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Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of cold roll forming process
Journal of Materials Processing Technology 177 (2006) 621–625
Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of cold roll forming process R. Boman, L. Papeleux, Q.V. Bui, J.P. Ponthot∗ LTAS-MC&T—Aerospace and Mechanics Department, University of Li`ege, Chemin des chevreuils 1, B4000 Li`ege, Belgium Received 10 February 2006; accepted 24 April 2006
Abstract In this paper, the Arbitrary Lagrangian Eulerian formalism is used to compute the steady state of a 3D U-shaped cold roll forming process. Compared to the Lagrangian case, this method allows to keep a refined mesh near the tools, thus allowing accurate contact prediction all along the computation with a limited number of elements. Mesh can also be kept refined in the bending zone, thus leading to accurate representation of the sheet with a limited computational time. The main problem of this kind of simulation lies in the rezoning of the nodes on the free surfaces of the sheet. A modified iterative isoparametric smoother is used to manage this geometrically complex and CPU expensive task. © 2006 Elsevier B.V. All rights reserved. Keywords: Cold roll forming; ALE formulation; Mesh smoothing; Large deformation; Finite elements
1. Introduction
2. ALE formalism
Being a progressive and continuous process, in which small amounts of forming are applied at each pass of rolls, cold roll forming is largely employed to bend a long strip of sheet blank into a desired cross-sectional profile via roller dies. Analytical and experimental approaches have been employed for the study of cold roll forming of simple profiles or pipe sections. In order to analyze more complex profiles and to obtain in details the distribution of stress and strains in the formed sheet, numerical methods need to be exploited. Following this trend, a 3D finite element analysis has been involved in a previous work [3] to study the deformation of the strip during and after the forming process. A classical Lagrangian formulation was successfully used to compare the simulation with experimental data. However, these 3D models become very CPU expensive due to the large number of finite elements that must be used to get an accurate solution. An alternative approach, considering the Arbitrary Lagrangian Eulerian (ALE) formulation [11,1,2,5], has been investigated. This formalism allows to obtain the stationary solution of the process with a minimal number of finite elements.
In order to find the solution of steady state processes by numerical simulation with the classical Lagrangian formulation, very large and useless meshes have to be considered. For example, when dealing with the roll forming simulation, the whole sheet has to be finely discretized along its length even if the results obtained between each set of tools do not require a refined mesh in the final stationary state. However, these small finite elements cannot be enlarged because a fine discretization is needed when the elements will reach a pair of rolls and go through it. This fine mesh is thus required in order to reach an accurate steady state solution. Moreover, each surface node is potentially in contact with the corresponding upper/lower rolls as the simulation progresses in time. Consequently, the CPU time soon becomes very large as the required accuracy increases. The Arbitrary Lagrangian Eulerian formulation was introduced to overcome these problems. The mesh can be automatically handled by the software, irrespective of the body motion. In such a formulation, each time step is divided into two phases: the first one is purely Lagrangian and the second one the ALE phase. The latter is a rezoning phase, where the nodes are moved according to mesh quality considerations, followed by a convective Eulerian phase, where the values stored at the Gauss points are updated. In order to avoid oscillations and instability, efficient convection algorithms have to be used [4,8].
∗
Corresponding author. Tel.: +32 4 366 93 10; fax: +32 4 366 91 41. E-mail address: [email protected] (J.P. Ponthot).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.04.120
622
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
3. Rezoning phase 3.1. Basic methods The main task concerning the creation of the ALE numerical model is the definition of the movement of the mesh with respect to the body motion. There exists as many methods as different mechanical problems. The rezoning strategies usually depend on the position of the node to be moved on the CAD geometry of the model. They can be classified according to the topological entity (vertex, edge, side, volume) it belongs to. In our approach, nodes laying on the vertices may move in a Lagrangian or Eulerian way. They may be fixed along one given direction. Nodes laying on the edges are remeshed using a spline curve that goes through the Lagrangian position of the nodes. The new position is computed from the initial curvilinear coordinate of the node (mesh topology is then kept unchanged during all the simulation irrespective of the edge deformation). The nodes inside the volume are moved using a transfinite mapping mesher or traditional volumic smoothing algorithms. 3.2. Plane surfaces Nodes laying on the free planar sides may be moved with the classical smoothing methods [9] that are commonly used for improving the mesh quality after a remeshing operation. Among these method, the Laplacian smoothing is the most famous one due to its simplicity. Unfortunately, in its simplest form, this smoother is not able to handle non uniform structured mesh (the algorithm tends to give each cell of the mesh the same area). In the case of the roll forming process, this kind of mesh is very useful. The structure of the mesh helps for a given number of elements to achieve a better accuracy on the results than an unstructured one. Moreover, it is more efficient to define small cells near the contact regions and larger cells far from them. Keeping this in mind, many other methods, like Giuliani’s method [6], the angle smoother [12], “area pull” smoother [10], have to be discarded. In the case of a structured mesh of graded elements, the isoparametric smoother [7] seems to be a good alternative. The basic idea is to reposition the node to the center of an isoparametric element associated with the neighboring nodes (see Fig. 1): xn+1 =
1 n x1 + x3n − x2n nel
Fig. 1. Isoparametric smoothing method.
