Recent and future developments in finite element metal ... · ICTP'2014 - JL Chenot et al 1 Recent...

ICTP'2014 - JL Chenot et al 1

Recent and future developments in finite element metal forming

simulationJ.-L. Chenota,b, M. Bernackib, P-O Bouchardb,

L. Fourmentb, E. Hachemb, E. PerchataaTransvalor, 694 Avenue du Dr. Maurice Donat, 06255 Mougins Cedex, France

bCEMEF, Mines Paristech, B.P. 207, 06904 Sophia Antipolis Cedex, France


Short historical introduction

FE simulation of metal forming processes: in 2D : end of 60’s and 70’s (Japan, USA, UK). in 3D : 80’s (France, Japan, USA).

Main steps 1995: automatic remeshing, 1996: element quality control, 1997: parallel computing, 2002: heat treatment, quenching, 2003: deformable tools, 2005: coupling with metallurgy , 2007: thermal regime in tools, multi body forming, 2009: process optimization, 2014: three-dimensional modeling of induction heating.


Summary

Mechanical and thermal formulations

Space and time discretization

Resolution procedures

Coupled approaches

Optimization and identification

Computation at the Micro Scale

Future challenges


Mechanical and thermal formulations

Updated Lagrangian, Eulerian or ALE

Constitutive modeling

Contact, friction and wear

Damage

Heat equation

Updated Lagrangian, Eulerian or ALE

Classical integral formulation:

For steady state processes (e. g. rolling, extrusion, wire drawing, etc.) A fixed domain Ω is considered, Free surface must be determined, Stress and memory variables are computed along

streamlines, with SUPG, etc.


c

v*dV dV v dS 0ργ σ ε τΩ Ω ∂Ω

+ − =∫ ∫ ∫: * *ɺ

Inertia terms can often be neglected:

For non steady state processes, Updated Lagrangian:

Parameters update :

ALE is a compromise Numerical velocity material velocity

ALE derivative:


t tc

t t t tdV v dS 0σ ε τ+∆ +∆

Ω ∂Ω

− =∫ ∫: * *ɺ

t t t t

t t t t

t t t t

x x tv

t

t

σ σ σε ε ε

+∆

+∆

+∆

= + ∆

= + ∆

= + ∆

ɺ

ɺ

ALEv v≠

( ) ( )ALEALE

dv v grad

dt

ε ε ε= + −ɺ

Constitutive modeling

Viscoplasticity (Perzyna):

Elastoviscoplasticity: Additive decomposition:

Elastic law:

Anisotropic behaviour: Hill, Barlat, etc.

Compressible materials: Gurson, Abouaf et al.…


11

ε 1 σ σp

mK R K− ′= −/ ( ) /ɺ

e pε ε ε= +ɺ ɺ ɺ

2e eJdtr( )

dt

σ λ ε µε= +ɺ ɺ

Contact, friction and wear

Contact condition between bodies a and b:

treated by penalty

Friction: example of Coulomb Norton law:

Wear: generalized Archard law:


( - ). . 0a bv v n v n∆= ≤

1 fpf f n v vτ µ σ= ∆ ∆ -

- ( ) /

W

nW W m

V

VK dt

H

σδ ⋅ ∆= ⋅ ⋅∫

Modelling Ductile fracture

Phenomenological approaches: Failure criteria maximum stress : Latham & Cockcroft,

triaxiality ratio : Oyane,

& Lode angle : Bao & Wierzbicki …

Continuum Damage Mechanics:

Kachanov, Lemaitre …

Micromechanical approaches: Void growth models:,

Porous plasticity modelsICTP'2014 - JL Chenot et al 9

Cup and cone defect

Heat equation


0t t t t

V ncTwdV kgrad T grad w dV q wdV wdSΩ Ω Ω Ω

ρ φ+ − + =∫ ∫ ∫ ∫ɺ ɺ( ) ( )

Non-linear and fully coupled with mechanical equations:

• ρ, c, k depend on T,

• Non-linear boundary conditions,

• is the plastic work,

• Constitutive equation depends on T.

