Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T03:12:36.619Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTISYMPLECTIC VARIATIONAL INTEGRATORS FOR NONSMOOTH LAGRANGIAN CONTINUUM MECHANICS

Part of: Symplectic geometry, contact geometry Elastic materials Numerical methods in calculus of variations and optimal control Equations of mathematical physics and other areas of application Numerical approximation and computational geometry Existence theories Special kinds of problems

Published online by Cambridge University Press:  08 July 2016

FRANÇOIS DEMOURES ,
FRANÇOIS GAY-BALMAZ  and
TUDOR S. RATIU
FRANÇOIS DEMOURES
Affiliation:
Department of Mathematics, Imperial College, London, UK; demoures@lmd.ens.fr LMD/IPSL, CNRS, École Normale Supérieure, PSL Research University, Université Paris-Saclay, Sorbonne Universités, Paris, France; francois.gay-balmaz@lmd.ens.fr
FRANÇOIS GAY-BALMAZ
Affiliation:
LMD/IPSL, CNRS, École Normale Supérieure, PSL Research University, Université Paris-Saclay, Sorbonne Universités, Paris, France; francois.gay-balmaz@lmd.ens.fr
TUDOR S. RATIU
Affiliation:
Department of Mathematics, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, 200240 Shanghai, China Section de Mathématiques, École Polytechnique Fédérale de Lausanne, CH 1015 Lausanne, Switzerland; tudor.ratiu@epfl.ch

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper develops the theory of multisymplectic variational integrators for nonsmooth continuum mechanics with constraints. Typical problems are the impact of an elastic body on a rigid plate or the collision of two elastic bodies. The integrators are obtained by combining, at the continuous and discrete levels, the variational multisymplectic formulation of nonsmooth continuum mechanics with the generalized Lagrange multiplier approach for optimization problems with nonsmooth constraints. These integrators verify a spacetime multisymplectic formula that generalizes the symplectic property of time integrators. In addition, they preserve the energy during the impact. In the presence of symmetry, a discrete version of the Noether theorem is verified. All these properties are inherited from the variational character of the integrator. Numerical illustrations are presented.

MSC classification

Primary: 65D99: None of the above, but in this section 49M30: Other methods 74B20: Nonlinear elasticity 74M20: Impact
Secondary: 49J52: Nonsmooth analysis 53D20: Momentum maps; symplectic reduction 35Q74: PDEs in connection with mechanics of deformable solids
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 4 , 2016 , e19
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2016

References

Alart, P., ‘Multiplicateurs augmentés et méthode de Newton généralisée pour contact avec frottement’, Report, Document LMA-DME-EPFL, Lausanne (1988).Google Scholar
Alart, P. and Curnier, A., ‘A mixed formulation for frictional contact problems prone to Newton like solution methods’, Comput. Methods Appl. Mech. Engrg. 92(3) (1991), 353–375.Google Scholar
Armero, F. and Petocz, E., ‘Formulation and analysis of conserving algorithms for frictionless dynamic contact/impact problems’, Comput. Methods Appl. Mech. Engrg. 158 (1998), 269–300.Google Scholar
Arrow, K. J. and Solow, R. M., ‘Gradient methods for constrained maxima with weakened assumptions’, inStudies in Linear and Nonlinear Progrmming (eds. Arrow, K. J., Hurwitz, L. and Uzawa, H.) (Stanford University Press, Stanford, 1958), 166–176.Google Scholar
