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rGOOSEFEM/docs/theory/examples/cartesian_linear-elasticity_B-matrix.py
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cartesian_linear-elasticity_B-matrix.py
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# ==================================================================================================
#
# (c - GPLv3) T.W.J. de Geus (Tom) | tom@geus.me | www.geus.me | github.com/tdegeus/GooseFEM
#
# ==================================================================================================
import matplotlib.pyplot as plt
import numpy as np
np.set_printoptions(linewidth=200)
# ==================================================================================================
# tensor products
# ==================================================================================================
def ddot22(A2, B2):
return np.einsum("...ij,...ji->...", A2, B2)
def ddot42(A4, B2):
return np.einsum("...ijkl,...lk->...ij", A4, B2)
def dyad22(A2, B2):
return np.einsum("...ij,...kl->...ijkl", A2, B2)
# ==================================================================================================
# mesh definition
# ==================================================================================================
# number of elements
nx = 20
ny = 20
# mesh dimensions
nelem = nx * ny # number of elements
nnode = (nx + 1) * (ny + 1) # number of nodes
nne = 4 # number of nodes per element
ndim = 2 # number of dimensions
ndof = nnode * ndim # total number of degrees of freedom
# out-of-plane thickness
thick = 1.0
# zero-initialise coordinates, displacements, and connectivity
coor = np.zeros((nnode, ndim), dtype="float")
disp = np.zeros((nnode, ndim), dtype="float")
conn = np.zeros((nelem, nne), dtype="int")
# coordinates
# - grid of points
x, y = np.meshgrid(np.linspace(0, 1, nx + 1), np.linspace(0, 1, ny + 1))
# - store from grid of points
coor[:, 0] = x.ravel()
coor[:, 1] = y.ravel()
# connectivity
# - grid of node numbers
inode = np.arange(nnode).reshape(ny + 1, nx + 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()
# - node sets
nodesLeft = inode[:, 0]
nodesRight = inode[:, -1]
nodesBottom = inode[0, :]
nodesTop = inode[-1, :]
# DOF-numbers per node
dofs = np.arange(nnode * ndim).reshape(nnode, ndim)
# ==================================================================================================
# quadrature definition
# ==================================================================================================
# integration point coordinates (local element coordinates)
Xi = np.array(
[
[-1.0 / np.sqrt(3.0), -1.0 / np.sqrt(3.0)],
[+1.0 / np.sqrt(3.0), -1.0 / np.sqrt(3.0)],
[+1.0 / np.sqrt(3.0), +1.0 / np.sqrt(3.0)],
[-1.0 / np.sqrt(3.0), +1.0 / np.sqrt(3.0)],
]
)
# integration point weights
W = np.array(
[
[1.0],
[1.0],
[1.0],
[1.0],
]
)
# number of integration points
nip = 4
# ==================================================================================================
# B-matrix at each integration point
# ==================================================================================================
# allocate
B = np.empty((nelem, nip, nne, 3, 3, 3))
vol = np.empty((nelem, nip))
# loop over nodes
for e, nodes in enumerate(conn):
# nodal coordinates
xe = coor[nodes, :]
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# shape function gradients (w.r.t. the local element coordinates)
dNdxi = np.array(
[
[-0.25 * (1.0 - xi[1]), -0.25 * (1.0 - xi[0])],
[+0.25 * (1.0 - xi[1]), -0.25 * (1.0 + xi[0])],
[+0.25 * (1.0 + xi[1]), +0.25 * (1.0 + xi[0])],
[-0.25 * (1.0 + xi[1]), +0.25 * (1.0 - xi[0])],
]
)
# Jacobian
Je = np.einsum("mi,mj->ij", dNdxi, xe)
invJe = np.linalg.inv(Je)
detJe = np.linalg.det(Je)
# shape function gradients (w.r.t. the global coordinates)
dNdxe = np.einsum("ij,mj->mi", invJe, dNdxi)
