Geijselaers, Hubertus J.M.

Huetink, Han

Koopman, A.J.

English

Even though Extrusion is often regarded as a semi stationary process, the defor- mations of the die at the beginning of the process can have great influence on the process later on. During filling of the die, the deformation of the die depends on the location of the flow front up to a point where parts of the profile will be opened or closed, especially in porthole dies. In this paper we present an accurate 2D method to simulate the filling of extrusion dies. The method is based on the pseudo concentration technique. We compare different options to model the pseudo material and chose the best

Geijselaers, H.J.M.

Huétink, J.

University of Twente Research Information

Flow front tracking in ALE/Eulerian formulation FEM simulations ofaluminium extrusion.A.J. Koopman1,a, H.J.M. Geijselaers1,b , J. Huétink1,c1University of Twente, P.O. Box 217, 7500 AE Enschede, The NetherlandsaA.J.Koopman@utwente.nl,bH.J.M.Geijselaers@utwente.nl,cJ.Huetink@utwente.nlKeywords: Simulation, Finite Element Method, Pseudo Concentration, FillingAbstract: Even though Extrusion is often regarded as a semi stationary process, the defor-mations of the die at the beginning of the process can have great influence on the process lateron. During filling of the die, the deformation of the die depends on the location of the flowfront up to a point where parts of the profile will be opened or closed, especially in portholedies. In this paper we present an accurate 2D method to simulate the filling of extrusion dies.The method is based on the pseudo concentration technique. We compare different options tomodel the pseudo material and chose the best.1 IntroductionDuring filling of the die at the startup of the extrusion process, the location and velocity of theflow front will have a great influence on the deformation of the die, especially in porthole dies.This deformation will affect the performance of the die in the rest of the process. Increasinglytighter requirements on complexity and geometric tolerances raise the demand for more insightin the deformations during the process. While an optimized extrusion process should createnot more then 15% scrap, it is in everyday practice not uncommon to have a scrap rates over25%.2 Modeling Aluminium Extrusion processesUsing ALE/Eulerian FEM simulations Lof [10] showed that this type of FEM is very suitableto make predictions about exit velocity and extrusion force. Using ALE/Eulerian formulatedFEM the process can be modeled without the expensive remeshing. Lof only simulated thesteady state flow through the die. Using Eulerian description to simulate the filling requires amethod that tracks the interface between air and aluminium, the flow front. In this paper wewill focus on interface tracking techniques using a fixed mesh on which a pseudo- concentrationfunction is defined and convected during simulation.In 1986 Thompson introduced the pseudo-concentration method [11], [12] to track the flowfront. Besides the material variables an extra pseudo-concentration variable C is convectedthrough a fixed mesh during the simulation. If C = 1 material is present, when C = 0no material is present and at C = 0.5 is the material-air interface. The discontinuity ofthe pseudo- concentration function around the flow front introduces inaccuracy during theconvection through the FEM mesh. Dvorkin [4] already notices that if the initial distributionof the concentration function is smooth enough, no smoothing algorithm is necessary. Dvorkinchooses to reduce the viscosity with a factor 1000, if the concentration function is greater thanzero. The boundary conditions are concentration function dependent, to prevent the effects ofincompressibility of the material that gets trapped. Between the material and the boundaries.Incompressibility effects of the material the gets trapped inside its self are not taken care of.3 Particle track methodAs pseudo-concentration function we chose the initially smooth function of the original coor-dinates. The method