Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-30T23:42:58.407Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical analysis of hemivariational inequalities in contact mechanics

Published online by Cambridge University Press:  14 June 2019

Weimin Han  and
Mircea Sofonea
Weimin Han
Affiliation:
Program in Applied Mathematical and Computational Sciences (AMCS), and Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA E-mail: [email protected]
Mircea Sofonea
Affiliation:
Laboratoire de Mathématiques et Physique, Université de Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860 Perpignan, France E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Contact phenomena arise in a variety of industrial process and engineering applications. For this reason, contact mechanics has attracted substantial attention from research communities. Mathematical problems from contact mechanics have been studied extensively for over half a century. Effort was initially focused on variational inequality formulations, and in the past ten years considerable effort has been devoted to contact problems in the form of hemivariational inequalities. This article surveys recent development in studies of hemivariational inequalities arising in contact mechanics. We focus on contact problems with elastic and viscoelastic materials, in the framework of linearized strain theory, with a particular emphasis on their numerical analysis. We begin by introducing three representative mathematical models which describe the contact between a deformable body in contact with a foundation, in static, history-dependent and dynamic cases. In weak formulations, the models we consider lead to various forms of hemivariational inequalities in which the unknown is either the displacement or the velocity field. Based on these examples, we introduce and study three abstract hemivariational inequalities for which we present existence and uniqueness results, together with convergence analysis and error estimates for numerical solutions. The results on the abstract hemivariational inequalities are general and can be applied to the study of a variety of problems in contact mechanics; in particular, they are applied to the three representative mathematical models. We present numerical simulation results giving numerical evidence on the theoretically predicted optimal convergence order; we also provide mechanical interpretations of simulation results.

Type
Research Article
Information
Acta Numerica , Volume 28 , 01 May 2019 , pp. 175 - 286
Copyright
© Cambridge University Press, 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES 1

Ainsworth, M. and Oden, J. T. (2000), A Posteriori Error Estimation in Finite Element Analysis, Wiley.Google Scholar
Alsaedi, A., Brezzi, F., Marini, L. and Russo, A. (2013), ‘Equivalent projectors for virtual element methods’, Comput. Math. Appl. 66, 376–391.Google Scholar
Arnold, D. N., Brezzi, F., Cockburn, B. and Marini, L. D. (2002), ‘Unified analysis of discontinuous Galerkin methods for elliptic problems’, SIAM J. Numer. Anal. 39, 1749–1779.Google Scholar
Atkinson, K. and Han, W. (2009), Theoretical Numerical Analysis: A Functional Analysis Framework, third edition, Springer.Google Scholar
Babuška, I. and Rheinboldt, W. C. (1978a), ‘Error estimates for adaptive finite element computations’, SIAM J. Numer. Anal. 15, 736–754.Google Scholar
Babuška, I. and Rheinboldt, W. C. (1978b), ‘ A posteriori error estimates for the finite element method’, Intern. J. Numer. Methods Engrg 12, 1597–1615.Google Scholar
