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The development of real time re-meshing technique for simulating cold-roll-forming using FE methods
Journal of Materials Processing Technology 147 (2004) 1–9
The development of real time re-meshing technique for simulating cold-roll-forming using FE methods A. Alsamhan a,∗ , I. Pillinger b , P. Hartely b b
a Mechanical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Received 26 October 2000; received in revised form 26 October 2000; accepted 17 February 2003
Abstract Cold-roll-forming (CRF) is an important sheet metal forming process. However, product design procedures, in terms of rolls design and rolls pass schedule, remains more an art than science. Finite-element (FE) computer simulation can be used to predict the deformation and final product geometry, hence, reduce the design time and cost. The main objective of this work is to develop a real time re-meshing technique that can be used to run and complete the computer simulation of CRF process. This will enable decreasing the computational time and predicting the final geometry of cold-roll-formed product. A real time re-meshing technique using dual meshes was developed and benchmarked by applying the technique on traditional flat stripe rolling with friction. © 2003 Elsevier B.V. All rights reserved. Keywords: Re-meshing; Finite-element; CRF
1. Introduction Cold-roll-forming (CRF) is an important sheet metal forming process, which is widely used to manufacture long sheet metal products with a constant cross-section thickness. The design procedure for CRF products, roll pass sequences and forming rolls remains more an art than a science. Therefore, to reduce the process, development efforts and forming defects, which result in improper process design, finite-element (FE)-computer simulation can be used to predict strain distributions and final geometry of cold-roll-formed products. Due to the complicated deformation of the CRF process, it was found that most of the FE models described in the literature are based on three approaches. The first two approaches are based on simulating strip rolling without friction, while the third approach is based on simulating strip bending between the forming stands. In first approach the initial deformation shape is obtained by pressing the deformed rolls towards the undeformed strip, followed by pulling the strip longitudinally from the strip leading edge through the roll gaps of the forming stands [1–3]. The second approach, involves predicting the deformation by pulling an undeformed strip from the leading edge, through the roll gaps of deformed roll sets of the forming stands ∗
Corresponding author.
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00469-2
[4,5]. The third approach, is based on predicting the deformation between the forming stands, without simulating the strip rolling and starting from undeformed stock [6]. This approach involves predicting the deformation by applying nodal linear displacements on the nodes located at the channel flange edge, of each forming stand. Nodal constraints on mesh thickness, are imposed on the web nodes located at the forming stand centre line plane. In the initial two approaches, the FE mesh is modelled from fine elements along the strip length [4,5]. This was adapted to continue and complete the strip rolling simulation, this may increase the computational time. To decrease the computational time, Off-line re-meshing technique can be used frequently, for each phase of the deformation [1–3]. Off-line can decrease the computational time, however, the deformation history of the FE mesh is lost after re-meshing. Hence, re-meshing the FE mesh using dual meshes, one for storing the deformation history, and another for FE computational will solve the two problems. A real time re-meshing techniques using dual meshes was developed and benchmarked are given and demonstrated in this paper. 2. Computation services and experimental work The utilised FE code is the EPFEP3 [7] program. It is a 3D implicit elastic–plastic FE program developed, initially to deal with bulk metal forming analysis by Pillinger [7].
