A priori error analysis of virtual element method for contact problem
Tóm tắt
As an extension of the finite element method, the virtual element method (VEM) can handle very general polygonal meshes, making it very suitable for non-matching meshes. In (Wriggers et al. in Comput. Mech. 58:1039–1050, 2016), the lowest-order virtual element method was applied to solve the contact problem of two elastic bodies on non-matching meshes. The numerical experiments showed the robustness and accuracy of the virtual element scheme. In this paper, we establish a priori error estimate of the virtual element method for the contact problem and prove that the lowest-order VEM achieves linear convergence order, which is optimal.
Tài liệu tham khảo
Andersson, L.E.: A quasistatic frictional problem with normal compliance. Nonlinear Anal. 16, 347–369 (1991)
Antonietti, P.F., Beirão da Veiga, L., Mora, D., Verani, M.: A stream function formulation of the Stokes problem for the virtual element method. SIAM J. Numer. Anal. 52, 386–404 (2014)
Antonietti, P.F., Beirão da Veiga, L., Scacchi, S., Verani, M.: A \(C^{1}\) virtual element method for the Cahn–Hilliard equation with polygonal meshes. SIAM J. Numer. Anal. 54, 34–56 (2016)
Atkinson, K., Han, W.: Theoretical Numerical Analysis: A Functional Analysis Framework, 3rd edn. Springer, New York (2009)
Beirão da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51, 794–812 (2013)
Benkhira, E.L.-H., Fakhar, R., Mandyly, Y.: Analysis and numerical approximation of a contact problem involving nonlinear Hencky-type materials with nonlocal Coulomb’s friction law. Numer. Funct. Anal. Optim. 40, 1291–1314 (2019)
Chen, L., Wei, H., Wen, M.: An interface-fitted mesh generator and virtual element methods for elliptic interface problems. J. Comput. Phys. 334, 327–348 (2017)
Deng, Y., Wang, F., Wei, H.: A posteriori error estimates of virtual element method for a simplified friction problem. J. Sci. Comput. 83, 52 (2020)
Djoko, J.K., Ebobisse, F., McBride, A.T., Reddy, B.D.: A discontinuous Galerkin formulation for classical and gradient plasticity – Part 1: formulation and analysis. Comput. Methods Appl. Mech. Eng. 196, 3881–3897 (2007)
Falk, R.: Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28, 963–971 (1974)
Feng, F., Han, W., Huang, J.: Virtual element methods for elliptic variational inequalities of the second kind. J. Sci. Comput. 80, 60–80 (2019)
Feng, F., Han, W., Huang, J.: Virtual element method for an elliptic hemivariational inequality with applications to contact mechanics. J. Sci. Comput. 81, 2388–2412 (2019)
Fischer, K.A., Wriggers, P.: Mortar based frictional contact formulation for higher order interpolations using the moving friction cone. Comput. Methods Appl. Mech. Eng. 195, 5020–5036 (2006)
Gain, A.L., Talischi, C., Paulino, G.H.: On the virtual element method for three-dimensional elasticity problems on arbitrary polyhedral meshes. Comput. Methods Appl. Mech. Eng. 282, 132–160 (2014)
Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
Gudi, T., Porwal, K.: A posteriori error control of discontinuous Galerkin methods for elliptic obstacle problems. Math. Compet. 83, 579–602 (2014)
Han, W.: On the numerical approximation of a frictional contact problem with normal compliance. Numer. Funct. Anal. Optim. 17, 307–322 (1996)
Han, W., Sofonea, M.: Analysis and numerical approximation of an elastic frictional contact problem with normal compliance. Appl. Math. 25, 415–435 (1999)
Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)
Klarbring, A., Mikelič, A., Schillor, M.: Frictional contact problems with normal compliance. Int. J. Eng. Sci. 26, 811–832 (1988)
Laborde, P., Renard, Y.: Fixed point strategies for elastostatic frictional contact problems. Math. Methods Appl. Sci. 31, 415–441 (2008)
Lee, C.Y., Oden, J.T.: Theory and approximation of quasistatic frictional contact problems. Comput. Methods Appl. Mech. Eng. 106, 407–429 (1993)
Ling, M., Wang, F., Han, W.: The nonconforming virtual element method for a stationary Stokes hemivariational inequality with slip boundary condition. J. Sci. Comput. 85, 56 (2020)
Martins, J.T., Oden, J.T.: Existence and uniqueness results for dynamics contact problems with nonlinear normal and friction interface laws. Nonlinear Anal. 11, 407–428 (1987)
Migorski, S., Gamorski, P.: A new class of quasistatic frictional contact problems governed by a variational–hemivariational inequality. Nonlinear Anal., Real World Appl. 50, 583–602 (2019)
Perugia, I., Pietra, P., Russo, A.: A plane wave virtual element method for the Helmholtz problem. ESAIM: M2AN 50, 783–808 (2016)
Popp, A., Gitterle, M., Gee, M., Wall, W.A.: A dual mortar approach for 3D finite deformation contact with consistent linearization. Int. J. Numer. Methods Eng. 83, 1428–1465 (2010)
Rochdi, M., Shillor, M., Sofonea, M.: Quasistatic viscoelastic contact with normal compliance and friction. J. Elast. 51, 105–126 (1998)
Sofonea, M.: A fixed point result with applications in the study of viscoplastic frictionless contact problems. Commun. Pure Appl. Anal. 7, 645–658 (2008)
Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. Cambridge University Press, Cambridge (2012)
Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving elliptic variational inequalities. SIAM J. Numer. Anal. 48, 708–733 (2010)
Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving Signorini problem. IMA J. Numer. Anal. 31, 1754–1772 (2011)
Wang, F., Han, W., Cheng, X.: Discontinuous Galerkin methods for solving the quasistatic contact problem. Numer. Math. 126, 771–800 (2014)
Wang, F., Han, W., Eichholz, J., Cheng, X.: A posteriori error estimates of discontinuous Galerkin methods for obstacle problems. Nonlinear Anal., Real World Appl. 22, 664–679 (2015)
Wang, F., Wei, H.: Virtual element method for simplified friction problem. Appl. Math. Lett. 85, 125–131 (2018)
Wang, F., Wei, H.: Virtual element methods for obstacle problem. IMA J. Numer. Anal. 40, 708–728 (2020)
Wang, F., Wu, B., Han, W.: The virtual element method for general elliptic hemivariational inequalities. J. Comput. Appl. Math. 389, 113330 (2021)
Wang, F., Zhao, J.: Conforming and nonconforming virtual element methods for a Kirchhoff plate contact problem. IMA J. Numer. Anal. 41, 1496–1521 (2021)
Wohlmuth, B.: A mortar finite element method using dual spaces for the Lagrange multiplier. SIAM J. Numer. Anal. 38, 989–1012 (2000)
Wohlmuth, B., Krause, R.: Monotone methods on nonmatching grids for non linear contact problems. SIAM J. Sci. Comput. 25, 324–347 (2004)
Wriggers, P.: Computational Contact Mechanics, 2nd edn. Springer, Berlin (2006)
Wriggers, P., Rust, W.T., Reddy, B.D.: A virtual element method for contact. Comput. Mech. 58, 1039–1050 (2016)
Zhao, J., Chen, S., Zhang, B.: The nonconforming virtual element method for plate bending problems. Math. Models Methods Appl. Sci. 26, 1671–1687 (2016)