is implemented in our In-house code DiekA. The original coordinate inintegration point i is calculated as in equation 1 and the total displacement U is calculated inan ALE/EULERIAN formulation.X i0 = Xi− U i (1)On the convective part of the total displacement global smoothing is applied. Therefore theoriginal coordinates are convected through the mesh together with the displacement and everystep the material stiffness is updated based on the original coordinates at the beginning of thestep. The total domain is split in two parts, in an aluminium domain and an air domain. Thedomains are split by an original coordinate function which has a value in every point of thedomain: ρval. The stiffness in an integration point is based on ρival = Xini − Xi0, with Xinithe location of the initial front. If ρival < 0 the integration point is in the air domain and thematerial stiffness needs to be reduced. How this is done is explained in the next section.To avoid the incompressibility effect of the material permeable boundary conditions are used.When in an element one of the integration points is in the aluminium domain the contributionfor all nodes to the stiffness matrix is in the order of the aluminium. Without permeableboundary conditions the domain would not fill completely. In a simulation this effect can beseen in layer of elements at the boundary which are filled for approximately 50%. Only usingpermeable boundary conditions however would be a problem as soon as air gets trapped insidethe aluminium, which can happen if porthole dies are used and aluminium has to weld togetherafter it has passed the legs. Therefore also the material in the air domain is adjusted so that isis compressible. In Fig. 1 is shown how the boundary conditions are updated every step basedt = t0 t = t1ρval > 0ρval > 0Figure 1: Permeable boundary conditionson the value of ρnval in the (boundary) nodes. The aluminium is modeled with the Sellers-TegartLaw:σy = smarcsinh((κ̇Aexp(QRT))1m)(2)In this equation κ̇ is the equivalent platsic strain rate. The other parameters are constantduring the simulation. Therefore the flow stress only depends on the equivalent plastic strainrate.4 Modeling the pseudo materialThe pseudo material can be modeled in many different ways. Here we explored 26 differentmodels. In simulation 0 (see table 1) the pseudo material is modeled as a viscous material witha low viscosity to ensure enough stiffness reducing and a low bulk modulus to make the pseudomaterial compressible. In the other 25 simulations the material model for the pseudo materialis the same as the aluminium model (equation 2). The stiffness reduction is done on differentlevels in DiekA. In figure 2 a simplified chart of the implementation of DiekA is given. Thestiffness reduction can be done in either Predictor, Corrector or in both steps. The latter isNext StepPredictor:Corrector:Unbalance:}{][}{][1_1FKuJDBBKipnriT−=== ∑JBFpsreturnRadialipnrinTnnn∑=++++σ=+=σ_11111}{:||}{}}{{|| FuKUnbal −=If Unbal>tolIf Unbal<tolFigure 2: simplified incremental implementationpreferred because of conservation of consistency between predictor and corrector. In the inputfile in Dieka five multiplication factors can be defined. If in an integration point the OCTVALis less than zero, the factors are applied in the following manner.• ρyldThe factor ρyld is multiplied with the yield stress related to the equivalent strain rate asdefined in equation 2. This impacts the material stiffness D used in the predictor step andthe stress calculated in the radial return routine in the corrector step.• ρblkThe factor ρblk is multiplied with the bulk modulus. This also impacts the material stiffnessD in the predictor step and it reduces the hydrostatic pressure part of the stress p in thecorrector step.• ρsigThe factor ρsig is multiplied only with the deviatoric stress s in the corrector step.• ρhydThe factor ρhyd is multiplied directly with the hydrostatic stress p in the corrector step.