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Numerical Simulation of Three-DimensionalExtrusion Process Using hp-Adaptive Finite Element

ModelI I I ,M. P. Reddy, T. A. Westermann, E. G. Schaub, and J. T. Oden-

'Computational Mechanics Company, Inc., Austin, TX, 787592nCAM, The University of Texas at Austin, Austin, TX 78712

Abstract

Extrusion of aluminum sections is analyzed with a solution hp-adaptivefinite element model. The analysis is based on the assumption that the elasticeffects are negligible and rigid-viscoplastic material behavior is acceptable.The equations governing the flow of incompressible non-Newtonian fluidsare solved using an iterative penalty finite element model. An errorestimation scheme based on the element residual method is used to obtainerror indicators that form a basis for mesh refinement. The results obtainedwith solution adapted meshes are compared with those obtained after a seriesof uniform refinements of the coarse mesh. Based on the analysis it isobserved that reliable a-posteriori estimates of local errors in the solutionprovide near optimal finite element meshes for solving complex problems.

1. Introduction

1.1 BackgroundExtrusion is a commercially important process for mass production of

metal goods. Demand for high-speed production rates, increased productsafety standards, and lower energy consumption have resulted in theincreased use of numerical methods to predict the flow of metal duringextrusion and other forming processes. A complete understanding of the flowfield, temperature, load distribution, and behavior of the metal under variousoperating conditions is essential in selecting and optimizing the processvariables such as the extrusion rate, initial billet temperature, and die design.

In the past, analytical methods such as the slip-line field methods [I] wereused to analyze the fol111ing processes. These techniques allow the analysis

of -fofc'es,nand flow in·nsteady-state processes for simple geometries.Nevertheless, forming processes are often unsteady and the geometries areoften complex. In addition, the extrusion process is highly nonlinear and thesimulation results are often sensitive to small changes in die geometry,temperahlre, die pressure, friction, and material characterization. Hence,classical analytical methods are seldom suitable for accurately determiningthe effects of various parameters on the metal flow.

It is now accepted that these shortcomings can be overcome by usingmodem numerical methods such as the finite element method. Numericalmethods allow one to conduct systematic studies of the effects of variousparameters on the forming processes while providing understanding of thefunctionality of the process, often better than experimental methods. Whenusing numerical methods, it is necessary to ask the following questions:• Can complex flow domains be realistically modeled?• How accurate are the numerical solutions?• Is the !:,rrid fine enough to capture and resolve the flow behavior

accurately?The answer to the first question depends on the choice of mesh generation

scheme. It should be emphasized that the numerical method being used toanalyze the problem must be grid and coordinate system independent andalso be able to automatically add or delete cells or grid points as needed toresolve the local solution errors. This can be achieved by using unstructured!:,rrids,which is not feasible with mesh generators that use a body-finedcoordinate system. The solution adaptive finite element methods satisfy thesecriteria and are well suited for modeling complex flow domains, and alsoenable the consistent imposition of gradient boundary conditions. Inaddition, the finite element method enables the user to obtain detailedsolutions of the mechanics in the deforming body, such as: velocities,stresses, shear rates, and temperature and pressure distributions.

The accuracy of the numerical solution depends on many factors, such asphysical and mathematical model of the flow phenomenon. and the quality ofthe numerical approximation. In metal extrusion, as in other formingprocesses, the material may undergo plastic deformation. The plastic strainsoften outweigh elastic strains and the idealization of rigid-plastic or rigid-viscoplastic material behavior is acceptable [2]. This assumption leads towhat is known as flow formulation. In flow formulation, one obtains the flO\vfield by solving the equations governing the flow of incompressible non-Newtonian fluids in a given domain with appropriate constitutiverelationship and boundary conditions. The finite element method has beenused successfully to analyze extrusion and various forming processes [2-6].

