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rGOOSEFEM/docs/theory/examples/axisymmetric_linear-elasticity.py
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axisymmetric_linear-elasticity.py
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import numpy as np
np.set_printoptions(linewidth=200)
# ==================================================================================================
# tensor products
# ==================================================================================================
def ddot43(A,B):
nd = A.shape[0]
C = np.zeros((nd,nd,nd))
for i in range(nd):
for j in range(nd):
for k in range(nd):
for l in range(nd):
for m in range(nd):
C[i,j,m] += A[i,j,k,l] * B[l,k,m]
return C
# --------------------------------------------------------------------------------------------------
def ddot33(A,B):
nd = A.shape[0]
C = np.zeros((nd,nd))
for i in range(nd):
for j in range(nd):
for k in range(nd):
for l in range(nd):
C[i,l] += A[i,j,k] * B[k,j,l]
return C
# --------------------------------------------------------------------------------------------------
def transpose3(A):
nd = A.shape[0]
C = np.zeros((nd,nd,nd))
for i in range(nd):
for j in range(nd):
for k in range(nd):
C[k,j,i] = A[i,j,k]
return C
# ==================================================================================================
# identity tensors
# ==================================================================================================
II = np.zeros((3,3,3,3))
I4 = np.zeros((3,3,3,3))
I4rt = np.zeros((3,3,3,3))
Is = np.zeros((3,3,3,3))
for i in range(3):
for j in range(3):
for k in range(3):
for l in range(3):
if ( i == j and k == l ):
II[i,j,k,l] = 1.
for i in range(3):
for j in range(3):
for k in range(3):
for l in range(3):
if ( i == l and j == k ):
I4[i,j,k,l] = 1.
for i in range(3):
for j in range(3):
for k in range(3):
for l in range(3):
if ( i == k and j == l ):
I4rt[i,j,k,l] = 1.
I4s = ( I4 + I4rt ) / 2.
I4d = ( I4s - II/3. )
# ==================================================================================================
# elasticity tensor
# ==================================================================================================
# bulk and shear modulus
K = 0.8333333333333333
G = 0.3846153846153846
# elasticity tensor (3d)
C4 = K * II + 2. * G * I4d
# ==================================================================================================
# mesh definition
# ==================================================================================================
# number of elements
nz = 20
nr = 16
# mesh dimensions
nelem = nz * nr # number of elements
nnode = (nz+1) * (nr+1) # number of nodes
nne = 4 # number of nodes per element
nd = 2 # number of dimensions
ndof = nnode * nd # total number of degrees of freedom
# coordinates and connectivity: zero-initialize
coor = np.zeros((nnode,nd ), dtype='float')
conn = np.zeros((nelem,nne), dtype='int' )
# coordinates: set
# - grid of points
z,r = np.meshgrid(np.linspace(0,1,nz+1), np.linspace(0,1,nr+1))
# - store from grid of points
coor[:,0] = z.ravel()
coor[:,1] = r.ravel()
# connectivity: set
# - grid of node numbers
inode = np.arange(nnode).reshape(nr+1, nz+1)
# - store from grid of node numbers
conn[:,0] = inode[ :-1, :-1].ravel()
conn[:,1] = inode[ :-1,1: ].ravel()
conn[:,2] = inode[1: ,1: ].ravel()
conn[:,3] = inode[1: , :-1].ravel()
# DOF-numbers per node
dofs = np.arange(nnode*nd).reshape(nnode,nd)
# ==================================================================================================
# quadrature definition
# ==================================================================================================
# integration point coordinates (local element coordinates)
Xi = np.array([
[-1./np.sqrt(3.), -1./np.sqrt(3.)],
[+1./np.sqrt(3.), -1./np.sqrt(3.)],
[+1./np.sqrt(3.), +1./np.sqrt(3.)],
[-1./np.sqrt(3.), +1./np.sqrt(3.)],
])
# integration point weights
W = np.array([
[1.],
[1.],
[1.],
[1.],
])
# ==================================================================================================
# assemble stiffness matrix
# ==================================================================================================
# allocate matrix
K = np.zeros((ndof, ndof))
# loop over nodes
for e in conn:
# - nodal coordinates
xe = coor[e,:]