Attouch, H. and Wets, R., ‘Approximation and convergence in nonlinear optimization’, inNonlinear Programming 4 (eds. Mangasarian, O., Meyer, R. and Robinson, S.) (Academic Press, New York–London, 1981), 367–394.CrossRefGoogle Scholar
Attouch, H. and Wets, R., ‘A convergence theory for saddle functions’, Trans. Amer. Math. Soc. 280 (1983), 1–41.Google Scholar
Attouch, H. and Wets, R., ‘Epigraphical analysis’, inAnalyse Non Linéaire (eds. Attouch, H., Aubin, J.-P., Clarke, F. and Ekeland, I.) (Gauthier-Villars, Paris, 1989), 73–100.Google Scholar
Beltrami, E. J., ‘On infinite-dimensional convex programs’, J. Comput. System Sci. 1(4) (1967), 323–329.Google Scholar
Bertsekas, D. P., ‘On penalty and multiplier methods for constrained minimization’, inNonlinear Programming 2 (eds. Mangasarian, O. L., Meyer, R. and Robinson, S.) (Academic Press, New York, 1975), 165–191.Google Scholar
Bertsekas, D. P., ‘Necessary and sufficient conditions for a penalty method to be exact’, Math. Program. 9 (1975), 87–99.Google Scholar
Bertsekas, D. P., ‘Combined primal-dual and penalty methods for constrained minimization’, SIAM J. Control 13(3) (1975), 521–544.Google Scholar
Bertsekas, D. P., ‘On the method of multipliers for convex programming’, IEEE Trans. Automat. Control (1975), 385–388.Google Scholar
Bouligand, G., Précis de mécanique rationnelle à l’usage des élèves des facultés des sciences (Libraire Vuibert, Paris, 1925).Google Scholar
Bridges, T. and Reich, S., ‘Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity’, Phys. Lett. A 284(4–5) (2001), 184–193.Google Scholar
Buys, J. D., ‘Dual algorithms for constrained optimization problems’, PhD Thesis, University of Leiden, 1972.Google Scholar
Carnot, L., Essai sur les machines en général (Dijon, Defay, 1783).Google Scholar
Cirak, F. and West, M., ‘Decomposition contact response (DCR) for explicit finite element dynamics’, Internat. J. Numer. Methods Engrg. 64 (2005), 1078–1110.Google Scholar
Clarke, F. H., Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, 5 (SIAM, Philadelphia, PA, 1990).Google Scholar
Clausius, R., ‘On the mean length of the paths described by the separate molecules of gaseous bodies on the occurrence of molecular motion: together with some other remarks upon the mechanical theory of heat’, Phil. Mag. 4(17) (1859), 81–91.Google Scholar
Courant, R., ‘Über direkte Methoden in der Variationsrechnung und über Verwandte Fragen’, Math. Ann. 97 (1927), 711–736.Google Scholar
Courant, R., ‘Variational methods for the solution of problems of equilibrium and vibration’, Bull. Amer. Soc. 49 (1943), 1–23.Google Scholar
Courant, R., Methods of Mathematical Physics, Vol. 2 (Wiley-Interscience, New York, 1962).Google Scholar
Cournot, A. A., ‘Sur les percussions entre deux corps durs, qui se choquent en plusieurs points’, Bull. Sci. Math., Astron., Phys. Chim. 7 (1827), 4–11.Google Scholar
Cournot, A. A., ‘Extension du principe des vitesses virtuelles au cas où les conditions de liaison du système sont exprimées par des inégalités’, Bull. Sci. Math., Astron., Phys. Chim. 8 (1827), 165–170.Google Scholar
Curnier, A., ‘Unilateral contact, mechanical modeling’, inNew Developments in Contact Problems (eds. Wriggers, P. and Panagiotopoulos, P.) CISM Courses and Lectures, 388 (International Center for Mechanical Sciences, Springer, Wien, 1999), 1–54.Google Scholar
Curnier, A., Contact Mechanics and Nonsmooth Tribology, (EPFL, 2011), in preparation.Google Scholar
