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A node-to-node scheme for three-dimensional contact problems using the scaled boundary finite element method. (English) Zbl 1440.74251
Summary: A node-to-node (NTN) scheme for modeling three-dimensional contact problems within a scaled boundary finite element method (SBFEM) framework is proposed. Polyhedral elements with an arbitrary number of faces and nodes are constructed using the SBFEM. Only the boundary of the polyhedral element is discretized to accommodate a higher degree of flexibility in mesh transitioning. Nonmatching meshes can be simply converted into matching ones by appropriate node insertions, thereby allowing the use of a favorable NTN contact scheme. The general three-dimensional frictional contact is explicitly formulated as a mixed complementarity problem (MCP). The inherent nonlinearity in the three-dimensional friction condition is treated accurately without requiring piecewise linearization. Contact constraints for non-penetration and stick-slide are enforced directly in a complementarity format. Numerical examples with 1st and 2nd order elements demonstrate the accuracy and robustness of the proposed scheme.

##### MSC:
 74M15 Contact in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
FEAPpv; MCPLIB
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##### References:
 [1] Lin, T.; Ou, H.; Li, R., A finite element method for 3D static and dynamic contact/impact analysis of gear drives, Comput. Methods Appl. Mech. Engrg., 196, 1716-1728 (2007) · Zbl 1173.74433 [2] Steif, P. S., Crack extension under compressive loading, Eng. Fract. Mech., 20, 463-473 (1984) [3] Jardine, R. J.; Potts, D.; Fourie, A.; Burland, J., Studies of the influence of non-linear stress-strain characteristics in soil-structure interaction, Geotechnique, 36, 377-396 (1986) [4] Ribeaucourt, R.; Baietto-Dubourg, M.-C.; Gravouil, A., A new fatigue frictional contact crack propagation model with the coupledX-FEM/LATIN method, Comput. Methods Appl. Mech. Engrg., 196, 3230-3247 (2007) · Zbl 1173.74385 [5] Baillet, L.; Sassi, T., Finite element method with lagrange multipliers for contact problems with friction, C. R. Math., 334, 917-922 (2002) · Zbl 1073.74047 [6] Fischer, K.; Wriggers, P., Frictionless 2D contact formulations for finite deformations based on the mortar method, Comput. Mech., 36, 226-244 (2005) · Zbl 1102.74033 [7] Kikuchi, N.; Song, Y. J., Penalty/finite-element approximations of a class of unilateral problems in linear elasticity, Quart. Appl. Math., 39, 1-22 (1981) · Zbl 0457.73097 [8] Zavarise, G.; Wriggers, P., A segment-to-segment contact strategy, Math. Comput. Modelling, 28, 497-515 (1998) · Zbl 1002.74564 [9] Zavarise, G.; De Lorenzis, L., The node-to-segment algorithm for 2D frictionless contact: classical formulation and special cases, Comput. Methods Appl. Mech. Engrg., 198, 3428-3451 (2009) · Zbl 1230.74237 [10] Kwak, B. M., Numerical implementation of three-dimensional frictional contact by a linear complementarity problem, KSME J., 4, 23 (1990) [11] Wriggers, P., Computational Contact Mechanics (2006), Springer · Zbl 1104.74002 [12] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method for Solid and Structural Mechanics (2005), Elsevier · Zbl 1084.74001 [13] Belytschko, T.; Liu, W. K.; Moran, B.; Elkhodary, K., Nonlinear Finite Elements for Continua and Structures (2013), John Wiley & Sons · Zbl 1279.74002 [14] Simo, J. C.; Wriggers, P.; Taylor, R. L., A perturbed lagrangian formulation for the finite element solution of contact problems, Comput. Methods