• Possibility of strain localization: e. g. adiabatic shear band

Vqɺ

Space and time discretization

Tetrahedral elements

Time discretization

Space-time finite element

Remeshing and adaptive remeshing

Anisotropic remeshing

Remapping



Tetrahedral elements

P1+P1 tetrahedral elements

Discretization in term of:

velocity v, or displacement increment ∆u,

Pressure p,

and temperature T

Non linear equations:R V P T 0

S T V 0

( , , )

( , )

==

pressure pvelocity v

Mechanical equation

Heat equation

Time discretization Classical incremental approach: First order Second order :

Runge et Kutta, “Crank and Nicholson” scheme:

“Acceleration method” for quasi static formulation Time differentiatation of the integral equation

-> linear equation in γ

Space-time elements Slices of space-time are discretized Possibility of local time steps

Necessitated 3D or 4D elements


2t t t t t tx x t( v v ) /+∆ +∆= + ∆ +

Remeshing and adaptive remeshing

“Geometric” remeshing by iterative method (T. Coupez et al.)

Iterative local improvements, Restore possible degeneracy and maximize element quality, Control contact or penetration / tools, Parallel implementation.

Adaptive remeshing (L. Fourment) : Error estimation: modified Zienkiewicz energy norm method:

Map of element size, Remeshing respecting size and quality, Use of iterative remeshing method with weighting factors, Prescribed accuracy or maximum number of nodes.


1/2

( ) : ( )h h h h dη σ σ ε εΩ

= − − Ω ∫ ɶɶ

Application to sheet blanking (L.Fourment et al.)


Anisotropic remeshing Basic principle:

Introduce a local metric defined by the tensor , Use the same remeshing algorithm.

Computation of the metric tensor: Error estimation contribution:

Skin adaptation:• Compute normal vectors,• define mesh size along• normals:


M

21

22

23

1 0 0

0 1 0

0 0 1

e

/ h

M / h

/ h

′ =

2

1s

s

M n nh

′ = ⊗

Curvatures adaptation:

Metric tensor:


212

22

0 0 01

0 1 0

0 0 1

cM / R

/ Rα

′ =

e s cM M M M= + +

Isotropic case: 138 840 nodes

Anisotropic case: 18 770 nodes=> divided by 7


Isotropic: 4 procs - 1h 20mn Anisotropic: 4 procs – 40 mn

=> divided by 2 Equivalent or better accuracy

Thickness without remeshing

Thickness with remeshing

Application to deep drawing


Application to rolling

Comparison: anisotropic 8 786 nodes / isotropic 61 474 nodes, => divided by 7,Equivalent accuracy close to the cylinders,Higher thickness accuracy.

21

( ) ( )k

f f Nk k

=∑ ɶx x

M

P0

f

M

P1

M2

P0

fɶ f

Mapping SPR

( ) kfk

=ɶ x P.a

Interpolation

Nodecentered

patch

( )

( )( )2

4,

( )

( )

, ( )

( )

kk k k

g k k g g

k g g

f fk k g k g

kfk

f f

P f f

f fMIN

∇

=

= +∇ −

∀ ∈ =

− ∑ −

ɶ ɶ

ɶ

ɶ ɶ ɶ

ɶ

ɶ

x

x P.a

x f x x

x x

x

x x

k

Use of Super convergent patch recovery method

Remapping (L. Fourment et al.)

ICTP'2014 - JL Chenot et al

Initialmesh

finalmesh

Resolution procedures

Linearization of the equations Localized contact with the tools Parallel computing Multi mesh Multigrid


Linearization of the equations

Integral equations , with

Newton-Raphson iterative procedure:

Several linear systems to solve with unknown

Possibility of minimization of the residual / α :


Z (U ,P)=0R( Z )=1n n nZ Z Zδ+ = +

( )1 0 0n n n nR( Z ) R( Z ) R( Z ) / Z Zδ+ = ⇒ + ∂ ∂ ≅nZδ

n nR( Z Z )αδ+

Localized contact with the tools

The distance to the tool must be positive or null:

Penalty form:

Improvements: Penalty can be imposed at each Newton-Raphson iteration, Second order smoothing of the surface : Nagata

Formulations: Master and slave, Quasi symmetric (L. Fourment et al)


( ) ( ) ( )2( ) ( ) .t t t t t t t t t tn n n n n tool n n

dM M M t O t v V n t

dt∆ ∆ δδ δ δ ∆ ∆ δ ∆+ += = + + ≈ + −

( ) ( )2

1.

2c

n pencontact n tool n n

n

V V v n StΩ

δ δϕ ρ

∆

+−

∈∂

+= − −

∑

25

Upper tool

Ω3Ω2

Ω1

Lower tool

Forge

Forge

Forge

P2

P3

P1

Forge Parallel

Interface data files

Parallel computing

Iterative solverLU Preconditionning Domain decomposition

26

Algorithm for parallel solving

kxA

Np

kk Ω

=∑=

1

xA .Product Matrix / Vector in parallel :

Computation on each domainCommunication for updating interfaces

Scalar Product in Parallel : ∑ ∑

⋅⋅=

DomainSub NodesnodesNodesNodes

wvuvu ),(

Heavy computation distributed on sub domainsLimited communication

Np : number of processors


Computation speed-up from 4 to 24 cores

Multi mesh (L. Fourment et al.)