Delassus, E., ‘Les diverses formes du principe de d’Alembert et les équations générales du mouvement des systémes soumis à des liaisons d’ordre quelconques’, Comptes Rendus T. CLVI (1913), 205–209.Google Scholar
Demoures, F., Gay-Balmaz, F., Kobilarov, M. and Ratiu, T. S., ‘Multisymplectic Lie algebra variational integrator for a geometrically exact beam in ℝ3 ’, Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 3492–3512.Google Scholar
Demoures, F., Gay-Balmaz, F. and Ratiu, T. S., ‘Multisymplectic variational integrator and space/time symplecticity’, Anal. Appl. 14(3) (2016), 341–391.Google Scholar
Demoures, F., Gay-Balmaz, F., Desbrun, M., Ratiu, T. S. and Aragon, A., ‘A multisymplectic integrator for elastodynamic frictionless impact problems’, Preprint, 2015.Google Scholar
Dontchev, A. L. and Rockafellar, R. T., Implicit Functions and Solution mappings. A View from Variational Analysis, 2nd edn, Springer Series in Operations Research and Financial Engineering (Springer, New York, 2014).Google Scholar
Duhem, P., Hydrodynamique, élasticité et acoustique (A. Hemann, Paris, 1891).Google Scholar
Ekeland, I. and Temam, R., Analyse convexe et problèmes variationnels, 2nd edn, Convex Analysis and Variational Problems (Dunod-Gauthier-Villars, 1974), 1976, Classics in Applied Mathematics, 28, SIAM, Philadelphia, PA, 1999.Google Scholar
Ellis, D., Gay-Balmaz, F., Holm, D.D., Putkaradze, V. and Ratiu, T. S., ‘Symmetry reduced dynamics of charged molecular strands’, Arch. Rat. Mech. Anal. 197(2) (2010), 811–902.Google Scholar
Euler, L., ‘Mechanica sive motus scientia analytice exposita’, Opera Omnia II(1) (1736).Google Scholar
Euler, L., ‘Methodus inveniendi lineas curvas’, Opera Omnia I(24) (1744).Google Scholar
Euler, L., ‘Découverte d’un nouveau principe de mécanique (1750)’, Mémoires l’Acad. Sci. Berlin 6 (1752), 185–217.Google Scholar
Farkas, J., ‘Die algebraische Grundlage der Anwendungen des machanischen Pricips von Fourier’, Math. Naturwiss. Ber. Ungarn 16 (1899), 154–157.Google Scholar
Farkas, J., ‘Theorie der einfachen Ungleichungen’, J. Reine Angew Math. 124 (1901), 1–27.Google Scholar
Fenchel, W., Convex Cones, Sets and Functions, Lecture Notes (Princeton University, Princeton, NJ, 1951).Google Scholar
Fetecau, R. C., Marsden, J. E. and West, M., ‘Variational multisymplectic formulations of nonsmooth continuum mechanics’, inPerspectives and Problems in Nonlinear Science (Springer, New York, 2003), 229–261.CrossRefGoogle Scholar
Fetecau, R. C., Marsden, J. E., Ortiz, M. and West, M., ‘Nonsmooth Lagrangian mechanics and variational collision integrators’, SIAM J. Appl. Dyn. Syst. 2 (2003), 381–416.Google Scholar
Fiacco, A. V. and McCormick, G. P., Nonlinear Programming, Classics in Applied Mathematics, 4 (SIAM, Philadelphia, PA, 1990), Reprint of the 1968 original.Google Scholar
Filippov, A. F., ‘Differential equations with discontinuous right-hand side (Russian)’, Mat. Sb. (N.S.) 51(93) (1960), 99–128. AMS Transl., Series 2, 42, 1964, pp. 199–231.Google Scholar
Filippov, A. F., ‘Classical solutions of differential equations with multivalued right-hand side’, SIAM J. Control 5 (1967), 609–621.Google Scholar
Filippov, A. F., Differential Equations with Discontinuous Right-Hand Sides (Kluwer Academic, Norwell, MA, 1988).Google Scholar
Forsgren, A., Gill, P. E. and Wright, M. H., ‘Interior methods for nonlinear optimization’, SIAM Rev. 44(4) (2002), 525–597.Google Scholar
