Abstract
This paper describes a fully implicit algorithm developed and optimized to simulate sheet metal forming processes. This algorithm was implemented in the in-house code DD3IMP. Attention is paid to the augmented lagrangian method adopted to treat the contact with friction problem. The global resolution of the coupled equilibrium and contact problem is performed in a single loop, with a static implicit iterative Newton-Raphson scheme. This demands particular attention in the contact search algorithm, which in this case adopts a parametric description of the tools. In order to highlight the adopted strategies a review of the state-of-the-art in sheet metal forming simulation is presented, with respect to models reliability and efficiency.
Similar content being viewed by others
References
Esche SK, Kinzel GL, Altan T (1997) Issues in convergence improvement for non-linear finite element programs. Int J Numer Methods Eng 40:4577–4594
Menezes LF, Teodosiu C (2000) Three dimensional numerical simulation of the deep drawing process using solid finite element. J Mater Process Technol 97:100–106
Oliveira MC, Alves JL, Menezes LF (2003) Improvement of a frictional contact algorithm for strongly curved contact problems. Int J Numer Methods Eng 58:2083–2101
Alves JL, Oliveira MC, Menezes LF (2004) Springback evaluation with several phenomenological yield criteria. Mater Sci Forum 455–456:732–736
Alves JL, Oliveira MC, Menezes LF, Bouvier S (2006) Influence of the yield criteria on the numerical results: the cross tool example. In: Juster N, Rosochowski A (eds) Proceedings of the ESAFORM’06, the 9th international ESAFORM conference on material forming, pp 887–890
Alves JL, Oliveira MC, Menezes LF (2004) An advanced constitutive model in sheet metal forming simulation: the Teodosiu microstructural model and the Cazacu Barlat yield criterion. In: Glosh S et al. (eds) Proceedings of the NUMIFORM’04, materials processing and design: modelling, simulation and applications. American Institute of Physics, Melville, p 1645
Chaparro BM, Oliveira MC, Alves JL, Menezes LF (2004) Work hardening models and the numerical simulation of the deep drawing process. Mater Sci Forum 455–456:717–722
Bouvier S, Alves JL, Oliveira MC, Menezes LF (2005) Modelling of anisotropic work-hardening behaviour of metallic materials subjected to strain path changes. Comput Mater Sci 32:301–315
Oliveira MC, Alves JL, Chaparro BM, Menezes LF (2007) Study on the influence of work-hardening modeling in springback prediction. Int J Plast 23:516–543
Yoon JW, Yang DY, Chung K, Barlat F (1999) A general elastoplastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming. Int J Plast 15:35–67
Geng L, Wagoner RH (2002) Role of plastic anisotropy and its evolution on springback. Int J Mech Sci 44:123–148
Hu J, Jonas JJ, Ishikawa T (1998) FEM simulation of the forming of texture aluminium sheets. Mat Sci Eng A-Struct A256:51–59
Duchene L, Godinas A, Cescotto S, Habraken AM (2002) Texture evolution during deep drawing processes. J Mater Process Technol 125:110–118
Alart P (1993) Contact avec frottement. Mémoire d’habilitation à diriger des recherches, Laboratoire de Mécanique et Génie Civil, Université Montpellier II, France
Pietrzak G, Curnier A (1999) Large deformation frictional contact mechanics: continuum formulation and augmented lagrangian treatment. Comput Methods Appl Math 177:351–381
Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. Studies in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia
Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Math 92-3:353–375
Simo JC, Laursen TA (1992) An augmented lagrangian treatment of contact problems involving friction. Comput Struct 42:97–116
Cao HL (1990) Modélisation mécanique et simulation numérique de l’emboutissage. PhD thesis, Institut National Polytechnique de Grenoble, France
