ddet, B = fe.stdm3NT(ri, si, coord)
eps = B*ul
sig = C*eps
eG[:, i] = eps[:]
sG[:, i] = sig[:]
grid_x, grid_y = np.mgrid[0:1:100j, 0:1:100j]
print('Strain field')
for j in range(3):
for i in range(3):
e[i] = eG[j, i]
plt.figure()
grid_z0 = griddata(r, e, (grid_x, grid_y), method='linear')
plt.imshow(grid_z0.T, aspect='equal', extent=(0.0, 1.0, 0.0, 1.0),
origin='lower')
plt.colorbar(orientation='vertical')
plt.grid()
print('Stress field')
for j in range(3):
for i in range(3):
s[i] = sG[j, i]
plt.figure()
grid_z0 = griddata(r, s, (grid_x, grid_y), method='linear')
plt.imshow(grid_z0.T, aspect='equal', extent=(0.0, 1.0, 0.0, 1.0),
origin='lower')
plt.colorbar(orientation='vertical')
plt.grid()
def locstrain3nT(ul, coord, enu, Emod):
"""
Plot strain and stress for a linear 3-noded element in the
natural space
Parameters
----------
ul : ndarray (float)
Array with the displacements for the element.
coord : ndarray (float)
Coordinates for the nodes of the element.
enu : float
Poisson coefficient in the range (-1, 0.5).
Emod : float
Young modulus, should be greater than zero.
"""
r = np.zeros([3, 2])
eG = np.zeros([3, 3])
sG = np.zeros([3, 3])
eps = np.zeros([3, 3])
e = np.zeros([3])
s = np.zeros([3])
C = np.zeros([3, 3])
r[0, 0] = 0.0
r[0, 1] = 0.0
r[1, 0] = 1.0
r[1, 1] = 0.0
r[2, 0] = 0.0
r[2, 1] = 1.0
C = fe.umat(enu, Emod)
for i in range(3):
ri = r[i, 0]
si = r[i, 1]
ddet, B = fe.stdm3NT(ri, si, coord)
eps = B * ul
sig = C * eps
eG[:, i] = eps[:]
sG[:, i] = sig[:]
grid_x, grid_y = np.mgrid[0:1:100j, 0:1:100j]
print('Strain field')
for j in range(3):
for i in range(3):
e[i] = eG[j, i]
plt.figure()
grid_z0 = griddata(r, e, (grid_x, grid_y), method='linear')
plt.imshow(grid_z0.T,
aspect='equal',
extent=(0.0, 1.0, 0.0, 1.0),
origin='lower')
plt.colorbar(orientation='vertical')
plt.grid()
print('Stress field')
for j in range(3):
for i in range(3):
s[i] = sG[j, i]
plt.figure()
grid_z0 = griddata(r, s, (grid_x, grid_y), method='linear')
plt.imshow(grid_z0.T,
aspect='equal',
extent=(0.0, 1.0, 0.0, 1.0),
origin='lower')
plt.colorbar(orientation='vertical')
plt.grid()