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On the control of time discretized dynamic contact problems. (English) Zbl 1383.49002

Summary: We consider optimal control problems with distributed control that involve a time-stepping formulation of dynamic one body contact problems as constraints. We link the continuous and the time-stepping formulation by a nonconforming finite element discretization and derive existence of optimal solutions and strong stationarity conditions. We use this information for a steepest descent type optimization scheme based on the resulting adjoint scheme and implement its numerical application.

MSC:

49J20 Existence theories for optimal control problems involving partial differential equations
49M25 Discrete approximations in optimal control
65K15 Numerical methods for variational inequalities and related problems
74H15 Numerical approximation of solutions of dynamical problems in solid mechanics
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[1] Ahn, J., Stewart, D.E.: Dynamic frictionless contact in linear viscoelasticity. IMA J. Numer. Anal. 29(1), 43-71 (2009). doi:10.1093/imanum/drm029 · Zbl 1155.74029
[2] Attouch, H., Buttazzo, G., Michaille, G.: Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. In: Society for Industrial and Applied Mathematics, MOS-SIAM Series on Optimization. (2014) ISBN 9781611973471. https://books.google.de/books?id=AxKuBAAAQBAJ · Zbl 1311.49001
[3] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE. Computing 82(2-3), 121-138 (2008) · Zbl 1151.65088
[4] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part I: abstract framework. Computing 82(2-3), 103-119 (2008) · Zbl 1151.65089
[5] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Fahlke, J., Gräser, C., Klöfkorn, R., Nolte, M., Ohlberger, M., Sander, O.: DUNE Web Page. http://www.dune-project.org (2011) · Zbl 1295.74072
[6] Betz, T.: Optimal Control of Two Variational Inequalities Arising in Solid Mechanics. PhD thesis, Technische Universitüt Dortmund (2015) · Zbl 1231.74470
[7] Blum, H., Rademacher, A., Schröder, A.: Space adaptive finite element methods for dynamic Signorini problems. Comput. Mech. 44(4), 481-491 (2009) · Zbl 1241.74034
[8] Chouly, F., Hild, P., Renard, Y.: A Nitsche finite element method for dynamic contact: 1. Space semi-discretization and time-marching schemes. ESAIM Math. Model. Numer. Anal. 49(2), 481-502 (2015). ISSN 0764-583X · Zbl 1311.74113
[9] Christof, C., Müller, G.: A note on the equivalence and the boundary behavior of a class of Sobolev capacities. https://epub.uni-bayreuth.de/3155/ (2017) · Zbl 1051.74045
[10] Cocu, M., Ricaud, J.-M.: Analysis of a class of implicit evolution inequalities associated to viscoelastic dynamic contact problems with friction. Int. J. Eng. Sci. 38(14), 1535-1552 (2000). ISSN 0020-7225 · Zbl 1210.74128
[11] Deuflhard, P., Krause, R., Ertel, S.: A contact-stabilized newmark method for dynamical contact problems. Int. J. Numer. Methods Eng. 73(9), 1274-1290 (2008). doi:10.1002/nme.2119. ISSN 1097-0207 · Zbl 1169.74053
[12] Doyen, D., Ern, A., Piperno, S.: Time-integration schemes for the finite element dynamic Signorini problem. SIAM J. Sci. Comput. 33(1), 223-249 (2011). ISSN 1064-8275 · Zbl 1315.74019
[13] Eck, C., Jarušek, J., Krbec, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems, Volume 270 of Pure and Applied Mathematics (Boca Raton). Chapman & Hall/CRC, Boca Raton (2005). ISBN 978-1-57444-629-6; 1-57444-629-0 · Zbl 1079.74003
[14] Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Textbooks in Mathematics, revised edn. CRC Press, Boca Raton (2015) · Zbl 1310.28001