Teodosiu C (1989) The plastic spin: microstructural origin and computational significance. In: Owen DRJ et al (eds) Proceedings of the 2nd international conference on computational plasticity. Barcelona, Spain
Menezes LF (1995) Modelação tridimensional e simulação numérica dos processos de enformação por deformação plástica: aplicação à estampagem de chapas metálicas. PhD thesis, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Portugal
Teodosiu C, Genevois P (1998) Modelling large deformation anisotropic plastic behaviour of mild steel sheet. In: Proceedings of the MECAMAT’98, international symposium on inelastic behaviour of solids: models and utilisation. Besançon, France, pp 211–224
Sidoroff H (1981) Formulation elastoplastique en grandes déformations. Rapport GRECO 29
Simo JC (1985) On the computational significance of the intermediate configuration and hyperelastic stress relations in finite deformation elastoplasticity. Mech Mater 4:439
Alves JL (2003) Simulação numérica do processo de estampagem de chapas metálicas: modelação mecânica e métodos numéricos. PhD thesis, Departamento de Engenharia Mecânica, Universidade do Minho, Portugal
Heegaard JH, Curnier A (1993) An augmented lagrangian method for discrete large slip contact problems. Int J Numer Methods Eng 36:569–593
Klarbring A (1986) A mathematics programming approach to three-dimensional contact problems with friction. Comput Methods Appl Math 58:175–200
Hallquist JO, Goudreau GL, Benson DJ (1985) Sliding interfaces with contact-impact in large-scale lagrangian computations. Comput Methods Appl Math 51:107–137
Heege A (1992) Simulation numerique 3D du contact avec frottement et application à la mise en forme. PhD thesis, Institut National Polythecnique de Grenoble, France
Moreau JJ (1979) Application of convex analysis to some problems of dry friction. In: Zorski H (ed) Trends of pure mathematics applied to mechanics, vol II. Pitman, London
Laursen TA (1992) Formulation and treatment of frictional contact problems using finite elements. PhD thesis, Stanford University, USA
Mijar AR, Arora JS (2000) Review of formulations for elastostatic frictional contact problems. Struct Multidiscipl Optim 20:167–189
Klarbring A (1995) Large displacement frictional contact: a continuum framework for finite element discretization. Eur J Mech A-Solids 14:237–253
Kloosterman G, Van Damme RMJ, Van den Boogaard AH, Huétink J (2001) A geometrical based contact algorithm using a barrier method. Int J Numer Methods Eng 51:865–882
Tekkaya AE (2000) State of the art of simulation of sheet metal forming. J Mater Process Technol 103:14–22
Rojek J, Zienkiewicz OC, Oñate E, Postek E (2001) Advances in FE explicit formulation for simulation of metal forming processes. J Mater Process Technol 119:41–47
Finn MJ, Galbraith PC, Wu L, Hallquist JO, Lum L, Lin T-L (1995) Use of a coupled explicit implicit solver for calculating springback in automotive body panels. J Mater Process Technol 50:395–406
Bathe KJ (2004) On the state of finite element procedures for forming processes. In: Glosh S et al. (eds) Proceedings of the NUMIFORM’04, materials processing and design: modelling, simulation and applications. American Institute of Physics, Melville, p 34
Menezes LF, Teodosiu C, Makinouchi A (1991) 3-D solid elasto plastic elements for simulating sheet metal forming processes by the finite element method. In: FE simulation of 3D sheet metal forming processes in automotive industry, VDI Berichte Nr 894, pp 381–403
Schönbach E, Glanzer G, Kubli W, Selig M (2004) Springback simulation: the last missing link for a complete forming simulation. In: Kergen R et al (eds) Proceedings of the IDDRG’04, international deep drawing research group conference, forming the future: global trends in sheet metal forming. Sildenfingen, Germany, pp 83–94
Sriram S, Wagoner RH (2000) Adding bending stiffness to 3-D membrane FEM programs. Int J Mech Sci 42:1753–1782