• ρdetThe factor ρdet is multiplied with the determinant of the Jacobian J in both predictorwhen calculation the contribution of an integration point to the stiffness matrix [K] andthe corrector step in the calculation of the Force vector in the corrector step.Table 1: SimulationsSimulation ρyld ρblk ρsig ρhyd ρdet run0 Viscous, µ = 1e − 6m2s−1, Cb = 1MPa OK1 0.1 1.0 1.0 1.0 1.0 Incomplete2 0.01 1.0 1.0 1.0 1.0 Fails3 0.001 1.0 1.0 1.0 1.0 Fails4 1.0 0.1 1.0 1.0 1.0 OK5 1.0 0.01 1.0 1.0 1.0 OK6 1.0 0.001 1.0 1.0 1.0 OK7 0.01 0.01 1.0 1.0 1.0 Incomplete8 0.01 0.001 1.0 1.0 1.0 OK9 0.01 0.00001 1.0 1.0 1.0 Fails10 1.0 1.0 1.0 1.0 0.1 OK11 1.0 1.0 1.0 1.0 0.01 Incomplete12 1.0 1.0 1.0 1.0 0.001 Incomplete13 1.0 1.0 0.667 1.0 1.0 OK14 1.0 1.0 0.333 1.0 1.0 OK15 1.0 1.0 0.000 1.0 1.0 Fails16 1.0 1.0 -0.333 1.0 1.0 Fails17 1.0 1.0 -0.667 1.0 1.0 Fails18 1.0 1.0 -1.000 1.0 1.0 Fails19 1.0 1.0 1.0 1.0 1.0 OK20 1.0 1.0 1.0 0.667 1.0 OK21 1.0 1.0 1.0 0.333 1.0 OK22 1.0 1.0 1.0 0.000 1.0 Fails23 1.0 1.0 1.0 -0.333 1.0 Fails24 1.0 1.0 1.0 -0.667 1.0 Fails25 1.0 1.0 1.0 -1.000 1.0 Fails5 Numerical ExperimentsSimulations with plane strain and axially symmetric Q4 elements are performed. The alu-minium is modeled isotropic with a Sellars-Tegart law. In the air domain different materialmodels and numerical tricks are used to make its stiffness 2- 3 orders lower than the stiffness ofthe aluminium. The different options are described in table 1. Simulation 0 is with the pseudomaterial modeled as a viscous material.10mm150mm100mmAlu. t=0Figure 3: 2D geometryIn the rest of the simulations the air is modeled with the Sellers - Tegart law adjusted as de-scribed in the previous section with the factors as in the table. The geometry for the test is a2D plane strain model see figure 3. The single crosshatched section is the domain inside thedie filled with air at the start of the simulation. The double crosshatched section is the domaininside the container already filled with aluminium. The container and die walls are assumedsticking, except for the bearing where frictionless slip is assumed. Again, when no Aluminiumis present the boundary conditions are released, therefore making the wall permeable for air.The isothermal simulation is performed at 823K. The system is solved with a direct solver withan unbalance tolerance of 0.001.6 ResultsThe best option will be chosen based on the results on volume conservation, flow front shape,ability to fill corners and the simulation times. First the simulations that had a step that didn’tconverge were ignored. In figure 4 the simulation times of completed simulations are shown.02004006008001000120014000 5 10 15 20 25UsedCPUtime[sec]Simulations”Figure 4: Used CPU time per simulationThe simulation with the pseudo-material modeled with a viscous material converges properly.Simulation 1 does run, but only until step 1524. This is the point where the die is almost filledexcept for the bearing and ”welding chamber”. Simulations 2, 3, and 9 reach the maximumiterations at the first step. The same thing happens for simulation 15 to 18 and 22 to 25. Thebad convergence behavior of these simulations can be easily explained by the fact that only inthe corrector step the stresses are updated according to the OCTV AL. Simulation 19 is thereference simulation in which all the factors are 1.0, meaning that both air and aluminium aremodeled as aluminium. In simulation 20 & 21 the hydrostatic stress is affected only mildly, sothe consistency between predictor and corrector is not disturbed to badly, but convergence isslow and time consuming.However the flow front is also not different form the flow front in simulation 19 as can beseen in figure 5. Only simulation 0 and simulation 8 give a good representation of the flow front.Simulation 11 also gives the same flow front, however this simulation only runs up to step 923and will therefore be ignored. In the figure 6 the volume conservation is plotted against thelevels[E2]column 22-1.0000.7502.500(a) Simulation 0 (b) Simulation 4 (c) Simulation 5 (d) Simulation 6 (e) Simulation 8(f) Simulation 10 (g) Simulation 13 (h) Simulation 14 (i) Simulation 