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ScienceDirect Materials Today: Proceedings 2 (2015) 4726 – 4731
Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB 2015
High efficiency in the simulation of complex extrusion processes using an advanced simulation method Longchang Tonga*, Christoph Beckera, Pavel Horaa a
ETH Zurich, Institute of Virtual Manufacturing, Tannenstrasse 3, 8092 Zurich, Switzerland
Abstract In order to simulate the entire process of the complex profile extrusion with remarkably reduced computational costs, a new procedure is proposed in this paper. The FE-mesh is generated previously in the pre-processing stage and the elements are activated according to the material flow during the FE computation. With this method the problem of instable boundary conditions that arises from the remeshing algorithm is avoided. The feasibility of this method is demonstrated with complex 3-D examples. Several important factors for successful simulation are also discussed in this paper. 2014Elsevier Elsevier Ltd. rights reserved. ©©2015 Ltd. AllAll rights reserved. Selection under responsibility of Conference Committee of Aluminium Two Thousand World Congress International Selectionand andPeer-review Peer-review under responsibility of Conference Committee of Aluminium Two Thousand Worldand Congress and Conference on Conference Extrusion andonBenchmark 2015 ExtrusionICEB and Benchmark ICEB 2015. International Keywords: Extrusion; Advanced numerical methods; Finite element simulation; Frictional Modelling
1. Introduction Aluminium alloys have been used to produce complex profiled bars at elevated temperature with extrusion processes. This method possesses high efficiency as well as high precision. Nevertheless, the design of the extrusion dies is still based mainly on the experience of the engineers despite the long history of this manufacturing technology. Try and error method is inevitable for new complex products. Finite Element Method (FEM) has been used to simulate the extrusion processes of profiled materials for more than 2 decades [1-3]. However, much less
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2214-7853 © 2015 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of Conference Committee of Aluminium Two Thousand World Congress and International Conference on Extrusion and Benchmark ICEB 2015 doi:10.1016/j.matpr.2015.10.005
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progress in the simulation of profile extrusion has been achieved in comparison with the simulation of sheet forming processes. The situation is mainly due to several special difficulties. First of all, the plastic deformation and the strain rate in the extrusion processes is extremely high. If the Updated Lagrangian Method (ULM) is adopted, the strong distortion of the FE mesh results in numerical errors and stops the further computation. In these cases the regeneration of FE mesh is indispensable. However, most remeshing algorithms use the deformed surface of the workpiece as the reference surface. The geometries of the tools are difficult to be involved. In many cases the new mesh possesses insufficient precision for the geometry of forming tools. Instable and wrong boundary conditions are caused during the computation. The results of the simulation is then absolutely wrong, as shown in Fig. 1.
Fig. 1. Numerical errors introduced by remeshing: a) Incorrect mapping of sharp edges b) Wrong velocity distribution due to mistakes in remeshing.
Secondly, the properties of aluminium alloys for the profile extrusion are also very complicated. As the extrusion processes are performed at high temperature, the material behavior is a function of more variables like strain, strain rate and temperature. To get the reasonable description of the flow function is not a trivial task. Besides, the behaviors of friction are much more complex than that in the sheet forming processes. The friction force can be calculated neither with the Coulomb model [4] nor the shear friction model [5]. If the friction is not properly applied, the distribution of the velocity will have remarkable deviation from the real process depending on the friction model and parameters. Last but not least, due to the complex structure of the extrusion tools, very large equation systems are inevitable. Although the computers are getting faster and faster, the efficiency of the simulation plays always an important role. There are several special purpose programs for the simulation of profile extrusion processes as well as docents of general purpose packages for the simulation of bulk forming processes available in the market. However, none of the general purpose packages can finish the computation for a complex extrusion process in reasonable time. From the practice we can define the reasonable computational time as a couple of hours for a profile with moderate complexity and no more than 24 hours for very complicated processes. Some simulation tools like HyperXtrude ignore the filling process and just calculate the forming of profile. Excellent results have been reported [6-7]. Anyway, we know that the material flow in the welding channels determines not only the temperature distribution at the end of the filling but also can cause the damage on the extrusion tools if the unbalance loads are induced in the filling process. In order to simulate the profile extrusion successfully, the problems mentioned above have to be solved. Based on the consideration discussed in previous sections, a new method is proposed in this work. The mesh is generated in the pre-processing stage and the elements are activated sequentially according to the velocity field during the computation. Stable boundary conditions are achieved and errors due to remeshing are entirely eliminated. Computation examples show the feasibility of this method. Meanwhile, the computation time is remarkably reduced. 2. Special ALE-meshing method A completely new concept has been developed to simulate the profile extrusion processes. Instead to perform remeshing repeatedly, the whole workpiece include billet, welding channels and a piece of profile is meshed previously in the pre-processing stage. The element size of the mesh in different zones is elaborately adjusted so that it can describe the geometries with sufficient precision while too large system should be avoided. Since it belongs to the pre-processing, the mesh can be iteratively modified without much computational costs.