Traditionally, the accuracy of the numerical method can be established bycomparing the solution with available experimental data, which is usuallyvery difficult to obtain for complex flow fields. In such instances, a finite

•

element mesh is built based on experience and engineering judgment, and apreliminary analysis is done. Then, doubling the mesh density, a secondanalysis is performed. If there is a significant difference in the two solutionsadditional refinements are required until a mesh-independent solution isobtained. Such numerical experiments are not always feasible when solvingthe complex problems encountered in common engineering practice becauseof the time involved in regenerating the finite element mesh and thecomputational cost in solving large problems.

These difficulties can be overcome using an automatic mesh refinementmethod that is based on local error estimates of the solution. The errorestimates provide a basis for directing mesh modifications to the regions ofthe domain where the solution errors are large. Using reliable informationand hp-adaptation techniques, the mesh in these critical regions can beredefined by either refining the elements or enriching the interpolation basisfor the elements.

This hp-adaptation and error-estimate strategy can remove the guess workfrom the mesh generation process and can provide the user with a highlysophisticated tool to analyze complex flow problems starting with a verycoarse mesh. Adaptive hp-finite element models have been used for theanalysis of significant classes of incompressible flow problems [7,8]. Thesetechniques automatically adjust the element size h and its interpolation orderp so as to deliver very high rates of convergence.

1.2 Present StudyIn the present work, we apply the mesh adaptive methodologies to

analyze steady-state extrusion of visco-plastic materials. To this end, wepresent results for axisymmetric, and three-dimensional problems. In eachcase, first we analyzed the problems using successive mesh refinements (upto three levels) starting from an initial coarse mesh. Next, starting with thesame initial coarse mesh, for each converged solution we estimated the localerror in the solution and adapted the mesh by refining the elements with largeerrors to produce an optimal mesh. The results from these refinement studiesare compared to determine the solution accuracy.

In the next Section we describe the governing equations and boundaryconditions for the aluminum extrusion process. In Section 3 we describe thefinite element model and adaptation strategies. Numerical results arepresented in Section 4.

2. Governing Equations

2.1 Conservation EquationsConservation of mass, momentum, and energy give the fundamental

equations that govern flow and heat transfer of incompressible viscous fluids.

These equations'- are written in terms of primitive variables (velocity,pressure, and temperature) with reference to an Eulerian reference frame, i.e.,a space-fixed system of coordinates through which the fluid flows. Let Q bea bounded domain of 9\0, n = 2 or 3 with a piecewise smooth boundary r.

Veu

p (u e V)upC pu e VT

=0=Ve(cr)

= V e (q) + <l>

In

In

In

nnn

(1)

(2)

(3)

where, u is the velocity vector, (J is the total stress tensor, and t is time. T isthe temperature, p is the mass density, Cp is the specific heat of the fluid atconstant pressure, q is the heat flux vector, and cD represents internal heatgeneration rate due to viscous dissipation.

2.2 Constitutive ModelFor viscous incompressible fluids, the components of total stress tensor CY

can be represented as the sum of the viscous (r), and the hydrostatic (p) part.cr=1:-pl (4)

where p denotes the pressure and 1 is the unit tensor. The constitutiverelations for the viscous stress tensor 1", and heat flux vector q, are given by

1: = 2µy (5)q = -kVT (6)

respectively. Where y = [(Vu) + (Vuy] /2 are the components of the rate ofthe deformation tensor, Tl is the viscosity of the fluid, which is a function ofthe temperature and rate of deformation, and k is the isotropic thermalconductivity of the material. In this analysis the viscosity of the fluid isassumed to obey a sine-hyperbolic inverse relation [9, I0].

(J

µ=---:-3E

(7)

where, CY is the flow stress, and c is the strain rate. The flow stress foraluminum depends on an internal state variable representing the materialmicrostructure [10]. An accurate description of the flow stress requiressolving an evolution equation, which governs the internal state variable.However, without loss of valuable information, the following empiricalrelations can approximate the flow stress:

. [2 ]112 . I {(Z lit"} .E = :]": 'Y ; (J = a sinh -I A ; Z = Ee(QIRT) (8)

where a is a constant, Q is the activation energy, A is the reciprocal strainfactor, II is the stress exponent, Z is the temperature dependent strain rate,and R is the universal gas constant.