[15] Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (1998) · Zbl 0902.35002
[16] Götschel, S., Weiser, M., Schiela, A.: Solving optimal control problems with the Kaskade 7 finite element toolbox. In: Dedner, A., Flemisch, B., Klöfkorn, R. (eds.) Advances in DUNE: Proceedings of the DUNE User Meeting, Held in October 6th-8th 2010 in Stuttgart, Germany, pp. 101-112. Springer Berlin Heidelberg (2012) . doi:10.1007/978-3-642-28589-9_8 · Zbl 1524.65006
[17] Hager, C., Hüeber, S., Wohlmuth, B.I.: A stable energy-conserving approach for frictional contact problems based on quadrature formulas. Int. J. Numer. Methods Eng. 73(2), 205-225 (2008). ISSN 0029-5981 · Zbl 1166.74050
[18] Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Volume 30 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence (2002). ISBN 0-8218-3192-5 · Zbl 1013.74001
[19] Hüber, S., Stadler, G., Wohlmuth, B.I.: A primal-dual active set algorithm for three-dimensional contact problems with Coulomb friction. SIAM J. Sci. Comput. 30(2), 572-596 (2008) · Zbl 1158.74045
[20] Jarušek, J., Eck, C.: Remark to dynamic contact problems for bodies with a singular memory. Comment. Math. Univ. Carol. 3 39(3), 545-550 (1998) · Zbl 0963.35137
[21] Jarušek, J., Eck, C.: Dynamic contact problems with small Coulomb friction for viscoelastic bodies. Existence of solutions. Math. Models Methods Appl. Sci. 9(1), 11-34 (1999). ISSN 0218-2025 · Zbl 0938.74048
[22] Kane, C., Repetto, E.A., Ortiz, M., Marsden, J.E.: Finite element analysis of nonsmooth contact. Comput. Method Appl. Mech. Eng. 180(1-2), 1-26 (1999). ISSN 0045-7825 · Zbl 0963.74061
[23] Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. Society for Industrial and Applied Mathematics, Philadelphia (1988) · Zbl 0685.73002
[24] Kilpeläinen, T., Malý, J.: Supersolutions to degenerate elliptic equation on quasi open sets. Commun. Partial Differ. Equ. 17(3-4), 371-405 (1992). doi:10.1080/03605309208820847. ISSN 0360-5302 · Zbl 0781.31009
[25] Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications, Volume 88 of Pure and Applied Mathematics. Academic Press, Inc., New York (1980). ISBN 0-12-407350-6 · Zbl 0457.35001
[26] Klapproth, C.: Adaptive Numerical Integration of Dynamical Contact Problems. PhD thesis, Freie Universits̈t, Berlin (2010) · Zbl 0682.49015
[27] Klapproth, C., Schiela, A., Deuflhard, P.: Consistency results on Newmark methods for dynamical contact problems. J. Numer. Math. 116(1), 65-94 (2010). ISSN 0029-599X · Zbl 1283.74045
[28] Klapproth, C., Schiela, A., Deuflhard, P.: Adaptive timestep control for the contact-stabilized Newmark method. J. Numer. Math. 119(1), 49-81 (2011). ISSN 0029-599X · Zbl 1391.74277
[29] Kornhuber, R., Krause, R.: Adaptive multigrid methods for Signorini’s problem in linear elasticity. Comput. Vis. Sci. 4, 9-20 (2001) · Zbl 1051.74045
[30] Kornhuber, R., Krause, R., Sander, O., Deuflhard, P., Ertel, S.: A monotone multigrid solver for two body contact problems in biomechanics. Comput. Vis. Sci. 11(1), 3-15 (2008). doi:10.1007/s00791-006-0053-6. ISSN 1432-9360
[31] Krause, R.: Monotone Multigrid Methods for Signorini’s Problem with Friction. PhD thesis, Freie Universität, Fachbereich Mathematik und Informatik, Berlin (2001)
[32] Krause, R., Walloth, M.: A time discretization scheme based on Rothe’s method for dynamical contact problems with friction. Comput. Method Appl. Mech. Eng. 199(1-4), 1-19 (2009). doi:10.1016/j.cma.2009.08.022. ISSN 0045-7825 · Zbl 1231.74470
[33] Krause, R., Walloth, M.: Presentation and comparison of selected algorithms for dynamic contact based on the newmark scheme. Appl. Numer. Math. 62(10), 1393-1410 (2012). ISSN 0168-9274 · Zbl 1295.74072
