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python_goose_ref.py
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#!/usr/bin/env python3
# -*- coding:utf-8 -*-
"""
file python_goose_ref.py
@author Till Junge <till.junge@altermail.ch>
@date 19 Jan 2018
@brief adapted scripts from GooseFFT, https://github.com/tdegeus/GooseFFT,
which are MIT licensed
@section LICENSE
Copyright © 2018 Till Junge
µSpectre is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public License as
published by the Free Software Foundation, either version 3, or (at
your option) any later version.
µSpectre is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
General Public License for more details.
You should have received a copy of the GNU Lesser General Public License
along with µSpectre; see the file COPYING. If not, write to the
Free Software Foundation, Inc., 59 Temple Place - Suite 330,
Boston, MA 02111-1307, USA.
Additional permission under GNU GPL version 3 section 7
If you modify this Program, or any covered work, by linking or combining it
with proprietary FFT implementations or numerical libraries, containing parts
covered by the terms of those libraries' licenses, the licensors of this
Program grant you additional permission to convey the resulting work.
"""
import numpy as np
import scipy.sparse.linalg as sp
import itertools
def get_bulk_shear(E, nu):
return E/(3*(1-2*nu)), E/(2*(1+nu))
class ProjectionGooseFFT(object):
def __init__(self, ndim, resolution, incl_size, E, nu, contrast):
"""
wraps the GooseFFT hyper-elasticity script into a more user-friendly
class
Keyword Arguments:
ndim -- number of dimensions of the problem, should be 2 or 3
resolution -- pixel resolution, integer
incl_size -- edge length of cubic hard inclusion in pixels
E -- Young's modulus of soft phase
nu -- Poisson's ratio
constrast -- ratio between hard and soft Young's modulus
"""
self.ndim = ndim
self.resolution = resolution
self.incl_size = incl_size
self.E = E
self.nu = nu
self.contrast = contrast
self.Kval, self.mu = get_bulk_shear(E, nu)
self.setup()
def setup(self):
ndim=self.ndim
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)
i = np.eye(ndim)
# identity tensors [grid of tensors]
shape = tuple((self.resolution for _ in range(ndim)))
oneblock = np.ones(shape)
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))
I = expand(i)
self.I = 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
N = self.resolution
freq = np.fft.fftfreq(N, 1/N) # coordinate axis -> freq. axis
Ghat4 = np.zeros([ndim,ndim,ndim,ndim,*shape]) # zero initialize
# - compute
for xyz in itertools.product(range(N), repeat=self.ndim):
q = np.array([freq[index] for index in xyz]) # frequency vector
index = tuple((*(slice(None) for _ in range(4)), *xyz))
Ghat4[index] = self.comp_ghat(q)
# (inverse) Fourier transform (for each tensor component in each direction)
fft = lambda x : np.fft.fftn (x, shape)
ifft = lambda x : np.fft.ifftn(x, shape)
# functions for the projection 'G', and the product 'G : K^LT : (delta F)^T'
G = lambda A2 : np.real( ifft( ddot42(Ghat4,fft(A2)) ) ).reshape(-1)
K_dF = lambda dFm: trans2(ddot42(self.K4,trans2(dFm.reshape(ndim,ndim,*shape))))
G_K_dF = lambda dFm: G(K_dF(dFm))
K_deps = lambda depsm: ddot42(self.C4,depsm.reshape(ndim,ndim,N,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)
incl = self.incl_size
phase = np.zeros(shape)
if self.ndim == 2:
phase[-incl:,:incl] = 1.
else:
phase[-incl:,:incl,-incl:] = 1.
