Aluminum Alloy

8/2/2019 Aluminum Alloy

1/18

ngineering Computationsmerald Article: Aluminum alloy profile extrusion simulation using finiteolume method on nonorthogonal structured grids

humei Lou, Guoqun Zhao, Rui Wang

rticle information:

cite this document: Shumei Lou, Guoqun Zhao, Rui Wang, (2010),"Aluminum alloy profile extrusion simulation using finite volume

ethod on nonorthogonal structured grids", Engineering Computations, Vol. 29 Iss: 1 pp. 31 - 47

rmanent link to this document:

p://dx.doi.org/10.1108/02644401211190555

ownloaded on: 01-04-2012

ferences: This document contains references to 13 other documents

copy this document: [email protected]

his document has been downloaded 110 times.

ccess to this document was granted through an Emerald subscription provided by RMIT UNIVERSITY

r Authors:

you would like to write for this, or any other Emerald publication, then please use our Emerald for Authors service.

formation about how to choose which publication to write for and submission guidelines are available for all. Additional help

r authors is available for Emerald subscribers. Please visit www.emeraldinsight.com/authors for more information.

bout Emerald www.emeraldinsight.com

ith over forty years' experience, Emerald Group Publishing is a leading independent publisher of global research with impact in

siness, society, public policy and education. In total, Emerald publishes over 275 journals and more than 130 book series, as


2/18

Aluminum alloy profile extrusionsimulation using finite volume

method on nonorthogonalstructured grids

Shumei LouDepartment of Mechanical and Electrical Engineering,

Shandong University of Science and Technology, Taian, China

Guoqun ZhaoEngineering Research Center for Mould & Die Technology,

Shandong University, Jinan, China, and

Rui WangLinyi Normal University, Linyi, China

Abstract

Purpose The paper aims to use the finite volume method widely used in computational fluiddynamics to avoid the serious remeshing and mesh distortion during aluminium profile extrusionprocesses simulation when using the finite element method. Block-structured grids are used to fit thecomplex domain of the extrusion. A finite volume method (FVM) model for aluminium extrusionnumerical simulation using non-orthogonal structured grids was established.

Design/methodology/approach The influences of the elements nonorthogonality on thegoverning equations discretization of the metal flow in aluminium extrusion processes were fullyconsidered to ensure the simulation accuracy. Volume-of-fluid (VOF) scheme was used to catch the free

surface of the unsteady flow. Rigid slip boundary condition was applied on non-orthogonal grids.

Findings This paper involved a simulation of a typical aluminium extrusion process by the FVMscheme. By comparing the simulation by the FVM model established in this paper with the onessimulated by the finite element method (FEM) software Deform-3D and the correspondingexperiments, the correctness and efficiency of the FVM model for aluminium alloy profile extrusionprocesses in this paper was proved.

Originality/value This paper uses the FVM widely used in CFD to calculate the aluminium profileextrusion processes avoiding the remeshing and mesh distortion during aluminium profile extrusionprocesses simulation when using the finite element method. Block-structured grids with the advantageof simple data structure, small storage and high numerical efficiency are used to fit the complexdomain of the extrusion.

Keywords Flow, Aluminium, Alloys, Aluminum profile extrusion, Finite volume method,Non-orthogonal block-structured grids, Volume-of-fluid scheme

Paper type Research paper

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0264-4401.htm

This research work is supported by National Natural Science Foundation for DistinguishedYoung Scholars of China (No. 50425517), and Promotive research fund for excellent young andmiddle-aged scientists of Shandong Province (No. BS2010ZZ008). Thanks for the cutting-edgeinformation at International Conference on Extrusion and Benchmark.

Aluminum alloyprofile extrusion

31

Engineering Computations

International Journal for

Computer-Aided Engineering and

Software

Vol. 29 No. 1, 2012

pp. 31-47

q Emerald Group Publishing Limited

0264-4401

DOI 10.1108/02644401211190555


3/18

1. IntroductionThe production of aluminum alloy profile now is mainly by extruding. Under highpressure, aluminum ingot-shaped billet with certain speed and temperature is squeezedinto die cavity, and then a profile having almost the same cross-section with the cavity

