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direct_comparison_small_strain.py
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Sat, Nov 23, 16:50

direct_comparison_small_strain.py
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## Most of the following is copied from https://github.com/tdegeus/GooseFFT (MIT license)
import numpy as np
import scipy.sparse.linalg as sp
import itertools
import os
import sys
sys.path.append(os.path.join(os.getcwd(), "language_bindings/python"))
import muSpectre as µ
# ----------------------------------- GRID ------------------------------------
ndim = 2 # number of dimensions
N = 31 #31 # number of voxels (assumed equal for all directions)
offset = 3 #9
ndof = ndim**2*N**2 # number of degrees-of-freedom
cell = µ.SystemFactory(µ.get_2d_cube(N),
µ.get_2d_cube(1.),
µ.Formulation.small_strain)
# ---------------------- PROJECTION, TENSORS, OPERATIONS ----------------------
# tensor operations/products: np.einsum enables index notation, avoiding loops
# e.g. ddot42 performs $C_ij = A_ijkl B_lk$ for the entire grid
trans2 = lambda A2 : np.einsum('ij... ->ji... ',A2 )
ddot42 = lambda A4,B2: np.einsum('ijkl...,lk... ->ij... ',A4,B2)
ddot44 = lambda A4,B4: np.einsum('ijkl...,lkmn...->ijmn...',A4,B4)
dot22 = lambda A2,B2: np.einsum('ij... ,jk... ->ik... ',A2,B2)
dot24 = lambda A2,B4: np.einsum('ij... ,jkmn...->ikmn...',A2,B4)
dot42 = lambda A4,B2: np.einsum('ijkl...,lm... ->ijkm...',A4,B2)
dyad22 = lambda A2,B2: np.einsum('ij... ,kl... ->ijkl...',A2,B2)
shape = tuple((N for _ in range(ndim)))
# identity tensor [single tensor]
i = np.eye(ndim)
def expand(arr):
new_shape = (np.prod(arr.shape), np.prod(shape))
ret_arr = np.zeros(new_shape)
ret_arr[:] = arr.reshape(-1)[:, np.newaxis]
return ret_arr.reshape((*arr.shape, *shape))
# identity tensors [grid of tensors]
I = expand(i)
I4 = expand(np.einsum('il,jk',i,i))
I4rt = expand(np.einsum('ik,jl',i,i))
I4s = (I4+I4rt)/2.
II = dyad22(I,I)
# projection operator [grid of tensors]
# NB can be vectorized (faster, less readable), see: "elasto-plasticity.py"
# - support function / look-up list / zero initialize
delta = lambda i,j: np.float(i==j) # Dirac delta function
freq = np.arange(-(N-1)/2.,+(N+1)/2.) # coordinate axis -> freq. axis
Ghat4 = np.zeros([ndim,ndim,ndim,ndim,N,N]) # zero initialize
# - compute
for i,j,l,m in itertools.product(range(ndim),repeat=4):
for x,y in itertools.product(range(N), repeat=ndim):
q = np.array([freq[x], freq[y]]) # frequency vector
if not q.dot(q) == 0: # zero freq. -> mean
Ghat4[i,j,l,m,x,y] = -(q[i]*q[j]*q[l]*q[m])/(q.dot(q))**2+\
(delta(j,l)*q[i]*q[m]+delta(j,m)*q[i]*q[l]+\
delta(i,l)*q[j]*q[m]+delta(i,m)*q[j]*q[l])/(2.*q.dot(q))
# (inverse) Fourier transform (for each tensor component in each direction)
fft = lambda x: np.fft.fftshift(np.fft.fftn (np.fft.ifftshift(x),shape))
ifft = lambda x: np.fft.fftshift(np.fft.ifftn(np.fft.ifftshift(x),shape))
# functions for the projection 'G', and the product 'G : K : eps'
G = lambda A2 : np.real( ifft( ddot42(Ghat4,fft(A2)) ) ).reshape(-1)
K_deps = lambda depsm: ddot42(K4,depsm.reshape(ndim,ndim,N,N))
G_K_deps = lambda depsm: G(K_deps(depsm))
# ------------------- PROBLEM DEFINITION / CONSTITIVE MODEL -------------------
# phase indicator: cubical inclusion of volume fraction (9**3)/(31**3)
E2, E1 = 75e10, 70e9
poisson = .33
hard = µ.material.MaterialHooke2d.make(cell, "hard",
E2, poisson)
soft = µ.material.MaterialHooke2d.make(cell, "soft",
E1, poisson)
#for pixel in cell:
# if ((pixel[0] >= N-offset) and
# (pixel[1] < offset)):
# hard.add_pixel(pixel)
# else:
# soft.add_pixel(pixel)
#
phase = np.zeros(shape); phase[-offset:,:offset,] = 1.
phase = np.zeros(shape); phase[0,:] = 1.
phase = np.zeros(shape);
for i, j in itertools.product(range(N), repeat=ndim):
c = N//2
if (i-c)**2 + (j-c)**2 < (N//5)**2:
#if j<10:
phase[i,j] = 1.