2.3 Boundary Conditions -To complete the set of equations, Equations (1) - (3) need to be

complemeted with an appropriate set of boundary conditions. Let r be theboundary of the domain n. For the momentum equation, the velocities or thesurface tractions must be specified along the boundary. Similarly, for theenergy equation the temperature or the heat flux must be specified along theboundary. Let ru denote the boundary on which the velocities are specified,and rt be the boundary on which the stresses are specified. We have r = ruEBrt. Similarly, for the energy equation, let r T and r q denote the boundarysegments on which temperature and fluxes are specified (r = rT EB rq),Mathematically, these boundary conditions are:

u = u 0 on r" or t = cr - n on r,T=To on rT or h=q-n on rq (9)where n denotes the outward unit normal vector on the boundary r. Notethat pressure enters the natural boundary conditions through the total stresscomponents.

3. Finite Element Model

3.1 Iterative Penalty Finite Element ModelThe governing equations (1-3) are solved using an iterative penalty finite

element model [I I, 12]. This approach results in replacing the pressure in themomentum equation with (10) and omitting the continuity equation.pll+1 = pll - A(V'. u) (10)

Here A. is the penalty parameter and n is the nonlinear iteration number.Equation (10) is substituted into Equation (2) and the resulting equation issolved to obtain the velocity field. Pressure is calculated using a residua/-least-square Pressure-Poisson scheme. Details of the algorithm are presentedin [13].

Variational FormulationBefore proceeding with the variational formulation of the governing

equations, we introduce appropriate functional spaces and associated norms.Let V, H, W, and Q denote the spaces of admissible velocities, subspace ofsolenoidal velocities, temperature, and the space of admissible fluidpressures:

V = {v = vex) E (HI (.0))" : v = Uo on [II c r}H = {v E V :V' • v = 0 in n}W = {w = w( x) E W (Q)" : w = Toon rT c r }

Q={q=q(X)EL1(n):lqdx=0} (lla)

Ilvll: =11vII~,n= L(vev+vv:vv)L~ ; IIql12 = L(q2}ix

Ilwll:. = Ilwll~.n= L(w w + Vw e Vw}k (lIb)

In the iterative penalty algorithm, we obtain u, p as the limits ofsequences un, pn starting with an arbitrary initial pressure distribution l E

e(o.). With pn known (n ~ 0, we iteratively solve the following threeproblems until the solution sequence converges:Find u (11+1) E V such that

C(U1n+IJ, u1n+l), v) + a(u1n+1), v) - b(p", v)

+A(Veul,,-I),Vev)=(t,v) 'if VEV (12)

Find pln+I) E Q such that

(pIJ"'IJ_p",q)-A(Veuln~l\q)=O '\j qEQ (l3)

Find T(II+I) E Q such that

aT(T(II+IJ,W)= g(w)+(Iz,w) 'if WEW (14)

The linear, bilinear, and trilinear forms are given by:

a: V X V ~ 9i a(u, v) = L2µY: Y dO.

b : Q X V ~ 9i b(q, v) = LqV e v dO.

c : V X V X V ~ 9i c( u, v, w) = Lu e Vu e w dO.

aT:W XW ~9i aT(T,w)= L(WPCpueVT+kvweVT) dO.

g :W ~ ~ g( w) = L~w dO.