Chapelle D, Bathe KJ (1998) Fundamental considerations for the finite element analysis of shell structures. Comput Struct 66:19–36
Kawka M, Makinouchi A (1995) Shell element formulation in the static explicit FEM code for the simulation of sheet metal forming. J Mater Process Technol 50:105–115
Alves JL, Menezes LF (2001) Application of tri-linear and tri-quadratic 3-D solid finite elements in sheet metal forming process simulations. In: Mori N (ed) Proceedings of the NUMIFORM’01, simulation of materials processing: theory, methods and applications, pp 639–644
Kawka M, Kakita T, Makinouchi A (1998) Simulation of multi-steep sheet metal forming processes by a static explicit FEM code. J Mater Process Technol 80-81:54–59
Li KP, Carden WP, Wagoner RH (2002) Simulation of springback. Int J Mech Sci 44:103–122
Areias PMA, César de Sá JMA, Conceição António CA, Fernandes AA (2003) Analysis of 3D problems using a new enhanced strain hexahedral element. Int J Numer Methods Eng 58:1382–1637
Wang J, Wagoner RH (2005) A practical large-strain solid finite element for sheet forming. Int J Numer Methods Eng 63:473–501
Reese S (2005) On a physically stabilized one point finite element formulation for three-dimensional finite elastoplasticity. Comput Methods Appl Math 194:4685–4715
Péric D, Vaz M Jr, Owen DRJ (1999) On adaptative strategies for large deformations of elasto plastic solids at finite strains: computational issues and industrial applications. Comput Methods Appl Math 176:279–312
Meinders T (2000) Developments in numerical simulations of the real life deep drawing process. PhD thesis, University of Twente. Ponsen & Looijen Wageningen, Netherlands, ISBN 90-36514002
Mc Meeking RM, Rice JR (1975) Finite element formulations for problems of large elastic plastic deformation. Int J Solids Struct 11:601–616
Yamada Y, Yoshimura N, Sakurai T (1968) Plastic stress strain matrix and its application for the solution of elastic plastic problems by the finite element method. Int J Mech Sci 10:343–354
Hughes TJR, Winget J (1980) Finite rotation effects in numerical integration of rate constitutive equations arising in large deformation analysis. Int J Numer Methods Eng 15:1862–1867
Simo JC, Taylor RL (1984) Consistent tangent operators for rate independent elastoplasticity. Comput Methods Appl Math 48:101–118
Hughes TJR (1980) Generalization of selective reduced integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15:1413–1418
Santos A (1993) Tool description and contact strategies used in the static explicit FEM code ITAS3D for simulation of 3-D sheet metal forming process. PhD thesis, University of Tokyo, Japan
Santos A, Makinouchi A (1995) Contact Strategies to deal with different tool descriptions in static explicit FEM of 3-D sheet-metal forming simulation. J Mater Process Technol 50:277–291
Hallquist JO, Wainscott B, Schweizerhof K (1995) Improved simulation of thin-sheet metal forming using LS-DYNA3D on parallel computers. J Mater Process Technol 50:144–147
Lin J, Ball AA, Zheng JJ (2001) Approximating circular arcs by Bézier curves and its application to modelling tooling for FE forming simulations. Int J Mach Tools Manuf 41:703–717
Belytschko T, Lin JI (1987) A three-dimensional impact-penetration algorithm with erosion. Comput Struct 25:95–104
Zhong Z-H, Nilsson L (1994) Automatic contact searching algorithm for dynamic finite element analysis. Comput Struct 52:187–197
Bergman G, Oldenburg M (2004) A Finite element model for thermomechanical analysis of sheet metal forming. Int J Numer Methods Eng 59:1167–1186
**ng HL, Fujimoto T, Makinouchi A, Nikishkov GP (1998) Static-explicit FE modeling of 3-D large deformation multibody contact problems on parallel computer. In: Huétink J (ed) Simulation of materials processing: theory, methods and applications. Balkema, Rotterdam, pp 207–212
Wang F, Cheng J, Yao Z (2001) FFS contact searching algorithm for dynamic finite element analysis. Int J Numer Methods Eng 52:655–672
