A study of regularization techniques of nondifferentiable optimization in view of application to hemivariational inequalities. (English) Zbl 1298.74083
Summary: This paper presents a study of regularization techniques of nondifferentiable optimization with focus to the application to a special class of hemivariational inequalities. We establish some convergence results for the regularization method of hemivariational inequalities. As a model example we consider the delamination problem for laminated composite structures and provide numerical experiments, which underline our regularization theory.
MSC:
| 74G15 | Numerical approximation of solutions of equilibrium problems in solid mechanics |
| 74M15 | Contact in solid mechanics |
| 74R99 | Fracture and damage |
| 74S05 | Finite element methods applied to problems in solid mechanics |
Keywords:
plus function; smoothing approximation; hemivariational inequalities; finite element discretization; delamination problemReferences:
| [1] | Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) · Zbl 0536.65054 · doi:10.1007/978-3-662-12613-4 |
| [2] | Glowinski, R., Lions, J.L., Trémoliéres, R.: Numerical Analysis of Variational Inequalities. North-Holland, Amsterdam (1981) · Zbl 0463.65046 |
| [3] | Huang, H., Han, W., Zhou, J.: The regularization method for an obstacle problem. Numer. Math. 69, 155-166 (1994) · Zbl 0817.65050 · doi:10.1007/s002110050086 |
| [4] | Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845 |
| [5] | Reddy, B.D., Griffin, T.B.: Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem. Numer. Math. 52, 101-117 (1988) · Zbl 0618.73034 · doi:10.1007/BF01401024 |
| [6] | Sofonea, M., Matei, A.: Variational Inequalities with Applications. Springer, New York (2009) · Zbl 1195.49002 |
| [7] | Panagiotopoulos, P.D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer, Berlin (1993) · Zbl 0826.73002 · doi:10.1007/978-3-642-51677-1 |
| [8] | Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Application. Convex and Nonconvex Energy Functions. Springer, Birkhäuser, Boston (1998) |
| [9] | Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities: Theory, Methods and Applications, vol. I: Unilateral Analysis and Unilateral Mechanics, vol. II: Unilateral Problems. Kluwer, Dordrecht (2003) · Zbl 1259.49002 · doi:10.1007/978-1-4419-8610-8 |
| [10] | Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities. Springer, Berlin (2007) · Zbl 1109.35004 · doi:10.1007/978-0-387-46252-3 |
| [11] | Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. Springer, Berlin (2013) · Zbl 1262.49001 · doi:10.1007/978-1-4614-4232-5 |
| [12] | Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995) · Zbl 0968.49008 |
| [13] | Migórski, S.: Dynamic hemivariational inequality modeling viscoelastic contact problem with normal damped response and friction. Appl. Anal. 84, 669-699 (2005) · Zbl 1081.74036 · doi:10.1080/00036810500048129 |
| [14] | Migórski, S.: Hemivariational inequality for a frictional contact problem in elasto-piezoelectricity. Discret. Cont. Dyn. Syst. 6, 1339-1356 (2006) · Zbl 1109.74039 |
| [15] | Migórski, S., Ochal, A.: A unified approach to dynamic contact problems in viscoelasticity. J. Elast. 83, 247-275 (2006) · Zbl 1138.74375 · doi:10.1007/s10659-005-9034-0 |
| [16] | Migórski, S., Ochal, A., Sofonea, M.: Integrodifferential hemivariational inequalities with applications to viscoelastic frictional contact. Math. Models Methods Appl. Sci. 18, 271-290 (2008) · Zbl 1159.47038 · doi:10.1142/S021820250800267X |
| [17] | Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Methods for Hemivariational Inequalities. Kluwer, Dordrecht (1999) · Zbl 0949.65069 · doi:10.1007/978-1-4757-5233-5 |
| [18] | Ovcharova, N.: Regularization methods and finite element approximation of hemivariational inequalities with applications to nonmonotone contact problems. PhD Thesis, Universität der Bundeswehr München, Cuvillier Verlag, Göttingen (2012) · Zbl 1032.49017 |
