Finite deformation contact based on a 3D dual mortar and semi-smooth Newton approach. (English) Zbl 1273.74544
Zavarise, Giorgio (ed.) et al., Trends in computational contact mechanics. Papers based on the presentations at the 1st international conference on computational contact mechanics, Lecce, Italy, September 16–18, 2009. Berlin: Springer (ISBN 978-3-642-22166-8/hbk; 978-3-642-22167-5/ebook). Lecture Notes in Applied and Computational Mechanics 58, 57-77 (2011).
Summary: This paper gives a review of the recently proposed dual mortar approach combined with a consistently linearized semi-smooth Newton scheme for 3D finite deformation contact analysis. Some implementation aspects are presented in detail and the most important extensions of the contact model including friction and the treatment of self contact are highlighted. The mortar finite element method, which is applied as discretization scheme, initially yields a mixed formulation with the nodal Lagrange multiplier degrees of freedom as additional primary unknowns. However, by using so-called dual shape functions for Lagrange multiplier interpolation, the global linear system of equations to be solved within each Newton step can be condensed and thus contains only displacement degrees of freedom. All types of nonlinearities, including finite deformations, nonlinear material behavior and contact itself (active set search) are handled within one single iterative solution scheme based on a consistently linearized semi-smooth Newton method. Some very demanding numerical examples are presented to show the high quality of results obtained with this approach as well as to illustrate its superior numerical efficiency and robustness.
For the entire collection see [Zbl 1225.74006].
For the entire collection see [Zbl 1225.74006].
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74M15 | Contact in solid mechanics |
References:
| [1] | Bernardi, C.; Maday, Y.; Patera, A. T.; Brezis, H.; Lions, J. L., A new nonconforming approach to domain decomposition: the mortar element method, Nonlinear Partial Differential Equations and Their Applications, 13-51 (1994), London, New Yark: Pitman/Wiley, London, New Yark · Zbl 0797.65094 |
| [2] | Christensen, P. W.; Klarbring, A.; Pang, J. S.; Strömberg, N., Formulation and comparison of algorithms for frictional contact problems, International Journal for Numerical Methods in Engineering, 42, 1, 145-173 (1998) · Zbl 0917.73063 · doi:10.1002/(SICI)1097-0207(19980515)42:1<145::AID-NME358>3.0.CO;2-L |
| [3] | Chung, J.; Hulbert, G. M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-alpha method, Journal of Applied Mechanics, 60, 371-375 (1993) · Zbl 0775.73337 · doi:10.1115/1.2900803 |
| [4] | Fischer, K. A.; Wriggers, P., Frictionless 2D contact formulations for finite deformations based on the mortar method, Computational Mechanics, 36, 3, 226-244 (2005) · Zbl 1102.74033 · doi:10.1007/s00466-005-0660-y |
| [5] | Franke, D.; Düster, A.; Nübel, V.; Rank, E., A comparison of the h-, p-, hp-, and rp-version of the FEM for the solution of the 2D Hertzian contact problem, Computational Mechanics, 45, 5, 513-522 (2010) · Zbl 1398.74330 · doi:10.1007/s00466-009-0464-6 |
| [6] | Gitterle, M.; Popp, A.; Gee, M. W.; Wall, W. A., Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization, International Journal for Numerical Methods in Engineering, 84, 5, 543-571 (2010) · Zbl 1202.74121 |
| [7] | Hartmann, S.: Kontaktanalyse dünnwandiger Strukturen bei grossen Deformationen. PhD Thesis, Institut für Baustatik und Baudynamik, Universität Stuttgart (2007) |
| [8] | Hartmann, S.; Brunssen, S.; Ramm, E.; Wohlmuth, B., Unilateral non-linear dynamic contact of thin-walled structures using a primal-dual active set strategy, International Journal for Numerical Methods in Engineering, 70, 8, 883-912 (2007) · Zbl 1194.74218 · doi:10.1002/nme.1894 |
| [9] | Hesch, C.; Betsch, P., A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems, International Journal for Numerical Methods in Engineering, 77, 10, 1468-1500 (2009) · Zbl 1156.74378 · doi:10.1002/nme.2466 |
| [10] | Hintermüller, M.; Ito, K.; Kunisch, K., The primal-dual active set strategy as a semismooth Newton method, SIAM Journal on Optimization, 13, 3, 865-888 (2002) · Zbl 1080.90074 · doi:10.1137/S1052623401383558 |
| [11] | Holzapfel, F., Nonlinear Solid Mechanics - A Continuum Approach for Engineering (2000), Chichester: John Wiley & Sons, Chichester · Zbl 0980.74001 |