Belytschko T, Neal MO (1991) Contact-impact by the pinball algorithm with penalty and lagrangian-methods. Int J Numer Methods Eng 31:547–572
Wang SP, Nakamachi E (1997) The inside-outside contact search algorithm for finite element analysis. Int J Numer Methods Eng 40:3665–3685
El-Abbasi N, Bathe K-J (2001) Stability and patch test performance of contact discretizations and a new solution algorithm. Comput Struct 79:1473–1486
Zavarise G, Wriggers P (1999) A superlinear convergent augmented lagrangian procedure for contact problems. Eng Comput 16:88–119
Chabrand P, Chertier O, Dubois F (2001) Complementarity methods for multibody friction contact problems in finite deformation. Int J Numer Methods Eng 51:553–578
Simo JC, Wriggers P, Taylor RL (1985) A perturbed lagrangian formulation for the finite element solution of contact problems. Comput Methods Appl Math 50:163–180
Belgacem FB, Hild P, Laborde P (1998) The Mortar finite element method for contact problems. Math Comput Model 28:253–271
McDevitt TW, Laursen TA (2000) A mortar-finite element formulation for frictional contact problems. Int J Numer Methods Eng 48:1525–1547
Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Math 193:601–629
Puso MA, Laursen TA (2002) A 3D contact smoothing method using Gregory patches. Int J Numer Methods Eng 54:1161–1194
Al-Dojayli M, Meguid SA (2002) Accurate modeling of contact using cubic splines. Finite Elem Anal Des 38:337–352
El-Abbasi N, Meguid SA, Czekanski A (2001) On the modelling of smooth contact surfaces using cubic splines. Int J Numer Methods Eng 50:953–967
Wriggers P, Krstulovic-Opara L, Korelc J (2001) Smooth C1-interpolations for two-dimensional frictional problems. Int J Numer Methods Eng 51:1469–1495
Stadler M, Holzapfel GA, Korelc J (2003) Cn continuous modelling of smooth contact surfaces using NURBS and application to 2D problems. Int J Numer Methods Eng 57:2177–2203
Belytschko T, Daniel WJT, Ventura G (2002) A monolithic smoothing-gap algorithm for contact-impact based on the signed distance function. Int J Numer Methods Eng 55:101–125
Kim T-J, Yang D-Y (2007) FE-analysis of sheet metal forming processes using continuous contact treatment. Int J Plast 23:544–560
Farin G (1993) Curves and surfaces for computer aided geometric design—a practical guide, 3rd edn. Academic, New York
Heege A, Alart P (1996) A frictional contact element for strongly curved contact problems. Int J Numer Methods Eng 39:165–184
Xu W, Di S, Thomson P (2003) Rigid-plastic/rigid-viscoplastic FE simulation using linear programming for metal forming. Int J Numer Methods Eng 56:487–506
Luenberg DG (1984) Linear and nonlinear programming, 2nd edn. Addison-Wesley, Reading
Laursen TA, Oancea VG (1994) Automation and assessment of augmented lagrangian algorithms for frictional contact problems. J Appl Mech-T ASME 61:956–963
Refaat MH, Meguid SA (1994) On the elastic solution of frictional contact problems using variational inequalities. Int J Mech Sci 36:329–342
Zhang HW, Xu WL, Di SL, Thomson PF (2002) Quadratic programming method in numerical simulation of metal forming process. Comput Methods Appl Math 191:5555–5578
Refaat MH, Meguid SA (1998) A new strategy for the solution of frictional contact problems. Int J Numer Methods Eng 43:1053–1068
Christensen PW, Klarbring A, Pang JS, Stromberg N (1998) Formulation and comparison of algorithms for frictional contact problems. Int J Numer Methods Eng 42:145–173
Kloosterman G (2002) Contact methods in finite element simulations. PhD thesis, Netherlands Institute for Metals Research, University of Twente, Netherlands
Nour-Omid B, Wriggers P (1987) A note on the optimum choice for penalty parameters. Commun Numer Methods Eng 3:518–585
Shanno DF, Breitfeld MG, Simantiraki EM (1995) Implementating barrier methods for nonlinear programming. Rutcor Research Report 39-95:1–14