Fortin, M. and Glowinski, R., Méthodes de lagrangien augmenté. Applications la résolution numérique de problèmes aux limites (Dunod, Paris, 1982).Google Scholar
Fourier, J., ‘Mémoire sur la statique contenant la démonstration du principe des vitesses virtuelles et la théorie des moments’, J. Ec. Polytech. (1798), V $^{e}$ Cahier, 20–64. Œuvres de Fourier, tome second, 477–521, Gauthier-Villars et Fils, Paris, 1840.Google Scholar
Fourier, J., ‘Solution d’une question particulière du calcul des inégalites’, Nouv. Bull. Sci. Soc. Philomatique Paris. Œuvres de Fourier (1826), Gauthier-Villars et Fils, 1840.Google Scholar
Gauß, C. F., ‘Über ein neues allgemeines Grundgesetz der Mechanik’, J. Reine Angew. Math. 4 (1829), 232–235.Google Scholar
Glocker, C., Set-Valued Force Laws Dynamics of Non-smooth Systems, Lecture Notes in Applied Mechanics, 1 (Springer, Berlin, 2001).Google Scholar
Glocker, C., ‘Concepts for modeling impacts without friction’, Acta Mech. 168 (2004), 1–19.Google Scholar
Glowinski, R. and Marocco, A., ‘On the solution of a class of non-linear Dirichlet problems by penalty-duality method and finite elements of order one’, C. R. Acad. Sci. Paris 278A (1974), 1649–1652.Google Scholar
Glowinski, R. and Le Tallec, P., Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics (SIAM, Philadelphia, PA, 1989).Google Scholar
Glowinski, R., Lions, J. L. and Tremolières, R., Analyse numérique des inéquations variationnelles, Tome 1 et 2, Méth. Math. Inf., 5 (Dunod, Paris, 1976).Google Scholar
Gotay, M. J., Isenberg, J. and Marsden, J. E., Momentum Maps and Classical Fields (Caltech, 2006).Google Scholar
Haarhoff, P. C. and Buys, J. D., ‘A new method for the optimization of a nonlinear function subject to nonlinear constraints’, Comput. J. 13 (1970), 178–184.Google Scholar
Hairer, E., Lubich, C. and Wanner, G., Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31 (Springer, Heidelberg, 2010).Google Scholar
Hamel, G., ‘Die Axiome der Mechanik’, inHandbuch der Physik (eds. Geiger, H . and Scheel, K.) Band 5, Grundlagen der Mechanik, Mechanik der Punkte und Starren Körper (Springer, Berlin, 1927).Google Scholar
Hamel, G., Theoretische Mechanik Eine einheitliche Einführung in die gesamte Mechanik, Grundlehren der Mathematischen Wissenschaften, 57 (Springer, Berlin–Göttingen–Heidelberg, 1949), Corrected reprint of the 1949 edition, Springer, Berlin–New York, 1978.Google Scholar
He, Q. C., Telega, J. J. and Curnier, A., ‘Unilateral contact of two solids subject to large deformations: formulation and existence results’, Math. Phys. Eng. Sci. 452(1955) (1996), 2691–2717.Google Scholar
Hefgaard, J. H. and Curnier, A., ‘An augmented Lagrangian method for discrete large-slip contact problems’, Internat. J. Numer. Methods Engrg. 36 (1993), 569–593.Google Scholar
Hertz, H., ‘Über die Berührung fester elastischer Körper’, J. Reine Angew. Math. 92 (1881), 156–171.Google Scholar
Hestenes, M. R., ‘Multiplier and gradient methods’, J. Optim. Theory Appl. 4(5) (1969), 303–320.Google Scholar
Hughes, T. J. R., Taylor, R. L., Sackman, J. L., Curnier, A. and Kanoknukulchai, W., ‘A finite element method for a class of contact-impact problems’, Comput. Methods Appl. Mech. Engrg. 8 (1975), 249–276.Google Scholar
Iserles, A., Munthe-Kaas, H. Z., Nørsett, S. P. and Zanna, A., ‘Lie-group methods’, Acta Numer. 9 (2000), 215–365.Google Scholar
Ito, K. and Kunisch, K., ‘An augmented Lagrangian technique for variational inequalities’, Appl. Math. Optim. 21(3) (1990), 223–241.Google Scholar
