Non-Linear Computational Mechanics ATHENS week March 2016...
Transcript of Non-Linear Computational Mechanics ATHENS week March 2016...
Identification of Material and Process
Parameters Michel Bellet
Non-Linear Computational Mechanics – ATHENS week – March 2016
NLCM - Michel Bellet 2016-03 2
Introduction to Parameter Identification
• Ingredients:
– Measurements carried out on instrumented experiments
– A numerical model of such experiments
– An optimization algorithm
• Method: example in the context of heat transfer
– Formulate the distance between measured and calculated values: "cost"
function or "objective" function
– Define a set of initial values for parameters: p(0)
– Calculate the optimal values of parameters, by use of an iterative algorithm
minimizing the cost function
p vector of Np parameters to be identified
S number of experiments
I(s) number of measurements per exp.
M. Rappaz, M. Bellet, M. Deville, Numerical modelling in materials science and engineering, Springer (2003)
S
s
sI
i
exp
is
cal
is TT1
)(
1
2
,, )-)(()( pp
NLCM - Michel Bellet 2016-03 3
Example: Gauss-Newton Algorithm
At optimum,
0)( pR
Method of Newton-Raphson type: at each iteration n,
)( )1(
)1(
n
n
pRpp
R
that is:
A simple method to evaluate sensitivity coefficients:
then: ppp nn )1()(
solve: Approached stiffness:
NB: at each iteration, Np+1 "direct" calculations: 1 to estimate , Np to estimate the
S
s
sI
i
exp
is
cal
is TT1
)(
1
2
,, )-()(p
0)-)((2,11
)(
1
,
,,
S
s
sI
i m
cal
isexp
is
cal
is
m
pp
TTT
pNm p
S
s
sI
i n
cal
is
m
cal
is
n
m
p
T
p
T
p
R
1
)(
1
,,2
p
pppTppppT
p
Tpp Nn
cal
isNn
cal
is
n
cal
is
),,,,(),,,,( 1,1,,
pT
NLCM - Michel Bellet 2016-03 4
Additional regularization
pN
qrefq
refqq
S
s
sI
iexpis
expis
calis
p
pp
T
TT
12
2
1
)(
12
,
2,,
)(
)-(
)(
)-()1()( p
Note:
- need to normalize the two terms (making them non-dimensional)
- weighting factor (original Gauss-Newton retrieved for = 0)
- minor impact on calculations of residual vector and stiffness matrix:
)(
)(
22
refmmref
mm
regulregulm pp
ppR
mnref
mn
regulm
pp
R
2)(
2
Tikhonov
Variants:
pN
qiterprev
q
iterprevqq
S
s
sI
iexpis
expis
calis
p
pp
T
TT
12_
2_
1
)(
12
,
2,,
)(
)-(
)(
)-()1()( p
0when0such that)(with
)(
)-(
)(
)-()1()(
12_
2_
1
)(
12
,
2,,
RR
p
iteriter
N
qiterprev
q
iterprevqqiter
S
s
sI
iexpis
expis
calisiter
f
p
pp
T
TT p
Levenberg
Levenberg-
Marquardt
Ph.D. A. Gavrus (Ecole des Mines de Paris, 1996) and Gavrus, Massoni, Chenot, J. Mater. Proc. Tech. 60 (1996) 447-454
NLCM - Michel Bellet 2016-03 5
Application: Rheological Identification from Hot Torsion Tests
• Hot torsion machine
– Characterization of metal behaviour at high temperature and large strains
– Steels, superalloys, aluminium alloys…
• Identification of parameters: not so easy, by hand
– High radial gradients of strains
– Radial and axial temperature gradients
• FEM inverse method
NLCM - Michel Bellet 2016-03 6
Reminder: Viscoplastic Constitutive Models
• Pure viscoplasticity, no elasticity
Law Khard W m Parameters Nb
A Tn00 e)(K
0 m0 m,,n,,K 00 5
C Tn00 e)(K
re1 Tmm 10
stst10
00
,K,m,m
,r,,n,,K
9
F
Tnnn
ee1K
10
T)(n0
0
0 Tmm 10
10
1000
m,m
,,n,n,,K 7
G
Tnnn
ee1K
10
T)(n0
0
Trrr
e1
10
r
Tmm 10
stst101
01000
,K,m,m,r
,r,,n,n,,K
11
mK
Tsatsat
sathard
sat
eKK
WWKWKK
0
10]1[
NLCM - Michel Bellet 2016-03 7
Application: Rheological Identification from Hot Torsion Tests
• Constitutive model for Ti-6%Al-4%V
– Viscoplastic model, law G, 8 parameters
Torq
ue [N
mm
]
Number of rotations [-]
Experimental measurements +++
FE computation -----
18 rpm, 800 C
180 rpm, 900 C
18 rpm, 900 C
Param.