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A. Alsamhan et al. / Journal of Materials Processing Technology 147 (2004) 1–9
The EPFEP3 program has proved successful techniques for modelling metal forming processes, such as boundary surface and friction layer techniques. All the computer simulations presented in this research were executed on HP 400 Apollo workstations, with 32 MB of memory and 600 MB of hard drive capacity for the data storage. The strip material used in the FE model as given in the experimental work of [8] was bright mild steel (JISG 3141 SPEEN-50) with yield strength of 160 MPa. These experiments were conducted on cold-rolled trapezoidal channel sections. It was found that the mild steel (AISI 1006) is the closest material property sites in the literatures [8,9]. The Poisson’s ratio was specified as 0.34 and the Young’s modules as 190 GPa. The flow stress–strain curve of AISI 1006 is obtained from Altan et al. [10] and given (in SI units) as (σ¯ = σy + Kεn ) where σ¯ is the stress, ε the plastic strain, σ y the yield stress equal to 160 MPa; K = 613.8 MPa, and exponent n = 0.31. The constitutive material equation employed by the EPFEP3 [7] program, uses logarithmic and exponential functions to model material strain-hardening behaviour at the room temperature and given as follows: εp + hp2 y(εp ) = hp0 + hp1 ln hp2 +hp3 exp(hp4 (εp − hp5 )) for εp ≤ 1.0 (1) y(εp ) = Y(εp ) +
dY(ε0 ) (εp − ε0 ) dεp
for εp > 1.0
(2)
The coefficients of the logarithmic function (hp0 , hp1 and hp2 ) and the exponential function (hp3 , hp4 and hp5 ), were obtained through trial and error procedure [9] and are given in Table 1. Fig. 1 illustrates the strain-hardening behaviour of the two material equations, (σ¯ = σy + Kεn ) and Eqs. (1) and
Table 1 The coefficients of the constitutive material equation of the EPFEP3 program hp0 (MPa) hp1 (MPa) hp2 hp3 (MPa) hp4 hp5 ε0
60.0 139.467 0.00602 322.6 −23.423 −0.05 0.705
(2). The figure illustrates a little variation between two flow curves. The friction layer in the EPFEP3 program is transitory and uses the friction factor [11] to imitate different lubricants. Friction factor of m = 0.5 was adopted in all the FE results presented here. It was found that, this value provided stable contact between rolls and FE mesh during the rolling simulation. Maintaining this contact during the rolling simulation is very important for the effectiveness of the friction layer technique, that assists mesh movement along the rolling direction.
3. Dual mesh and re-meshing techniques It was found that, the best re-meshing technique could be achieved using a special re-meshing technique adopted from [12], to simulate ring-rolling press as bulk forming process. In this technique dual meshes utilised are a computational mesh and a fine mesh. The computational mesh has fine elements in the main deformation zone, and coarse elements elsewhere, and would be used for the FE analysis. The fine mesh has fine longitudinal elements structure throughout its
Fig. 1. Actual and fitted flow stress–strain curves for plastic strain range from 0 to 0.1.
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length, and is used to store the state variables of the FE model, which are updated real time after each phase of deformation. The computational mesh is constructed from a selection of nodes in the fine mesh, at each phase of the deformation. The state variables, such as stresses and strains (including the membrane strains), are transferred directly between the dual meshes for those nodes, that belong to both meshes in the deformation zone, while they are interpolated for the nodes that do not correspond to any in the computational mesh. The CRF program is a FORTRAN program developed by the authors, to execute the FE model with the dual mesh technique, for roll-formed channel sections. The program is used in conjunction with the FE analysis program, EPFEP3, to simulate the CRF process, for any prescribed section. Its main objective is to establish a suitable FE mesh and to carry out re-meshing and data transfer between the dual meshes, to continue the computer simulation for the process. The
3
program can be used to simulate any CRF product or any flat rolling, by modifying the sub-programs that generate the initial dual meshes. During the FE analysis and data transfer between the dual meshes, different files were generated to store the model files and to display the FE results for the dual meshes. Details of the CRF program and its functions can be found in [9]. To established the initial contact between friction boundary surfaces and FE mesh, the initial deformed fine mesh is set up using the bend angle curve [13], with fine elements along the strip length and through all the forming stands. The general configurations and number of elements and their lengths along the strip width, thickness and the longitudinal length, are all variables that can be specified through the CRF program data file. Furthermore, the number of the forming stands, their folding angles and guiding stand consideration, are also variables that can be specified by the user through the CRF program data file. Fig. 2 illustrates an
Fig. 2. Fine mesh modelling for two forming stands of channel section with folding angle sequence 0–20–40◦ using bend angle curve given in [13].
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Fig. 3. Fine mesh and computational mesh structure, interpolation and data transfer between the dual meshes.