Appl. Mech. Engrg., 50, 163-180 (1985) · Zbl 0552.73097 [15] Arrow, K. J.; Solow, R. M., Gradient methods for constrained maxima, with weakened assumptions, Stud. Linear Nonlinear Programm., 166-176 (1958) [16] Tin-Loi, F.; Xia, S., Nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem, Comput. Methods Appl. Mech. Engrg., 190, 4551-4568 (2001) · Zbl 1059.74063 [17] Tin-Loi, F.; Xia, S., An iterative complementarity approach for elastoplastic analysis involving frictional contact, Int. J. Mech. Sci., 45, 197-216 (2003) · Zbl 1032.74686 [18] Dirkse, S. P.; Ferris, M. C., MCPLIB: a collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5, 319-345 (1995) [19] Francavilla, A.; Zienkiewicz, O., A note on numerical computation of elastic contact problems, Internat. J. Numer. Methods Engrg., 9, 913-924 (1975) [20] Hughes, T. J.; Taylor, R. L.; Sackman, J. L.; Curnier, A.; Kanoknukulchai, W., A finite element method for a class of contact-impact problems, Comput. Methods Appl. Mech. Engrg., 8, 249-276 (1976) · Zbl 0367.73075 [21] Taylor, R. L.; Papadopoulos, P., On a patch test for contact problems in two dimensions, Comput. Methods Nonlinear Mech., 690-702 (1991) [22] Puso, M. A.; Laursen, T. A., A mortar segment-to-segment contact method for large deformation solid mechanics, Comput. Methods Appl. Mech. Engrg., 193, 601-629 (2004) · Zbl 1060.74636 [23] Bernardi, C.; Maday, Y.; Patera, A., A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and Their Applications, 13-51 (1994) · Zbl 0797.65094 [24] Wohlmuth, B. I., A mortar finite element method using dual spaces for the lagrange multiplier, SIAM J. Numer. Anal., 38, 989-1012 (2000) · Zbl 0974.65105 [25] El-Abbasi, N.; Bathe, K.-J., Stability and patch test performance of contact discretizations and a new solution algorithm, Comput. Struct., 79, 1473-1486 (2001) [26] Lamichhane, B. P.; Stevenson, R. P.; Wohlmuth, B. I., Higher order mortar finite element methods in 3D with dual lagrange multiplier bases, Numer. Math., 102, 93-121 (2005) · Zbl 1082.65120 [27] Popp, A., Mortar Methods for Computational Contact Mechanics and General Interface Problems (2012), Technische Universität München, (Ph.D thesis) [28] Lamichhane, B. P.; Wohlmuth, B. I., Higher order dual lagrange multiplier spaces for mortar finite element discretizations, Calcolo, 39, 219-237 (2002) · Zbl 1168.65414 [29] Nistor, I.; Guiton, M.; Massin, P.; Moës, N.; Geniaut, S., An X-FEM approach for large sliding contact along discontinuities, Internat. J. Numer. Methods Engrg., 78, 1407-1435 (2009) · Zbl 1183.74301 [30] Bitencourt Jr, L. A.; Manzoli, O. L.; Prazeres, P. G.; Rodrigues, E. A.; Bittencourt, T. N., A coupling technique for non-matching finite element meshes, Comput. Methods Appl. Mech. Engrg., 290, 19-44 (2015) [31] Kim, J. H.; Lim, J. H.; Lee, J. H.; Im, S., A new computational approach to contact mechanics using variable-node finite elements, Internat. J. Numer. Methods Engrg., 73, 1966-1988 (2008) · Zbl 1195.74180 [32] Sohn, D.; Jin, S., Polyhedral elements with strain smoothing for coupling hexahedral meshes at arbitrary nonmatching interfaces, Comput. Methods Appl. Mech. Engrg., 293, 92-113 (2015) [33] Liu, G.; Dai, K.; Nguyen, T. T., A smoothed finite element method for mechanics problems, Comput. Mech., 39, 859-877 (2007) · Zbl 1169.74047 [34] Jin, S.; Sohn, D.; Im, S., Node-to-node scheme for three-dimensional contact mechanics using polyhedral type variable-node elements, Comput. Methods Appl. Mech. Engrg., 304, 217-242 (2016) [35] Wriggers, P.; Rust, W.; Reddy, B., A virtual element