(t, v) = r t e v dr (h, w) = r h w drJrl Jrq

Error EstimationThe error estimation technique used in this analysis is based on the elementresidual method [14]. Let (II, p, 1) be the tme solution of the governingequations (1) - (3) and (uh

• p", 111) be the finite element approximation of thesolution of equations (12) - (14), The discretization elTor (ii, eP, eT

) E V XQ X W is defined bye"=u-u" ;eP=p_ph ;eT=T-T" (15)

Now consider the tuple (~, \11,6) E V X Q X W satisfying the problem:

a(<jJ,v)=a(eU,v)-b(v,eP)+c(uh,Uh,V) 'r;/VEV

d('V,q) = -b(eU,q) 'r;/ q E Q (16)

ace, w) =ar(er, w) 'r;/ WE W

where, d(p,q) = Ipq dO; aCT, w) = lkVT e Vw dO

Next we define the following energy-like norm [8]., , , ,III(eU,eP,eT)III~=II<jJt +1'411;+Iel: (17)

Equation (17) is solved locally over each element K in the mesh. A globalestimate of the error is obtained by summing element indicators, which arethe solutions to local problems with the element residual as data. Thesolution to the local problem (~K, \jIK, 8K) is obtained by solving:

aK(~K , V K) = -a KCu ~ . V K) - C K(u ~ ,u ~ , V K) + b( v K' p; )

+ f~~.~necrCu~, p~)t ev Kd[ 'if v K E VK C V

aK(8K,wK)=gK(wK)-aTl\CT:',WK) + fJ~'~nehKLwKd[

'ifwKEWKCW

\jI =-Veu"K K (I8)

Here < >a denotes a weighted average of the fluxes across the elementboundary. The solution to the local residual problem over each element iscalculated using quadratic polynomial approximation. The local errorindicator 11K for the element K and the global error indicator 11over the flowdomain are given by

11~ =aKC¢K,¢K)+lcveuK)2dO +aKC8K,8K)Kn

(19)

The mesh adaptation is controlled by a pair of user specifiedrefinement/unrefinement cut off parameters. The element errors in the meshare normalized by the maximum value of the element error. The cut-offparameters are compared against this normalized error and adaptationdecisions are made[ I 5]. Solution gradients are used to select the preferreddirections for anisotropic adaptation. Due to the strong directional nature ofthe solution, anisotropic adaptation can provide significant computationalsavings by keeping the number of degrees-of-freedom to a minimum.

The problem is-automatically re-solved several times, and each steady-statesolution step is followed by error estimation and automatic adaptation. Thesimulation is concluded whenever the solution error is significantly smalland the key solution variables such as velocity, pressure, and temperatureshow sufficiently small variation with respect to mesh changes.

4. Numerical ResultsIn this section we present results for two sample problems. The first one is anaxisymmetric rod extrusion. The second problem is extrusion of aU-section.The material properties for aluminum given Table 1 are used in this study.

Table I: Material properties for aluminum.Specific HeatThermal ConductivitDensitAa

...QIRN

4.1 Axisymmetric Rod Extrusion:As mentioned in the Introduction, the aim of this study is to apply meshadaptive methodologies to analyze the steady-state extrusion of visco-plasticmaterials. To iIJustrate the advantages of using adaptation techniques with ana-posteriori error estimate, we first solve the problem using a series ofuniformly refined meshes. Next, we solve the same problem using theadaptation techniques explained above. The relative accuracy of thenumerical solution is determined by comparing two sets of results (die force,centerline velocity, and temperature).

Here the diameter of the billet is O.I524m and the length of the billet 0.254m.The die opening is equal 0.0254m. This gives an area reduction ratio equal to36: 1. The billet is forced through the die opening by a ram at a constantspeed of 0.002 mls. The inlet temperature of the billet is held constant at6450 K. The cylinder and die are insulated and allow no slip.

Figure 1 shows the initial mesh, consisting of 27 bilinear elements and 40nodes. Starting with this coarse mesh, the mesh is uniformly refined threetimes to obtain three meshes with 108.432, and 1728 elements, respectively.The successive finer meshes are obtained by dividing each element in theprevious coarse mesh into four smaller elements of same aspect ratio. Inother words, the 108 elements mesh is obtained by dividing each element inthe 27 element mesh into four smaller elements of equal area.

'.. '~...-.:.

Figure I. Initial coarse mesh (27 elements; 40 nodes).

