Abstract
This work addresses the problems arising in the finite element simulation of contact problems undergoing large deformation. The frictional contact problem is formulated in the continuum framework, introducing the interface laws for the normal and tangential stress components in the contact area. The variational formulation is presented, considering different methods to enforce the contact constraints. The spatial discretization within the finite element method is applied, as well as the temporal discretization required to solve the three sources of nonlinearities: geometric, material and frictional contact. The discretization of contact surfaces is discussed in detail, including different surface smoothing procedures. This numerical strategy allows to solve the difficulties associated with the discontinuities in the contact surface geometry introduced by finite element discretization, which leads to nonphysical oscillations of the contact force for large sliding problems. The geometrical accuracy of different interpolation methods is evaluated, paying particular attention to the Nagata patch interpolation recently proposed. In this framework, the Node-to-Nagata contact elements are developed using the augmented Lagrangian method to regularize the variational frictional contact problem. The techniques used to search for contact in case of large deformations are discussed, including self-contact phenomena. Several numerical examples are presented, comprising both the contact between deformable and rigid obstacles and the contact between deformable bodies. The results show that the accuracy and robustness of the numerical simulations is improved when the contact surface is smoothed with Nagata patches.
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References
Zhong Z-H (1993) Finite element procedures for contact-impact problems. Oxford University Press, Oxford
Kikuchi N, Oden JT (1988) Contact problems in elasticity: a study of variational inequalities and finite element methods. SIAM, Philadelphia
Wriggers P (2006) Computational contact mechanics. Springer, Berlin
Hertz H (1881) Über die Berührung fester elastische Körper. J für die reine und Angew Math 92:156–171. doi:10.1243/PIME_PROC_1982_196_039_02
Johnson KL (1982) One hundred years of hertz contact. Proc Inst Mech Eng 196:363–378. doi:10.1243/PIME_PROC_1982_196_039_02
Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals. Finite Elem Method Basis Fundam. doi:10.1016/B978-1-85617-633-0.00019-8
Signorini A (1933) Sopra alcune questioni di elastostatica. Atti della Soc Ital per Prog delle Sci
Campos LT, Oden JT, Kikuchi N (1982) A numerical analysis of a class of contact problems with friction in elastostatics. Comput Methods Appl Mech Eng 34:821–845. doi:10.1016/0045-7825(82)90090-1
Wriggers P, Van Vu T, Stein E (1990) Finite element formulation of large deformation impact-contact problems with friction. Comput Struct 37:319–331. doi:10.1016/0045-7949(90)90324-U
Laursen TA, Simo JC (1993) A continuum-based finite element formulation for the implicit solution of multibody, large deformation-frictional contact problems. Int J Numer Methods Eng 36:3451–3485. doi:10.1002/nme.1620362005
Hallquist JO, Goudreau GL, Benson DJ (1985) Sliding interfaces with contact-impact in large-scale Lagrangian computations. Comput Methods Appl Mech Eng 51:107–137. doi:10.1016/0045-7825(85)90030-1
Zienkiewicz OC (1995) Origins, milestones and directions of the finite element method—a personal view. Arch Comput Methods Eng 2:1–48. doi:10.1007/BF02736188
Wilson EA, Parsons B (1970) Finite element analysis of elastic contact problems using differential displacements. Int J Numer Methods Eng 2:387–395. doi:10.1002/nme.1620020307
Chan SK, Tuba IS (1971) A finite element method for contact problems of solid bodies—Part I. Theory and validation. Int J Mech Sci 13:615–625. doi:10.1016/0020-7403(71)90032-4