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This method is shown schematically in Fig. 2. A basic mesh is generated before the calculation is started (Fig. 2 a). The criterion for a well generated mesh ensures the accurate description of the geometries and the sufficient number of elements in the sections of the profiles. Meanwhile, the number of total elements can be limited according to the capacity of the computers. At the beginning of the simulation, only the elements for the billet are activated. The velocity field is then calculated. Further elements are put into usage according to the velocity distribution (Fig. 2 b). The Surface of the material flow can also be well traced with the velocity distribution. As soon as all pre-meshed elements are activated, the program generated new elements at the end position of the premeshed profile (Fig 2. c). The newly generated elements are updated to provide the information of the movement of the extruded material.
a)
pre-meshing of the workpiece
b) activated elements
c) further generated elements
Fig. 2. Schematic explanation of the new method
There are several advantages using this method. First of all, since the mesh gives very exact description for the geometries of the forming tools, the sharp edges and corners can be well recognized to ensure very stable definitions of boundary conditions. The numerical problems that comes from the “instable” boundary conditions are then entirely eliminated. Although very small time increments have to be adopted to ensure the convergence of the solution, the velocity distribution does not change abruptly between small increments. In this method, new elements are activated as the material flows into the volume occupied by the elements. The incremental volume and corresponding time step can be calculated as, n
't
¦ 'V
i
i 1 punch
v
(1)
A
where the 'Vi is the volume of the i-th new activated element, vpunch denotes the velocity of the extrusion punch and A represents the section area of the billet. Usually the time increment obtained with Eq. (1) is relatively large. Sub-increment are used to keep the numerical process stable. However, the velocity field can be treated as constant in the increment. Only plastic deformation, temperature and the convections of the variables are calculated in the sub-increments. This procedure achieves remarkable acceleration of the computation. 3. Element formulation and direct solver It is very difficult to generate a mesh to fill a complex geometry with hexahedral elements. For this reason most of the simulation packages for bulk forming simulation adopt the tetrahedral element formulation. As well know, strong locking effect leads to wrong results if the linear function for the velocity is adopted because of the incompressible condition in the bulk forming processes. In order to overcome the difficulty, the mini element with a bubble function introduced by a middle node can be used [8], as shown in Fig. 3. The interpolating functions are
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Fig. 3. A non-linear tetrahedral element. 4
u
¦Lu
256 L1 L2 L3 L4u m
i i
(2)
i 1 4
p
¦L p i
i
i 1
here the Li denotes the linear interpolating function of the i-th node. The whole interpolation satisfies the requirement of the isoparameteric conditions with higher polynomial expression. The locking effect is eliminated consequently. Because the middle node is isolated from other elements, it is an internal node and can be eliminated explicitly from the variables. The equation of an element can be therefore expressed as ª K uu K up º u ½ F ½ . «sym. C » ® ¾ ® ¾ pp ¼ ¯ p ¿ ¯0¿ ¬
(3)
A disadvantage of this element formulation is that the system is non-positive definite as the diagonal terms possess negative values. Therefore the high efficient conjugate gradient iterative solver fails to solve the equation system. Fortunately, the direct solvers have been well developed with high efficiency and the possibility of parallel computation. For example, we adopt the PARDISO parallel direct sparse solver from Intel MKL [9]. High computational performance has been achieved. As one of the examples, Table 1 lists the computing parameters of a moderate complicated extrusion process shown in Fig 2. Table 1: The computational parameters for the example in Fig. 2 Number of elements Number of CPUs CPU-frequency
CPU-Time
up to 985,000
3.5 hours
6 Intel Xeon E5-1650
3.2GH
4. Implementation of friction model One of the most important and most sensitive factors for successful simulation of extrusion processes is the friction model. In comparison with the tribological conditions in the sheet forming processes, the friction stress in the extrusion processes is much higher and shows much stronger dependency on multi-factors like surface pressure, temperature and the relative velocity. It is very difficult to measure the friction force directly under the real process conditions. Besides, the friction force possesses strong non-linear property. It is induced by the relative movement of the material and can jump abruptly if the contact condition changes slightly. Significant progress has been achieved recently [10]. The method to measure and to evaluate the friction stress is proposed in that work. The dependency of pressure, temperature and relative velocities is discussed and the function to evaluate the friction stresses is well established based on the experiments. Although this description still cannot distinguish different contact conditions at the smooth surfaces of the tools and at the sharp edges of the extrusion dies where the deformation rate is extremely high, computational examples verified the validity of the model. Special attention should be paid to apply the evaluated friction force in the simulation. The friction force is a kind of passive force. It is induced by the relative movement. If the movement decreases to a sticking behavior, neither the magnitude nor the directions of the friction force can be determined. Improperly applied strong friction force might
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result in wrong distribution of the velocity field. Even oscillation of the nodes between iterations can be caused as soon as the friction force is overestimated. An alternative method has been adopted for the profile extrusion simulation to avoid the numerical problem caused by high friction forces in present work. A virtual nonlinear spring is introduced on each node which get friction force from forming tools to stabilize the computation, as shown in Fig. 4. The stiffness of the spring depends on the friction force and the relative displacement at the node as WA / 'u ('u ! 'uthr ) ® WA / 'uthr ¯
K
where
(4)
'uthr is a threshold value to avoid numerical errors. It is set as 10-4 to ensure the reasonable results. tool
FE mesh
W
vr
nonlinear spring
Fig. 4. The nonlinear spring model for the friction behaviour.
