Summary
In this paper we review some recent results in the field of numerical simulation of extrusion and other forming processes obtained by the authors by using a meshless approach, together with a wide review of the existing bibliography on the topic. Three main alternatives exist in the literature, namely (updated) Lagrangian, Eulerian and arbitrary Lagrangian-Eulerian (ALE) methods. A review of the most important characteristics of each of these three approaches is here presented and their possible advantages are pointed out. Finally, an updated Lagrangian approach over a meshless approximation, based on a class of methods globally coined as natural element methods (also as natural neighbour Galerkin methods) is analysed and its relative advantages studied. Some numerical examples are included that clearly show the potential capabilities of the proposed method.
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References
I. Alfaro, D. Bel, E. Cueto, M. Doblaré and F. Chinesta (2004). Three-dimensional simulation of Aluminium extrusion by the α-shape based Natural Element Method.Computer Methods in Applied Mechanics and Engineering, submitted.
I. Alfaro, J. Yvonnet, E. Cueto, F. Chinesta and M. Doblaré (2004). Meshless methods with application to metal forming.Computer Methods in Applied Mechanics and Engineering, in press.
E.H. Atzema and J. Huétink (1995). Finite Element Analysis of Forward/Backward Extrusion using ALE Techniques. In S.F. Shen and P.R. Dawson, editors,proceedings of the Numiform 95 conference, pages 383–388.
I. Babuška (1973). The Finite Element Method with Lagrange multipliers.Numerische Mathematik,20, 179–192.
I. Babuška, U. Banerjee and J.E. Osborn (2003). Survey of meshless and generalized finite element methods: A unified approach.Acta Numerica,12, 1–125.
I. Babuška and J.M. Melenk (1996). The partition of unity finite element method: Basic theory and applications.Comp. Meth. in Appl. Mech. and Eng.,4, 289–314.
K.J. Bathe (1996).Finite Element procedures. Prentice-Hall.
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming and P. Krysl (1998). Meshless methods: An overview and recent developments.Computer Methods in Applied Mechanics and Engineering,139, 3–47.
T. Belytschko, Y.Y. Lu and L. Gu (1994). Element-Free Galerkin Methods.International Journal for Numerical Methods in Engineering,37, 229–256.
J. Bonet (1994). The incremental flow formulation for the numerical analysis of plane stress and thin sheet forming processes.Computer Methods in Applied Mechanics and Engineering,114, 103–122.
J. Bonet and S. Kulasegaram (2000). Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations.International Journal for Numerical Methods in Engineering,47, 1189–1214.
J. Bonet and T.S.L. Lok (1999). Variational and momentum preservation aspects of smooth particle hydrodynamic formulations.Computer Methods in Applied Mechanics and Engineering,180, 97–115.
F. Brezzi (1974). On the existence, uniqueness and approximation of saddle-point problems arising from lagrange multipliers.Revue Française d'Automatique Informatique Recherche Operationelle, Analyse Numérique,8, 129–151.
D. Chapelle and K.J. Bathe(1993). The inf-sup test.Computers and Structures,47(4–5), 537–545.
J.-S. Chen, C. Pan, C.M.O.L. Roque and H.-P. Wang (1998). A lagrangian reproducing kernel particle method for metal forming analysis.Computational Mechanics,22, 289–307.
Jiun-Shyan Chen, Cheng-Tang Wu, Sangpil Yoon and Yang You (2001). A stabilized conforming nodal integration for galerkin mesh-free methods.International Journal for Numerical Methods in Engineering,50, 435–466.
F. Chinesta, E. Cueto, P. Quintela and J. Paredes (2004). Induced anisotropy in foams forming processes: modelling and simulation.Journal of Materials Processing Technology, in press.
F. Chinesta, E. Cueto, D. Ryckelynck and A. Ammar (2004). α-NEM and model reduction: two new and powerful numerical techniques to describe flows involving short fibers suspensions.Revue Europeenne de Elements Finis, submitted.
E. Cueto, B. Calvo and M. Doblaré (2002). Modeling three-dimensional piece-wise homogeneous domains using the α-shape based Natural Element Method.International Journal for Numerical Methods in Engineering,54, 871–897.