can be got. During aluminum alloy extrusion, the deformation is strenuous, and themetal flow is complicated. To predict the metal flow, strain and stress distribution,numerical simulations are very useful to avoid extrusion defects, such as stressconcentration and bending. The simulation can decrease the product development timeand improve the product development quality. Many numerical methods are used tosimulate the aluminum alloy extrusion, among which the finite element method (FEM) isthe most popular one. Zhou etal. (2003), Lof and Blockhuis (2002) and Krause et al. (2004)simulated complex and thin-walled aluminum extrusion processes using the FEMapproach. Some influential conferences like International Conference on Extrusion andBenchmark (www.ice-b.net) and ET conference series (www.etfoundation.org) provideextrusion benchmarks mostly simulated by FEM codes. Since the deformation ratio ofthe aluminum alloy extrusion is relatively large, simulation using FEM can unavoidablyencounter remeshing processes, which would cause much time consuming and seriousvolume lost, and even simulation interruption due to mesh distortion.

In these years, the finite volume method (FVM) which was widely used in thecomputational fluid field is applied into aluminum alloy profile extrusion by Zhou and Su(2003) and Lou et al. (2008). However, the grids they used are orthogonal, and stepwiseapproximation is applied to fit cylindrical or complex boundaries. To decrease thesimulation error, the grids on the border often need partial grid refinement as shown inFigure 1(a). This would increase the number of the grids and thus influence thecomputational speed, and furthermore could not avoid the numerical error fundamentally.What is more important is that stepwise approximation makes it difficult to applyboundary conditions. To improve the grids fitness ability of the complex or curved

boundary, the most effective method is to use body-fitted grids. Nowadays, body-fittedgrids generation methods generally conclude two different types. One is coordinationtranslation method, which transforms irregular geometries of physical domain into simpleand regular geometries of computational domain, as demonstrated by Thompson (1980).But the coordinate transformation can lead to rather complex governing equations andambiguous physicalmeaning of terms. The other is the non-orthogonal grid methodraisedby Ferziger and Peric (2002). According to the relationship of the cells central nodes,this method is divided into structured and unstructured girds. Structured girds areorderly in arrangement, and the position relationships of the neighbor nodes are clear.In unstructured grids, the positions of the central nodes are disordered. Compared withthe structured grids, unstructured grids can fit irregular border better, even as well as theFEM approach can do. But the immense job of unstructured grids generation and the low

solving speed of the discreted governing equations are the bottlenecks of unstructuredgrids method. Structured grids method has the advantages of simple data structure,small storage, high numerical efficiency, etc. To improve the fitness of the structure gridsto complicated border, in this paper, block-structured grids were used, as shown inFigure 1(b). The typical aluminum profile extrusion field was discreted by non-orthogonalblock-structured grids. On the interface of the blocks, additional treatments to connect thecalculations in the blocks are needed. By using the block-structured grids, the complexboundaries of general aluminum profile extrusion could be fitted well.

EC29,1

32


4/18

2. Numerical model2.1 Constitutive equationsIn the aluminum profile extrusion, plastic deformation is very large. Elastic deformationsin extrusion processes are usually negligible compared with plastic deformation. So thedeformation material is considered as rigid visco-plastic material. Since the extrusion is aprocess with forced flow, material during hot profile extrusions is assumed isotropic.Therefore, it is described as a non-linear Newtonian fluid material, and the relationship ofthe stressand the strain is expressed by Faghrietal. (1984)and Asako and Faghri (1986)asfollows:

s0ij 2m _1 _1ij 1

where, m is the dynamic viscosity coefficient, _1 stands for the equivalent strain rate. _1ijrepresents the strain rate and is expressed as:

_1ij 1

2

ui

xjuj

xi

;

and the dynamic viscosity coefficient is expressed as:

m 1

3

s

_12

Figure 1.The grids used to discrete

flowing fields

x z

y

x z

y

Mesh refinement x dimension

Notes: (a) Non-orthogonal grids; (b) orthogonal grids

Block3Block2

(a)

(b)

Block1

Meshrefine

ment

inydimension


33


5/18

where, sis the equivalent stress. To make the equation (2) valid, the equivalent strain rate_1 cannot be zero. Otherwise, the material behaves as a rigid body. In order to extend themodel to the rigid zones, a lower, limiting value of_10 must be defined, that is, when _1 # _10,the control volume (CV) is assumed as rigid body (elastic region in nature), and _1 is

assumed to be 102

3 s2

1.During metal plastic forming, the constitutive equation can be expressed as

equation (3):

s s 1; _1; T 3

From equations (1)-(3), it can be seen that the dynamic viscosity coefficient m can bepresented as a function of equivalent strain 1, equivalent strain rate _1 and temperatureT. Equation (3) can be expressed by different material types according to the materialproperties, the deformation temperature, etc. (Aukrust, 1997).

