Numerical simulation of forming processes: present achievements and

future challenges

Thierry CoupezCEMEF - CIM

Ecole des Mines de ParisUmr CNRS 7635

Plan• Forming process simulation :

– Large deformations : forging, stamping,…– Free surface flow : Injection molding, casting– Multi-modeling : flow, deformation, heat transfer, liquid solid transition

• Computational techniques :– EF solver : mixed FE, incompressibility, viscoelasticity – EF Lagranian, remeshing, – EF Eulerian, vof, levelset

• New chalenge : structure prediction– Multiscale modeling– Multiphase Example :

• Foam : form nuclation, bubble growth, and cell construction• Fibers reinforced polymer : suspension to long fiber high concentration

– Physic property:• Polymer : macromolecular orientation in polymer• Cristalysation

– Computational Chalenges :• Multiphases calculation : liquid, solid , gas • Transition : critalynity, mixture solid liquid (dendrite, spherolite)• Macroscopic descriptor

Computational Material forming

• Solid material :– High temperature : viscoplasticity,– Low temperature : plasticity, elastoplasticity

• Fluid material : – Low viscosity :liquid metal (foundry), Newtonian

incompressible liquid (turbulence)– Low viscosity : Newtonian, reactive material,

thermoset, – High viscosity : Pseudopalsticity, viscoelasticity

thermoplastic polymer• Liquid solid transition

Mechanical approaches• FE for both solid and fluid problems

– Implicit– Iterative solver (linear system), parallel (Petsc)– Stable (Brezzi Babuska) Mixed Finite Element (incompressibility) (P1+/P1)

• Large deformations (Forge3 : forging ): – Lagrangian description

• Flow formulation (velocity)• Unilateral contact condition• Remeshing

• Flow (Rem3D : injection moulding)– Stokes and Navier Stokes solver (velocity, pressure)– Transport equation solver (Space time discontinuous Galerkin method)

• Heat transfer coupling– Rheology temperature dependent– Convection diffusion (Dicontinuous Galerkin method)– Thermal shock– Phase change, structural coupling

Forging example :Large deformationsLagrangian FE FormulationKey issue : remeshing

FORGE3®

TRANSVALOR

Industrial remeshing : • complex forging• cutting

Adaptive remeshing and error estimation

free surface flow

• Polymer injection moulding (Rem3D)

• Metal casting

• Filling process

• Mixing

• Foaming

• Material Liquid state to solid state

• Gas liquid solid…

MOVING FREE SURFACES AND INTERFACES

Eulerian approach

– the diffuse interface approach

– Transport equation solver

– Capture of interfaces

– Space time finite element method

– Mesh adaptation• R-adaptivity (~ALE)

• Conservative scheme

fluid air

Freesurface

)()( tt airfluid

Free surface = Interface fluid / empty space (air)

0.

))(2.(

v

gpv

A fluid column crushing under its own weight. High Reynolds.

Incompressible Navier Stokes and moving free surface

Mesh adaptation: interface tracking

3D Crushing column of liquida rectangular box

3D Navier Stokes + moving free surfaces +Mesh adaptation + Space time FE

Instability of ahoney falling drop

Electrical device

Courtesy of Schneider Electric Rem3D

Material : Polysulfure de phénylène (PPS, thermoplastique semi-cristallin)

Carreau law + arrhenius :K = 588 Pa.Sm= 0.7E= 33 kJ/molek= 0.3 W/m °C = 1.64 10 Kg/m^3

Multiscale modelling in material forming

• Examples : – Foam, – Fibre reinforced polymer, – constitutive equation based on the macromolecule orientation

• Structure descriptors : microscale to macroscale – Microscale : modelling by direct multidomain simulation of moving

bubbles or fibres in a sample volume of liquid– Macroscale :

• Concentration, gas rate • Distribution of bubble size, fibre shape factor, • Orientation tensor: fibres, macromolecules, …

– Flow oriented structure : micro-macro • Evolution equation of the orientation tensor : closure approximation• Interaction description (fibre fibre, entangled polymer, bubble density)• Influence of the structure on the rheology• End use property