| [12] | Hüeber, S.; Stadler, G.; Wohlmuth, B. I., A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction, SIAM Journal on Scientific Computing, 30, 2, 572-596 (2008) · Zbl 1158.74045 · doi:10.1137/060671061 |
| [13] | Hüeber, S.; Wohlmuth, B. I., A primal-dual active set strategy for non-linear multibody contact problems, Computer Methods in Applied Mechanics and Engineering, 194, 27-29, 3147-3166 (2005) · Zbl 1093.74056 · doi:10.1016/j.cma.2004.08.006 |
| [14] | Lamichhane, B. P.; Stevenson, R. P.; Wohlmuth, B. I., Higher order mortar finite element methods in 3D with dual Lagrange multiplier bases, Numerische Mathematik, 102, 1, 93-121 (2005) · Zbl 1082.65120 · doi:10.1007/s00211-005-0636-z |
| [15] | Laursen, T. A., Computational Contact and Impact Mechanics (2002), Heidelberg: Springer-Verlag, Heidelberg · Zbl 0996.74003 |
| [16] | Mayer, U.; Popp, A.; Gerstenberger, A.; Wall, W., 3D fluid-structure-contact interaction based on a combined XFEM FSI and dual mortar contact approach, Computational Mechanics, 46, 1, 53-67 (2010) · Zbl 1301.74018 · doi:10.1007/s00466-010-0486-0 |
| [17] | Papadopoulos, P.; Taylor, R. L., A mixed formulation for the finite element solution of contact problems, Computer Methods in Applied Mechanics and Engineering, 94, 3, 373-389 (1992) · Zbl 0743.73029 · doi:10.1016/0045-7825(92)90061-N |
| [18] | Popp, A.; Gee, M. W.; Wall, W. A., A finite deformation mortar contact formulation using a primal-dual active set strategy, International Journal for Numerical Methods in Engineering, 79, 11, 1354-1391 (2009) · Zbl 1176.74133 · doi:10.1002/nme.2614 |
| [19] | Popp, A.; Gitterle, M.; Gee, M. W.; Wall, W. A., A dual mortar approach for 3D finite deformation contact with consistent linearization, International Journal for Numerical Methods in Engineering, 83, 11, 1428-1465 (2010) · Zbl 1202.74183 · doi:10.1002/nme.2866 |
| [20] | Puso, M. A., A 3D mortar method for solid mechanics, International Journal for Numerical Methods in Engineering, 59, 3, 315-336 (2004) · Zbl 1047.74065 · doi:10.1002/nme.865 |
| [21] | Puso, M. A.; Laursen, T. A.; Solberg, J., A segment-to-segment mortar contact method for quadratic elements and large deformations, Computer Methods in Applied Mechanics and Engineering, 197, 6-8, 555-566 (2008) · Zbl 1169.74627 · doi:10.1016/j.cma.2007.08.009 |
| [22] | Puso, M. A.; Laursen, T. A., A mortar segment-to-segment contact method for large deformation solid mechanics, Computer Methods in Applied Mechanics and Engineering, 193, 6-8, 601-629 (2004) · Zbl 1060.74636 · doi:10.1016/j.cma.2003.10.010 |
| [23] | Puso, M. A.; Laursen, T. A., A mortar segment-to-segment frictional contact method for large deformations, Computer Methods in Applied Mechanics and Engineering, 193, 45-47, 4891-4913 (2004) · Zbl 1112.74535 · doi:10.1016/j.cma.2004.06.001 |
| [24] | Qi, L.; Sun, J., A nonsmooth version of Newton’s method, Mathematical Programming, 58, 1, 353-367 (1993) · Zbl 0780.90090 · doi:10.1007/BF01581275 |
| [25] | Timoshenko, S. P.; Goodier, J. N., Theory of Elasticity (1970), New York: McGraw-Hill, New York · Zbl 0266.73008 |
| [26] | Wall, W.A., Gee, M.W.: Baci - A multiphysics simulation environment. Technical Report, Technische Universität München (2009) |
| [27] | Wohlmuth, B. I., A mortar finite element method using dual spaces for the Lagrange multiplier, SIAM Journal on Numerical Analysis, 38, 3, 989-1012 (2000) · Zbl 0974.65105 · doi:10.1137/S0036142999350929 |
| [28] | Wohlmuth, B. I., Discretization Methods and Iterative Solvers Based on Domain Decomposition (2001), Berlin: Springer, Berlin · Zbl 0966.65097 · doi:10.1007/978-3-642-56767-4 |
| [29] | Wriggers, P., Computational Contact Mechanics (2002), Chichester: John Wiley & Sons, Chichester |
| [30] | Yang, B.; Laursen, T. A.; Meng, X., Two dimensional mortar contact methods for large deformation frictional sliding, International Journal for Numerical Methods in Engineering, 62, 9, 1183-1225 (2005) · Zbl 1161.74497 · doi:10.1002/nme.1222 |
| [31] | Yang, B.; Laursen, T., A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations, Computational Mechanics, 41, 2, 189-205 (2008) · Zbl 1162.74481 · doi:10.1007/s00466-006-0116-z |
| [32] | Yang, B.; Laursen, T. A., A large deformation mortar formulation of self contact with finite sliding, Computer Methods in Applied Mechanics and Engineering, 197, 6-8, 756-772 (2008) · Zbl 1169.74513 · doi:10.1016/j.cma.2007.09.004 |
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