Babuška, I. and Strouboulis, T. (2001), The Finite Element Method and its Reliability, Oxford University Press.Google Scholar
Baiocchi, C. and Capelo, A. (1984), Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems, Wiley.Google Scholar
Barboteu, M., Bartosz, K., Han, W. and Janiczko, T. (2015), ‘Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact’, SIAM J. Numer. Anal. 53, 527–550.Google Scholar
Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D. and Russo, A. (2013), ‘Basic principles of virtual element methods’, Math. Models Methods Appl. Sci. 23, 1–16.Google Scholar
Ben Belgacem, F., Bernardi, C., Blouza, A. and Vohralík, M. (2012), ‘On the unilateral contact between membranes, 2: a posteriori analysis and numerical experiments’, IMA J. Numer. Anal. 32, 1147–1172.Google Scholar
Bostan, V. and Han, W. (2009), Adaptive finite element solution of variational inequalities with application in contact problems. In Advances in Applied Mathematics and Global Optimization (Gao, D. Y. and Sherali, H. D., eds), Springer, pp. 25–106.Google Scholar
Brenner, S. C. and Scott, L. R. (2008), The Mathematical Theory of Finite Element Methods, third edition, Springer.Google Scholar
Brézis, H. (1972), ‘Problèmes unilatéraux’, J. Math. Pures Appl. 51, 1–168.Google Scholar
Capatina, A. (2014), Variational Inequalities and Frictional Contact Problems, Vol. 31 of Advances in Mechanics and Mathematics, Springer.Google Scholar
Ciarlet, P. G. (1978), The Finite Element Method for Elliptic Problems, North-Holland.Google Scholar
Ciarlet, P. G. (1988), Mathematical Elasticity, I: Three Dimensional Elasticity, Vol. 20 of Studies in Mathematics and its Applications, North-Holland.Google Scholar
Clarke, F. H. (1975), ‘Generalized gradients and applications’, Trans. Amer. Math. Soc. 205, 247–262.Google Scholar
Clarke, F. H. (1983), Optimization and Nonsmooth Analysis, Wiley-Interscience.Google Scholar
Cockburn, B., Karniadakis, G. E. & Shu, C.-W., eds (2000), Discontinuous Galerkin Methods. Theory, Computation and Applications, Vol. 11 of Lecture Notes in Computational Science and Engineering, Springer.Google Scholar
Denkowski, Z., Migórski, S. and Papageorgiou, N. S. (2003a), An Introduction to Nonlinear Analysis: Theory, Kluwer Academic/Plenum.Google Scholar
Denkowski, Z., Migórski, S. and Papageorgiou, N. S. (2003b), An Introduction to Nonlinear Analysis: Applications, Kluwer Academic/Plenum.Google Scholar
Drouet, G. and Hild, P. (2015), ‘Optimal convergence for discrete variational inequalities modelling Signorini contact in 2D and 3D without additional assumptions on the unknown contact set’, SIAM J. Numer. Anal. 53, 1488–1507.Google Scholar
Drozdov, A. D. Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids, World Scientific.Google Scholar
Duvaut, G. and Lions, J.-L. (1976), Inequalities in Mechanics and Physics, Springer.Google Scholar
Eck, C., Jarušek, J. and Krbec, M. (2005), Unilateral Contact Problems: Variational Methods and Existence Theorems, Vol. 270 of Pure and Applied Mathematics, Chapman & Hall/CRC.Google Scholar
Ekeland, I. and Temam, R. (1976), Convex Analysis and Variational Problems, North-Holland.Google Scholar
Falk, R. S. (1974), ‘Error estimates for the approximation of a class of variational inequalities’, Math. Comp. 28, 963–971.Google Scholar
Feng, F., Han, W. and Huang, J. (2019), ‘Virtual element methods for elliptic variational inequalities of the second kind’, J. Sci. Comput. doi:10.1007/s10915-019-00929-y Google Scholar
Fichera, G. (1964), ‘Problemi elastostatici con vincoli unilaterali, II: Problema di Signorini con ambique condizioni al contorno’, Mem. Accas. Naz. Lincei, Ser. VIII, Vol. VII, Sez. I 5, 91–140.Google Scholar