Initial
value
Identified
value
K0 10 322,2
n0 10 314,9
n1 0 0,323
r0 0 -11,55
r1 0 0,0104
8000 9444
m0 0,1 0,358
m1x1000 0 -0,254
Iterations - 20
0,27 0,03
Ph.D. A. Gavrus (Ecole des Mines de Paris, 1996) and Gavrus, Massoni, Chenot, J. Mater. Proc. Tech. 60 (1996) 447-454
NLCM - Michel Bellet 2016-03 8
Identification from Plane Compression Test
• Cu-40%Zn-2%Pb. Constitutive model: law C
• Friction model: Tresca
• Temperature: 550, 600, 650, 700 C
• Nominal strain rate: 0.1 s-1 and 5 s-1
• 2 thickness values: 5 and 10 mm
• Observable: compression force
Experiment FEM: code FORGE®
tg
tg
tg mv
vT
3
1cm
or
0.5
cm
Ph.D. R. Forestier (Ecole des Mines, 2004) and R. Forestier, Y. Chastel, E. Massoni, Inv. Prob. Engng 11 (2003) 255-271
NLCM - Michel Bellet 2016-03 9
Strain-rate 0.1 s-1 / Two
thickness values
Strain-rate 5 s-1 / Two
thickness values
Initial values Identified values
7 iterations
NLCM - Michel Bellet 2016-03 10
Alternative Method: Metamodels
Optimization
algorithm
Calculation of
through full model (i.e. FEM)
Optimization
algorithm
Calculation of
through full model (i.e. FEM)
Evaluation of
through simplified modelling
Metamodel
NLCM - Michel Bellet 2016-03 11
Building a Metamodel by Kriging
• "Kriging" comes from the name of a mining engineer, Daniel Krige, from South Africa. The
method was formalised by Georges Matheron (1930-2000), researcher in geostatistics at
Ecole des Mines.
• A method to build a metamodel through minimum requirements to the full calculation of
– Objective: propose an estimation of the cost function over the whole optimization domain
– is known for a certain number of "master points", for which is known through a full FEM
calculation
– Assumption that is a random Gaussian field:
– Main task: add a new master point to the existing data base,
in order to improve the metamodel
')( p
Long range trend Local random deviation
(Gaussian type)
NLCM - Michel Bellet 2016-03 12
Building a Metamodel by Kriging
• Dimension of the optimization domain: n
• Number of existing master points: m
• Least squares method:
– C is the correlation matrix:
– with usual correlation functions:
mm
m
ICI
yCI1T
1T
mn
m
ppP ,,1
m
m )(,),( 1 ppy
n
mm IyCIy
1T
2
mjiC jiij ,1,),corr( pp
),(exp),,,corr( jiji q pppp
Example n = 1
large
small
2
D.R. Jones, J Global Optimization 21 (2001) 345-383
E. Roux, PhD Thesis, Mines ParisTech, 2012
n
k
kk baq
1
2
),(
ba
NLCM - Michel Bellet 2016-03 13
Building a Metamodel by Kriging
• Kriging predictor:
• Variance of this estimator:
Master points
miim ,1
1T ),corr()()(ˆ
ppCIyp
mm
vmvv
ICI
pcCIpcCpcp
1T
21T
1T22 )(1)()(1)(ˆ
)(pcv
Example n = 1
NLCM - Michel Bellet 2016-03 14
Building a Metamodel by Kriging
• Enrichment of the data base
– Point for which the predictor is minimum ("exploitation")
– Point for which the variance of the predictor is maximum ("exploration")
– Solution of those minimization problems: different methods can be used
(genetic algorithm ok, because the calculation of is very fast)
)(ˆ p
NLCM - Michel Bellet 2016-03 15
Examples of Rheological Characterization
• Project "Cracracks" on hot tearing of steels in solidification processes
– ArcelorMittal, Industeel, Ascometal, CTIF, Transvalor, CEMEF
• Objective: identification of the behaviour of steels at high temperature
– From 900 C to solidus
– Small deformations (< 0.10)
– Low strain rates 10-5 – 10-3 s-1
– Elastic-viscoplastic behaviour models
• Approach
– Combined Traction-Relaxation tests with resistive heating (Joule effect
• GLEEBLE machine (ASCOMETAL)
• TABOO machine (CEMEF)
– Development of non contact extensometry
– Application of a method for automatic identification (FE inverse analysis)
– Application to 4 steel grades
• DP780 (CEMEF)
• ST52 (CEMEF)
• 100Cr6 (ASCOMETAL)
• 40CrMnNiMo8-6-4 (ASCOMETAL)
NLCM - Michel Bellet 2016-03 16
Tensile Tests with Joule Heating
GLEEBLE 3800
(ASCOMETAL)
TABOO
(CEMEF)
– Fast heating and cooling
– Existence of temperature gradients
– Non-uniformity of strain
NLCM - Michel Bellet 2016-03 17