example of a fine mesh modelled for two forming stands with fold angle of 20◦ , first forming stand, and with fold angle of 40◦ for the second forming stand, using the bend angle curve prescribed in [13]. The computational mesh between forming stands generally consists of two main zones (Fig. 3) shows the uniform zone and the deformed zone. Since the uniform zone is remote from the deformation, very coarse longitudinal elements were used. The second zone is mainly through the deformation length and modelled using graded coarse elements. Fine elements were also modelled, using behind and along the roll gap. Once the number of the longitudinal elements and their lengths are defined, the total number of the longitudinal elements and their lengths, in terms of contained fine elements are defined along the strip length. The data transfer between the two meshes is carried out in two phases. The first phase includes data transfer in the fine mesh. Since the location of nodes and elements are identical in both meshes, the state variables of the FE model are transferred directly without any interpolation. The second phase,
includes data transfer in the coarse and very coarse mesh zones of the computational mesh, where interpolation is essential. In this case, some of the nodes in both meshes will be in the same position, e.g. node N1 , N2 , N3 and N4 in Fig. 3, where the state variables also transferred directly without interpolation. However, the majority of nodes will not be coincident, e.g. node P in Fig. 3, and interpolation is essential to obtain the state variables for those nodes. Referring again to Fig. 3 and the node P, if a conventional re-meshing technique was used, the position of the prescribed node P is obtained, at the position P by interpolating its position from the positions of the four nodes N1 , N2 , N3 and N4 , assuming a two-dimensional interpolation function. This illustrates that the interpolated position, will lie along the chord between N2 and N4 and not the actual deformation path on the arc between N2 and N4 . The data transfer will be further in error when the bending in this zone is larger, and the selected computational element is very coarse. Therefore, to minimise the accumulated errors, the position of the point P is determined by updating its co-ordinates from the
A. Alsamhan et al. / Journal of Materials Processing Technology 147 (2004) 1–9
incremental displacement vector (V), which is interpolated from the corresponding values of the four nodes. Furthermore, the predicted error can be minimised by utilising finer longitudinal elements in the secondary deformation zone, so that the arc joining N2 and N4 will be small. The prescribed error will be nearly eliminated for those very coarse elements located away from the deformation zones. Re-meshing the computational mesh was considered along the rolling direction. Initially, the longitudinal element lengths of the computational mesh, in terms of the number of the longitudinal fine elements, were calculated using the initial computational mesh generator. The frequent re-meshing of the computational mesh depends mainly on adjusting the longitudinal element lengths for the prescribed mesh. The decision to re-mesh was determined by comparing the longitudinal position of the fine mesh nodes to the
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longitudinal position of the centre line of the last forming stand. For example, consider the fine mesh overlaid on the computational mesh given in Fig. 4 for two forming stands. As illustrated in Fig. 4, monitoring the longitudinal position of the fine mesh node N11 determines the 1st re-meshing decision. Re-meshing of the computational mesh is applied when the longitudinal nodal position of the prescribed node is greater or equal to the longitudinal position of the centre line of the last forming stand (Y). Accordingly, the longitudinal element lengths will be updated, hence the new computational mesh structure will be updated using the updated computational element lengths. The next re-meshing decision will be considered by monitoring the longitudinal position of the fine mesh node N10 . This re-meshing procedure will continue until the fine mesh passes the centre line of the last forming stand rolls, and fully deformed longitudinally.
Fig. 4. Adjusting coarse element lengths of the computational mesh during re-mesh.
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4. Benchmarking the dual mesh and re-meshing techniques The main objective of the benchmark test program, is to test the dual mesh FE model before moving to more sophisticated CRF process modelling. It is also important to check the program functions during the CRF program development, e.g. the interpolation function, data transfer between the dual meshes, membrane strain calculation etc. For these reasons a small-scale problem was utilised in the
benchmark test, where small number of elements and nodes were used for the dual meshes. Furthermore, the simulation results of the dual meshes FE model were compared to the FE results of the conventional FE model, using the same fine mesh used in the dual meshes model for the FE computation. The benchmark test is modelled as a traditional flat strip rolling process, but with very little thickness reduction. The upper and lower rolls were represented by two cylindrical boundary surfaces, and used to roll a strip of sheet metal of dimensions 6.4 mm (length) by 18.0 mm (width) by 0.6 mm
Fig. 5. Dual meshes FE model (benchmark), computational mesh and fine mesh configurations.