zero. Consequently, the nodes cannot be moved on the facets without changing the total volume of the 3D mesh and destroying the global shape of the surface. The extension of the one dimensional “spline remesher” used for the edges is then proposed to deal with this problem. At each time step, during the rezoning phase, a spline surface is built over each side of the mesh. The spline is a set of bi-linear patches constructed on each facets, considering cubic shaped edges. The resulting surface is not a full cubic patch (twist terms are missing) but the curvature is well approximated and it is really faster. The shape of the cubic edges are driven by the approximated normal vector defined at each node. The surface rezoning is then an iterative process. For each node, an approximation of the normal vector is computed (see Fig. 2): n=
nbf
ni
i=1
where nbf is the number of surface facets neighboring the node. ni = d1 ∧ d2 , the normal vector at the considered node on the patch i. The vectors d1 and d2 are the edges of the patch i starting from the node.
nel
(1)
i=1
where nel is the number of cells containing the node. This iterative method is usually speeded-up by a successive over-relaxation (SOR) algorithm. 3.3. Curved surfaces One of the biggest problem dealing with the three dimensional ALE formalism is the rezoning of the nodes laying on curved boundary sides of the mesh. Due to discretization, the faceted surface is not smooth as soon as the curvature is not
(2)
Fig. 2. Smoothing on the approximate tangent plane.
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
623
the most perpendicular edge starting from this node and the boundary. 4. Numerical simulation of cold roll forming process 4.1. Model definition
Fig. 3. Projection on the interpolation spline.
Fig. 2 shows a surface mesh around a node before the rezoning phase. Once the normal is computed, classical 2D smoothers may be used on the tangent plane. The resulting position is then projected on the spline described above (Fig. 3). The projection algorithm must also be able to search the projection outside the neighboring facets if the rezoning method moves the node far from its original position. 3.4. Eulerian boundaries When dealing with stationary processes, a space region must usually be defined (by boundary planes that we call “Eulerian boundaries”) in order to prevent the mesh from going outside this specified region. Great care must be taken with the nodes that may cross an Eulerian boundary during the Lagrangian step. A direct projection of the node on the boundary is not enough to prevent mesh distortions from occurring. That is why a specific rezoning strategy has been created for them: the node that has crossed the boundary is placed at the intersection of
A symmetrical U-channel is formed by a process divided in six pairs of forming rolls (diameter = 210 mm) separated by 500 mm. On each roll, the forming angle increases along the forming direction (i.e. 15, 30, 45, 60, 75 and 90◦ ). The dimension of the sheet is 3500 mm × 200 mm× 1 mm and only one half of the process is modeled due to symmetry. The material is elastoplastic (E = 210 GPa and ν = 0.3) with a linear isotropic hardening rule σ e = σ0e + hpl with σ0e = 150 MPa and h = 500 MPa. Frictionless rigid rolls are considered and the motion of the sheet is imposed on the symmetry plane (actually, in the real process, the rolls are rotating and friction forces the sheet to advance). This simplification avoids to consider the convection of friction in the ALE algorithm. Contact is taken into account using the penalty method. Dynamic effects are neglected (implicit quasi-static algorithm). As far as only the final stationary state of the sheet is interesting, the first step of the forming process is modified in such a way that the computation is faster (but far from the reality). The lower rolls are initially separated from the upper ones and the sheet is initially placed between them). As the sheet is already in the good position according to the rolling direction, the first step can be computed using the ALE formalism too. The lower rolls go up one by one bending the sheet of metal (this actually looks like a deep drawing process). Using this method, the mesh can be refined near the contact areas (lengths vary from 25 mm far from the rolls to 5 mm close to them) a limited number of contact elements can thus be defined. In the perpendicular direction, smaller element sizes are
Fig. 4. Free surface movement—snapshot #1.