Wriggers P (1995) Finite element algorithms for contact problems. Arch Comput Methods Eng 2:1–49. doi:10.1007/BF02736195
Mijar AR, Arora JS (2000) Review of formulations for elastostatic frictional contact problems. Struct Multidiscipl Optim 20:167–189. doi:10.1007/s001580050147
Zavarise G, De Lorenzis L (2009) The node-to-segment algorithm for 2D frictionless contact: classical formulation and special cases. Comput Methods Appl Mech Eng 198:3428–3451. doi:10.1016/j.cma.2009.06.022
Taylor R, Papadopoulos P (1991) On a patch test for contact problems in two dimensions. In: Wriggers P, Wagner W (eds) Computer methods nonlinear mechanics. Springer, Berlin, pp 690–702
Laursen TA (2002) Computational contact and impact mechanics: fundamentals of modeling interfacial phenomena in nonlinear finite element analysis. Springer, Berlin
Hansson E, Klarbring A (1990) Rigid contact modelled by CAD surface. Eng Comput 7:344–348. doi:10.1108/eb023821
Heege A, Alart P (1996) A frictional contact element for strongly curved contact problems. Int J Numer Methods Eng 39:165–184. doi:10.1002/(SICI)1097-0207(19960115)39:1<165::AID-NME846>3.0.CO;2-Y
Heegaard JH, Curnier A (1996) Geometric properties of 2D and 3D unilateral large slip contact operators. Comput Methods Appl Mech Eng 131:263–286. doi:10.1016/0045-7825(95)00977-9
Pietrzak G, Curnier A (1999) Large deformation frictional contact mechanics: continuum formulation and augmented Lagrangian treatment. Comput Methods Appl Mech Eng 177:351–381. doi:10.1016/S0168-874X(00)00029-9
Padmanabhan V, Laursen TA (2001) A framework for development of surface smoothing procedures in large deformation frictional contact analysis. Finite Elem Anal Des 37:173–198
Wriggers P, Krstulovic-Opara L, Korelc J (2001) Smooth C1-interpolations for two-dimensional frictional contact problems. Int J Numer Methods Eng 51:1469–1495. doi:10.1002/nme.227
Al-Dojayli M, Meguid SA (2002) Accurate modeling of contact using cubic splines. Finite Elem Anal Des 38:337–352. doi:10.1016/S0168-874X(01)00088-9
Krstulovic-Opara L, Wriggers P, Korelc J (2002) A C1-continuous formulation for 3D finite deformation frictional contact. Comput Mech 29:27–42. doi:10.1007/s00466-002-0317-z
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. doi:10.1002/nme.776
Puso MA, Laursen TA (2002) A 3D contact smoothing method using Gregory patches. Int J Numer Methods Eng 54:1161–1194. doi:10.1002/nme.466
Lengiewicz J, Korelc J, Stupkiewicz S (2011) Automation of finite element formulations for large deformation contact problems. Int J Numer Methods Eng 85:1252–1279. doi:10.1002/nme.3009
Corbett CJ, Sauer RA (2014) NURBS-enriched contact finite elements. Comput Methods Appl Mech Eng 275:55–75. doi:10.1016/j.cma.2014.02.019
Parisch H, Lübbing C (1997) A formulation of arbitrarily shaped surface elements for three-dimensional large deformation contact with friction. Int J Numer Methods Eng 40:3359–3383. doi:10.1002/(SICI)1097-0207(19970930)40:18<3359::AID-NME217>3.0.CO;2-5
Mijar AR, Arora JS (2000) Study of variational inequality and equality formulations for elastostatic frictional contact problems. Arch Comput Methods Eng 7:387–449. doi:10.1007/BF02736213
Bonet J, Wood RD (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge
Laursen TA (1994) The convected description in large deformation frictional contact problems. Int J Solids Struct 31:669–681. doi:10.1016/0020-7683(94)90145-7
Klarbring A (1995) Large displacement frictional contact: a continuum framework for finite element discretization. Eur J Mech A Solids 14:237–253
Agelet de Saracibar C (1997) A new frictional time integration algorithm for large slip multi-body frictional contact problems. Comput Methods Appl Mech Eng 142:303–334. doi:10.1016/S0045-7825(96)01133-4