Numerical Modelling of TABOO and GLEEBLE Tests
• Development of a coupled numerical
model: electric-thermal-mechanical
(base FORGE)
• Application to TABOO and GLEEBLE
to design specimens
– Validation: comparison with welded
thermocouples and IR images
– Characterization of a non-uniform
deformation
– TABOO: T ~ 30 C in centre zone (on
10 mm)
– GLEEBLE: Taxial ~ 30 C in centre
zone
Tradial ~ 50 C at 1300 C on F10 mm
Ph.D. of Changli Zhang (2010) and Christophe Pradille (2011)
C. Zhang, M. Bellet, M. Bobadilla, H. Shen, B. Liu, Metall. Mater. Trans. A 41 (2010) 2304-2317
C. Zhang, M. Bellet, M. Bobadilla, H. Shen, B. Liu, Inverse Problems Sci. Engng. 19 (2011) 485-508
An accurate identification of parameters
requires a finite element inverse model, both
for TABOO and GLEEBLE
NLCM - Michel Bellet 2016-03 18
Measurement of Displacement Field
Ch. Pradille, M. Bellet, Y. Chastel, Applied Mechanics and
Materials 24 (2010) 135-140
Tensile test 1250 C, 0.05 mm/s under
gas protection Ar + 5%H2
Digital Image Correlation by Laser
Speckles Interferometry
[mm]
NLCM - Michel Bellet 2016-03 19
TABOO Tests: Identification Methodology
• Tests with successive tractions/relaxations
• Identification of parameters by inverse
analysis based on FE modelling
• Elastic-viscoplastic constitutive model
Set of parameters
Minimization by kriging algorithm
(MOOPI platform, developed in Cemef)
Corrections
Initialization
Error Function
exp
1 12exp
,
2exp
,
num
,
1
2
exp
,
exp
,
num
,
exp
1N
j
M
i ij
ijij
j
M
i ij
ijij
j
jj
PF
FF
MNΨ
u
uu
NLCM - Michel Bellet 2016-03 20
NLCM - Michel Bellet 2016-03 21
0
100
200
300
400
500
600
700
800
900
0 50 100
Temps s
Forc
e N
essai T=900°C
modèle numérique
0
100
200
300
400
500
600
700
0 50 100 150
temps s
Forc
e N
essai T=1000°C
modèle numérique
0
50
100
150
200
250
0 50 100 150
temps (s)
Forc
e (N
)
modèle numérique
essai 1200°C
0
20
40
60
80
100
120
0 50 100 150
temps s
Forc
e N
modèle numérique
essai 1350°C
• Identification
for two steel
grades
– DP780
– ST52
1000°C 900°C
1200°C 1350°C
mnT
Y Ae
NLCM - Michel Bellet 2016-03 22
GLEEBLE – Cst strain-rate tests – 100Cr6
1000 °C - V=0,05 S-1
0
400
800
1200
1600
2000
0 200 400 600 800Allongement (microns)
Effort
(N
)
Essai Gleeble
Forge avec calage CREAS
1000 °C - V=0,01 S-1
0
300
600
900
1200
1500
0 200 400 600 800Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1000 °C - 0,001S-1
0
250
500
750
1000
0 200 400 600 800 1000Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C - V=0,05S-1
0
1000
2000
3000
0 200 400 600 800
Allongement (microns)
Effort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C - V=0,01 S-1
0
500
1000
1500
2000
2500
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C- V=0,001 S-1
0
400
800
1200
1600
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900°C – 0.01s-1 900°C – 0.05s-1
1000°C – 0.001s-1 1000°C – 0.01s-1 1000°C – 0.05s-1
900°C – 0.001s-1
1000 °C - V=0,05 S-1
0
400
800
1200
1600
2000
0 200 400 600 800Allongement (microns)
Effort
(N
)
Essai Gleeble
Forge avec calage CREAS
1000 °C - V=0,01 S-1
0
300
600
900
1200
1500
0 200 400 600 800Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1000 °C - 0,001S-1
0
250
500
750
1000
0 200 400 600 800 1000Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C - V=0,05S-1
0
1000
2000
3000
0 200 400 600 800
Allongement (microns)
Effort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C - V=0,01 S-1
0
500
1000
1500
2000
2500
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900 °C- V=0,001 S-1
0
400
800
1200
1600
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
900°C – 0.01s-1 900°C – 0.05s-1
1000°C – 0.001s-1 1000°C – 0.01s-1 1000°C – 0.05s-1
900°C – 0.001s-1
1100 °C - V=0,05 S-1
0
300
600
900
1200
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100 °C - V=0,01 S-1
0
200
400
600
800
1000
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100 °C - V=0,001 S-1
0
100
200
300
400
500
600
0 100 200 300 400 500 600 700Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100°C – 0.001s-1 1100°C – 0.01s-1 1100°C – 0.05s-11100 °C - V=0,05 S-1
