Abstract:
Duality is one of the oldest and most beautiful concepts in human knowledge with a simple origin from the oriental philosophy tracing back 5000 years. Canonical duality theory [1,2] is a newly developed, breakthrough powerful methodology, which can be used to model complex systems with a unified solution to a wide class of discrete and continuous problems in engineering sciences. The associated triality theory reveals an interesting multi-scale duality pattern in complex systems, which can be used to identify both global and local extrema and to design powerful algorithms for solving challenging problems in computational mechanics.
In this talk, the speaker will first present some fundamental principles for modeling complex systems. Based on the definitions of objectivity and canonical duality in continuum physics, he will show why the complex systems can be modelled within a unified framework, how the canonical duality theory is naturally developed and the fundamental reasons that lead to challenging problems in different fields, including chaotic dynamics, phase transitions of solids, multi-solutions in post-buckling analysis and NP-hard problems in computational sciences. By using the phase transitions of the Ericksen's bar, he will show magic to obtain a unified analytic solution for general finite deformation problems and to identify both global and local optimality conditions from infinitely many local solutions. A movie will illustrate a truth that for many nonconvex potential variational problems, the global optimal solutions are usually nonsmooth, and cannot be captured by any traditional Newton-type direct approaches[3]. Applications will be illustrated by certain well-known challenging problems in finite deformation theory (such as phase transitions and control of chaotic systems) as well as NP-hard problems in computational mechanics (such as topology optimization and post-buckling of large deformed beam). A set of complete analytical solutions to 3-D nonlinear elasticity will be presented [4,5]. Finally, some open problems and possible methodologies will be addressed.
This talk will bring some fundamentally new insights into modern mechanics, complex systems, and computational science.
References:
[1] Gao, D.Y. (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications. Kluwer Academic Publishers, Boston/Dordrecht/London, 2000, xviii+454pp.
[2] Gao, D.Y., Ruan, N., and Latorre, V. (2015). Canonical duality-triality: Bridge between nonconvex analysis/mechanics and global optimization, Math. Mech. Solids.
[3] Gao, D.Y. and Ogden, R.W. (2008) Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation, Quarterly J. Mech. Appl. Math. . 61 (4), 497-522
[4] Gao, D.Y. (2015). Analytic solutions to general anti-plane shear problems in finite elasticity, Continuum Mech. Thermodyn
[5] Gao, D.Y. and Hajilarov, E. On analytic solutions to 3-d finite deformation problems governed by St Venant–Kirchhoff material. Math. Mech. Solids (2015)
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