Basic principle: - refined mesh for heat equation: TM, - less refined mesh for mechanical equation: MM, - mesh for other parameters (equivalent strain, etc.). - transfer data between meshes for coupling.

Benefit: Adapt mesh refinement to problems, Save computer time. Compatibility with parallel computing.


29

Application to Cogging

Thermal Mesh

Mechanical Mesh

29

Configuration Nb elementsThermal Mesh

Nb elementsMechanical Mesh

Derefinement rate Speed-up

Beginning 53 500 13 800 3.9 9.2

End 66 300 13 900 4.8 18.1

Test of efficiency (single processor)


6th ISPF – J-L Chenot 30

Multigrid method (K. Mocellin, L. Fourment)

2 or 3 levels of grids: Automatic de-refinement by a local operator.

Data transfer between grids. For ring rolling: speed up about 7.

61 987 nodes 6 061 nodes 790 nodes10,2τ = 7,6τ =

Multigrid new developments (in progress): More general algorithm, Adaptation to parallel computation.


Coupled approaches

Thermal and mechanical coupling Multi material coupling Induction heating Coupling with micro structure evolution


Thermal and mechanical coupling Mechanical and heat equations :

Solve mechanical and thermal eqs. separately;

First :

Then :

Possibility of iteration : fix point.


t t t t t t

t t t t

R V P T 0

S T V 0

+∆ +∆ +∆

+∆ +∆

=

=

( , , )

( , )

t t t t t t t t tR V P T 0 V P+∆ +∆ +∆ +∆= →( , , ) ,

t t t t t tS T V 0 T+∆ +∆ +∆= →( , )

For strong coupling (e. g. Adiabatic Shear Band)

iterative scheme may not converge,

solve a global system - Newton-Raphson method

on V, P, T.

Example of Cutting

Equivalent strain


Multi material coupling

See presentation:

Finite Element simulation of multi material metal forming

By J.-L. Chenot, C. Béraudo, M. Bernacki, L. Fourment


Induction heating (F. Bay, J. Barlier et al.)

Maxwell equations are simplified:

ρ density of charge

V potential

A : magnetic vector potential with magnetic field

µ : magnetic permeability

Heating: with electric current density


10

1 1 1

( V )

A( A) V

t

ρ

ρ µ ρ

∇ ∇ =

∂ + ∇× ∇× = ∇∂

B A= ∇×

2EM elecw jσ=ɺ j V= ∇

Induction heating FE solution of the electromagnetic equations:

Classical discretisation for potential V,

Special tetrahedral edge elements for A,

Necessity of coupling with heat equation,

And metallurgical evolution.

Global mesh for:

Inductor,

Workpiece,

Air.


Application to heating of a crank shaft


Kinematics Austenite

Quenching


% martensite

Hardness


Dis

loca

tion

Den

sity

Gra

in R

adiu

sR

ecry

stal

lizat

ion

ratio

Flo

w S

tres

s

Determination of the flow stress as afunction of the mean dislocation density

⋅

⋅⋅⋅⋅

+⋅⋅⋅⋅=m

mbmp

AArg

bc

TKMbM

ρρ

ρµασ 52

9

sinh

( ) XX xdm ⋅+−⋅= ρρρ 1

⋅⋅−⋅−⋅−⋅+=

d

ddddd A

ArgAAAAAdt

d

ρρρρρρ 52/5

42/5

2210 sinh

Dislocation evolution model

Temperature = 1100 °C – Strain-Rate = 1 s-1

Recrystallization ratio and grain sizes

Developed at Aachen University for microalloyed steels

Precipitation model for V(C,N)

Coupling with micro structure evolutionExample of the mean-field approach: AFP model

Example : Reducer-rolling simulation

FORGE simulation

MEAN-FIELD APPROACH – AFP MODELX dm

ρm σ

Optimization and identification

Process optimization

Identification of material parameters


Process optimization (L. Fourment et al.)

Deterministic optimization Genetic algorithm + Surface response


1st operation with optimized die

Second operation

(3)

no defect

piping defect

Identification of material parameters

Data is obtained obtained from: Laboratory experiments, Industrial measurements, And define the set of observables.

Parameters to be identified: Constitutive equations, Friction laws, Damage, Metallurgical laws,…

Minimization of the mean square difference: Genetic algorithm, Kriging meta-model.