Foaming modelling by direct computation of bubble growthstructure parameters :

• density (gas rate) (10% G 99.5%)

• size (number) and shape of cells

Computation ingredients :

•Multidomains (individual bubble) (transport equation solver STDG, VoF, r-adaptation)

•Compressile gas in incompressible liquid (stable MFE method )

• from nuclei to bubble and cells

Fluid domain f

n gas bubbles gi

The sample domain

n

igf i

1

)(

Inflation of a large number of bubbles in a representative volume

Interaction by direct calculation of the expansion of several bubbles : validation : retrieve ideal structure cubic bubble

6 + 1 bubbles configuration

Inflated configuration

Cubical shape of trapped central bubble

Mesh : 98 000 nodes

550 000 elements

Foam structuration:

400 bubbles random nucleation

G=6%, V=1 G=16%, V=1.1G=31%, V=1.36

G=50%, V=1.8 G=58%, V=2.1

G=75%, V=4.8

Orientation : - Fibre reinforced polymer - viscoelasticity by molecular orientation

• Flow oriented structure:– Macroscale descriptor : orientation tensor– Orientation evolution (rigid fibre):

• Physical model : – Closure approximation

– Interaction modelling

– Orientation and stretch• Macromolecule orientation modelling

N

i

iiii

N

i

ii

ppppN

a

ppN

a

14

12

1

1

Fibres fibres interaction

Closure approximation

)(2):2(242222

2 anICaaaaaDt

aDdI

Macroscopic modelling :Equation model for a2 evolution : Closure approximation : a4 from a2

Interaction between fibres (concentration)

Microscale simulation :Direct computation of the flow of N fibres in a viscous fluidExact calculation of a2 and a4 from a statistical representative volume of fluid

oriented Isotropic

00

012a

5.00

05.02a

Direct simulation of the flow of a polymeric fluid with fiber

Periodical boundary condition

Simple shear flow

Flow modification

Impact of the fibre on the flow (vertical velocity component)

Flow with 64 fibres

•MFE flow solver •Interaction by Vof for each fibre•Fibre motion by bi-particle tracking

Concentration :

8%

15%

Concentration :

6%

12%

MATERIAL MODELLING

VISCOELASTICITY : a molecular approach POMPOM MODEL: REPTATION THEORY BASIS

One chain interacts with other chains, but is transversely blocked, even though it finds no obstacles in

its path

REPTATION

TUBE MODEL

The chain is still in the tube and has arms

The arms allow the stretch of the chain

Reptation of the arms

Stretch of the chainReptation of the chain when the arms penetrate in the tube

Stretch is the other variable of the pompom model

mequilibriuat lengthndeformatioafter length

MATERIAL BEHAVIOUR MODELLING: VISCOELASTICITY

POMPOM MODEL: EVOLUTION EQUATIONS

Determination of molecular orientation: dtdS

Tvv

SS

variation due to macroscopic flow

31 Iλb

S

relaxation

SS v

:2

diffusion

Determination of chain stretch:dtd

Elastic force

S:vε

Arm force

1

0

11 υ

s

eλ

Extra-stress explicit computation:

NM

i

is τvεηpσ1

2

I

ISτ 23 iii G

Stress explicit computation:

And conservation of momentum...

02 τγ pvη

VALIDATION AND APPLICATION TO SIMPLE GEOMETRIES

2D FILLING OF A PLATE

Orientation Stretch

3D COMPLEX INDUSTRIAL PARTS

ORIENTATION AND STRESSES

Stress normal to flow axis

Shearing

Conclusion• Forming process simulation :

– Large deformation and Lagrangian approach : forging, rolling, deep-drawing, machining

– Flow and Eulerian approach : injection moulding of polymer, casting, mixing

– Numerical techniques : Stable Mixed Finite Element method (incompressibility), Meshing technique (h-adaptation, r-adaptation, remeshing, anisotropic mesh), Transport solution, level set, Volume of Fluid, parallel computing

• Futures challenges :– Complex material : structure and morphology– Multiphase: liquid solid, liquid gas– Multiscale computing – Phase transition – End use property and microstructure prediction