Walter H (1999) Modelisation 3D par elements finites du contact avec frottement et del l’endommagement du beton: application a létude de fixations ancrees dans une structure de beton. PhD thesis, L’Institut National des Sciences Appliquées de Lyon, France
Laursen TA, Maker BN (1995) An augmented lagrangian quasi-Newton solver for constrained nonlinear finite element applications. Int J Numer Methods Eng 38:3571–3590
Cheng X-L, Han W (2002) Inexact Uzawa algorithms for variational inequalities of the second kind. Reports on Computational Mathematics of the Numerical Computing Group at the University of Iowa
Jones RE, Papadopoulos P (2001) A novel three-dimensional contact finite element based on smooth pressure interpolations. Int J Numer Methods Eng 51:791–811
Zavarise G, Wriggers P, Schrefler BA (1998) A method for solving contact problems. Int J Numer Methods Eng 42:473–498
Bathe KJ, Bouzinov PA (1997) On the constraint function method for contact problems. Comput Struct 64:1069–1085
Kawka M, Bathe KJ (2001) A new effective and reliable implicit scheme for the simulation of sheet metal forming processes. In: Mori N (ed) Proceedings of the NUMIFORM’01, simulation of materials processing: theory, methods and applications, pp 615–619
Belytschko T, Fleming M (1999) Smoothing, enrichment and contact in the element free Galerkin method. Comput Struct 71:173–195
Kim NH, Choi KK, Chen JS, Park YH (2000) Meshless shape design sensitivity analysis and optimization for contact problem with friction. Comput Mech 25:157–168
Zhong ZH, Nilsson L (1996) A unified contact algorithm based on the territory concept. Comput Methods Appl Math 130:1–16
Shanno DF, Breitfeld MG, Evangelia MS (1995) Implementating barrier methods for nonlinear programming. Rutcor Research Report 39-95, Rutgers University, 14
Kloosterman G, Van Damme RMJ, Van den Boogaard AH, Huétink J (2001) A geometrical based contact algorithm using a barrier method. Int J Numer Methods Eng 51:865–882
Giannakopoulos AE (1989) The radial map** method for the integration of friction constitutive relations. Comput Struct 6:281–290
Ling W, Stolarski HK (1997) A contact algorithm for problems involving quadrilateral approximation of surface. Comput Struct 63:963–975
Ju SH, Stone JJ, Rowlands RE (1995) A new symmetric contact element stiffness matrix for frictional contact problems. Comput Struct 54:289–301
Zavarise G, Wriggers P, Schrefler BA (1995) On augmented lagrangian algorithms for thermomechanical contact problems with friction. Int J Numer Methods Eng 38:2929–2949
Oliveira MC (2005) Algoritmos e estratégias de gestão do problema de contacto com atrito em grandes deformações: aplicação à estampagem de chapas metálicas. PhD thesis, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Portugal, ISBN 972-99204-7-8
Auricchio F, Sacco E (1996) Augmented lagrangian finite elements for plate contact problems. Int J Numer Methods Eng 39:4141–4158
Alart P (1992) A simple contact algorithm applied to large sliding and anisotropic friction. In: Contac mechanics international symposium. Presses Polytechniques Romades, Lausanne
Knibloe JR, Wagoner RH (1989) Experimental investigation and finite-element modeling of hemispherically stretched steel sheet. Metall Trans A—Phys Metall Mater Sci 20:1509–1521
Fromentina S, Martiny M, Ferron G, Tourkic Z, Moreira LP, Ferran G (2001) Finite element simulations of sheet-metal forming processes for planar-anisotropic materials. Int J Mech Sci 43:1833–1852
Stupkiewicz S (2001) Extension of the node-to-segment contact element for surface-expansion-dependent contact laws. Int J Numer Methods Eng 50:739–759
Hughes DA, Weingarten LI, Dawson DB (1985) Numerical simulation and experimental observations of initial friction transients. In: Shen SF, Dawson PR (eds) Simulation of materials processing: theory, methods and applications. Balkema, Rotterdam, pp 265–270