19 (j) Simulation 20(k) Simulation 21Figure 5: Flow front shapessteps. Only simulation 0, 8 and 19 are plotted.The oscillations in the beginning of the simulations can be explained by the behavior of theunder integrated 4 nodes bilinear elements. At the walls the filling is far from smooth. Firstfilling lags behind in these elements then in a few steps the elements fills completely. Thiseffect is a combination of the permeable boundary conditions and the element that has nostabilization against spurious modes from the under integration. This leads to the non physicalstick- slip behavior at the wall. Besides that it can be concluded that all the simulation areaccurate in volume conservation.-2-1.5-1-0.500.511.520 200 400 600 800 1000 1200 1400 1600Volumeconservationerror[%]Time increments [-]LegendSimulation 0Simulation 8Simulation 11Simulation 19Figure 6: Volume conservation7 DiscussionIn this paper we show that the Particle track method is suitable to make efficient simulationsof the filling of the dies. The volume conservation is accurate and the flow front looks good,however no comparisons with experiments have been made yet. Besides that a stabilizationshould be applied for this element to prevent the oscillatory behavior. Based on the resultsshown the method will be implemented for 3D simulations and simulations of aluminum flowin porthole dies can be performed.References[1] Cavaliere, M. A., M. B. Goldschmit, et al. (2001). ”Finite element analysis of steel rolling processes.”Computers & Structures 79(22-25): 2075-2089.[2] Cruchaga, D. C., Tezduyar, T.E., (2002). ”Computation of mould filling processes with a moving Lagrangianinterface technique.” Communications in Numerical Methods in Engineering 18(7): 483-493.[3] Devals, C., Heniche, M., Bertrand, F., Hayes, R. E., Tanguy, P. A., (2005). ”A finite element strategy forthe solution of interface tracking problems.” International Journal for Numerical Methods in Fluids 49(12):1305-1327.[4] Dvorkin, E. N.,Petöcz, E.G. (1993). ”An effective technique for modeling 2D metal forming processes usingan Eulerian technique.” Engineering Computations 10(4): 323-336.[5] Dvorkin, E. N., Cavaliere, M. A., et al. (1995). ”A three field element via Augmented Lagrangian formodeling bulk metal forming processes.” Computational Mechanics 17(1-2): 2-9.[6] Dvorkin, E. N., Goldschmit, M. B., et al. (1997). ”2D finite element parametric studies of the flat-rollingprocess.” Journal of Materials Processing Technology 68(1): 99-107.[7] Dvorkin, E. N., Toscano, R. G. (2003). ”A new rigid-viscoplastic model for simulating thermal strain effectsin metal-forming processes.” International Journal for Numerical Methods in Engineering 58(12): 1803-1816.[8] Haagh, G., Van De Vosse, F. N. (1998). ”Simulation of three-dimensional polymer mould filling processesusing a pseudo-concentration method.” International Journal for Numerical Methods in Fluids 28(9): 1355-1369.[9] Lewis, R. W., Ravindran, K. (2000). ”Finite element simulation of metal casting.” International Journal forNumerical Methods in Engineering 47(1-3): 29-59.[10] Lof, J. (2000). ”Developments in finite element simulations of aluminium extrusion.” PhD Thesis, Universityof Twente, Enschede.[11] Thompson, E. (1986). ”Use of Pseudo-concentrations to follow creeping viscous flows during transientanalysis.” International Journal for Numerical Methods in Fluids 6(10): 749-761.[12] Thompson, E. (1988). ”Transient analysis of forging operations by the pseudo-concentration method.”International Journal for Numerical Methods in Engineering 25(1): 177-189.[13] Tezduyar, T.E. (2006). ”Interface-tracking and interface-capturing techniques for finite element computa-tion of moving boundaries and interfaces.” Comput. Methods Appl. Engrg., Vol. 195 (2006), 2983-3000

Flow front tracking in ALE/Eulerian Formulation FEM simulations of Aluminium

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Flow front tracking in ALE/Eulerian Formulation FEM simulations of Aluminium

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