Johnson, G., Leyendecker, S. and Ortiz, M., ‘Discontinuous variational time integrators for complex multibody collisions’, Internat. J. Numer. Methods Engrg. 100 (2014), 871–913.Google Scholar
Kane, C., Repetto, E. A., Ortiz, M. and Marsden, J. E., ‘Finite element analysis of nonsmooth contact’, Comput. Methods Appl. Mech. Engrg. 180 (1999), 1–26.Google Scholar
Karush, W., ‘Minima of functions of several variables with inequalities as side constraints’, M.Sc. Thesis, Department of Mathematics, University of Chicago, USA, 1939.Google Scholar
Klarbring, A., ‘A mathematical programming approach to three-dimensional contact problems with friction’, Comput. Methods Appl. Mech. Engrg. 58(2) (1986), 175–200.Google Scholar
Kuhn, H. W. and Tucker, A. W., ‘Nonlinear Programming’, inProceedings of 2nd Berkeley Symposium (University of California Press, Berkeley, 1951), 481–492.Google Scholar
de Lagrange, J. L., Méchanique Analitique, chez la Veuve Desaint, Libraire, rue du Foin S. Jacques, Paris, 1788; reprinted by Éditions Jacques Gabay, Paris, 1989.Google Scholar
de Lagrange, J. L., Théorie des Fonctions Analytiques (Imprim. Rép., Paris, 1797).Google Scholar
Laursen, T. A., Computational Contact and Impact Mechanics (Springer, Berlin, 2002).Google Scholar
Legendre, A. M., Nouvelles Méthodes Pour la d’Étermination des Orbites de Comètes (Courcier, Paris, 1806).Google Scholar
Leine, R. I. and Glocker, C., ‘A set-valued force law for spatial Coulomb–Contensou friction’, Eur. J. Mech. A Solids 22(2) (2003), 193–216.Google Scholar
Leine, R. I. and Nijmeijer, H., Dynamics and Bifurcations of Non-smooth Mechanical Systems, Lecture Notes in Applied and Computational Mechanics, 18 (Springer, Berlin, 2004).Google Scholar
Lew, A., Marsden, J. E., Ortiz, M. and West, M., ‘Asynchronous variational integrators’, Arch. Ration. Mech. Anal. 167(2) (2003), 85–146.CrossRefGoogle Scholar
Lew, A., Marsden, J. E., Ortiz, M. and West, M., ‘Variational time integrators’, Internat. J. Numer. Methods Engrg. 60(1) (2004), 153–212.Google Scholar
Marsden, J. E., Patrick, G. W. and Shkoller, S., ‘Mulltisymplectic geometry, variational integrators and nonlinear PDEs’, Comm. Math. Phys. 199 (1998), 351–395.Google Scholar
Marsden, J. E. and West, M., ‘Discrete mechanics and variational integrators’, Acta Numer. 10 (2001), 357–514.Google Scholar
Martinet, B., ‘Regularisation d’inéquations variationnelles par approximations successives’, ESAIM Math. Model. Numer. Anal. 4(R3) (1970), 154–158.Google Scholar
Moreau, J.-J., ‘Fonctions convexes duales et points proximaux dans un espace hilbertien’, C. R. Acad. Sci. Paris 255 (1962), 2897–2899.Google Scholar
Moreau, J.-J., ‘Les liaisons unilatérales et le principe de Gauss’, C. R. Acad. Sci. Paris 256 (1963), 871–874.Google Scholar
Moreau, J.-J., ‘Théorèmes “inf-sup,”’, C. R. Acad. Sci. Paris 258 (1964), 2720–2722.Google Scholar
Moreau, J.-J., ‘Proximité et dualité dans un espace hilbertien’, Bull. Soc. Math. France 93 (1965), 273–299.Google Scholar
Moreau, J.-J., Fonctionnelles Convexes, Sém EDP (Collège de France, Paris, 1966).Google Scholar
Moreau, J.-J., ‘La notion de sur-potentiel et les liaisons unilatérales en élastostatique’, C. R. Acad. Sci. Paris Ser. A–B 267 (1968), A954–A957.Google Scholar
Moreau, J.-J., ‘Systèmes élastoplastiques de liberté finie’, Travaux du Séminaire d’Analyse Convexe 3(12) (1973), 33 pp. Secrétariat des Math., Publ. No. 125. U.E.R. Math., Univ. Sci. Tech. Languedoc, Montpellier, 1973.Google Scholar