Fichera, G. (1972), Boundary value problems of elasticity with unilateral constraints. In Linear Theories of Elasticity and Thermoelasticity (Truesdell, C., ed.), Springer, pp. 391–424.Google Scholar
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer.Google Scholar
Glowinski, R., Lions, J.-L. and Trémolières, R. (1981), Numerical Analysis of Variational Inequalities, North-Holland.Google Scholar
Gudi, T. and Porwal, K. (2014), ‘ A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems’, Math. Comp. 83, 579–602.Google Scholar
Gudi, T. and Porwal, K. (2016), ‘ A posteriori error estimates of discontinuous Galerkin methods for the Signorini problem’, J. Comput. Appl. Math. 292, 257–278.Google Scholar
Han, W. (2005), A Posteriori Error Analysis via Duality Theory, with Applications in Modeling and Numerical Approximations, Springer.Google Scholar
Han, W. (2018), ‘Numerical analysis of stationary variational–hemivariational inequalities with applications in contact mechanics’, Math. Mech. Solids 23, 279–293.Google Scholar
Han, W., Huang, Z., Wang, C. and Xu, W. (2019), ‘Numerical analysis of elliptic hemivariational inequalities for semipermeable media’, J. Comput. Math. 37, 543–560.Google Scholar
Han, W., Migórski, S. and Sofonea, M. (2014), ‘A class of variational–hemivariational inequalities with applications to frictional contact problems’, SIAM J. Math. Anal. 46, 3891–3912.Google Scholar
Han, W., Migórski, S. & Sofonea, M., eds (2015), Advances in Variational and Hemivariational Inequalities: Theory, Numerical Analysis, and Applications, Springer.Google Scholar
Han, W. and Reddy, B. D. (1999), ‘Convergence analysis of discrete approximations of problems in hardening plasticity’, Comput. Methods Appl. Mech. Engrg 171, 327–340.Google Scholar
Han, W. and Reddy, B. D. (2000), ‘Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity’, Numer. Math. 87, 283–315.Google Scholar
Han, W. and Reddy, B. D. (2013), Plasticity: Mathematical Theory and Numerical Analysis, second edition, Springer.Google Scholar
Han, W. and Sofonea, M. (2002), Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Vol. 30 of Studies in Advanced Mathematics, American Mathematical Society/International Press.Google Scholar
Han, W., Sofonea, M. and Barboteu, M. (2017), ‘Numerical analysis of elliptic hemivariational inequalities’, SIAM J. Numer. Anal. 55, 640–663.Google Scholar
Han, W., Sofonea, M. and Danan, D. (2018), ‘Numerical analysis of stationary variational–hemivariational inequalities’, Numer. Math. 139, 563–592.Google Scholar
Haslinger, J. and Hlaváček, I. (1980), ‘Contact between two elastic bodies, I: Continuous problems’, Applikace Math. 25, 324–347.Google Scholar
Haslinger, J. and Hlaváček, I. (1981a), ‘Contact between two elastic bodies, II: Finite element analysis’, Applikace Math. 26, 263–290.Google Scholar
Haslinger, J. and Hlaváček, I. (1981b), ‘Contact between two elastic bodies, III: Dual finite element analysis’, Applikace Math. 26, 321–344.Google Scholar
Haslinger, J., Hlaváček, I. and Nečas, J. (1996), Numerical methods for unilateral problems in solid mechanics. (Ciarlet, P. G. and Lions, J.-L., eds), Handbook of Numerical Analysis, Vol. IV, North-Holland, pp. 313–485.Google Scholar
Haslinger, J., Miettinen, M. and Panagiotopoulos, P. D. (1999), Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic.Google Scholar
Hertz, H. (1882), ‘Über die Berührung fester Elastischer Körper’, J. Math. (Crelle) 92.Google Scholar