method for contact, Comput. Mech., 58, 1039-1050 (2016) · Zbl 1398.74420 [36] Xing, W.; Song, C.; Tin-Loi, F., A scaled boundary finite element based node-to-node scheme for 2D frictional contact problems, Comput. Methods Appl. Mech. Engrg., 333, 114-146 (2018) [37] Ooi, E. T.; Song, C.; Tin-Loi, F.; Yang, Z., Polygon scaled boundary finite elements for crack propagation modelling, Internat. J. Numer. Methods Engrg., 91, 319-342 (2012) · Zbl 1246.74062 [38] Song, C., The Scaled Boundary Finite Element Method: Introduction to Theory and Implementation (2018), John Wiley & Sons [39] Ooi, E. T.; Song, C.; Tin-Loi, F., A scaled boundary polygon formulation for elasto-plastic analyses, Comput. Methods Appl. Mech. Engrg., 268, 905-937 (2014) · Zbl 1295.74102 [40] He, K.; Song, C.; Ooi, E. T., A novel scaled boundary finite element formulation with stabilization and its application to image-based elastoplastic analysis, Internat. J. Numer. Methods Engrg., 115, 956-985 (2018) [41] Behnke, R.; Mundil, M.; Birk, C.; Kaliske, M., A physically and geometrically nonlinear scaled-boundary-based finite element formulation for fracture in elastomers, Internat. J. Numer. Methods Engrg., 99, 966-999 (2014) · Zbl 1352.74278 [42] Xu, H.; Zou, D.; Kong, X.; Hu, Z.; Su, X., A nonlinear analysis of dynamic interactions of CFRD-compressible reservoir system based on FEM-SBFEM, Soil Dyn. Earthq. Eng., 112, 24-34 (2018) [43] Ooi, E.; Song, C.; Natarajan, S., A scaled boundary finite element formulation for poroelasticity, Internat. J. Numer. Methods Engrg., 114, 905-929 (2018) [44] Song, C.; Wolf, J. P., The scaled boundary finite-element method - alias consistent infinitesimal finite-element cell method - for elastodynamics, Comput. Methods Appl. Mech. Engrg., 147, 329-355 (1997) · Zbl 0897.73069 [45] Song, C., A matrix function solution for the scaled boundary finite-element equation in statics, Comput. Methods Appl. Mech. Engrg., 193, 2325-2356 (2004) · Zbl 1067.74586 [46] Wanxie, Z.; Suming, S., A finite element method for elasto-plastic structures and contact problems by parametric quadratic programming, Internat. J. Numer. Methods Engrg., 26, 2723-2738 (1988) · Zbl 0673.73056 [47] Klarbring, A., A mathematical programming approach to three-dimensional contact problems with friction, Comput. Methods Appl. Mech. Engrg., 58, 175-200 (1986) · Zbl 0595.73125 [48] Kwak, B. M., Complementarity problem formulation of three-dimensional frictional contact, J. Appl. Mech., 58, 134-140 (1991) · Zbl 0756.73086 [49] Dirkse, S. P.; Ferris, M. C., The path solver: a nommonotone stabilization scheme for mixed complementarity problems, Optim. Methods Softw., 5, 123-156 (1995) [51] Dirkse, S. P.; Ferris, M. C., Crash techniques for large-scale complementarity problems, Complementarity Variational Probl.: State Art, 92, 40 (1997) · Zbl 0886.90150 [52] Burke, J. V.; Moré, J. J., On the identification of active constraints, SIAM J. Numer. Anal., 25, 1197-1211 (1988) · Zbl 0662.65052 [53] Calamai, P. H.; Moré, J. J., Projected gradient methods for linearly constrained problems, Math. Program., 39, 93-116 (1987) · Zbl 0634.90064 [54] Torstenfelt, B., Contact problems with friction in general purpose finite element computer programs, Comput. Struct., 16, 487-493 (1983) · Zbl 0499.73055 [56] Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals (2005), Butterworth-Heinemann · Zbl 1307.74005 [57] Crisfield, M., Re-visiting the contact patch test, Internat. J. Numer. Methods Engrg., 48, 435-449 (2000) · Zbl 0969.74062
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