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R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
Fig. 5. Free surface movement—snapshot #2.
Fig. 6. Free surface movement—snapshot #3 (stationary solution).
also used close to the expected position of the fold (lengths vary from 10 to 2 mm). Consequently, the resulting mesh is structured but non uniform). 4.2. Results During the first step, the rolls have bent the sheet and are in the correct position compared to the industrial process. The movement of the sheet is carried on until the solution reaches a steady state. At the beginning of this second step, the front of the sheet is almost straight (see Fig. 4). Due to the ALE convection, the free surface mesh deforms (Fig. 5) propagating the bending from the rolls until the front. Later on, the U-shaped sheet is obtained when the stationary solution is reached (Fig. 6).
5. Conclusions and future work A first simulation of a 3D cold roll forming process using the Arbitrary Lagrangian Eulerian has been presented. The ability of managing efficiently the free surfaces of the sheet has been proven. However, this model can be improved: friction could be taken into account and the motion of the sheet could result from the rotation of the rolls. This simulation will then be compared to a Lagrangian one and to physical experiments. To reach this goal, a special procedure must be created in order to compute efficiently the longitudinal elongation of the sheet. This result is easily available in the Lagrangian case but needs an incremental updating procedure in the case of the ALE mesh.
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
Another interesting aspect is the computation of the springback when the sheet is formed and leaves the tools. In this case, the rolls could be removed using a third Lagrangian step. Acknowledgment This research work was carried out under grant number 01/1/4710 (PROMETA) from the Walloon Region which is gratefully acknowledged. References [1] R. Boman, J.-P. Ponthot, ALE methods for determining stationary solutions of metal forming processes, in: E. Onate (Ed.), Proceedings of ECCOMAS 2000, Barcelona, Spain, 2000 on CDROM. [2] R. Boman, J.-P. Ponthot, Finite element simulation of lubricated contact in rolling using Arbitrary Lagrangian Eulerian formulation, Compt. Methods Appl. Mech. Eng. 193 (39–41) (2004) 4323–4353. [3] Q.V. Bui, L. Papeleux, R. Boman, J.P. Ponthot, P. Wouters, R. Kergen, G. Daolio, Numerical simulation of cold roll forming process, in: Proceedings of the ESAFORM2005, Cluj-Napoca, Romania, 2005 April, pp. 141– 144.
625
[4] F. Casadei, J. Donea, A. Huerta, Arbitrary lagrangian eulerian finite elements in non-linear fast transient continuum mechanics, Technical report EUR 16327 EN, University of Catalunya, Barcelona, Spain, 1995. [5] J. Donea, A. Huerta, J.-P. Ponthot, A. Rodriguez-Ferran, E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Encyclopedia of Computational Mechanics, vol. 1, 2004, pp. 413–437 (Chapter 14). [6] S. Giuliani, An algorithm for continuous rezoning of the hydrodynamic grid in Arbitrary Lagrangian-Eulerian computer codes, Nucl. Eng. Design 72 (2) (1982) 205–212. [7] L.R. Hermann, Laplacian isoparametric grid generation scheme, J. Eng. Div. 102 (1976) 749–756. [8] A. Huerta, F. Casadei, J. Don´ea, ALE stress update in transient plasticity problems, in: E. O˜nate, D.R.J. Owen (Eds.), in: Proceedings of the International Conference on Computational Plasticity, Barcelona, Spain, 1995, pp. 1865–1876. [9] S. Hyun, L.-E. Lindgren, Smoothing and adaptive remeshing schemes for graded elements, Commun. Numer. Methods Eng. 17 (2001) 1–17. [10] R.E. Jones, A self-organizing mesh generation program, J. Pressure Vessel Technol., Trans. Am. Soc. Mech. Eng. 96 (3) (1974) 193–199. [11] J.-P. Ponthot, Traitement unifi´e de la M´ecanique des Milieux Continus solides en grandes transformations par la m´ethode des e´ l´ements finis (in French), Ph.D. thesis, Universit´e de Li`ege, Li`ege, Belgium, 1995. [12] T. Zhou, K. Shimada, An angle-based approach to two-dimensional mesh smoothing, in: Proceedings of the Ninth International Meshing Roundtable, New Orleans, Louisiana, USA, 2000 October, pp. 373–384.