Alart P, Curnier A (1991) A mixed formulation for frictional contact problems prone to Newton like solution methods. Comput Methods Appl Mech Eng 92:353–375. doi:10.1016/0045-7825(91)90022-X
Heegaard J-H, Curnier A (1993) An augmented Lagrangian method for discrete large-slip contact problems. Int J Numer Methods Eng 36:569–593. doi:10.1002/nme.1620360403
Buczkowski R, Kleiber M (2009) Statistical models of rough surfaces for finite element 3D-contact analysis. Arch Comput Methods Eng 16:399–424. doi:10.1007/s11831-009-9037-2
Yastrebov VA, Anciaux G, Molinari J-F (2014) The contact of elastic regular wavy surfaces revisited. Tribol Lett 56:171–183. doi:10.1007/s11249-014-0395-z
Simo JC, Laursen TA (1992) An augmented lagrangian treatment of contact problems involving friction. Comput Struct 42:97–116. doi:10.1016/0045-7949(92)90540-G
Heege A, Alart P, Oñate E (1995) Numerical modelling and simulation of frictional contact using a generalised coulomb law. Eng Comput 12:641–656. doi:10.1108/02644409510799820
Areias P, Rabczuk T, Queirós de Melo FJM, César de Sá J (2014) Coulomb frictional contact by explicit projection in the cone for finite displacement quasi-static problems. Comput Mech 55:57–72. doi:10.1007/s00466-014-1082-5
Oden JT, Pires EB (1984) Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws. Comput Struct 19:137–147. doi:10.1016/0045-7949(84)90212-8
Hjiaj M, Feng Z-Q, de Saxcé G, Mróz Z (2004) On the modelling of complex anisotropic frictional contact laws. Int J Eng Sci 42:1013–1034. doi:10.1016/j.ijengsci.2003.10.004
Persson B (2000) Sliding friction: physical principles and applications. Springer, Berlin
Refaat MH, Meguid SA (1998) A new strategy for the solution of frictional contact problems. Int J Numer Methods Eng 43:1053–1068. doi:10.1002/(SICI)1097-0207(19981130)43:6<1053::AID-NME460>3.0.CO;2-L
Agelet de Saracibar C (1998) Numerical analysis of coupled thermomechanical frictional contact problems. Computational model and applications. Arch Comput Methods Eng 5:243–301. doi:10.1007/BF02897875
Luenberger DG, Ye Y (2008) Linear and nonlinear programming, 3rd edn. doi:10.1007/978-0-387-74503-9
Yastrebov VA (2013) Numerical methods in contact mechanics. Wiley, Hoboken
Courtney-Pratt JS, Eisner E (1957) The effect of a tangential force on the contact of metallic bodies. Proc R Soc A Math Phys Eng Sci 238:529–550. doi:10.1098/rspa.1957.0016
Hüeber S, Stadler G, Wohlmuth BI (2008) A primal-dual active set algorithm for three-dimensional contact problems with coulomb friction. SIAM J Sci Comput 30:572–596. doi:10.1137/060671061
Popp A, Gee MW, Wall WA (2009) A finite deformation mortar contact formulation using a primal-dual active set strategy. Int J Numer Methods Eng 79:1354–1391. doi:10.1002/nme.2614
Hestenes MR (1969) Multiplier and gradient methods. J Optim Theory Appl 4:303–320. doi:10.1007/BF00927673
Powell M (1969) A method for nonlinear constraints in minimization problems. In: Fletcher R (ed) Optimization. Academic Press, New York, pp 283–298
Cavalieri FJ, Cardona A (2015) Numerical solution of frictional contact problems based on a mortar algorithm with an augmented Lagrangian technique. Multibody Syst Dyn. doi:10.1007/s11044-015-9449-8
Alart PP (1997) Méthode de Newton généralisée en mécanique du contact. J Math Pures Appl 76:83–108. doi:10.1016/S0021-7824(97)89946-1
Mijar AR, Arora JS (2004) An augmented Lagrangian optimization method for contact analysis problems, 1: formulation and algorithm. Struct Multidiscipl Optim 28:99–112. doi:10.1007/s00158-004-0423-y
Mijar AR, Arora JS (2004) An augmented Lagrangian optimization method for contact analysis problems, 2: numerical evaluation. Struct Multidiscipl Optim 28:113–126. doi:10.1007/s00158-004-0424-x
Yoon J (1999) A general elasto-plastic finite element formulation based on incremental deformation theory for planar anisotropy and its application to sheet metal forming. Int J Plast 15:35–67. doi:10.1016/S0749-6419(98)00059-X
