Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Yang Gao David 2001Complementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systemsPhil. Trans. R. Soc. A.3592347–2367http://doi.org/10.1098/rsta.2001.0855SectionRestricted accessComplementarity, polarity and triality in non‐smooth, non–convex and non–conservative Hamilton systems David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author David Yang Gao David Yang Gao Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA () Google Scholar Find this author on PubMed Search for more papers by this author Published:15 December 2001https://doi.org/10.1098/rsta.2001.0855AbstractThis paper presents a unified critical–point theory in non–smooth, non–convex and dissipative Hamilton systems. The canonical dual/polar transformation methods and the associated bi–duality and triality theories proposed recently in non–convex variational problems are generalized into fully nonlinear dissipative dynamical systems governed by non–smooth constitutive laws and boundary conditions. It is shown that, by this method, non–smooth and non–convex Hamilton systems can be reformulated into certain smooth dual, complementary and polar variational problems. Based on a newly proposed polar Hamiltonian, a nice bipolarity variational principle is established for three–dimensional non–smooth elastodynamical systems, and a potentially powerful complementary variational principle can be used for solving unilateral variational inequality problems governed by non–smooth boundary conditions. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Qiu Z and Xia H (2021) Symplectic perturbation series methodology for non-conservative linear Hamiltonian system with damping, Acta Mechanica Sinica, 10.1007/s10409-021-01076-0, 37:6, (983-996), Online publication date: 1-Jun-2021. Gao D and Ali E (2019) A Novel Canonical Duality Theory for Solving 3-D Topology Optimization Problems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_13, (209-246), . Gao D (2019) Canonical Duality-Triality Theory: Unified Understanding for Modeling, Problems, and NP-Hardness in Global Optimization of Multi-Scale Systems Advances in Mathematical Methods and High Performance Computing, 10.1007/978-3-030-02487-1_1, (3-50), . 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Pfeiffer Article InformationDOI:https://doi.org/10.1098/rsta.2001.0855Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online15/12/2001Published in print15/12/2001 License: Citations and impact Keywordsnon–convex variational problemsnon–smooth elastodynamicstrialitypolaritynon–conservative Hamilton systemscomplementarity