E. Cueto, J. Cegoñino, B. Calvo and M. Doblaré (2003). On the imposition of essential boundary conditions in Natural Neighbour Galerkin methods.Communications in Numerical Methods in Engineering,19(5), 361–376.
E. Cueto, M. Doblaré and L. Gracia (2000). Imposing essential boundary conditions in the Natural Element Method by means of density-scaled α-shapes.International Journal for Numerical Methods in Engineering,49, 519–546.
E. Cueto, N. Sukumar, B. Calvo, M.A. Martínez, J. Cegoñino and M. Doblaré (2003). Overview and recent advances in Natural Neighbour Galerkin methods.Archives of Computational Methods in Engineering,10(4), 307–384.
S. De and K. J. Bathe (2001). Towards an efficient meshless computational technique: the method of finite spheres.Engineering Computations,18, 170–192.
B. Delaunay (1934). Sur la Sphère Vide. A la memoire de Georges Voronoi.Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskii i Estestvennyka Nauk,7, 793–800.
H. Edelsbrunner, D.G. Kirkpatrick and R. Seidel (1983). On the shape of a set of points in the plane.IEEE Transactions on Information Theory, IT-29(4), 551–559.
H. Edelsbrunner and E. Mücke (1994). Three dimensional alpha shapes.ACM Transactions on Graphics,13, 43–72.
D. Gonzalez, E. Cueto, M.A. Martinez and M. Doblare (2004). Numerical integration in Natural Neighbour Galerkin methods.International Journal for Numerical Methods in Engineering,60(12), 2077–2104.
D. González, E. Cueto and M. Doblaré (2004). Volumetric locking in Natural Neighbour Galerkin methods.International Journal for Numerical Methods in Engineering, in press.
H. Hiyoshi and K. Sugihara (1999). Two generalizations of an interpolant based on Voronoi diagrams.International Journal of Shape Modeling,5(2), 219–231.
S.R. Idelsohn, E. Oñate, N. Calvo and F. Del Pin (2003). The meshless finite element method.International Journal for Numerical Methods in Engineering,58, 893–912.
W.K. Liu, S. Jun, S. Li, J. Adee and T. Belytschko (1995). Reproducing kernel particle methods.International Journal for Numerical Methods in Engineering,38, 1655–1679.
J. Lof (2000)Developments in finite element simulations of aluminium extrusion. PhD thesis, Department of Mechanical Engineering, Applied Mechanics Section. University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands.
J. Lof (2001). Elasto-viscoplastic fem simulation of the aluminium flow in the bearing area for extrusion of thin-walled sections.Journal of Materials Processing Technology,114, 174–183.
J. Lof and Y. Blokhuis (2002). FEM simulations of the extrusion of complex thin-walled aluminium sections.Journal of Materials Processing Technology,122, 344–354.
J. Lof and J. Huétink (1999). Numerical simulation of the aluminium extrusion process. In J.A. Covas, editor,Proceedings of the 2nd ESAFORM Conference on Material Forming, pages 29–32.
J. López, F. Faura, J. Hernández and P. Gómez (2003). On the critical plunger speed and three-dimensional effects in high-pressure die casting injection chambers.Journal of Manufacturing Science and Engineering. Transactions of the ASME,125, 529–537.
J.C. Martin and W.J. Moyce(1952). An experimental study of the collapse of liquid columns on a rigid horizontal plane.Philosophical Transactions of the Royal Society of London,244, 312.
M.A. Martínez, E. Cueto, I. Alfaro, M. Doblare and F. Chinesta (2004). Updated Lagrangian free surface flow simulations with Natural Neighbour Galerkin methods.International Journal for Numerical Methods in Engineering,60(13), 2105–2129.
M.A. Martínez, E. Cueto, M. Doblaré and F. Chinesta (2003). Fixed mesh and meshfree techniques in the numerical simulation of injection processes involving short fiber suspensions.Journal of Non-Newtonian Fluid Mechanics,115, 51–78.
H.G. Mooi and J. Huétink (1995). Simulation of complex aluminium extrusion using an arbitrary eulerian lagrangian formulation. In S.F. Shen and P.R. Dawson, editors,proceedings of the Numiform 95 conference, pages 869–874.