Sensitivity Analysis Sensitivity of observables wrt input parameters

PARAMETERS

OBSERVAB L ES

Input Output

Layer 1

MOdular software dedicated to Optimisationand Parameters Identification (P-O Bouchard et al.)

M IPOO

Layer 0

Direct model – FE software

Software

Postprocessing : Computation of the cost function

OptimizationLayer 2

Optimal parameters

Inverse Analysis Identified set of parameters

Layer 3


( )0n

y Kσ ε σ ε= + If

elseTension test6 Parameters: Observables from a tension test.

Necessity of enrichment of observables: Full field measurement

Load vs. Displacement Necking vs. Displacement

Example of parameters identification / MOOPIStrain hardening Damage

Digital Image Correlation

Displacementfield

0y D,K ,n,b,S ,σ ε


Computation at the micro-scale

General approach

Identification of macro laws using finite element micro modeling


Principle of micro scale computation(M. Bernacki et al.)

Definition of a representative elementary volume.

F. E. mesh of the physical entities.

Initialization of the Level Set describing interfaces

between entities.

Introduce physical laws at the micro scale.

Simulation of the evolution of the entities.


Example: recrystallization modeling

θ3

G1 G3

G2

Γ23θ2

θ1

v23 = c23κ23 + e3 − e2

Interface mobilityGi / Gj

Grain boundaryenergy

between Gi & Gj

Curvatureof Γij

PrimarySecondary

Difference of bulk energies

Gi / Gj


• GG modelling

(200 grains)

• Uniform mobility

and grain boundary

energy,

2D example of secondary recrystallization

2D grain growth with germs


New developments

Zener pinning, Dynamic Recrystallization, Crystal plasticity, Twinning, GB anisotropy, Phase transformation,…

Identification of mean field models.


Future challenges

Process stability

Introduction of stochastic phenomena

Model reduction

Toward simulation of the whole process chain


Process stability and introduction of stochastic phenomena: application to robust optimization

Various sources of uncertainties: on process parameters, … on material data: behaviour,

friction, boundary conditions, on function evaluation (numerical noise)


54

Φ(p)

p

noisy functionΦ(p)

p

noisy function

smoothed

functionfalseoptimum reliable

optimum

piping defect

55

Smooth metamodel = minimize Hessian of metamodel

( ) ( ) ( )= +Φ δ Φ δ ωϕ δ

( )( ) ( ) 22min , f d

δΦ δ Φ δ ω

Ω

= ∇∫ ɶ

( )( ) ( )2

1

jmaxj

mj ,n

min , e

δδ

δϕ δ ϕ δ

=

=∑

+ among possible solutions, take the most probable

balanced average:

1, , m j j jj n f f δ∀ = = −

Metamodel is smoothed by correcting master points values:


Model reduction

Basic principle:

Computation of the more important eigen modes,

Approximation of the solution on these modes,

as linear combinations -> smaller system to solve

New solutions with different parameters are less

expensive in principle,

Remark: the evolution of the domain must not depend

too much on the parameters.


Preliminary work on sheet bulging (L. Fourment)


Imposed pressure

Rigid tool

sheet

Classical simulation (vertical velocity distribution)

Computation of eigen modes


mesh Mode 1

Mode 5Mode 4Mode 3

Mode 2

Toward simulation of the whole process chain

Next objective of the simulation:


HeatingPlastic deformation:Rolling, forging, etc.

Heat treatments In servicebehavior

Evolution of shape, microstructure, mechanical properties

GLOBAL OPTIMIZATION

First approach:Optimization of the strength of a clinched joint

(P-O Bouchard et al.)

Initial design

Forming Tension test

Analysis of the influence of the geometry of the tools:→ 2 parameters are important

AMPT Octobre 2010 - 61

Optimization / 2 parametersPunch radius Rpand die depth Pm

Configuration Strength (N)

Reference 737

Optimized 894

Improvement of 21%

on strength in tension

(b) Die

Pm

Lm

(a) Punch

w1

Rp

Rcp

w2

AMPT Octobre 2010 -62

After optimization

Forming Tension test


Conclusions Metal forming simulation is based on a complex

balance of developments between : Mechanics, Physics of materials, Numerical analysis, Computer science, Knowledge of the technology of industrial processes,…

New requests from industry in development: Realistic computation of µ structure evolutions, Prediction of the final properties, Introduction of stochastic phenomena, etc.


Thank you for

your kind attention

Recent and future developments in finite element metal ... · ICTP'2014 - JL Chenot et al 1 Recent...

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