2.2 Non-orthogonal structured gridThe metal flow fields during the aluminum profile extrusion need to be meshed.

To improve the fitness of the mesh to the complex region in actual aluminum profileextrusion, and avoid the numerical error and the boundary condition imposingdifficulty brought by the stepwise approximation, this paper used non-orthogonalhexahedron structured grids to discrete the flow regions of the aluminum profileextrusion. An arbitrary hexahedron structured grid is shown in Figure 2.

In the methods using orthogonal grids, staggered arrangement (Lou et al., 2008) cansuccessfully deal with the coupling of the velocity and the pressure and avoid erasing theshocks of the pressure field. But when the meshes are improved from two-dimensional tothree-dimensional, and simulation region is more complex,the disadvantages of staggeredarrangement, that is, complicated and inappropriate program, become more prominent.So another variable arrangement method, collocated arrangement, is widely used forcalculation in three-dimensionalnon-orthogonal or curved coordinate. In collocated grid,all variables are stored on the central nodes of the FVM cell, as shown in Figure 2. It can beseen that not only the scalars but also the vector components u, v and w are stored on thecentral nodes.

2.3 Governing equationsNo matter what grids are used, the metal flow would satisfy the governing equations,such as continuity equation, momentum equation and energy equation. The generalform of the governing equations is:

Figure 2.Non-orthogonal gridsand variable arrangementin collocated grids

u

vw

u

vw

x

yz

Node

Velocity

P E

EC29,1

34


6/18

trf divruf divGfgradf Sf 4

where, fis the general variable representing components of velocity vector u, v, w and

temperature T, etc. Gf is the general diffusion coefficient. When f represents avariable, Gf represents the corresponding physical meaning. For example, when frepresents the velocity components, Gf is dynamic viscosity m, while f denotes thetemperature T, Gf is thermal conductivity k. Sf is general source corresponding todifferent meaning off. u is the velocity vector on the CV face.

The integration of equation (4) in the CV V on the time step Dt is:ZV

ZtDtt

trfdt

dV

ZtDtt

ZS

n rufdS

dt

ZtDtt

ZS

n GfgradfdS

dt

ZtDtt

ZV

SfdVdt

5

where, S is the area of the CV face.

2.4 Discrete of the governing equationsTerms of the integrated form of the governing equation (5) on the non-orthogonal gridare discreted. The discrete procedure is similar with the orthogonal grid. But it isimportant to consider the non-orthogonality of the grid fully. In this paper, the discreteprocesses of the terms of the governing equations on the east CV face Se of an arbitrarynon-orthogonal hexahedral cell are studied, as shown in Figure 3. The discretizationsof the equations on the other five CV faces, that is, west, north, south, top and bottomfaces (w, n, s, t, b), are in the same way. P is assumed to be the central node of thecurrent volume. Its vector is rP. E, N and EN are the central node of the east, north andnorth-east volumes, respectively. Point e0 is the intersection point of the line P-E and

the face Se. Point e is the centre point of face Se.ne is the normal vector of Se. At thesame time, it is one of the dimensions in the local coordinate (n, t, s) assumed for

convenience. The direction of the line between the node P and E is defined to be j.Because of the non-orthogonality of the grid, j is not necessary consistent to thesurface vector ne, and point e

0 also not necessary coincides with the center e. Soauxiliary points P0 and E0 are used to calculate the normal gradient on the face.Here, point P0 is the intersection point of the surface vector ne and the face which goesthrough the line P-N and is parallel to Se. Point E

0 is the intersection points ofne and theface which includes the line E-NE and is parallel to Se.

Figure 3.East CV surface of the

non-orthogonal gridx

yz

e'

P'

P Ee

n

pr

e

xS

E

NCell Nodes

AuxiliaryNodes

EN


35


7/18

(1) Approximation of the change of time term. The first term of equation (5), which isthe change of time term, is discreted by using Euler implicit method:

d

dtZV rfdV

Documents

Aluminum Alloy