Method of solution composition in contact problems with friction of deformable bodies

A. S. Karavaev   ; S. P. Kopysov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 4, pp. 659-674

Résumé

A new iterative method for solving static contact problems of two deformable bodies is proposed. The method is based on alternately solving the unilateral contact problem for the first body and the linear elasticity problem with natural boundary conditions for the second body. Fulfillment of Coulomb's friction law involves correction of tangential nodal forces in the sliding area and setting kinematic boundary conditions in the sticking area for the contact boundary of the first body. The goal of solving the linear elasticity problem for the second body is to gradually equalize contact loads on the interacting surfaces. The advantages of the method are demonstrated by solving a number of model examples, including unilateral contact of a linear-elastic plate with a solid foundation, bilateral contact of pressing a deformable block into the foundation, the Hertz problem of contact of two deformable cylinders etc. The method can solve problems on flat and curvilinear contact boundaries.

Détail

Keywords: contact problem, finite element method
Mots-clés : Coulomb's friction law

Bibliographie

[1] Bonari J., Paggi M., Dini D., “A new finite element paradigm to solve contact problems with roughness”, International Journal of Solids and Structures, 253 (2022), 111643 | DOI

[2] Zimmerman B.K., Ateshian G.A., “A surface-to-surface finite element algorithm for large deformation frictional contact in FEBIO”, Journal of Biomechanical Engineering, 140:8 (2018), 081013 | DOI

[3] Wang Dongze, de Boer G., Neville A., Ghanbarzadeh A., “A review on modelling of viscoelastic contact problems”, Lubricants, 10:12 (2022), 358 | DOI

[4] Laursen T.A., Simo J.C., “A continuum-based finite element formulation for the implicit solution of multibody, large deformation–frictional contact problems”, International Journal for Numerical Methods in Engineering, 36:20 (1993), 3451–3485 | DOI | MR | Zbl

[5] Antoni N., “A further analysis on the analogy between friction and plasticity in Solid Mechanics”, International Journal of Engineering Science, 121 (2017), 34–51 | DOI | MR | Zbl

[6] Liu Dongyu, van den Boom S.J., Simone A., Aragón A.M., “An interface-enriched generalized finite element formulation for locking-free coupling of non-conforming discretizations and contact”, Computational Mechanics, 70:3 (2022), 477–499 | DOI | MR | Zbl

[7] Vodička R., Mantič V., Roubíček T., “Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM”, Journal of Computational and Applied Mathematics, 315 (2017), 249–272 | DOI | MR | Zbl

[8] Nakamura K., Matsumura S., Mizutani T., “Particle-to-surface frictional contact algorithm for material point method using weighted least squares”, Computers and Geotechnics, 134 (2021), 104069 | DOI

[9] Benkhira El-H., Essoufi El-H., Fakhar R., “On convergence of the penalty method for a static unilateral contact problem with nonlocal friction in electro-elasticity”, European Journal of Applied Mathematics, 27:1 (2016), 1–22 | DOI | MR | Zbl

[10] Burman E., Hansbo P., Larson M.G., “The augmented Lagrangian method as a framework for stabilised methods in computational mechanics”, Archives of Computational Methods in Engineering, 30:4 (2023), 2579–2604 | DOI | MR

[11] Negrut D., Serban R., Tasora A., “Posing multibody dynamics with friction and contact as a differential complementarity problem”, Journal of Computational and Nonlinear Dynamics, 13:1 (2018), 014503 | DOI

[12] Zhao Yidong, Choo Jinhyun, Jiang Yupeng, Li Minchen, Jiang Chenfanfu, Soga Kenichi, “A barrier method for frictional contact on embedded interfaces”, Computer Methods in Applied Mechanics and Engineering, 393 (2022), 114820 | DOI | MR | Zbl

[13] Bayada G., Sabil J., Sassi T., “A Neumann–Neumann domain decomposition algorithm for the Signorini problem”, Applied Mathematics Letters, 17:10 (2004), 1153–1159 | DOI | MR | Zbl

[14] Karavaev A.S., Kopysov S.P., “Space semidiscrete formulation of contact algorithm based on the Schwarz’s decomposition method for deformable bodies”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 27:3 (2017), 396–413 (in Russian) | DOI | MR | Zbl

[15] Lebedev V.I., Functional analysis and computational mathematics, Fizmatlit, Moscow, 2005 | MR

[16] Stankevich I.V., “Mathematical modeling of contact problems of elasticity theory with continuous unilateral contact”, Mathematics and Mathematical Modeling, 2015, no. 5, 83–96 (in Russian) | DOI

[17] Oden J.T., Pires E.B., “Algorithms and numerical results for finite element approximations of contact problems with non-classical friction laws”, Computers and Structures, 19:1–2 (1984), 137–147 | DOI

[18] Feng Z.-Q., “Some test examples of 2D and 3D contact problems involving Coulomb friction and large slip”, Mathematical and Computer Modelling, 28:4–8 (1998), 469–477 | DOI

[19] Olukoko O.A., Becker A.A., Fenner R.T., “Three benchmark examples for frictional contact modelling using finite element and boundary elements methods”, The Journal of Strain Analysis for Engineering Design, 28:4 (1993), 293–301 | DOI

[20] Lee Seok-Soon, “A computational method for frictional contact problem using finite element method”, International Journal for Numerical Methods in Engineering, 37:2 (1994), 217–228 | DOI | MR | Zbl

[21] 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 (2010), 543–571 | DOI | MR | Zbl

[22] Akula B.R., Vignollet J., Yastrebov V.A., “MorteX method for contact along real and embedded surfaces: coupling X-FEM with the mortar method”, 2019, arXiv: 1902.04000v1 [cs.CE] | DOI