# compute B-matrix and integration-point volume and store for later use
for m in range(nne):
Be = np.zeros((3, 3, 3))
Be[0, 0, 0] = dNdxe[m, 0]
Be[0, 1, 1] = dNdxe[m, 0]
Be[1, 0, 0] = dNdxe[m, 1]
Be[1, 1, 1] = dNdxe[m, 1]
B[e, q, m, :, :, :] = Be
vol[e, q] = w * detJe * thick
# ==================================================================================================
# stiffness tensor at each integration point (provides constitutive response and 'tangent')
# ==================================================================================================
# identity tensors (per integration point)
i = np.eye(3)
I2 = np.einsum("ij ,...", i, np.ones([nelem, nip]))
I4 = np.einsum("ijkl,...->...ijkl", np.einsum("il,jk", i, i), np.ones([nelem, nip]))
I4rt = np.einsum("ijkl,...->...ijkl", np.einsum("ik,jl", i, i), np.ones([nelem, nip]))
I4s = (I4 + I4rt) / 2.0
II = dyad22(I2, I2)
I4d = I4s - II / 3.0
# bulk and shear modulus (homogeneous)
K = 0.8333333333333333 * np.ones((nelem, nip))
G = 0.3846153846153846 * np.ones((nelem, nip))
# elasticity tensor (per integration point)
C4 = np.einsum("eq,eqijkl->eqijkl", K, II) + 2.0 * np.einsum("eq,eqijkl->eqijkl", G, I4d)
# ==================================================================================================
# strain from nodal displacements, stress from constitutive response
# ==================================================================================================
# allocate strain tensor per integration point
Eps = np.empty((nelem, nip, 3, 3))
# loop over nodes
for e, nodes in enumerate(conn):
# nodal displacements in 3-d
# u_z = 0
ue = np.zeros((nne, 3))
ue[:, 0] = disp[nodes, 0]
ue[:, 1] = disp[nodes, 1]
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# alias integration point values
Be = B[e, q, :, :, :, :]
# displacement gradient
gradu = np.einsum("mijk,mk->ij", Be, ue)
# compute strain tensor, and store per integration point
Eps[e, q] = 0.5 * (gradu + gradu.T)
# constitutive response: strain tensor -> stress tensor (per integration point)
Sig = ddot42(C4, Eps)
# ==================================================================================================
# internal force from stress
# ==================================================================================================
# allocate internal force
fint = np.zeros(ndof)
# loop over nodes
for e, nodes in enumerate(conn):
# allocate element internal force
fe = np.zeros(nne * ndim)
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# alias integration point values
sig = Sig[e, q, :, :]
Be = B[e, q, :, :, :, :]
dV = vol[e, q]
# assemble to element internal force
# (plane strain: force in z-direction irrelevant)
# Bm = B[e,q,m,:,:,:]
# fm = ddot32(transpose3(Bm), sig) * dV
# fe[[m*ndim+0, m*ndim+1]] = [fm[m,2], fm[m,0]]
fq = np.einsum("mijk,ij->mk", Be, sig) * dV
fe += fq[:, :2].reshape(nne * ndim)
# assemble to global stiffness matrix
iie = dofs[nodes, :].ravel()
fint[iie] += fe
# ==================================================================================================
# stiffness matrix from 'tangent'
# ==================================================================================================
# allocate stiffness matrix
K = np.zeros((ndof, ndof))
# loop over nodes
for e, nodes in enumerate(conn):
# allocate element stiffness matrix
Ke = np.zeros((nne * ndim, nne * ndim))
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# alias integration point values
c4 = C4[e, q, :, :, :, :]
Be = B[e, q, :, :, :, :]
dV = vol[e, q]
# assemble to element stiffness matrix
# Bm = B[e,q,m,:,:,:]
# Bn = B[e,q,n,:,:,:]
# Kmn = ddot33(transpose3(Bm),ddot43(C4,Bn)) * dV
# Ke[[m*ndim+0,m*ndim+1], [n*ndim+0,n*ndim+1]] += Kmn[[2,0], [2,0]]
Kq = np.einsum("mabc,abde,nedf->mcnf", Be, c4, Be) * dV
Ke += Kq[:, :2, :, :2].reshape(nne * ndim, nne * ndim)