Application of the Arbitrary Lagrangian Eulerian formulation to the numerical simulation of cold roll forming process R. Boman, L. Papeleux, Q.V. Bui, J.P. Ponthot∗ LTAS-MC&T—Aerospace and Mechanics Department, University of Li`ege, Chemin des chevreuils 1, B4000 Li`ege, Belgium Received 10 February 2006; accepted 24 April 2006
Abstract In this paper, the Arbitrary Lagrangian Eulerian formalism is used to compute the steady state of a 3D U-shaped cold roll forming process. Compared to the Lagrangian case, this method allows to keep a refined mesh near the tools, thus allowing accurate contact prediction all along the computation with a limited number of elements. Mesh can also be kept refined in the bending zone, thus leading to accurate representation of the sheet with a limited computational time. The main problem of this kind of simulation lies in the rezoning of the nodes on the free surfaces of the sheet. A modified iterative isoparametric smoother is used to manage this geometrically complex and CPU expensive task. © 2006 Elsevier B.V. All rights reserved. Keywords: Cold roll forming; ALE formulation; Mesh smoothing; Large deformation; Finite elements
1. Introduction
2. ALE formalism
Being a progressive and continuous process, in which small amounts of forming are applied at each pass of rolls, cold roll forming is largely employed to bend a long strip of sheet blank into a desired cross-sectional profile via roller dies. Analytical and experimental approaches have been employed for the study of cold roll forming of simple profiles or pipe sections. In order to analyze more complex profiles and to obtain in details the distribution of stress and strains in the formed sheet, numerical methods need to be exploited. Following this trend, a 3D finite element analysis has been involved in a previous work [3] to study the deformation of the strip during and after the forming process. A classical Lagrangian formulation was successfully used to compare the simulation with experimental data. However, these 3D models become very CPU expensive due to the large number of finite elements that must be used to get an accurate solution. An alternative approach, considering the Arbitrary Lagrangian Eulerian (ALE) formulation [11,1,2,5], has been investigated. This formalism allows to obtain the stationary solution of the process with a minimal number of finite elements.
In order to find the solution of steady state processes by numerical simulation with the classical Lagrangian formulation, very large and useless meshes have to be considered. For example, when dealing with the roll forming simulation, the whole sheet has to be finely discretized along its length even if the results obtained between each set of tools do not require a refined mesh in the final stationary state. However, these small finite elements cannot be enlarged because a fine discretization is needed when the elements will reach a pair of rolls and go through it. This fine mesh is thus required in order to reach an accurate steady state solution. Moreover, each surface node is potentially in contact with the corresponding upper/lower rolls as the simulation progresses in time. Consequently, the CPU time soon becomes very large as the required accuracy increases. The Arbitrary Lagrangian Eulerian formulation was introduced to overcome these problems. The mesh can be automatically handled by the software, irrespective of the body motion. In such a formulation, each time step is divided into two phases: the first one is purely Lagrangian and the second one the ALE phase. The latter is a rezoning phase, where the nodes are moved according to mesh quality considerations, followed by a convective Eulerian phase, where the values stored at the Gauss points are updated. In order to avoid oscillations and instability, efficient convection algorithms have to be used [4,8].
∗
Corresponding author. Tel.: +32 4 366 93 10; fax: +32 4 366 91 41. E-mail address: [email protected] (J.P. Ponthot).
0924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2006.04.120
622
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
3. Rezoning phase 3.1. Basic methods The main task concerning the creation of the ALE numerical model is the definition of the movement of the mesh with respect to the body motion. There exists as many methods as different mechanical problems. The rezoning strategies usually depend on the position of the node to be moved on the CAD geometry of the model. They can be classified according to the topological entity (vertex, edge, side, volume) it belongs to. In our approach, nodes laying on the vertices may move in a Lagrangian or Eulerian way. They may be fixed along one given direction. Nodes laying on the edges are remeshed using a spline curve that goes through the Lagrangian position of the nodes. The new position is computed from the initial curvilinear coordinate of the node (mesh topology is then kept unchanged during all the simulation irrespective of the edge deformation). The nodes inside the volume are moved using a transfinite mapping mesher or traditional volumic smoothing algorithms. 3.2. Plane surfaces Nodes laying on the free planar sides may be moved with the classical smoothing methods [9] that are commonly used for improving the mesh quality after a remeshing operation. Among these method, the Laplacian smoothing is the most famous one due to its simplicity. Unfortunately, in its simplest form, this smoother is not able to handle non uniform structured mesh (the algorithm tends to give each cell of the mesh the same area). In the case of the roll forming process, this kind of mesh is very useful. The structure of the mesh helps for a given number of elements to achieve a better accuracy on the results than an unstructured one. Moreover, it is more efficient to define small cells near the contact regions and larger cells far from them. Keeping this in mind, many other methods, like Giuliani’s method [6], the angle smoother [12], “area pull” smoother [10], have to be discarded. In the case of a structured mesh of graded elements, the isoparametric smoother [7] seems to be a good alternative. The basic idea is to reposition the node to the center of an isoparametric element associated with the neighboring nodes (see Fig. 1): xn+1 =
1 n x1 + x3n − x2n nel
Fig. 1. Isoparametric smoothing method.