With this implementation the overestimated friction forces enhance the sticking behaviors of the nodes therefore stabilized the numerical procedures. An experiment was set in order to verify the effect of the implementation for the friction forces. Four conical channels with the same positions and same diameters but different angles are used in the experiment as shown in Fig. 5 a). The simulation result with this implementation of friction model is also shown in Fig. 5 b) and c). Compare with the experiment data, satisfactory agreement with the measurement has been achieved.
a)
The setup of the experiment;
b) Simulation result;
c) Comparison with experiment
Fig. 5. The experiment to test the implementation of friction.
5. Computational example An extrusion process is chosen to demonstrate the possibility using this numerical tool to investigate the influence of slight modification of the geometry of the extrusion tools. The aluminium alloy used in the extrusion process is A6082 and the flow function can be expressed using a modified Zener-Hollomon model [11] as
VY
A exp(Q / RT )H m *{1 D exp[c(H H 0 ) 2 ]}[1 E exp( NH n )] .
(5)
The parameters for A6082 are obtained using the experimental data as listed in Table 2. The results from the simulation are shown in Fig. 6. The curvature of the profile indicates the uneven distribution of the velocity (Fig. 6 a). As the extrusion die is modified with conical form at the corners, improvement can be seen
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directly from Fig. 6 b). Fig. 6 c) shows the local distribution of the velocity. Since the conical shape has more brake effect than a vertical bearing it causes the dead zones at the corners. This slight modification improves the velocity distribution and correct the curvature to certain degree.
a)
The simulation of a profile extrusion; b)
The result with modification
c) Dead zones caused by the modification
Fig. 6. Demonstration of the simulation of a real process Table 2: The parameters for the flow function of A6082 A Q c D 0.4902 28177.0 10. 0.
H0 0.
m 0.0839
E 0.0893
N 38.445
n 1.
6. Conclusions The method presented in this paper realizes the simulation of the complex profile extrusion processes and accelerates the simulation remarkably. The computation can be performed on ordinary PCs within reasonable computational time. The results provide systematic information and can help the optimization of the processes in many circumstances. Acknowledgement The authors would like to express their sincere thanks to the companies WEFA and Honsel for providing the examples of testing computation. The long term cooperation between FZS, TU Berlin and IVP, ETH Zurich is also well appreciated. References [1] L. Tong, PhD thesis, Institut für Umformtechnik, ETH Zürich, Nr. 11107, (1995). [2] L. Donati, Proceedings of the Conference Latest Advances in Extrusion Technology and Simulation in Europe, Bologna, (2007) 89-95. [3] A. E. Tekkaya, S. Kavakli, Steel Res. 66 (1995) 377-383. [4] F. Parvizian, T. Kayser, C. Hortig, B. Svendsen, J. Mater. Process. Technol. 209 (2009) 876-883. [5] A. Güzel, A. Jäger, F. Parvizian, H.G. Lambers, A. Tekkaya, B. Svendsen, J. Mater. Process. Technol. 212 (2012) 323-330. [6] M. Schikorra, M. Kleiner, CIRP Annals-Manufacturing Technology 1 (2007) 317-320. [7] G. Liu, K. Huang, J. Zhou, J. Duszczyk, Comp. Meth. Mater. Sci. 2 (2011) 259-264. [8] R. H. Wagoner and J. L. Chenot, Metal Forming Analysis, Cambridge University Press, (2001). [9] O. Schenk, K. Gärtner, W. Fichtner, A. Stricker, Future Gener. Comp. Sy. 18. (2001) 69-78. [10] C. Becker, P. Hora, J. Maier, S. Müller, Key Eng. Mater. 585 (2014) 25-32. [11] L. Tong, S. Stahel, P. Hora, Numisheet 2005, 778 (2005) 625-629.