hard.add_pixel([i,j])
else:
soft.add_pixel([i,j])
# material parameters + function to convert to grid of scalars
param = lambda M0,M1: M0*np.ones(shape)*(1.-phase)+M1*np.ones(shape)*phase
# K = param(0.833,8.33) # bulk modulus [grid of scalars]
# mu = param(0.386,3.86) # shear modulus [grid of scalars]
K2, K1 = (E/(3*(1-2*poisson)) for E in (E2, E1))
m2, m1 = (E/(2*(1+poisson)) for E in (E2, E1))
K = param(K1, K2)
mu = param(m1, m2)
# stiffness tensor [grid of tensors]
K4 = K*II+2.*mu*(I4s-1./3.*II)
# ----------------------------- NEWTON ITERATIONS -----------------------------
# initialize stress and strain tensor [grid of tensors]
sig = np.zeros([ndim,ndim,N,N])
eps = np.zeros([ndim,ndim,N,N])
# set macroscopic loading
DE = np.zeros([ndim,ndim,N,N])
DE[0,1] += 0.01
DE[1,0] += 0.01
delEps0 = DE[:ndim, :ndim, 0, 0]
µDE = np.zeros([ndim**2, cell.size()])
cell.evaluate_stress_tangent(µDE);
µDE[:,:] = DE[:,:,0,0].reshape([-1, 1])
# initial residual: distribute "DE" over grid using "K4"
b = -G_K_deps(DE)
G_K_deps2 = lambda de: cell.directional_stiffness(de.reshape(µDE.shape)).ravel()
b2 = -G_K_deps2(µDE).ravel()
print("b2.shape = {}".format(b2.shape))
eps += DE
En = np.linalg.norm(eps)
iiter = 0
# iterate as long as the iterative update does not vanish
class accumul(object):
def __init__(self):
self.counter = 0
def __call__(self, x):
self.counter += 1
print("at step {}: ||Ax-b||/||b|| = {}".format(
self.counter,
np.linalg.norm(G_K_deps(x)-b)/np.linalg.norm(b)))
acc = accumul()
cg_tol = 1e-8
maxiter = 60
solver = µ.solvers.SolverCGEigen(cell, cg_tol, maxiter, verbose=True)
solver = µ.solvers.SolverCG(cell, cg_tol, maxiter, verbose=True)
try:
r = µ.solvers.newton_cg(cell, delEps0, solver, 1e-5, verbose=True)
except Exception as err:
print(err)
while True:
depsm,_ = sp.cg(tol=cg_tol,
A = sp.LinearOperator(shape=(ndof,ndof),matvec=G_K_deps,dtype='float'),
b = b,
callback=acc
) # solve linear system using CG
#depsm2,_ = sp.cg(tol=1.e-8,
# A = sp.LinearOperator(shape=(ndof,ndof),matvec=G_K_deps2,dtype='float'),
# b = b2,
# callback=acc
#) # solve linear system using CG
eps += depsm.reshape(ndim,ndim,N,N) # update DOFs (array -> tens.grid)
sig = ddot42(K4,eps) # new residual stress
b = -G(sig) # convert residual stress to residual
print('%10.2e'%(np.max(depsm)/En)) # print residual to the screen
print(np.linalg.norm(depsm)/np.linalg.norm(En))
if np.linalg.norm(depsm)/En<1.e-5 and iiter>0: break # check convergence
iiter += 1
print("nb_cg: {0}".format(acc.counter))
import matplotlib.pyplot as plt
s11 = sig[0,0, :,:]
s22 = sig[1,1, :,:]
s21_2 = sig[0,1, :, :]*sig[1,0,:, :]
vonM1 = np.sqrt(.5*((s11-s22)**2) + s11**2 + s22**2 + 6*s21_2)
#vonM1 = np.sqrt(sig[0, 0, :, :]**2 + sig[1, 1, :, :]**2 - sig[0, 0, :, :] * sig[1, 1, :, :] + 3 * sig[0,1]**2)
plt.figure()
img = plt.pcolormesh(vonM1)#eps[0,1,:,:])
plt.title("goose")
plt.colorbar(img)
try:
print(r.stress.shape)
arr = r.stress.T.reshape(N, N, ndim, ndim)
s11 = arr[:,:,0,0]
s22 = arr[:,:,1,1]
s21_2 = arr[:,:,0,1]*arr[:,:,1,0]
vonM2 = np.sqrt(.5*((s11-s22)**2) + s11**2 + s22**2 + 6*s21_2)
plt.figure()
plt.title("µSpectre")
img = plt.pcolormesh(vonM2)#eps[0,1,:,:])
plt.colorbar(img)
print("err = {}".format (np.linalg.norm(vonM1-vonM2)))
print("err2 = {}".format (np.linalg.norm(vonM1-vonM1.T)))
print("err3 = {}".format (np.linalg.norm(vonM2-vonM2.T)))
plt.figure()
plt.title("diff")
img = plt.pcolormesh(vonM1-vonM2.T)#eps[0,1,:,:])
plt.colorbar(img)
except Exception as err:
print(err)
plt.show()

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