# material parameters + function to convert to grid of scalars
param = lambda M0,M1: M0*oneblock*(1.-phase)+M1*oneblock*phase
K = param(self.Kval, self.contrast*self.Kval)
mu = param(self.mu, self.contrast*self.mu)
# constitutive model: grid of "F" -> grid of "P", "K4" [grid of tensors]
self.C4 = K*II+2.*mu*(I4s-1./3.*II)
def constitutive(F):
C4 = self.C4
S = ddot42(C4,.5*(dot22(trans2(F),F)-I))
P = dot22(F,S)
K4 = dot24(S,I4)+ddot44(ddot44(I4rt,dot42(dot24(F,C4),trans2(F))),I4rt)
self.K4 = K4
self.P = P
return P,K4
self.constitutive = constitutive
self.G = G
self.G_K_dF = G_K_dF
self.Ghat4 = Ghat4
self.G_K_deps = G_K_deps
class FiniteStrainProjectionGooseFFT(ProjectionGooseFFT):
def __init__(self, ndim, resolution, incl_size, E, nu, contrast):
super().__init__(ndim, resolution, incl_size, E, nu, contrast)
def comp_ghat(self, q):
temp = np.zeros((self.ndim, self.ndim, self.ndim, self.ndim))
delta = lambda i,j: np.float(i==j) # Dirac delta function
if not q.dot(q) == 0: # zero freq. -> mean
for i,j,l,m in itertools.product(range(self.ndim),repeat=4):
temp[i, j, l, m] = delta(i,m)*q[j]*q[l]/(q.dot(q))
return temp
def run(self):
ndim = self.ndim
shape = tuple((self.resolution for _ in range(ndim)))
# ----------------------------- NEWTON ITERATIONS -----------------------------
# initialize deformation gradient, and stress/stiffness [grid of tensors]
F = np.array(self.I,copy=True)
P,K4 = self.constitutive(F)
# set macroscopic loading
zer_shap = (ndim, ndim, *shape)
DbarF = np.zeros(zer_shap); DbarF[0,1] += 1.0
# initial residual: distribute "barF" over grid using "K4"
b = -self.G_K_dF(DbarF)
F += DbarF
Fn = np.linalg.norm(F)
iiter = 0
# iterate as long as the iterative update does not vanish
class accumul(object):
def __init__(self):
self.counter = 0
def __call__(self, dummy):
self.counter += 1
acc = accumul()
while True:
dFm,_ = sp.cg(tol=1.e-8,
A = sp.LinearOperator(shape=(
F.size,F.size),matvec=self.G_K_dF,dtype='float'),
b = b,
callback=acc
) # solve linear cell using CG
F += dFm.reshape(ndim,ndim,*shape) # update DOFs (array -> tens.grid)
P,K4 = self.constitutive(F) # new residual stress and tangent
b = -self.G(P) # convert res.stress to residual
print('%10.2e'%(np.linalg.norm(dFm)/Fn)) # print residual to the screen
if np.linalg.norm(dFm)/Fn<1.e-5 and iiter>0: break # check convergence
iiter += 1
print("nb_cg: {0}".format(acc.counter))
class SmallStrainProjectionGooseFFT(ProjectionGooseFFT):
def __init__(self, ndim, resolution, incl_size, E, nu, contrast):
super().__init__(ndim, resolution, incl_size, E, nu, contrast)
def comp_ghat(self, q):
temp = np.zeros((self.ndim, self.ndim, self.ndim, self.ndim))
delta = lambda i,j: np.float(i==j) # Dirac delta function
if not q.dot(q) == 0: # zero freq. -> mean
for i,j,l,m in itertools.product(range(self.ndim),repeat=4):
temp[i, j, l, m] = -(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))
return temp
def tangent_stiffness(self, field):
return self.constitutive(F)[0]
def run(self):
ndim = self.ndim
shape = tuple((self.resolution for _ in range(ndim)))
# ----------------------------- NEWTON ITERATIONS -----------------------------
# initialize stress and strain tensor [grid of tensors]
sig = np.zeros([ndim,ndim,N,N,N])
eps = np.zeros([ndim,ndim,N,N,N])
# set macroscopic loading
DE = np.zeros([ndim,ndim,N,N,N])
DE[0,1] += 0.01
DE[1,0] += 0.01
# initial residual: distribute "barF" over grid using "K4"
b = -self.G_K_deps(DE)
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, dummy):
self.counter += 1
acc = accumul()
while True:
depsm,_ = sp.cg(tol=1.e-8,
A = sp.LinearOperator(shape=(
eps.size,eps.size),matvec=self.G_K_deps,dtype='float'),
b = b,
callback=acc
) # solve linear cell using CG
eps += depsm.reshape(ndim,ndim,*shape) # update DOFs (array -> tens.grid)
sig = ddot42(self.C4, eps) # new residual stress and tangent
b = -self.G(sig) # convert res.stress to residual
print('%10.2e'%(np.linalg.norm(depsm)/En)) # print residual to the screen
if np.linalg.norm(depsm)/en<1.e-5 and iiter>0: break # check convergence
iiter += 1
print("nb_cg: {0}".format(acc.counter))

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