# assemble to global stiffness matrix
iie = dofs[nodes, :].ravel()
K[np.ix_(iie, iie)] += Ke
# ==================================================================================================
# partition and solve
# ==================================================================================================
# prescribed external force: zero on all free DOFs
# (other DOFs ignored in solution, the reaction forces on the DOFs are computed below)
fext = np.zeros(ndof)
# DOF-partitioning: ['u'nknown, 'p'rescribed]
# - prescribed
iip = np.hstack(
(
dofs[nodesBottom, 1],
dofs[nodesLeft, 0],
dofs[nodesRight, 0],
)
)
# - unknown
iiu = np.setdiff1d(dofs.ravel(), iip)
# fixed displacements
up = np.hstack(
(
0.0 * np.ones(len(nodesBottom)),
0.0 * np.ones(len(nodesLeft)),
0.1 * np.ones(len(nodesRight)),
)
)
# residual force
r = fext - fint
# 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)]
# - residual force
ru = r[iiu]
# solve for unknown displacement DOFs
uu = np.linalg.solve(Kuu, ru - Kup.dot(up))
# assemble
u = np.empty(ndof)
u[iiu] = uu
u[iip] = up
# convert to nodal displacements
disp = u[dofs]
# ==================================================================================================
# strain from nodal displacements, stress from constitutive response
# ==================================================================================================
# allocate strain tensor per integration point
Eps = np.empty((nelem, nip, 3, 3))
# loop over nodes
for e, nodes in enumerate(conn):
# nodal displacements in 3-d
# u_z = 0
ue = np.zeros((nne, 3))
ue[:, 0] = disp[nodes, 0]
ue[:, 1] = disp[nodes, 1]
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# alias integration point values
Be = B[e, q, :, :, :, :]
# displacement gradient
gradu = np.einsum("mijk,mk->ij", Be, ue)
# compute strain tensor, and store per integration point
Eps[e, q] = 0.5 * (gradu + gradu.T)
# constitutive response: strain tensor -> stress tensor (per integration point)
Sig = ddot42(C4, Eps)
# ==================================================================================================
# internal force from stress, reaction (external) force from internal force on fixed nodes
# ==================================================================================================
# allocate internal force
fint = np.zeros(ndof)
# loop over nodes
for e, nodes in enumerate(conn):
# allocate element internal force
fe = np.zeros(nne * ndim)
# loop over integration points
for q, (w, xi) in enumerate(zip(W, Xi)):
# alias integration point values
sig = Sig[e, q, :, :]
Be = B[e, q, :, :, :, :]
dV = vol[e, q]
# assemble to element internal force
# (plane strain: force in z-direction irrelevant)
# Bm = B[e,q,m,:,:,:]
# fm = ddot32(transpose3(Bm), sig) * dV
# fe[[m*ndim+0, m*ndim+1]] = [fm[m,2], fm[m,0]]
fq = np.einsum("mijk,ij->mk", Be, sig) * dV
fe += fq[:, :2].reshape(nne * ndim)
# assemble to global stiffness matrix
iie = dofs[nodes, :].ravel()
fint[iie] += fe
# reaction force
fext[iip] = fint[iip]
# ==================================================================================================
# plot
# ==================================================================================================
fig, axes = plt.subplots(ncols=2, figsize=(16, 8))
# reconstruct external force as nodal vectors
fext = fext[dofs]
# plot original nodal positions + displacement field as arrows
axes[0].scatter(coor[:, 0], coor[:, 1])
axes[0].quiver(coor[:, 0], coor[:, 1], disp[:, 0], disp[:, 1])
# plot new nodal positions + external (reaction) force field as arrows
axes[1].scatter((coor + disp)[:, 0], (coor + disp)[:, 1])
axes[1].quiver((coor + disp)[:, 0], (coor + disp)[:, 1], fext[:, 0], fext[:, 1], scale=0.1)
# fix axes limits
for axis in axes:
axis.set_xlim([-0.4, 1.5])
axis.set_ylim([-0.4, 1.5])
plt.show()
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