Cardoso RPR, Yoon J-W (2005) One point quadrature shell elements for sheet metal forming analysis. Arch Comput Methods Eng 12:3–66. doi:10.1007/BF02736172
Tekkaya AE, Martins PAF (2009) Accuracy, reliability and validity of finite element analysis in metal forming: a user’s perspective. Eng Comput 26:1026–1055. doi:10.1108/02644400910996880
Alart P, Lebon F (1995) Solution of frictional contact problems using ILU and coarse/fine preconditioners. Comput Mech 16:98–105. doi:10.1007/BF00365863
Saad Y (2003) Iterative methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia
Schenk O, Gärtner K (2004) Solving unsymmetric sparse systems of linear equations with PARDISO. Future Gener Comput Syst 20:475–487. doi:10.1016/j.future.2003.07.011
Gould NIM, Scott JA, Hu Y (2007) A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations. ACM Trans Math Softw 33:10-es. doi:10.1145/1236463.1236465
Chow E, Saad Y (1997) Experimental study of ILU preconditioners for indefinite matrices. J Comput Appl Math 86:387–414. doi:10.1016/S0377-0427(97)00171-4
Menezes LF, Neto DM, Oliveira MC, Alves JL (2011) Improving computational performance through HPC techniques: case study using DD3IMP in-house code. AIP Conf Proc 1353:1220–1225. doi:10.1063/1.3589683
Intel (2014) Intel math kernel library reference manual
Crisfield MA (2000) Re-visiting the contact patch test. Int J Numer Methods Eng 48:435–449. doi:10.1002/(SICI)1097-0207(20000530)48:3<435::AID-NME891>3.0.CO;2-V
Santos A, Makinouchi A (1995) Contact strategies to deal with different tool descriptions in static explicit FEM for 3-D sheet-metal forming simulation. J Mater Process Technol 50:277–291. doi:10.1016/0924-0136(94)01391-D
Hachani M, Fourment L (2013) A smoothing procedure based on quasi-C1 interpolation for 3D contact mechanics with applications to metal forming. Comput Struct 128:1–13. doi:10.1016/j.compstruc.2013.05.008
Hama T, Nagata T, Teodosiu C et al (2008) Finite-element simulation of springback in sheet metal forming using local interpolation for tool surfaces. Int J Mech Sci 50:175–192. doi:10.1016/j.ijmecsci.2007.07.005
Shim H, Suh E (2000) Contact treatment algorithm for the trimmed NURBS surface. J Mater Process Technol 104:200–206. doi:10.1016/S0924-0136(00)00555-0
Landon RL, Hast MW, Piazza SJ (2009) Robust contact modeling using trimmed NURBS surfaces for dynamic simulations of articular contact. Comput Methods Appl Mech Eng 198:2339–2346. doi:10.1016/j.cma.2009.02.022
Wang SP, Nakamachi E (1997) The inside-outside contact search algorithm for finite element analysis. Int J Numer Methods Eng 40:3665–3685. doi:10.1002/(SICI)1097-0207(19971015)40:19<3665::AID-NME234>3.0.CO;2-K
Farouki RT (1999) Closing the gap between cad model and downstream application. SIAM News 32:303–319
Zhu X-F, Hu P, Ma Z-D et al (2013) A new surface parameterization method based on one-step inverse forming for isogeometric analysis-suited geometry. Int J Adv Manuf Technol 65:1215–1227. doi:10.1007/s00170-012-4251-8
Chamoret D, Saillard P, Rassineux A, Bergheau J-M (2004) New smoothing procedures in contact mechanics. J Comput Appl Math 168:107–116. doi:10.1016/j.cam.2003.06.007
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. doi:10.1002/nme.568
Francavilla A, Zienkiewicz OC (1975) A note on numerical computation of elastic contact problems. Int J Numer Methods Eng 9:913–924. doi:10.1002/nme.1620090410
Jin S, Sohn D, Lim JH, Im S (2015) A node-to-node scheme with the aid of variable-node elements for elasto-plastic contact analysis. Int J Numer Methods Eng 102:1761–1783. doi:10.1002/nme.4862
Simo JC, Wriggers P, Taylor RL (1985) A perturbed Lagrangian formulation for the finite element solution of contact problems. Comput Methods Appl Mech Eng 50:163–180. doi:10.1016/0045-7825(85)90088-X