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(thickness). The rolling direction was selected to be along the strip length. Due to the vertical symmetry of the process, half of the strip width was modelled, i.e. 9.0 mm. The roll radius of both the upper and lower rolls was selected to be 51.5 mm. The fine mesh was modelled from 486 nodes and 160 elements, as illustrated in Fig. 5. The fine mesh was modelled from a single layer of elements through the mesh thickness. Two elements were utilised along the mesh width having 3 and 6 mm width. The fine mesh was created with 80 fine
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elements along the strip length, each element having a longitudinal length of 80 m. The longitudinal fine element length was selected to restrict the very small plastic strain during rolling, in which the roll penetration was 0.25 m. The computational mesh was created with 198 nodes and 64 elements as illustrated in Fig. 5. Ten fine elements were utilised near the mesh leading edge and also at the trailing edge. The roll contact area covered nearly four longitudinal fine elements. A series of longitudinal coarse elements, varying in size, were used before the roll contact area and
Fig. 6. The variation of the computational mesh configuration for different rolling increments of the benchmark model.
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the main deformation zone, as shown in Fig. 5. Three very coarse elements were used remote from the main deformation area. Ten fine elements near the trailing edge of the FE mesh, which are initially remote from the deformation area, which will gradually move towards and into the main deformation zone as re-meshing is implemented. The material of the strip was again mild steel AISI 1006. The rolling increment was selected to be 0.04◦ (fixed) applied on both rolls. The computer simulation was carried out for 120 rolling increments using the dual mesh FE model. The computational time was 19.4 s for each increment executed by the EPFEP3 program. The data transfer between the two meshes required 13.0 s, for the same increment. The computer simulation results were documented for specific rolling increments (i.e. 1st, 17th, 30th, 40th, 50th, 70th, 100th and 120th
increments). The general configurations of the computational mesh for the prescribed rolling increments, including roll contact areas are shown in Fig. 6. The figure illustrates that the configuration of the fine elements, refer increment 1, located near the mesh trailing edge, was gradually changed to be after the main deformation zone during re-meshing of the computational mesh (see increment 40, for example, of Fig. 6). Furthermore, the three very coarse elements were also moved gradually from a position before the main deformation area, increment 30, to a position after the main deformation area, where steady deformation was obtained (see increment 120 in the same figure). The computer simulation was executed for 120 rolling increments, using the same fine mesh in the FE analysis, conventional FE model. The computational time was 44 s,
Fig. 7. Generalised plastic strain contours of dual mesh FE model and conventional FE model for increment 70 of the benchmark model.
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for each increment using the fine mesh compare to 19.4 s, with the dual mesh model. In general the computational time with the dual meshes, including data transfer between the meshes is 50% less, compared to the conventional FE model. Fig. 7 illustrates the generalised strain contours for the dual mesh FE model. The generalised plastic strain contours for the dual mesh FE model and the conventional FE model look more different than they really are. This is due to the fact that the range of the contour colours selector employed by the program is dissimilar. The generalised plastic strain contours for the dual mesh FE model are identical especially in the main deformation zone (fine elements zone). Furthermore, the generalised plastic strain contours are also nearly identical for the fine meshes used in both models. These minor differences can be attributed to the interpolation and averaging procedures. 5. Conclusions From the benchmark test of the dual mesh FE model, it can be concluded that the computational time is much less when compared with the conventional FE model. Also, the accuracy of the FE results predicted by the dual mesh model is acceptable compared with the results predicted by the conventional FE model.
References [1] S. Geyer, Process simulation for profile rolling, The MARC News letter, MARC Analysis Research Corp., January 1993, pp. 6–11.
