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**Thierry Coupez CEMEF - CIM Ecole des Mines de Paris Umr CNRS 7635**

Numerical simulation of forming processes: present achievements and future challenges Thierry Coupez CEMEF - CIM Ecole des Mines de Paris Umr CNRS 7635

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**Plan Forming process simulation : Computational techniques :**

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

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**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

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**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

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**FORGE3® Forging example : Large deformations Lagrangian FE Formulation**

Key issue : remeshing FORGE3®

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**Industrial remeshing :**

complex forging cutting TRANSVALOR

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**Adaptive remeshing and error estimation**

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**free surface flow Polymer injection moulding (Rem3D) Metal casting**

Filling process Mixing Foaming Material Liquid state to solid state Gas liquid solid…

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**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 Free surface Free surface = Interface fluid / empty space (air)

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**Incompressible Navier Stokes and moving free surface**

A fluid column crushing under its own weight. High Reynolds. Mesh adaptation: interface tracking

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**Instability of a honey falling drop 3D Crushing column of liquid**

a rectangular box 3D Navier Stokes + moving free surfaces + Mesh adaptation + Space time FE Instability of a honey falling drop

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**Electrical device Courtesy of Schneider Electric**

Material : Polysulfure de phénylène (PPS, thermoplastique semi-cristallin) Carreau law + arrhenius : K = 588 Pa.S m= 0.7 E= 33 kJ/mole k= 0.3 W/m °C r = Kg/m^3 Rem3D Courtesy of Schneider Electric

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**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

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**Foaming modelling by direct computation of bubble growth**

structure 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 Inflation of a large number of bubbles in a representative volume

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Interaction by direct calculation of the expansion of several bubbles : validation : retrieve ideal structure cubic bubble Cubical shape of trapped central bubble 6 + 1 bubbles configuration Inflated configuration

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**Foam structuration: 400 bubbles random nucleation**

Mesh : nodes elements

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**G=31%, V=1.36 G=16%, V=1.1 G=6%, V=1 G=75%, V=4.8 G=58%, V=2.1**

Fraction gazeuse G, rapport des volumes V. Dernière image : limites du maillage. . Formes des bulles et tailles variées. . 58% - 65% en P3 bonne description : même volume de liquide grâce à l'expansion de domaine. . Mais la majeur partie du liquide reste dans les coins du domaine : au centre : fraction gazeuse plus importante => voir suite.

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**Flow oriented structure:**

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

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**Microscale simulation :**

Direct computation of the flow of N fibres in a viscous fluid Exact calculation of a2 and a4 from a statistical representative volume of fluid oriented Isotropic Macroscopic modelling : Equation model for a2 evolution : Closure approximation : a4 from a2 Interaction between fibres (concentration) Closure approximation Fibres fibres interaction

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**Direct simulation of the flow of a polymeric fluid with fiber**

Flow with 64 fibres Simple shear flow Periodical boundary condition Flow modification Impact of the fibre on the flow (vertical velocity component) MFE flow solver Interaction by Vof for each fibre Fibre motion by bi-particle tracking

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Concentration : 8% 15%

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Concentration : 6% 12%

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**VISCOELASTICITY : a molecular approach**

MATERIAL MODELLING VISCOELASTICITY : a molecular approach POMPOM MODEL: REPTATION THEORY BASIS The chain is still in the tube and has arms One chain interacts with other chains, but is transversely blocked, even though it finds no obstacles in its path REPTATION TUBE MODEL The arms allow the stretch of the chain Stretch is the other variable of the pompom model Reptation of the arms Stretch of the chain Reptation of the chain when the arms penetrate in the tube

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**MATERIAL BEHAVIOUR MODELLING: VISCOELASTICITY**

POMPOM MODEL: EVOLUTION EQUATIONS diffusion Determination of molecular orientation: relaxation variation due to macroscopic flow Elastic force Arm force Determination of chain stretch: Extra-stress explicit computation: Stress explicit computation: And conservation of momentum...

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**VALIDATION AND APPLICATION TO SIMPLE GEOMETRIES**

2D FILLING OF A PLATE Orientation Stretch

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**3D COMPLEX INDUSTRIAL PARTS**

ORIENTATION AND STRESSES Stress normal to flow axis Shearing

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**Conclusion Forming process simulation : Futures challenges :**

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

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