Sauer RA, De Lorenzis L (2015) An unbiased computational contact formulation for 3D friction. Int J Numer Methods Eng 101:251–280. doi:10.1002/nme.4794
Zavarise G, De Lorenzis L (2009) A modified node-to-segment algorithm passing the contact patch test. Int J Numer Methods Eng 79:379–416. doi:10.1002/nme.2559
Puso MA, Laursen TA (2004) A mortar segment-to-segment frictional contact method for large deformations. Comput Methods Appl Mech Eng 193:4891–4913. doi:10.1016/j.cma.2004.06.001
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. doi:10.1016/S0045-7949(01)00048-7
Zavarise G, Wriggers P (1998) A segment-to-segment contact strategy. Math Comput Model 28:497–515. doi:10.1016/S0895-7177(98)00138-1
Bernardi C, Debit N, Maday Y (1990) Coupling finite element and spectral methods: first results. Math Comput 54:21–39. doi:10.1090/S0025-5718-1990-0995205-7
Wohlmuth BI (2001) Discretization methods and iterative solvers based on domain decomposition. doi:10.1007/978-3-642-56767-4
Belgacem FB, Hild P, Laborde P (1998) The mortar finite element method for contact problems. Math Comput Model 28:263–271. doi:10.1016/S0895-7177(98)00121-6
McDevitt TW, Laursen TA (2000) A mortar-finite element formulation for frictional contact problems. Int J Numer Methods Eng 48:1525–1547. doi:10.1002/1097-0207(20000810)48:10<1525::AID-NME953>3.0.CO;2-Y
Puso MA, Laursen TA (2004) A mortar segment-to-segment contact method for large deformation solid mechanics. Comput Methods Appl Mech Eng 193:601–629. doi:10.1016/j.cma.2003.10.010
Yang B, Laursen TA (2008) A large deformation mortar formulation of self contact with finite sliding. Comput Methods Appl Mech Eng 197:756–772. doi:10.1016/j.cma.2007.09.004
Puso MA, Laursen TA, Solberg J (2008) A segment-to-segment mortar contact method for quadratic elements and large deformations. Comput Methods Appl Mech Eng 197:555–566. doi:10.1016/j.cma.2007.08.009
Laursen TA, Puso MA, Sanders J (2012) Mortar contact formulations for deformable–deformable contact: past contributions and new extensions for enriched and embedded interface formulations. Comput Methods Appl Mech Eng 205–208:3–15. doi:10.1016/j.cma.2010.09.006
Farah P, Popp A, Wall WA (2014) Segment-based vs. element-based integration for mortar methods in computational contact mechanics. Comput Mech 55:209–228. doi:10.1007/s00466-014-1093-2
Wohlmuth BI (2000) A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J Numer Anal 38:989–1012. doi:10.1137/S0036142999350929
Christensen PW, Klarbring A, Pang JS, Strömberg N (1998) Formulation and comparison of algorithms for frictional contact problems. Int J Numer Methods Eng 42:145–173. doi:10.1002/(SICI)1097-0207(19980515)42::1<145:AID-NME358>3.0.CO;2-L
Batailly A, Magnain B, Chevaugeon N (2012) A comparative study between two smoothing strategies for the simulation of contact with large sliding. Comput Mech 51:581–601. doi:10.1007/s00466-012-0737-3
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. doi:10.1002/1097-0207(20010210)50:4<953::AID-NME64>3.0.CO;2-P
Stadler M, Holzapfel GA (2004) Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D. Int J Numer Methods Eng 60:1161–1195. doi:10.1002/nme.1001
Qian X, Yuan H, Zhou M, Zhang B (2014) A general 3D contact smoothing method based on radial point interpolation. J Comput Appl Math 257:1–13. doi:10.1016/j.cam.2013.08.014
Farin G (2002) Curves and surfaces for CAGD. Curves Surf CAGD. doi:10.1016/B978-1-55860-737-8.50030-2
Piegl L, Tiller W (1997) The NURBS book. doi:10.1007/978-3-642-59223-2
Hughes TJR, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194:4135–4195. doi:10.1016/j.cma.2004.10.008
De Lorenzis L, Wriggers P, Hughes TJR (2014) Isogeometric contact: a review. GAMM Mitt 37:85–123. doi:10.1002/gamm.201410005
Piegl L (1991) On NURBS: a survey. IEEE Comput Graph Appl 11:55–71. doi:10.1109/38.67702