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[2] S. Geyer, N. Stricker, Neue Perspektiven fur die Umformtechnik, 5. Umformtechnisches Kolloquium, Inst. For Production Technol., University of Darmstadt, Germany, 1994, p. 16.1. [3] F. Heislitz, H. Livatyali, M.A. Ahmetoglu, Simulation of roll forming process with the 3D FEM code PAM-STAMP, J. Mater. Process. Technol. (59) (1996) 59–67. [4] N. Rebelo, J.C. Nagtegaal, L.M. Taylor, Comparison of implicit finite-element methods in the simulation of metal forming processes, in: C. Wood, Zienkiewicz (Eds.), Numerical Methods in Industrial Forming Processes, 1992, pp. 99–108. [5] C.K. McClure, H. Li, Roll forming simulation using finite-element analysis, Manuf. Rev. 8 (1995) 114. [6] R.S. Senanayake, I.M. Cole, S. Thiruvarudchelvan, The application of computational and experimental techniques to metal deformation in cold-roll-forming, J. Mater. Process. Technol. (45) (1994) 155–160. [7] I. Pillinger, The prediction of metal flow and properties in threedimensional forging using the finite-element method, Ph.D. Thesis, School of Manufacturing and Mechanical Engineering, The University of Birmingham, 1984. [8] K.F. Chiang, Cold roll forming, M.Sc. Thesis, Mechanical Engineering Department, University of Auckland, August 1984. [9] A. Alsamhan, The development of finite-element models and remeshing techniques in the computer simulation of cold-roll-forming, Ph.D. Thesis, School of Manufacturing and Mechanical Engineering, The University of Birmingham, 1999. [10] T. Altan, F.W. Boulger, J.R. Becker, N. Akgerman, H.J. Henning, Forging equipment, material and practices, Battle Columbus Laboratories, Metal Working Division, Air Force Material Laboratory, Wright-Patterson, Air Force Bace, OH, 1973. [11] G.W. Rowe, C.E.N. Sturgess, P. Hartley, I. Pillinger, Finite-element plasticity and metal forming analysis, Cambridge University Press, Cambridge, 1991. [12] Z. Hu, Finite element modelling of the ring-rolling process, Ph.D. Thesis, The School of Manufacturing and Mechanical Engineering, The University of Birmingham, 1995. [13] S.M. Panton, S.D. Zhu, J.L. Duncan, Geometric constraints on the forming path in roll forming channel sections, in: Proceedings of the Institution of Mechanical Engineers, J. Eng. Manuf. 206 (1992) 113–118.
The development of real time re-meshing technique for simulating cold-roll-forming using FE methods A. Alsamhan a,∗ , I. Pillinger b , P. Hartely b b
a Mechanical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia School of Manufacturing and Mechanical Engineering, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Received 26 October 2000; received in revised form 26 October 2000; accepted 17 February 2003
Abstract Cold-roll-forming (CRF) is an important sheet metal forming process. However, product design procedures, in terms of rolls design and rolls pass schedule, remains more an art than science. Finite-element (FE) computer simulation can be used to predict the deformation and final product geometry, hence, reduce the design time and cost. The main objective of this work is to develop a real time re-meshing technique that can be used to run and complete the computer simulation of CRF process. This will enable decreasing the computational time and predicting the final geometry of cold-roll-formed product. A real time re-meshing technique using dual meshes was developed and benchmarked by applying the technique on traditional flat stripe rolling with friction. © 2003 Elsevier B.V. All rights reserved. Keywords: Re-meshing; Finite-element; CRF
1. Introduction Cold-roll-forming (CRF) is an important sheet metal forming process, which is widely used to manufacture long sheet metal products with a constant cross-section thickness. The design procedure for CRF products, roll pass sequences and forming rolls remains more an art than a science. Therefore, to reduce the process, development efforts and forming defects, which result in improper process design, finite-element (FE)-computer simulation can be used to predict strain distributions and final geometry of cold-roll-formed products. Due to the complicated deformation of the CRF process, it was found that most of the FE models described in the literature are based on three approaches. The first two approaches are based on simulating strip rolling without friction, while the third approach is based on simulating strip bending between the forming stands. In first approach the initial deformation shape is obtained by pressing the deformed rolls towards the undeformed strip, followed by pulling the strip longitudinally from the strip leading edge through the roll gaps of the forming stands [1–3]. The second approach, involves predicting the deformation by pulling an undeformed strip from the leading edge, through the roll gaps of deformed roll sets of the forming stands ∗
Corresponding author.
0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00469-2
[4,5]. The third approach, is based on predicting the deformation between the forming stands, without simulating the strip rolling and starting from undeformed stock [6]. This approach involves predicting the deformation by applying nodal linear displacements on the nodes located at the channel flange edge, of each forming stand. Nodal constraints on mesh thickness, are imposed on the web nodes located at the forming stand centre line plane. In the initial two approaches, the FE mesh is modelled from fine elements along the strip length [4,5]. This was adapted to continue and complete the strip rolling simulation, this may increase the computational time. To decrease the computational time, Off-line re-meshing technique can be used frequently, for each phase of the deformation [1–3]. Off-line can decrease the computational time, however, the deformation history of the FE mesh is lost after re-meshing. Hence, re-meshing the FE mesh using dual meshes, one for storing the deformation history, and another for FE computational will solve the two problems. A real time re-meshing techniques using dual meshes was developed and benchmarked are given and demonstrated in this paper. 2. Computation services and experimental work The utilised FE code is the EPFEP3 [7] program. It is a 3D implicit elastic–plastic FE program developed, initially to deal with bulk metal forming analysis by Pillinger [7].
