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


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


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


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


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[


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


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


Strain-rate 0.1 s-1 / Two

thickness values

Strain-rate 5 s-1 / Two

thickness values

Initial values Identified values

7 iterations


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


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)



• 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



• 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



• 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


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)


Tensile Tests with Joule Heating

GLEEBLE 3800

(ASCOMETAL)

TABOO

(CEMEF)

– Fast heating and cooling

– Existence of temperature gradients

– Non-uniformity of strain


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


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]


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


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


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


Eff

ort

(N

)

Essai Gleeble


1000 °C - 0,001S-1

0

250

500

750

1000

0 200 400 600 800 1000Allongement (microns)

Eff

ort

(N

)

Essai Gleeble


900 °C - V=0,05S-1

0

1000

2000

3000

0 200 400 600 800

Allongement (microns)

Effort

(N

)

Essai Gleeble


900 °C - V=0,01 S-1

0

500

1000

1500

2000

2500

0 200 400 600 800


Eff

ort

(N

)

Essai Gleeble


900 °C- V=0,001 S-1

0

400

800

1200

1600

0 100 200 300 400 500 600


Eff

ort

(N

)

Essai Gleeble


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


Effort

(N

)

Essai Gleeble


1000 °C - V=0,01 S-1

0

300

600

900

1200

1500


Eff

ort

(N

)

Essai Gleeble


1000 °C - 0,001S-1

0

250

500

750

1000

0 200 400 600 800 1000Allongement (microns)

Eff

ort

(N

)

Essai Gleeble


900 °C - V=0,05S-1

0

1000

2000

3000

0 200 400 600 800


Effort

(N

)

Essai Gleeble


900 °C - V=0,01 S-1

0

500

1000

1500

2000

2500

0 200 400 600 800


Eff

ort

(N

)

Essai Gleeble


900 °C- V=0,001 S-1

0

400

800

1200

1600

0 100 200 300 400 500 600


Eff

ort

(N

)

Essai Gleeble


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


Eff

ort

(N

)

Essai Gleeble


1100 °C - V=0,01 S-1

0

200

400

600

800

1000

0 200 400 600 800


Eff

ort

(N

)

Essai Gleeble


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


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


Eff

ort

(N

)

Essai Gleeble


1100 °C - V=0,01 S-1

0

200

400

600

800

1000

0 200 400 600 800


Eff

ort

(N

)

Essai Gleeble


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


1100°C – 0.001s-1 1100°C – 0.01s-1 1100°C – 0.05s-1


GLEEBLE – Cst strain-rate tests – 100Cr6

1200 °C- VB

0

50

100

150

200

250

300

350

0 200 400 600 800 1000 1200


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


Eff

ort

(N

)


Essai Gleeble

1200 °C - VH

-100

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700


Eff

ort

(N

)


Essai Gleeble

1300 °C- VB

-50

0

50

100

150

200

250

300

350

0 200 400 600 800


Eff

ort

(N

)


Essai Gleeble 1270

1300 °C - VM

0

50

100

150

200

250

300

350

400

0 200 400 600 800 1000


Eff

ort

(N

)


Essai Gleeble 1280

1300 °C - VH

0

50

100

150

200

250

300

350

400

450

500

0 200 400 600 800 1000 1200


Eff

ort

(N

)


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


Eff

ort

(N

)


Essai Gleeble

1200 °C - VM

0

100

200

300

400

500

600

0 100 200 300 400 500 600


Eff

ort

(N

)


Essai Gleeble

1200 °C - VH

-100

0

100

200

300

400

500

600

700

0 100 200 300 400 500 600 700


Eff

ort

(N

)


Essai Gleeble

1300 °C- VB

-50

0

50

100

150

200

250

300

350

0 200 400 600 800


Eff

ort

(N

)


Essai Gleeble 1270

1300 °C - VM

0

50

100

150

200

250

300

350

400

0 200 400 600 800 1000


Eff

ort

(N

)


Essai Gleeble 1280

1300 °C - VH

0

50

100

150

200

250

300

350

400

450

500

0 200 400 600 800 1000 1200


Eff

ort

(N

)


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


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)


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


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

non-linear computational mechanics athens week march 2016...

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