0
300
600
900
1200
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100 °C - V=0,01 S-1
0
200
400
600
800
1000
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100 °C - V=0,001 S-1
0
100
200
300
400
500
600
0 100 200 300 400 500 600 700Allongement (microns)
Eff
ort
(N
)
Essai Gleeble
Forge avec calage CREAS
1100°C – 0.001s-1 1100°C – 0.01s-1 1100°C – 0.05s-1
NLCM - Michel Bellet 2016-03 23
GLEEBLE – Cst strain-rate tests – 100Cr6
1200 °C- VB
0
50
100
150
200
250
300
350
0 200 400 600 800 1000 1200
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1200 °C - VM
0
100
200
300
400
500
600
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1200 °C - VH
-100
0
100
200
300
400
500
600
700
0 100 200 300 400 500 600 700
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1300 °C- VB
-50
0
50
100
150
200
250
300
350
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1270
1300 °C - VM
0
50
100
150
200
250
300
350
400
0 200 400 600 800 1000
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1280
1300 °C - VH
0
50
100
150
200
250
300
350
400
450
500
0 200 400 600 800 1000 1200
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1280
Essai Gleeble 1270
1200°C – 0.001s-1 1200°C – 0.01s-1 1200°C – 0.05s-1
1270°C – 0.001s-1 1270°C – 0.01s-1 1270°C – 0.05s-1
1200 °C- VB
0
50
100
150
200
250
300
350
0 200 400 600 800 1000 1200
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1200 °C - VM
0
100
200
300
400
500
600
0 100 200 300 400 500 600
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1200 °C - VH
-100
0
100
200
300
400
500
600
700
0 100 200 300 400 500 600 700
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble
1300 °C- VB
-50
0
50
100
150
200
250
300
350
0 200 400 600 800
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1270
1300 °C - VM
0
50
100
150
200
250
300
350
400
0 200 400 600 800 1000
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1280
1300 °C - VH
0
50
100
150
200
250
300
350
400
450
500
0 200 400 600 800 1000 1200
Allongement (microns)
Eff
ort
(N
)
Forge avec loi Thercast
Essai Gleeble 1280
Essai Gleeble 1270
1200°C – 0.001s-1 1200°C – 0.01s-1 1200°C – 0.05s-1
1270°C – 0.001s-1 1270°C – 0.01s-1 1270°C – 0.05s-1
)()( TmTnTK
NLCM - Michel Bellet 2016-03 24
BMFZ S
k
S
cal
kt
BM
S
k
cal
kt
L
FZ
S
s
sI
i
exp
is
cal
isS
s
TtTS
tTTS
TT
sI
1
2
1
2
1
)(
1
2
,,
1
),(max1
),(max1
)1(
)-)((
)(
)(
pp
pp
• Set of parameters to be identified:
• "Objective" function, based on:
– temperature measures
– metallurgical observations
Identification of Heat Sources in Arc Welding
UIWW dropssurf
UIUIUI dd )1(
2
2
2 )tan(
3exp
)tan(
3),(
d
r
d
Wrdq
surf
kk keff
d
r
Thermocouples, InfraRed Camera
0
100
200
300
400
500
600
180 230 280 330 380 430 480
temps s
tem
péra
ture
(°C
)
Température1 (°C)
Température2 (°C)
Température3 (°C)
Température4 (°C)
Température5 (°C)
Température6 (°C)
Température1 (°C)
Température2 (°C)
Température3 (°C)
Température4 (°C)
Température5 (°C)
Température6 (°C)
NLCM - Michel Bellet 2016-03 25
0
100
200
300
400
500
600
700
800
900
1000
0 50 100 150 200Temps (s)
Te
mp
era
ture
(°C
)
Tc1: NumériqueTc2: NumériqueTc3: NumériqueTc1 : ExpTc2 : ExpTc3 : Exp
Optimal set of
parameters:
4.17
82.0tan
35.0
85.0
k
d
opt
β
316LN, 29 V, 360 A
width [mm] height [mm] depth [mm]
Exp. 13.0 2.4 4.8
Calc. 12.8 2.5 5.1
Time (s)
M. Bellet, M. Hamide, Int. J. Num. Meth. Heat Fluid Flow (2012) accepted, to be published
NLCM - Michel Bellet 2016-03 26
Conclusion
• Optimization algorithms make possible the automatic identification of
parameters for
– Complex constitutive equations
– Complex mechanical tests (in which the complexity induced by physics is
considered)
– Complex boundary conditions in processes
• Different strategies can be developed according to the cost of the direct
simulation of the tests
• In final, automatic identification provides the engineer with an efficient tool
for a more reliable knowledge of material behaviour and process
conditions