Cox MG (1972) The numerical evaluation of B-splines. IMA J Appl Math 10:134–149. doi:10.1093/imamat/10.2.134
De Boor C (1972) On calculating with B-splines. J Approx Theory 6:50–62. doi:10.1016/0021-9045(72)90080-9
Nagata T (2005) Simple local interpolation of surfaces using normal vectors. Comput Aided Geom Des 22:327–347. doi:10.1016/j.cagd.2005.01.004
Neto DM, Oliveira MC, Menezes LF, Alves JL (2014) Applying Nagata patches to smooth discretized surfaces used in 3D frictional contact problems. Comput Methods Appl Mech Eng 271:296–320. doi:10.1016/j.cma.2013.12.008
Sekine T, Obikawa T (2010) Normal-unit-vector-based tool path generation using a modified local interpolation for ball-end milling. J Adv Mech Des Syst Manuf 4:1246–1260. doi:10.1299/jamdsm.4.1246
Boschiroli M, Fünfzig C, Romani L, Albrecht G (2011) A comparison of local parametric C0 Bézier interpolants for triangular meshes. Comput Graph 35:20–34. doi:10.1016/j.cag.2010.09.011
Neto DM, Oliveira MC, Menezes LF, Alves JL (2013) Improving Nagata patch interpolation applied for tool surface description in sheet metal forming simulation. Comput Aided Des 45:639–656. doi:10.1016/j.cad.2012.10.046
Neto DM, Oliveira MC, Menezes LF, Alves JL (2013) Nagata patch interpolation using surface normal vectors evaluated from the IGES file. Finite Elem Anal Des 72:35–46. doi:10.1016/j.finel.2013.03.004
IGES (1996) Initial graphics exchange specification, IGES 5.3. IGES/PDES Organization
Todd PH, McLeod RJY (1986) Numerical estimation of the curvature of surfaces. Comput Des 18:33–37. doi:10.1016/S0010-4485(86)80008-2
Meek DS, Walton DJ (2000) On surface normal and Gaussian curvature approximations given data sampled from a smooth surface. Comput Aided Geom Des 17:521–543. doi:10.1016/S0167-8396(00)00006-6
OuYang D, Feng H-Y (2005) On the normal vector estimation for point cloud data from smooth surfaces. Comput Des 37:1071–1079. doi:10.1016/j.cad.2004.11.005
Page DL, Sun Y, Koschan AF et al (2002) Normal vector voting: crease detection and curvature estimation on large, noisy meshes. Graph Models 64:199–229. doi:10.1006/gmod.2002.0574
Jin S, Lewis RR, West D (2005) A comparison of algorithms for vertex normal computation. Vis Comput 21:71–82. doi:10.1007/s00371-004-0271-1
Ubach P-A, Estruch C, Garcia-Espinosa J (2013) On the interpolation of normal vectors for triangle meshes. Int J Numer Methods Eng 96:247–268. doi:10.1002/nme.4567
Neto DM, Oliveira MC, Menezes LF, Alves JL (2016) A contact smoothing method for arbitrary surface meshes using Nagata patches. Comput Methods Appl Mech Eng. doi:10.1016/j.cma.2015.11.011
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. doi:10.1016/S0890-6955(00)00100-0
Kamran K, Rossi R, Oñate E (2012) A contact algorithm for shell problems via Delaunay-based meshing of the contact domain. Comput Mech 52:1–16. doi:10.1007/s00466-012-0791-x
Yang B, Laursen TA (2008) A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations. Comput Mech 41:189–205. doi:10.1007/s00466-006-0116-z
Aragón AM, Yastrebov VA, Molinari J-F (2013) A constrained-optimization methodology for the detection phase in contact mechanics simulations. Int J Numer Methods Eng 96:323–338. doi:10.1002/nme.4561
Areias PMA, César de Sá JMA, Conceição António CA (2004) Algorithms for the analysis of 3D finite strain contact problems. Int J Numer Methods Eng 61:1107–1151. doi:10.1002/nme.1104
Zhi-Hua Z, Nilsson L (1989) A contact searching algorithm for general contact problems. Comput Struct 33:197–209. doi:10.1016/0045-7949(89)90141-7
Benson DJ, Hallquist JO (1990) A single surface contact algorithm for the post-buckling analysis of shell structures. Comput Methods Appl Mech Eng 78:141–163. doi:10.1016/0045-7825(90)90098-7
Oldenburg M, Nilsson L (1994) The position code algorithm for contact searching. Int J Numer Methods Eng 37:359–386. doi:10.1002/nme.1620370302