| [19] | Baniotopoulos, C.C., Haslinger, J., Morávková, Z.: Contact problems with nonmonotone friction: discretization and numerical realization. Comput. Mech. 40, 157-165 (2007) · Zbl 1163.74038 · doi:10.1007/s00466-006-0092-3 |
| [20] | Baniotopoulos, C.C., Haslinger, J., Morávková, Z.: Mathematical modeling of delamination and nonmonotone friction problems by hemivariational inequalities. Appl. Math. 50(1), 1-25 (2005) · Zbl 1099.74021 · doi:10.1007/s10492-005-0001-7 |
| [21] | Hintermüller, M., Kovtunenko, V.A., Kunisch, K.: Obstacle problems with cohesion: a hemivariational inequality approach and its efficient numerical solution. SIAM J. Optim. 21(2), 491-516 (2011) · Zbl 1228.49012 · doi:10.1137/10078299 |
| [22] | Kovtunenko, V.A.: A hemivariational inequality in crack problems. Optimization 60(8-9), 1071-1089 (2011) · Zbl 1242.49016 |
| [23] | Czepiel, J.: Proximal bundle method for a simplifed unilateral adhesion contact problem of elasticity. Schedae Inform. 20, 115-136 (2011) |
| [24] | Chen, X.: Smoothing methods for nonsmooth, nonconvex minimization. Math. Program. 134, 71-99 (2012) · Zbl 1266.90145 · doi:10.1007/s10107-012-0569-0 |
| [25] | Chen, C., Mangasarian, O.L.: A class of smoothing functions for nonlinear and mixed complementarity problems. Comput. Optim. Appl. 5, 97-138 (1996) · Zbl 0859.90112 · doi:10.1007/BF00249052 |
| [26] | Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. I, II. Springer, Berlin (2003) · Zbl 1062.90002 |
| [27] | Qi, L., Sun, D.: Smoothing functions and a smoothing Newton method for complementarity and variational inequality problems. J. Optim. Theory Appl. 113(1), 121-147 (2002) · Zbl 1032.49017 · doi:10.1023/A:1014861331301 |
| [28] | Sun, D., Qi, L.: Solving variational inequality problems via smoothing-nonsmooth reformulations. J. Comput. Appl. Math. 129, 37-62 (2001) · Zbl 0987.65059 · doi:10.1016/S0377-0427(00)00541-0 |
| [29] | Zang, I.: A smoothing-out technique for min-max optimization. Math. Program. 19, 61-77 (1980) · Zbl 0468.90064 · doi:10.1007/BF01581628 |
| [30] | Chen, C., Mangasarian, O.L.: Smoothing methods for convex inequalities and linear complementarity problems. Math. Program. 71, 51-69 (1995) · Zbl 0855.90124 · doi:10.1007/BF01592244 |
| [31] | Bertsekas, D.P.: Nondifferentiable optimization via approximation. Math. Program. Study 3, 1-25 (1975) · Zbl 0383.49025 · doi:10.1007/BFb0120696 |
| [32] | Clarke, F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983) · Zbl 0582.49001 |
| [33] | Chen, X., Qi, L., Sun, D.: Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Math. Comp. 67, 519-540 (1998) · Zbl 0894.90143 · doi:10.1090/S0025-5718-98-00932-6 |
| [34] | Pinar, M.Ç., Zenios, S.A.: On smoothing exact penalty functions for convex constrained optimization. SIAM J. Optim. 4(3), 486-511 (1994) · Zbl 0819.90072 · doi:10.1137/0804027 |
| [35] | Gwinner, J.: Nichtlineare Variationsungleichungen mit Anwendungen. PhD Thesis, Universität Mannheim (1978) · Zbl 0393.49001 |
| [36] | Zeidler, E.: Nonlinear Functional Analysis and its Applications, II/B. Springer, Berlin (1990) · Zbl 0684.47029 · doi:10.1007/978-1-4612-0981-2 |
| [37] | Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations. Springer, Berlin (2004) · Zbl 1072.35001 |
| [38] | Gwinner, J.: hp-FEM convergence for unilateral contact problems with Tresca friction in plane linear elastostatics. J. Comput. Appl. Math. 254, 175-184 (2013) · Zbl 1290.74041 · doi:10.1016/j.cam.2013.03.013 |
| [39] | Miettinen, M., Haslinger, J.: Approximation of optimal control problems of hemivariational inequalities. Numer. Funct. Anal. Optim. 13(1-2), 43-68 (1992) · Zbl 0765.49007 · doi:10.1080/01630569208816460 |
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