2
A. Alsamhan et al. / Journal of Materials Processing Technology 147 (2004) 1–9
The EPFEP3 program has proved successful techniques for modelling metal forming processes, such as boundary surface and friction layer techniques. All the computer simulations presented in this research were executed on HP 400 Apollo workstations, with 32 MB of memory and 600 MB of hard drive capacity for the data storage. The strip material used in the FE model as given in the experimental work of [8] was bright mild steel (JISG 3141 SPEEN-50) with yield strength of 160 MPa. These experiments were conducted on cold-rolled trapezoidal channel sections. It was found that the mild steel (AISI 1006) is the closest material property sites in the literatures [8,9]. The Poisson’s ratio was specified as 0.34 and the Young’s modules as 190 GPa. The flow stress–strain curve of AISI 1006 is obtained from Altan et al. [10] and given (in SI units) as (σ¯ = σy + Kεn ) where σ¯ is the stress, ε the plastic strain, σ y the yield stress equal to 160 MPa; K = 613.8 MPa, and exponent n = 0.31. The constitutive material equation employed by the EPFEP3 [7] program, uses logarithmic and exponential functions to model material strain-hardening behaviour at the room temperature and given as follows: εp + hp2 y(εp ) = hp0 + hp1 ln hp2 +hp3 exp(hp4 (εp − hp5 )) for εp ≤ 1.0 (1) y(εp ) = Y(εp ) +
dY(ε0 ) (εp − ε0 ) dεp
for εp > 1.0
(2)
The coefficients of the logarithmic function (hp0 , hp1 and hp2 ) and the exponential function (hp3 , hp4 and hp5 ), were obtained through trial and error procedure [9] and are given in Table 1. Fig. 1 illustrates the strain-hardening behaviour of the two material equations, (σ¯ = σy + Kεn ) and Eqs. (1) and
Table 1 The coefficients of the constitutive material equation of the EPFEP3 program hp0 (MPa) hp1 (MPa) hp2 hp3 (MPa) hp4 hp5 ε0
60.0 139.467 0.00602 322.6 −23.423 −0.05 0.705
(2). The figure illustrates a little variation between two flow curves. The friction layer in the EPFEP3 program is transitory and uses the friction factor [11] to imitate different lubricants. Friction factor of m = 0.5 was adopted in all the FE results presented here. It was found that, this value provided stable contact between rolls and FE mesh during the rolling simulation. Maintaining this contact during the rolling simulation is very important for the effectiveness of the friction layer technique, that assists mesh movement along the rolling direction.
3. Dual mesh and re-meshing techniques It was found that, the best re-meshing technique could be achieved using a special re-meshing technique adopted from [12], to simulate ring-rolling press as bulk forming process. In this technique dual meshes utilised are a computational mesh and a fine mesh. The computational mesh has fine elements in the main deformation zone, and coarse elements elsewhere, and would be used for the FE analysis. The fine mesh has fine longitudinal elements structure throughout its
Fig. 1. Actual and fitted flow stress–strain curves for plastic strain range from 0 to 0.1.
A. Alsamhan et al. / Journal of Materials Processing Technology 147 (2004) 1–9
length, and is used to store the state variables of the FE model, which are updated real time after each phase of deformation. The computational mesh is constructed from a selection of nodes in the fine mesh, at each phase of the deformation. The state variables, such as stresses and strains (including the membrane strains), are transferred directly between the dual meshes for those nodes, that belong to both meshes in the deformation zone, while they are interpolated for the nodes that do not correspond to any in the computational mesh. The CRF program is a FORTRAN program developed by the authors, to execute the FE model with the dual mesh technique, for roll-formed channel sections. The program is used in conjunction with the FE analysis program, EPFEP3, to simulate the CRF process, for any prescribed section. Its main objective is to establish a suitable FE mesh and to carry out re-meshing and data transfer between the dual meshes, to continue the computer simulation for the process. The
3
program can be used to simulate any CRF product or any flat rolling, by modifying the sub-programs that generate the initial dual meshes. During the FE analysis and data transfer between the dual meshes, different files were generated to store the model files and to display the FE results for the dual meshes. Details of the CRF program and its functions can be found in [9]. To established the initial contact between friction boundary surfaces and FE mesh, the initial deformed fine mesh is set up using the bend angle curve [13], with fine elements along the strip length and through all the forming stands. The general configurations and number of elements and their lengths along the strip width, thickness and the longitudinal length, are all variables that can be specified through the CRF program data file. Furthermore, the number of the forming stands, their folding angles and guiding stand consideration, are also variables that can be specified by the user through the CRF program data file. Fig. 2 illustrates an
Fig. 2. Fine mesh modelling for two forming stands of channel section with folding angle sequence 0–20–40◦ using bend angle curve given in [13].