[34] Kröner, A., Kunisch, K., Vexler, B.: Semismooth newton methods for optimal control of the wave equation with control constraints. SIAM J. Control Optim. 49(2), 830-858 (2011) · Zbl 1218.49035
[35] Kuttler, K., Shillor, M.: Dynamic contact with Signorini’s condition and slip rate dependent friction. Electron. J. Differ. Equ., 83, 21 (2004). ISSN 1072-6691 · Zbl 1081.74035
[36] Laursen, T.A., Chawla, V.: Design of energy conserving algorithms for frictionless dynamic contact problems. Int. J. Numer. Methods Eng. 40(5), 863-886 (1997). ISSN 0029-5981 · Zbl 0886.73067
[37] Laursen, T.A.: Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis. Springer, Berlin (2013) · Zbl 0996.74003
[38] Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20(3), 493-519 (1967) · Zbl 0152.34601
[39] Mignot, F.: Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22(2), 130-185 (1976) · Zbl 0364.49003
[40] Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85(3), 67-94 (1959)
[41] Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory. Applications and Numerical Results. Nonconvex Optimization and Its Applications. Springer, New York (2013). ISBN 9781475728255 · Zbl 0947.90093
[42] Schatzman, M.: A class of nonlinear differential equations of second order in time. Nonlinear Anal. 2(3), 355-373 (1978). ISSN 0362-546X · Zbl 0382.34003
[43] Schiela, A.: A flexible framework for cubic regularization algorithms for non-convex optimization in function space. Technical University of Hamburg-Harburg (2014)
[44] Shapiro, A.: On concepts of directional differentiability. J. Optim. Theory Appl. 66(3), 477-487 (1990). ISSN 0022-3239 · Zbl 0682.49015
[45] Signorini, A.: Sopra alcune questioni di elastostatic. Atti Società Italiana per il Progresso delle Scienze (1933) · JFM 59.1413.02
[46] Stadler, G.: Path-following and augmented Lagrangian methods for contact problems in linear elasticity. J. Comput. Appl. Math. 203(2), 533-547 (2007) · Zbl 1119.49028
[47] Stollmann, P.: Closed ideals in dirichlet spaces. Potential Anal. 2(3), 263-268 (1993) · Zbl 0784.31009
[48] Trimble, T.: Is the preimage of the closure the closure of the preimage under a quotient map? http://mathoverflow.net/questions/74415/is-the-preimage-of-the-closure-the-closure-of-the-preimage-under-a-quotient-map (2011). Accessed: 12 10 2015
[49] Ulbrich, M., Ulbrich, S., Koller, D.: A Multigrid Semismooth Newton Method for Contact Problems in Linear Elasticity. Technical report, Department of Mathematics, TU Darmstadt. Submitted (2013) · Zbl 0382.34003
[50] Wachsmuth, G.: Mathematical programs with complementarity constraints in banach spaces. J. Optim. Theory Appl. 1-28 (2014). ISSN 0022-3239 · Zbl 0886.73067
[51] Wachsmuth, G.: A guided tour of polyhedric sets. Technical University of Chemnitz (2016) · Zbl 1412.52006
[52] Wehrstedt, J.C.: Formoptimierung mit Variationsungleichungen als Nebenbedingung und eine Anwendung in der Kieferchirurgie. Phd thesis, TU München (2007) · Zbl 0152.34601
[53] Werner, D.: Funktionalanalysis, 3rd edn. Springer, Heidelberg (2000) · Zbl 0964.46001
[54] Yosida, K.: Functional Analysis. Classics in Mathematics. Springer, Berlin (1995). ISBN 3-540-58654-7. Reprint of the sixth (1980) edition · Zbl 0435.46002
[55] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A. Springer, New York (1990). ISBN 0-387-96802-4. doi:10.1007/978-1-4612-0985-0. Linear monotone operators. Translated from the German by the author and Leo F. Boron · Zbl 0684.47028
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