As expected, the qualityof the solution in termsof the computedvelocity, pressure, andtemperature distributionis improved with eachrefinement. For clarityand ease of explanation,we refer to the initialmesh as MeshO. Meshl,Mesh2, and Mesh3 are Figure 2: hp-adapted finite element meshthe meshes obtained (58 elements: 324 nodes)

after one-, two-, andthree- uniformrefinements of MeshO.Figures 2 shows thefinite element mesh Fi!!:ure3: Mesh after 3 uniform refinementsobtained after three (1728 elements: 1825 nodes)levels of hp-adaptation.The hp-adapted mesh for the consists of 58 elements and 324 nodes. In thismeshes, the element density is higher in regions where large variations in thesolution take place (such as the die entrance and other singularities). Thefinite element mesh after three uniform refinements (Mesh3) is shown inFigure 3. Comparing mesh in Figure2 with the one shown in Figure 3indicates that in Mesh3 large number of elements are placed in regions oflittle importance. It should be pointed out that for complex geometries, itwould be difficult to manually generate meshes similar to those shown inFigures 2,

The predicted values of normal force acting on the die face for variousmeshes is presented in Table 2. The difference in normal force predicted byadapted mesh and Mesh3 is less than I. I % Similar behavior was observedfor the other variables (maximum velocity, temperature distribution, ramforce, and energy balance). It is clear from the results that reliable a-posteriori estimates of local errors in the numerical solution can be used toprovide a near optimal finite element mesh to solve complex problems.

Table 2: Normal force on the dieElements Nodes Force (MN/m) Remarks27 40 4.6086 MeshO (Initial mesh)108 133 4.0264 Meshl (I Uniform Refinement)432 481 3.0522 Mesh2 (I Uniform Refinement)1728 1825 2.7516 Meshl (I Uniform Refinement)

i 58 1324 2.7820 hp-adapted mesh I

4.2 Extrusion of a V-SectionHere we analyze the extrusion of a U-section. The diameter of the billet isO.2m and its length is O.32m. The dimensions of the horizontal and verticallegs of U-section are O.08m x O.005m and O.075m x 0.005m, respectively.The billet is forced through the die opening by a ram at a constant speed of0.002 mls. The inlet temperature of the billet is held constant at 6450 K. Thecylinder and die are insulated and allow no slip. A bearing length of 0.005mis assumed at the die exit and no slip conditions are imposed at thebearing/profile interface. Initial finite element mesh show'n in Figure 4consists of 1354 hexahedral elements with 1903 nodes. Starting with thisinitial coarse mesh we solved the problem using the adaptive processexplained in previous sections. The final adapted mesh is shown in Figure 5and consists of 10,958 elements and 13,6 I5 nodes. This figure shows that themajority of the refinement has taken place near the die exit. Computed valueof normal force exerted by the ram is 7.187 MN.

Figure 4: Extrusion ofU-section:Initial coarse mesh.

Figure 5: Extrusion ofU-section:Adapted final mesh

Figure 6: Extrusion ofU-section:Iso-surfaces of temperature.

~...~.~,'~l

Figure 7: Extrusion ofU-section:Particle traces

Temperature iso-surfaces are plotted in Figure 6 and they show thetemperature increase inside the container due to viscoLls dissipation. Particle

trajectories are plortedin Figure 7. These lines show the time history ofparticles entering the flow domain.

5. ConclusionsThe main objective of this effort is to demonstrate the applicability of hp-adaptive finite element techniques to simulate steady state extrusionprocesses. We solved several extrusion problems with different degrees ofcomplexities. Based on the analyses, it is observed that the present errorestimation and adaptation strategy is capable of handling highly nonlinearproblems such as modeling of aluminum extrusion. The element residualmethod has been found to be capable of identifying regions with largesolution errors. The main advantage in using this method is that it eliminatesthe time spent in performing several mesh generation and analysis cycles thatare common with conventional numerical tools.

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