zero. Consequently, the nodes cannot be moved on the facets without changing the total volume of the 3D mesh and destroying the global shape of the surface. The extension of the one dimensional “spline remesher” used for the edges is then proposed to deal with this problem. At each time step, during the rezoning phase, a spline surface is built over each side of the mesh. The spline is a set of bi-linear patches constructed on each facets, considering cubic shaped edges. The resulting surface is not a full cubic patch (twist terms are missing) but the curvature is well approximated and it is really faster. The shape of the cubic edges are driven by the approximated normal vector defined at each node. The surface rezoning is then an iterative process. For each node, an approximation of the normal vector is computed (see Fig. 2): n=
nbf
ni
i=1
where nbf is the number of surface facets neighboring the node. ni = d1 ∧ d2 , the normal vector at the considered node on the patch i. The vectors d1 and d2 are the edges of the patch i starting from the node.
nel
(1)
i=1
where nel is the number of cells containing the node. This iterative method is usually speeded-up by a successive over-relaxation (SOR) algorithm. 3.3. Curved surfaces One of the biggest problem dealing with the three dimensional ALE formalism is the rezoning of the nodes laying on curved boundary sides of the mesh. Due to discretization, the faceted surface is not smooth as soon as the curvature is not
(2)
Fig. 2. Smoothing on the approximate tangent plane.
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
623
the most perpendicular edge starting from this node and the boundary. 4. Numerical simulation of cold roll forming process 4.1. Model definition
Fig. 3. Projection on the interpolation spline.
Fig. 2 shows a surface mesh around a node before the rezoning phase. Once the normal is computed, classical 2D smoothers may be used on the tangent plane. The resulting position is then projected on the spline described above (Fig. 3). The projection algorithm must also be able to search the projection outside the neighboring facets if the rezoning method moves the node far from its original position. 3.4. Eulerian boundaries When dealing with stationary processes, a space region must usually be defined (by boundary planes that we call “Eulerian boundaries”) in order to prevent the mesh from going outside this specified region. Great care must be taken with the nodes that may cross an Eulerian boundary during the Lagrangian step. A direct projection of the node on the boundary is not enough to prevent mesh distortions from occurring. That is why a specific rezoning strategy has been created for them: the node that has crossed the boundary is placed at the intersection of
A symmetrical U-channel is formed by a process divided in six pairs of forming rolls (diameter = 210 mm) separated by 500 mm. On each roll, the forming angle increases along the forming direction (i.e. 15, 30, 45, 60, 75 and 90◦ ). The dimension of the sheet is 3500 mm × 200 mm× 1 mm and only one half of the process is modeled due to symmetry. The material is elastoplastic (E = 210 GPa and ν = 0.3) with a linear isotropic hardening rule σ e = σ0e + hpl with σ0e = 150 MPa and h = 500 MPa. Frictionless rigid rolls are considered and the motion of the sheet is imposed on the symmetry plane (actually, in the real process, the rolls are rotating and friction forces the sheet to advance). This simplification avoids to consider the convection of friction in the ALE algorithm. Contact is taken into account using the penalty method. Dynamic effects are neglected (implicit quasi-static algorithm). As far as only the final stationary state of the sheet is interesting, the first step of the forming process is modified in such a way that the computation is faster (but far from the reality). The lower rolls are initially separated from the upper ones and the sheet is initially placed between them). As the sheet is already in the good position according to the rolling direction, the first step can be computed using the ALE formalism too. The lower rolls go up one by one bending the sheet of metal (this actually looks like a deep drawing process). Using this method, the mesh can be refined near the contact areas (lengths vary from 25 mm far from the rolls to 5 mm close to them) a limited number of contact elements can thus be defined. In the perpendicular direction, smaller element sizes are
Fig. 4. Free surface movement—snapshot #1.