Fujun W, Jiangang C, Zhenhan Y (2000) A contact searching algorithm for contact-impact problems. Acta Mech Sin 16:374–382. doi:10.1007/BF02487690
Konyukhov A, Schweizerhof K (2008) On the solvability of closest point projection procedures in contact analysis: analysis and solution strategy for surfaces of arbitrary geometry. Comput Methods Appl Mech Eng 197:3045–3056. doi:10.1016/j.cma.2008.02.009
Belytschko T, Neal MO (1991) Contact-impact by the pinball algorithm with penalty and Lagrangian methods. Int J Numer Methods Eng 31:547–572. doi:10.1002/nme.1620310309
Konyukhov A, Schweizerhof K (2005) Covariant description for frictional contact problems. Comput Mech 35:190–213. doi:10.1007/s00466-004-0616-7
Parisch H (1989) A consistent tangent stiffness matrix for three-dimensional non-linear contact analysis. Int J Numer Methods Eng 28:1803–1812. doi:10.1002/nme.1620280807
Klarbring A, Bjöourkman G (1992) Solution of large displacement contact problems with friction using Newton’s method for generalized equations. Int J Numer Methods Eng 34:249–269. doi:10.1002/nme.1620340116
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. doi:10.1002/nme.1620382103
Renard Y (2013) Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity. Comput Methods Appl Mech Eng 256:38–55. doi:10.1016/j.cma.2012.12.008
Alart P, Heege A (1995) Consistent tangent matrices of curved contact operators involving anisotropic friction. Rev Eur des Éléments Finis 4:183–207. doi:10.1080/12506559.1995.10511173
Menezes LF, Teodosiu C (2000) Three-dimensional numerical simulation of the deep-drawing process using solid finite elements. J Mater Process Technol 97:100–106. doi:10.1016/S0924-0136(99)00345-3
Oliveira MC, Alves JL, Menezes LF (2008) Algorithms and strategies for treatment of large deformation frictional contact in the numerical simulation of deep drawing process. Arch Comput Methods Eng 15:113–162. doi:10.1007/s11831-008-9018-x
Oliveira MC, Alves JL, Chaparro B, Menezes LF (2007) Study on the influence of work-hardening modeling in springback prediction. Int J Plast 23:516–543. doi:10.1016/j.ijplas.2006.07.003
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. doi:10.1016/0020-7403(68)90001-5
Tur M, Fuenmayor FJ, Wriggers P (2009) A mortar-based frictional contact formulation for large deformations using Lagrange multipliers. Comput Methods Appl Mech Eng 198:2860–2873. doi:10.1016/j.cma.2009.04.007
NUMISHEET’93 (1993) Proceedings of the 2nd international conference numerical simulation of 3-D sheet metal forming processes
Neto DM, Oliveira MC, Alves JL, Menezes LF (2015) Comparing faceted and smoothed tool surface descriptions in sheet metal forming simulation. Int J Mater Form 8:549–565. doi:10.1007/s12289-014-1177-8
Oliveira MC, Menezes LF (2004) Automatic correction of the time step in implicit simulations of the stamping process. Finite Elem Anal Des 40:1995–2010. doi:10.1016/j.finel.2004.01.009
Klang M (1979) On interior contact under friction between cylindrical elastic bodies. Linköping University, Linköping
Hammer ME (2012) Frictional mortar contact for finite deformation problems with synthetic contact kinematics. Comput Mech 51:975–998. doi:10.1007/s00466-012-0780-0
Acknowledgments
The authors gratefully acknowledge the financial support of the Portuguese Foundation for Science and Technology (FCT) via the project PTDC/EMS-TEC/1805/2012. The first author is also grateful to the FCT for the postdoctoral Grant SFRH/BPD/101334/2014.
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Neto, D.M., Oliveira, M.C. & Menezes, L.F. Surface Smoothing Procedures in Computational Contact Mechanics. Arch Computat Methods Eng 24, 37–87 (2017). https://doi.org/10.1007/s11831-015-9159-7
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DOI: https://doi.org/10.1007/s11831-015-9159-7