Hild, P. and Lleras, V. (2009), ‘Residual error estimators for Coulomb friction’, SIAM J. Numer. Anal. 47, 3550–3583.Google Scholar
Hlaváček, I., Haslinger, J., Nečas, J. and Lovíšek, J. (1988), Solution of Variational Inequalities in Mechanics, Springer.Google Scholar
Hüeber, S. and Wohlmuth, B. (2005a), ‘A primal–dual active set strategy for non-linear multibody contact problems’, Comput. Meth. Appl. Mech. Engrg 194, 3147–3166.Google Scholar
Hüeber, S. and Wohlmuth, B. (2005b), ‘An optimal a priori estimate for non-linear multibody contact problems’, SIAM J. Numer. Anal. 43, 157–173.Google Scholar
Kikuchi, N. and Oden, J. T. (1988), Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM.Google Scholar
Kinderlehrer, D. and Stampacchia, G. (2000), An Introduction to Variational Inequalities and Their Applications, Vol. 31 of Classics in Applied Mathematics, SIAM.Google Scholar
Khludnev, A. M. and Sokolowski, J. (1997), Modelling and Control in Solid Mechanics, Birkhäuser.Google Scholar
Kornhuber, R. and Krause, R. (2001), ‘Adaptive multigrid methods for Signorini’s problem in linear elasticity’, Comput Visual. 4, 9–20.Google Scholar
Kurdila, A. J. and Zabarankin, M. (2005), Convex Functional Analysis, Birkhäuser.Google Scholar
Laursen, T. A. (2002), Computational Contact and Impact Mechanics, Springer.Google Scholar
Lions, J.-L. and Stampacchia, G. (1967), ‘Variational inequalities’, Comm. Pure Appl. Math. 20, 493–519.Google Scholar
Marcus, M. and Mizel, V. (1972), ‘Absolute continuity on tracks and mappings of Sobolev space’, Arch. Rat. Mech. Anal. 45, 294–302.Google Scholar
Matei, A., Sitzmann, S., Willner, K. and Wohlmuth, B. I. (2017), ‘A mixed variational formulation for a class of contact problems in viscoelasticity’, Appl. Anal. 97, 1340–1356.Google Scholar
Migórski, S., Ochal, A. and Sofonea, M. (2010), ‘Variational analysis of static frictional contact problems for electro-elastic materials’, Math. Nachr. 283, 1314–1335.Google Scholar
Migórski, S., Ochal, A. and Sofonea, M. (2013), Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Vol. 26 of Advances in Mechanics and Mathematics, Springer.Google Scholar
Migórski, S., Ochal, A. and Sofonea, M. (2017), ‘A class of variational–hemivariational inequalities in reflexive Banach spaces’, J. Elasticity 127, 151–178.Google Scholar
Naniewicz, Z. and Panagiotopoulos, P. D. (1995), Mathematical Theory of Hemivariational Inequalities and Applications, Dekker.Google Scholar
Nečas, J. and Hlaváček, I. (1981), Mathematical Theory of Elastic and Elastico-Plastic Bodies: An Introduction, Elsevier.Google Scholar
Oden, J. T. and Martins, J. A. C. (1985), ‘Models and computational methods for dynamic friction phenomena’, Comput. Methods Appl. Mech. Engrg 52, 527–634.Google Scholar
Panagiotopoulos, P. D. (1985), Inequality Problems in Mechanics and Applications, Birkhäuser.Google Scholar
Panagiotopoulos, P. D. (1993), Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer.Google Scholar
Renon, N., Montmitonnet, P. and Laborde, P. (2005), ‘A 3D finite element model for soil/tool interaction in large deformation’, Eng. Comput. 22, 87–109.Google Scholar
Scholz, C. H. (1990), The Mechanics of Earthquakes and Faulting, Cambridge University Press.Google Scholar
Signorini, A. (1933), ‘Sopra alcune questioni di elastostatica’, Atti della Società Italiana per il Progresso delle Scienze.Google Scholar
Shillor, M., Sofonea, M. and Telega, J. J. (2004), Models and Analysis of Quasistatic Contact, Vol. 655 of Lecture Notes in Physics, Springer.Google Scholar