624
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
Fig. 5. Free surface movement—snapshot #2.
Fig. 6. Free surface movement—snapshot #3 (stationary solution).
also used close to the expected position of the fold (lengths vary from 10 to 2 mm). Consequently, the resulting mesh is structured but non uniform). 4.2. Results During the first step, the rolls have bent the sheet and are in the correct position compared to the industrial process. The movement of the sheet is carried on until the solution reaches a steady state. At the beginning of this second step, the front of the sheet is almost straight (see Fig. 4). Due to the ALE convection, the free surface mesh deforms (Fig. 5) propagating the bending from the rolls until the front. Later on, the U-shaped sheet is obtained when the stationary solution is reached (Fig. 6).
5. Conclusions and future work A first simulation of a 3D cold roll forming process using the Arbitrary Lagrangian Eulerian has been presented. The ability of managing efficiently the free surfaces of the sheet has been proven. However, this model can be improved: friction could be taken into account and the motion of the sheet could result from the rotation of the rolls. This simulation will then be compared to a Lagrangian one and to physical experiments. To reach this goal, a special procedure must be created in order to compute efficiently the longitudinal elongation of the sheet. This result is easily available in the Lagrangian case but needs an incremental updating procedure in the case of the ALE mesh.
R. Boman et al. / Journal of Materials Processing Technology 177 (2006) 621–625
Another interesting aspect is the computation of the springback when the sheet is formed and leaves the tools. In this case, the rolls could be removed using a third Lagrangian step. Acknowledgment This research work was carried out under grant number 01/1/4710 (PROMETA) from the Walloon Region which is gratefully acknowledged. References [1] R. Boman, J.-P. Ponthot, ALE methods for determining stationary solutions of metal forming processes, in: E. Onate (Ed.), Proceedings of ECCOMAS 2000, Barcelona, Spain, 2000 on CDROM. [2] R. Boman, J.-P. Ponthot, Finite element simulation of lubricated contact in rolling using Arbitrary Lagrangian Eulerian formulation, Compt. Methods Appl. Mech. Eng. 193 (39–41) (2004) 4323–4353. [3] Q.V. Bui, L. Papeleux, R. Boman, J.P. Ponthot, P. Wouters, R. Kergen, G. Daolio, Numerical simulation of cold roll forming process, in: Proceedings of the ESAFORM2005, Cluj-Napoca, Romania, 2005 April, pp. 141– 144.
625
[4] F. Casadei, J. Donea, A. Huerta, Arbitrary lagrangian eulerian finite elements in non-linear fast transient continuum mechanics, Technical report EUR 16327 EN, University of Catalunya, Barcelona, Spain, 1995. [5] J. Donea, A. Huerta, J.-P. Ponthot, A. Rodriguez-Ferran, E. Stein, R. de Borst, T.J.R. Hughes (Eds.), Encyclopedia of Computational Mechanics, vol. 1, 2004, pp. 413–437 (Chapter 14). [6] S. Giuliani, An algorithm for continuous rezoning of the hydrodynamic grid in Arbitrary Lagrangian-Eulerian computer codes, Nucl. Eng. Design 72 (2) (1982) 205–212. [7] L.R. Hermann, Laplacian isoparametric grid generation scheme, J. Eng. Div. 102 (1976) 749–756. [8] A. Huerta, F. Casadei, J. Don´ea, ALE stress update in transient plasticity problems, in: E. O˜nate, D.R.J. Owen (Eds.), in: Proceedings of the International Conference on Computational Plasticity, Barcelona, Spain, 1995, pp. 1865–1876. [9] S. Hyun, L.-E. Lindgren, Smoothing and adaptive remeshing schemes for graded elements, Commun. Numer. Methods Eng. 17 (2001) 1–17. [10] R.E. Jones, A self-organizing mesh generation program, J. Pressure Vessel Technol., Trans. Am. Soc. Mech. Eng. 96 (3) (1974) 193–199. [11] J.-P. Ponthot, Traitement unifi´e de la M´ecanique des Milieux Continus solides en grandes transformations par la m´ethode des e´ l´ements finis (in French), Ph.D. thesis, Universit´e de Li`ege, Li`ege, Belgium, 1995. [12] T. Zhou, K. Shimada, An angle-based approach to two-dimensional mesh smoothing, in: Proceedings of the Ninth International Meshing Roundtable, New Orleans, Louisiana, USA, 2000 October, pp. 373–384.