# - allocate element stiffness matrix
Ke = np.zeros((nne*nd, nne*nd))
# - numerical quadrature
for w, xi in zip(W, Xi):
# -- shape functions
N = np.array([
[.25 * (1.-xi[0]) * (1.-xi[1])],
[.25 * (1.+xi[0]) * (1.-xi[1])],
[.25 * (1.+xi[0]) * (1.+xi[1])],
[.25 * (1.-xi[0]) * (1.+xi[1])],
])
# -- shape function gradients (w.r.t. the local element coordinates)
dNdxi = np.array([
[-.25*(1.-xi[1]), -.25*(1.-xi[0])],
[+.25*(1.-xi[1]), -.25*(1.+xi[0])],
[+.25*(1.+xi[1]), +.25*(1.+xi[0])],
[-.25*(1.+xi[1]), +.25*(1.-xi[0])],
])
# -- Jacobian
J = np.zeros((nd, nd))
for m in range(nne):
for i in range(nd):
for j in range(nd):
J[i,j] += dNdxi[m,i] * xe[m,j]
Jinv = np.linalg.inv(J)
Jdet = np.linalg.det(J)
# -- shape function gradients (w.r.t. the global coordinates)
dNdx = np.zeros((nne,nd))
for m in range(nne):
for i in range(nd):
for j in range(nd):
dNdx[m,i] += Jinv[i,j] * dNdxi[m,j]
# -- global coordinates of the integration point
xk = np.zeros((nd))
for n in range(nne):
for i in range(nd):
xk[i] += N[n] * xe[n,i]
rk = xk[1]
# -- assemble to element stiffness matrix
for m in range(nne):
Bm = np.zeros((3,3,3))
Bm[0,0,0] = dNdx[m,1] # B(m, r r r )
Bm[0,1,1] = -1./rk * N[m] # B(m, r \theta \theta )
Bm[0,1,1] = dNdx[m,1] # B(m, r \theta \theta )
Bm[0,2,2] = dNdx[m,1] # B(m, r z z )
Bm[1,0,0] = 0. # B(m, \theta r r ) - axisymmetric
Bm[1,1,0] = +1./rk * N[m] # B(m, \theta \theta r )
Bm[1,1,1] = 0. # B(m, \theta \theta \theta ) - axisymmetric
Bm[1,2,2] = 0. # B(m, \theta z z ) - axisymmetric
Bm[2,0,0] = dNdx[m,0] # B(m, z r r )
Bm[2,1,1] = dNdx[m,0] # B(m, z \theta \theta )
Bm[2,2,2] = dNdx[m,0] # B(m, z z z )
for n in range(nne):
Bn = np.zeros((3,3,3))
Bn[0,0,0] = dNdx[n,1] # B(n, r r r )
Bn[0,1,1] = -1./rk * N[n] # B(n, r \theta \theta )
Bn[0,1,1] = dNdx[n,1] # B(n, r \theta \theta )
Bn[0,2,2] = dNdx[n,1] # B(n, r z z )
Bn[1,0,0] = 0. # B(n, \theta r r ) - axisymmetric
Bn[1,1,0] = +1./rk * N[n] # B(n, \theta \theta r )
Bn[1,1,1] = 0. # B(n, \theta \theta \theta ) - axisymmetric
Bn[1,2,2] = 0. # B(n, \theta z z ) - axisymmetric
Bn[2,0,0] = dNdx[n,0] # B(n, z r r )
Bn[2,1,1] = dNdx[n,0] # B(n, z \theta \theta )
Bn[2,2,2] = dNdx[n,0] # B(n, z z z )
Kmn = ddot33(transpose3(Bm),ddot43(C4,Bn))
iim = np.array([ m*nd+0, m*nd+1 ])
iin = np.array([ n*nd+0, n*nd+1 ])
Ke[np.ix_(iim,iin)] += Kmn[np.ix_([2,0], [2,0])] * w * Jdet * rk * 2. * np.pi
# - assemble to global stiffness matrix
iie = dofs[e,:].ravel()
K[np.ix_(iie,iie)] += Ke
# ==================================================================================================
# partition and solve
# ==================================================================================================
# zero-initialize displacements and forces
f = np.zeros((ndof))
u = np.zeros((ndof))
# fixed displacements
# - zero-initialize
iip = np.empty((0),dtype='int' )
up = np.empty((0),dtype='float')
# - 'r = 0' : 'u_r = 0'
idof = dofs[inode[0,:], 1]
iip = np.hstack(( iip, idof ))
up = np.hstack(( up , 0.0 * np.ones(idof.shape) ))
# - 'z = 0' : 'u_z = 0'
idof = dofs[inode[:,0], 0]
iip = np.hstack(( iip, idof ))
up = np.hstack(( up , 0.0 * np.ones(idof.shape) ))
# - 'z = 1' : 'u_z = 0.1'
idof = dofs[inode[:,-1], 0]
iip = np.hstack(( iip, idof ))
up = np.hstack(( up , 0.1 * np.ones(idof.shape) ))
# free displacements
iiu = np.setdiff1d(dofs.ravel(), iip)
# partition
# - stiffness matrix
Kuu = K[np.ix_(iiu, iiu)]
Kup = K[np.ix_(iiu, iip)]
Kpu = K[np.ix_(iip, iiu)]
Kpp = K[np.ix_(iip, iip)]
# - external force
fu = f[iiu]
# solve
uu = np.linalg.solve(Kuu, fu - Kup.dot(up))
fp = Kpu.dot(uu) + Kpp.dot(up)
# assemble
u[iiu] = uu
u[iip] = up
f[iip] = fp
# convert to nodal displacement and forces
disp = u[dofs]
fext = f[dofs]
# ==================================================================================================
# plot
# ==================================================================================================
import matplotlib.pyplot as plt
fig, axes = plt.subplots(ncols=2, figsize=(16,8))
axes[0].scatter(coor[:,0], coor[:,1])
axes[0].quiver (coor[:,0], coor[:,1], disp[:,0], disp[:,1])
axes[1].scatter((coor+disp)[:,0], (coor+disp)[:,1])
axes[1].quiver ((coor+disp)[:,0], (coor+disp)[:,1], fext[:,0], fext[:,1], scale=.4)
for axis in axes:
axis.set_xlim([-0.4, 1.5])
axis.set_ylim([-0.4, 1.5])
plt.show()
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