Sofonea, M., Avramescu, C. and Matei, A. (2008), ‘A fixed point result with applications in the study of viscoplastic frictionless contact problems’, Comm. Pure Appl. Anal. 7, 645–658.Google Scholar
Sofonea, M., Han, W. and Barboteu, M. (2017), A variational–hemivariational inequality in contact mechanics. In Mathematical Modelling in Solid Mechanics, (dell’Isola, F. et al. , eds), Vol. 69 of Advanced Structured Materials, Springer, pp. 251–264.Google Scholar
Sofonea, M. and Matei, A. (2011), ‘History-dependent quasivariational inequalities arising in contact mechanics’, Euro. J. Appl. Math. 22, 471–491.Google Scholar
Sofonea, M. and Matei, A. (2012), Mathematical Models in Contact Mechanics, Vol. 398 of London Mathematical Society Lecture Note Series, Cambridge University Press.Google Scholar
Sofonea, M. and Migórski, S. (2018), Variational–hemivariational Inequalities with Applications, Pure and Applied Mathematics, Chapman & Hall/CRC.Google Scholar
Sofonea, M., Renon, N. and Shillor, M. (2004), ‘Stress formulation for frictionless contact of an elastic-perfectly-plastic body’, Appl. Anal. 83, 1157–1170.Google Scholar
Sofonea, M. and Xiao, Y. (2016), ‘Fully history-dependent quasivariational inequalities in contact mechanics’, Appl. Anal. 95, 2464–2484.Google Scholar
Temam, R. and Miranville, A. (2001), Mathematical Modeling in Continuum Mechanics, Cambridge University Press.Google Scholar
Verfürth, R. (2013), A Posteriori Error Estimation Techniques for Finite Element Methods, Oxford University Press.Google Scholar
Wang, F., Han, W. and Cheng, X.-L. (2010), ‘Discontinuous Galerkin methods for solving elliptic variational inequalities’, SIAM J. Numer. Anal. 48, 708–733.Google Scholar
Wang, F., Han, W. and Cheng, X.-L. (2011), ‘Discontinuous Galerkin methods for solving the Signorini problem’, IMA J. Numer. Anal. 31, 1754–1772.Google Scholar
Wang, F., Han, W. and Cheng, X.-L. (2014), ‘Discontinuous Galerkin methods for solving a quasistatic contact problem’, Numer. Math. 126, 771–800.Google Scholar
Wang, F. and Wei, H. (2018a), ‘Virtual element methods for the obstacle problem’, IMA J. Numer. Anal. doi:10.1093/imanum/dry055 Google Scholar
Wang, F. and Wei, H. (2018b), ‘Virtual element method for simplified friction problem’, Appl. Math. Lett. 85, 125–131.Google Scholar
Wohlmuth, B. (2011), Variationally consistent discretization schemes and numerical algorithms for contact problems. In Acta Numerica, Vol. 20, Cambridge University Press, pp. 569–734.Google Scholar
Wohlmuth, B. and Krause, R. (2003), ‘Monotone methods on non-matching grids for nonlinear contact problems’, SIAM J. Sci. Comput. 25, 324–347.Google Scholar
Wriggers, P. (2006), Computational Contact Mechanics, second edition, Springer.Google Scholar
Wriggers, P. and Fischer, K. (2005), ‘Frictionless 2D contact formulations for finite deformations based on the mortar method’, Comput. Mech. 36, 226–244.Google Scholar
Wriggers, P. and Laursen, T. (2007), Computational Contact Mechanics, Vol. 298 of CISM Courses and Lectures, Springer.Google Scholar
Wriggers, P., Rust, W. T. and Reddy, B. D. (2016), ‘A virtual element method for contact’, Comput. Mech. 58, 1039–1050.Google Scholar
Xu, W., Huang, Z., Han, W., Chen, W. and Wang, C. (2019), ‘Numerical analysis of history-dependent variational–hemivariational inequalities with applications in contact mechanics’, J. Comput. Appl. Math. 351, 364–377.Google Scholar
Zeidler, E. (1985), Nonlinear Functional Analysis and its Applications, I: Fixed-point Theorems, Springer.Google Scholar
Zeidler, E. (1990), Nonlinear Functional Analysis and its Applications, II/B: Nonlinear Monotone Operators, Springer.Google Scholar