Haar R (1996) Friction in sheet metal forming: the influence of (local) contact conditions and deformation. PhD thesis, University of Twente, Netherlands
Martinet F, Chabrand P (2000) Application of ALE finite elements method to a lubricated friction model in sheet metal forming. Int J Solids Struct 37:4005–4031
Magny C (2002) Lois de frottement évolutives destinées à la simulation numérique de l’emboutissage. Rev Metall 145:156–2002
Agelet de Saracibar C, Chiumenti M (1999) On the numerical modeling of frictional wear design. Comput Methods Appl Math 177:401–426
Heege A, Alart P, Oñate E (1995) Numerical modelling and simulation of frictional contact using a generalised Coulomb law. Eng Comput 12:641–656
Laursen TA, Oancea VG (1997) On the constitutive modeling and finite element computation of rate-depedent frictional sliding in large deformation. Comput Methods Appl Math 143:197–227
Behrens A, Schafstall H (1998) 2D and 3D simulation of complex multistage forging processes by use of adaptive friction coefficient. J Mater Process Technol 80-81:298–303
Bandeira AA, Wriggers P, Pimenta PM (2004) Numerical derivation of contact mechanics interface laws using a finite element approach for large 3D deformation. Int J Numer Methods Eng 59:173–195
Levaillant C, Chenot JL (1992) Physical modelling and numerical prediction of defects in sheet metal forming. J Mater Process Technol 32:383–397
Wouters P, Daniel D, Magny C (2002) Selection and identification of friction models for the 3DS materials-identification of the frictional behaviour. Digital Die Design Systems (3DS) IMS 1999 000051, Work Package 3, Task 1
Areias PMA, César de Sá JM, Conceição António CA (2004) Algorithms for the analysis of 3D finite strain contact problems. Int J Numer Methods Eng 61:1107–1151
Baptista AJ (2007) Modelação mecânica e simulação numérica do processo de estampagem multi-etapas: aplicação ao processo de estampagem de chapas soldadas. PhD thesis, Faculdade de Ciências e Tecnologia, Universidade de Coimbra, Portugal, ISBN 972-8954-08-5
Tanner JA (1996) Computational methods for frictional contact with applications to the space shuttle orbiter nose-gear tire: development of frictional contact algorithm. NASA Technical Paper 3574
Oliveira MC, Alves JL, Menezes LF (2006) Optimizing the description of forming tools with Bézier surfaces in the numerical simulation of the deep drawing process. In: Mota Soares CA et al. (eds) Proceedings of the ECCM’06, III European conference on computational mechanics: solids, structures and coupled problems in engineering. Springer, Dordrecht, p 332
Laboratoire des Propriétés Mécaniques et Thermodynamiques des Matériaux (2001) Selection and identification of elastoplastic models for the materials used in the benchmarks. Digital Die Design Systems (3DS) IMS 1999 000051, Work Package 3, Task1, 18-Month Progress Report, University Paris 13
Rohleder M, Roll K, Menezes LF, Oliveira MC, Andersson A, Krantz F (2002) Standardization of input output data for benchmark tests. Digital Die Design Systems (3DS) IMS 1999 000051, Work Package 2, Task 3
Digital Die Design Systems (3DS) IMS 1999 000051 (2004) Final Technical Report
Oliveira MC, Alves JL, Menezes LF (2003) One step springback strategies in sheet metal forming. In: Owen DRJ et al. (eds) Proceedings of the COMPLAS’02. International Center for Numerical Methods in Engineering, Barcelona, p 87
Kase K, Makinouchi A, Nakagawa T, Suzuki H, Kimura F (1999) Shape error evaluation method of free-form surfaces. Comput Aided Des 31:495–505
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oliveira, M.C., Alves, J.L. & Menezes, L.F. Algorithms and Strategies for Treatment of Large Deformation Frictional Contact in the Numerical Simulation of Deep Drawing Process. Arch Computat Methods Eng 15, 113–162 (2008). https://doi.org/10.1007/s11831-008-9018-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11831-008-9018-x