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Fig. 3. Fine mesh and computational mesh structure, interpolation and data transfer between the dual meshes.
example of a fine mesh modelled for two forming stands with fold angle of 20◦ , first forming stand, and with fold angle of 40◦ for the second forming stand, using the bend angle curve prescribed in [13]. The computational mesh between forming stands generally consists of two main zones (Fig. 3) shows the uniform zone and the deformed zone. Since the uniform zone is remote from the deformation, very coarse longitudinal elements were used. The second zone is mainly through the deformation length and modelled using graded coarse elements. Fine elements were also modelled, using behind and along the roll gap. Once the number of the longitudinal elements and their lengths are defined, the total number of the longitudinal elements and their lengths, in terms of contained fine elements are defined along the strip length. The data transfer between the two meshes is carried out in two phases. The first phase includes data transfer in the fine mesh. Since the location of nodes and elements are identical in both meshes, the state variables of the FE model are transferred directly without any interpolation. The second phase,
includes data transfer in the coarse and very coarse mesh zones of the computational mesh, where interpolation is essential. In this case, some of the nodes in both meshes will be in the same position, e.g. node N1 , N2 , N3 and N4 in Fig. 3, where the state variables also transferred directly without interpolation. However, the majority of nodes will not be coincident, e.g. node P in Fig. 3, and interpolation is essential to obtain the state variables for those nodes. Referring again to Fig. 3 and the node P, if a conventional re-meshing technique was used, the position of the prescribed node P is obtained, at the position P by interpolating its position from the positions of the four nodes N1 , N2 , N3 and N4 , assuming a two-dimensional interpolation function. This illustrates that the interpolated position, will lie along the chord between N2 and N4 and not the actual deformation path on the arc between N2 and N4 . The data transfer will be further in error when the bending in this zone is larger, and the selected computational element is very coarse. Therefore, to minimise the accumulated errors, the position of the point P is determined by updating its co-ordinates from the
A. Alsamhan et al. / Journal of Materials Processing Technology 147 (2004) 1–9
incremental displacement vector (V), which is interpolated from the corresponding values of the four nodes. Furthermore, the predicted error can be minimised by utilising finer longitudinal elements in the secondary deformation zone, so that the arc joining N2 and N4 will be small. The prescribed error will be nearly eliminated for those very coarse elements located away from the deformation zones. Re-meshing the computational mesh was considered along the rolling direction. Initially, the longitudinal element lengths of the computational mesh, in terms of the number of the longitudinal fine elements, were calculated using the initial computational mesh generator. The frequent re-meshing of the computational mesh depends mainly on adjusting the longitudinal element lengths for the prescribed mesh. The decision to re-mesh was determined by comparing the longitudinal position of the fine mesh nodes to the
5
longitudinal position of the centre line of the last forming stand. For example, consider the fine mesh overlaid on the computational mesh given in Fig. 4 for two forming stands. As illustrated in Fig. 4, monitoring the longitudinal position of the fine mesh node N11 determines the 1st re-meshing decision. Re-meshing of the computational mesh is applied when the longitudinal nodal position of the prescribed node is greater or equal to the longitudinal position of the centre line of the last forming stand (Y). Accordingly, the longitudinal element lengths will be updated, hence the new computational mesh structure will be updated using the updated computational element lengths. The next re-meshing decision will be considered by monitoring the longitudinal position of the fine mesh node N10 . This re-meshing procedure will continue until the fine mesh passes the centre line of the last forming stand rolls, and fully deformed longitudinally.
Fig. 4. Adjusting coarse element lengths of the computational mesh during re-mesh.