Moreau, J.-J., On Unilateral Constraints, Friction and Plasticity, Centro Internaz. Mat. Estivo (C.I.M.E.), II Ciclo Bressanone, 1973, 171–322, Edizioni Cremonese, Roma, 1974.Google Scholar
Moreau, J.-J., ‘Liaisons unilatérales sans frottement et chocs inélastiques’, C. R. Acad. Sci. Paris Ser. II Méc. Phys. Chim. Sci. Univers Sci. Terre 296(19) (1983), 1473–1476.Google Scholar
Moreau, J.-J., ‘Une formulation du contact à frottement sec; application au calcul numérique’, C. R. Acad. Sci. Paris Sér. II 302 (1986), 799–801.Google Scholar
Moreau, J.-J., ‘Unilateral contact and dry friction in finite freedom dynamics’, inNonsmooth Mechanics and Applications, (eds. Moreau, J.-J. and Panagiotopoulos, P. D.) CISM Courses and Lectures, 302 (International Center for Mechanical Sciences, Springer, 1988), 1–82.Google Scholar
Moreau, J.-J., ‘Jump functions of a real interval to a Banach space’, Ann. Fac. Sci. Toulouse Math. (6) 5(suppl.) (1989), 77–91.Google Scholar
Moreau, J.-J., ‘Contact et frottement en dynamique des systèmes de corps rigides’, Revue Européenne des Éléments Finis 9 (2000), 9–28.Google Scholar
Moreau, J.-J. and Panagiotopoulos, P. D., Non-Smooth Mechanics and Applications, 2nd edn, CISM Courses and Lectures, 302 (International Center for Mechanical Sciences, Springer, Wien, New York, 1988).Google Scholar
Ostrogradsky, M. V., ‘Considérations générales sur les momen(t)s des forces’, Mém. Acad. Imp. Sci. Saint-Petersbourg, Sixième Série, Sciences Math. et Phys., 1835–1838 1 (1834), 129–150. Submitted November 7, 1834.Google Scholar
Ostrogradsky, M. V., ‘Mémoire sur les déplacemen(t)s instantanés des systèmes assujettis à des conditions variables’, Mém. Acad. Imp. Sci. St. Petersbourg, Sixième Série, Sciences Math. et Phys., 1835–1838 1 (1838), 565–600. Submitted April 20, 1938.Google Scholar
Painlevé, P., ‘Sur les lois du frottement de glissemment’, C. R. Acad. Sci. Paris 121 (1895), 112–115.Google Scholar
Pandolfi, A., Kane, C., Marsden, J. E. and Ortiz, M., ‘Time-discretized variational formulation of non-smooth frictional contact’, Internat. J. Numer. Methods Engrg. 53 (2002), 1801–1829.Google Scholar
Papadopoulos, P., ‘A nonlinear programming approach to the unilateral contact and friction-boundary value problem in the theory of elasticity’, Ing. Arch. 44 (1975), 421–432.Google Scholar
Papadopoulos, P. and Taylor, R. L., ‘A mixed formulation for the finite element solution of contact problems’, Comput. Methods Appl. Mech. Engrg. 94 (1992), 373–389.Google Scholar
Powell, M. J. D., ‘A method for nonlinear constraints in minimization problems’, in1969 Optimization (Sympos., Univ. Keele, Keele, 1968) (ed. Fletcher, R.) (Academic Press, New York, NY, 1969), 283–298.Google Scholar
Pietrzak, G., ‘Continuum mechanics modelling and augmented Lagrangian formulation of large deformation frictional contact problems’, PhD Thesis, EPFL, 1997.Google Scholar
Prékopa, A., ‘On the development of optimization theory’, Amer. Math. Monthly 87(7) (1980), 527–542.Google Scholar
Rockafellar, R. T., ‘Convex functions and dual extremum problems’, PhD Thesis, Harvard University, Cambridge, MA, 1963.Google Scholar
Rockafellar, R. T., ‘Characterization of the subdifferentials of convex functions’, Pacific J. Math. 17 (1966), 497–510.Google Scholar
