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rMUSPECTRE µSpectre
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 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 General Public License
along with GNU Emacs; see the file COPYING. If not, write to the
Free Software Foundation, Inc., 59 Temple Place - Suite 330,
Boston, MA 02111-1307, USA.
"""
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 system 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 system 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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