B. Nayroles, G. Touzot and P. Villon (1992). Generalizing the finite element method: Diffuse approximation and diffuse elements.Computational Mechanics,10, 307–318.
O.A. Ladyzhenskaya (1969).The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, London.
P. Perzyna (1966). Fundamental problems in visco-plasticity. InRecent advances in Applied Mechanics. Academic Press, New York.
C. M. Sellars and W.J.M. Tegart (1972). Hot workability.International Metallurgical Review,17, 1–24.
R. Sibson (1980). A Vector Identify for the Dirichlet Tesselation.Mathematical Proceedings of the Cambridge Philosophical Society,87, 151–155.
R. Sibson (1981). A brief description of natural neighbour interpolation. InInterpreting Multivariate Data. V. Barnett (Editor), pages 21–36. John Wiley.
N. Sukumar, J. Dolbow, A. Devan, J. Yvonnet, F. Chinesta, D. Ryckelynck, P. Lorong, I. Alfaro, M.A. Martínez, E. Cueto and M. Doblaré (2004). Meshless Methods and Partition of Unity Finite Elements.International Journal of Forming Processes, to appear.
N. Sukumar, B. Moran and T. Belytschko (1998). The Natural Element Method in Solid Mechanics.International Journal for Numerical Methods in Engineering,43(5), 839–887.
N. Sukumar, B. Moran, A. Yu Semenov and V.V. Belikov (2001). Natural Neighbor Galerkin Methods.International Journal for Numerical Methods in Engineering,50(1), 1–27.
A. H. Thiessen (1911). Precipitation averages for large areas.Monthly Weather Report,39, 1082–1084.
J. van de Langkruis, J. Lof, W.H. Kool, S. van der Zwaag and J. Huétink (2000). Comparison of experimental AA6063 extrusion trials to 3D numerical simulations, using a general solute-dependent constitutive model.Computational Materials Science, pages 381–392.
G. M. Voronoi (1908). Nouvelles Applications des Paramètres Continus à la Théorie des Formes Quadratiques. Deuxième Memoire: Recherches sur les parallélloèdres Primitifs.J. Reine Angew. Math.,134, 198–287.
Sangpil Yoon and Jiun-Shyan Chen (2002). Accelerated meshfree method for metal forming simulation.Finite Elements in Analysis and Design,38, 937–948.
J. Yvonnet, D. Ryckelynck, P. Lorong and F. Chinesta (2004). A new extension of the Natural Element method for non-convex and discontnuous problems: the Constrained Natural Element method.International Journal for Numerical Methods in Enginering,60(8), 1452–1474.
J. Zhou, L. Li and J. Duszczyk. (2003). 3 D FEM simulation of the whole cycle of aluminium extrusion throughout the transient state and the steady state using the updated Lagrangian approach.Journal of Materials Processing Technology,134, 383–397.
O.C. Zienkiewicz (1984). Flow formulation for numerical solution of forming processes. In R.D. Wood, J.F.T. Pittman, O.C. Zienkiewicz and J.M. Alexander (Eds.),Numerical analysis of forming processes, pages 1–44. John Wiley & Sons.
O.C. Zienkiewicz and P.N. Godbolet (1974). Flow of plastic and visco-plastic solids with special reference to extrusion and forming processes.International Journal for Numerical Methods in Engineering,8, 3–16.
O.C. Zienkiewicz, E. Oñate and J.C. Heinrich (1978). Plastic flow in metal forming. (I) Coupled thermal (II) Thin sheet forming. InApplications of Numerical Methods to Forming Processes. AMD-vol 28, pages 107–120.
O.C. Zienkiewicz, P.C. Pain and E. Oñate (1978). Flow of solids during forming and extrusion: some aspects of numerical solutions.International Journal of Solids and Structures,14, 15–38.
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Alfaro, I., González, D., Bel, D. et al. Recent advances in the meshless simulation of aluminium extrusion and other related forming processes. Arch Computat Methods Eng 13, 3–43 (2006). https://doi.org/10.1007/BF02905930
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DOI: https://doi.org/10.1007/BF02905930