Rockafellar, R. T., ‘Convex functions, monotone operators and variational inequalities’, inTheory and Applications of Monotone Operators (ed. Ghizzetti, A.) (Tipografia Oderisi Editrice, Gubbio, Italy, 1968), 34–65.Google Scholar
Rockafellar, R. T., Convex Analysis (Princeton, NJ, 1970).Google Scholar
Rockafellar, R. T., ‘Integrals which are convex functionals. II’, Pacific J. Math. 39 (1971), 439–469.Google Scholar
Rockafellar, R. T., ‘Penalty methods and augmented Lagrangians in nonlinear programming’, inFifth Conference on Optimization Techniques (eds. Conti, R. and Ruberti, A.) (Springer, Berlin, 1973), 518–525.Google Scholar
Rockafellar, R. T., ‘Augmented Lagrange multiplier functions and duality in nonconvex programming’, SIAM J. Control 12 (1974), 268–285.Google Scholar
Rockafellar, R. T., ‘Monotone operators and the proximal point algorithm’, SIAM J. Control Optim. 14 (1976a), 877–898.Google Scholar
Rockafellar, R. T., ‘Augmented Lagrangians and applications of the proximal point algorithm in convex programming’, Math. Oper. Res. 1 (1976b), 97–116.Google Scholar
Rockafellar, R. T., ‘Monotone operators and augmented Lagrangian methods in nonlinear programming’, inNonlinear Programming 3 (ed. Mangasarian, O. L. et al. ) (Academic Press, New York–London, 1978), 1–25.Google Scholar
Rockafellar, R. T., ‘Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization’, Math. Oper. Res. 6 (1981), 427–437.Google Scholar
Rockafellar, R. T., ‘Lagrange multipliers and optimality’, SIAM Rev. 35(2) (1993), 183–238.Google Scholar
Rockafellar, R. T. and Wets, R. J.-B., Variational Analysis, Grundlehren der Mathematischen Wissenschaften, 317 (Springer, Berlin, 1998).Google Scholar
Selig, J. N., ‘Cayley maps for SE (3)’, inProceedings of the 12th IFToMM World Congress, Besançon, June 18–21, 2007 (eds. Merlet, J.-P. and Dahan, M.) (2007), paper A270.Google Scholar
Signorini, S., ‘Sopra alcune questioni di elastostatica’, Atti della Societa Italiana per il Progresso delle Scienze (1933).Google Scholar
Simó, J. C., ‘A finite strain beam formulation. The three-dimensional dynamic problem. Part I’, Comput. Methods Appl. Mech. Engrg. 49(1) (1985), 55–70.Google Scholar
Simó, J. C., Marsden, J. E. and Krishnaprasad, P. S., ‘The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods, and plates’, Arch. Ration. Mech. Anal. 104(2) (1988), 125–183.Google Scholar
Studer, C., Leine, R. I. and Glocker, C., ‘Step size adjustment and extrapolation for time-stepping schemes in non-smooth dynamics’, Internat. J. Numer. Methods Engrg. 76 (2008), 1747–1781.Google Scholar
Taylor, R. L. and Papadopoulos, P., ‘On a finite element method for dynamic contact/impact problems’, Internat. J. Numer. Methods Engrg. 36 (1993), 2123–2140.Google Scholar
Thomson, W. and Tait, P. G., Treatise on Natural Philosophy, Vol. 1 (Oxford University Press, Oxford, 1867).Google Scholar
Vankerschaver, J., Liao, C. and Leok, M., ‘Generating functionals and Lagrangian partial differential equations’, J. Math. Phys. 54(8) (2013), 082901, 22 pp.Google Scholar
Wierzbicki, A. P., ‘A penalty function shifting method in constrained static optimization and its convergence properties’, Archiw. Autom. Telemech. 16 (1971), 395–416.Google Scholar
Wriggers, P., Computational Contact Mechanics, 2nd edn (Springer, Berlin, Heidelberg, 2006).Google Scholar
Wriggers, P. and Laursen, T. A. (Eds.), Computational Contact Mechanics, CISM Courses and Lectures, 498 (Springer, Wien–New York, 2007).Google Scholar
Yosida, K., Functional Analysis (Springer, New York, 1965).Google Scholar