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4. Benchmarking the dual mesh and re-meshing techniques The main objective of the benchmark test program, is to test the dual mesh FE model before moving to more sophisticated CRF process modelling. It is also important to check the program functions during the CRF program development, e.g. the interpolation function, data transfer between the dual meshes, membrane strain calculation etc. For these reasons a small-scale problem was utilised in the
benchmark test, where small number of elements and nodes were used for the dual meshes. Furthermore, the simulation results of the dual meshes FE model were compared to the FE results of the conventional FE model, using the same fine mesh used in the dual meshes model for the FE computation. The benchmark test is modelled as a traditional flat strip rolling process, but with very little thickness reduction. The upper and lower rolls were represented by two cylindrical boundary surfaces, and used to roll a strip of sheet metal of dimensions 6.4 mm (length) by 18.0 mm (width) by 0.6 mm
Fig. 5. Dual meshes FE model (benchmark), computational mesh and fine mesh configurations.
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(thickness). The rolling direction was selected to be along the strip length. Due to the vertical symmetry of the process, half of the strip width was modelled, i.e. 9.0 mm. The roll radius of both the upper and lower rolls was selected to be 51.5 mm. The fine mesh was modelled from 486 nodes and 160 elements, as illustrated in Fig. 5. The fine mesh was modelled from a single layer of elements through the mesh thickness. Two elements were utilised along the mesh width having 3 and 6 mm width. The fine mesh was created with 80 fine
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elements along the strip length, each element having a longitudinal length of 80 m. The longitudinal fine element length was selected to restrict the very small plastic strain during rolling, in which the roll penetration was 0.25 m. The computational mesh was created with 198 nodes and 64 elements as illustrated in Fig. 5. Ten fine elements were utilised near the mesh leading edge and also at the trailing edge. The roll contact area covered nearly four longitudinal fine elements. A series of longitudinal coarse elements, varying in size, were used before the roll contact area and
Fig. 6. The variation of the computational mesh configuration for different rolling increments of the benchmark model.
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the main deformation zone, as shown in Fig. 5. Three very coarse elements were used remote from the main deformation area. Ten fine elements near the trailing edge of the FE mesh, which are initially remote from the deformation area, which will gradually move towards and into the main deformation zone as re-meshing is implemented. The material of the strip was again mild steel AISI 1006. The rolling increment was selected to be 0.04◦ (fixed) applied on both rolls. The computer simulation was carried out for 120 rolling increments using the dual mesh FE model. The computational time was 19.4 s for each increment executed by the EPFEP3 program. The data transfer between the two meshes required 13.0 s, for the same increment. The computer simulation results were documented for specific rolling increments (i.e. 1st, 17th, 30th, 40th, 50th, 70th, 100th and 120th
increments). The general configurations of the computational mesh for the prescribed rolling increments, including roll contact areas are shown in Fig. 6. The figure illustrates that the configuration of the fine elements, refer increment 1, located near the mesh trailing edge, was gradually changed to be after the main deformation zone during re-meshing of the computational mesh (see increment 40, for example, of Fig. 6). Furthermore, the three very coarse elements were also moved gradually from a position before the main deformation area, increment 30, to a position after the main deformation area, where steady deformation was obtained (see increment 120 in the same figure). The computer simulation was executed for 120 rolling increments, using the same fine mesh in the FE analysis, conventional FE model. The computational time was 44 s,
Fig. 7. Generalised plastic strain contours of dual mesh FE model and conventional FE model for increment 70 of the benchmark model.
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for each increment using the fine mesh compare to 19.4 s, with the dual mesh model. In general the computational time with the dual meshes, including data transfer between the meshes is 50% less, compared to the conventional FE model. Fig. 7 illustrates the generalised strain contours for the dual mesh FE model. The generalised plastic strain contours for the dual mesh FE model and the conventional FE model look more different than they really are. This is due to the fact that the range of the contour colours selector employed by the program is dissimilar. The generalised plastic strain contours for the dual mesh FE model are identical especially in the main deformation zone (fine elements zone). Furthermore, the generalised plastic strain contours are also nearly identical for the fine meshes used in both models. These minor differences can be attributed to the interpolation and averaging procedures. 5. Conclusions From the benchmark test of the dual mesh FE model, it can be concluded that the computational time is much less when compared with the conventional FE model. Also, the accuracy of the FE results predicted by the dual mesh model is acceptable compared with the results predicted by the conventional FE model.
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