Dynamically loaded rigid-plastic analysis under large deformation

Gao, Yang

doi:10.1090/qam/1079916

Cited by 10 publications

(6 citation statements)

References 18 publications

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“…The concepts of geometrical and physical nonlinearities are well-known in continuum physics [30][31][32][33][34][35][36]143], but not in abstract analysis and optimization. This leads to many confusions.…”

Section: Definition 2 (Canonical Function and Canonical Transformation)mentioning

confidence: 99%

Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System

Gao

Ruan

Latorre

2017

Advances in Mechanics and Mathematics

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Canonical duality-triality is a breakthrough methodological theory, which can be used not only for modeling complex systems within a unified framework, but also for solving a wide class of challenging problems from real-world applications. This paper presents a brief review on this theory, its philosophical origin, physics foundation, and mathematical statements in both finite-and infinite-dimensional spaces. Particular emphasis is placed on its role for bridging the gap between nonconvex analysis/mechanics and global optimization. Special attentions are paid on unified understanding the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization, as well as the theorems, methods, and algorithms for solving these challenging problems. Misunderstandings and confusion on some basic concepts, such as objectivity, nonlinearity, Lagrangian, and generalized convexities are discussed and classified. Breakthrough from recent challenges and conceptual mistakes by M. Voisei, C. Zȃlinescu and his coworker are addressed. The paper is ended with some open problems and future works in global optimization and nonconvex mechanics.

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Section: Definition 2 (Canonical Function and Canonical Transformation)mentioning

confidence: 99%

Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System

Gao

Ruan

Latorre

2017

Advances in Mechanics and Mathematics

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“…For example, in modeling of hysteresis, phase transitions, shape-memory alloys, and super-conducting materials, the free energy functions are usually nonconvex due to certain internal variables [28][29][30]. In large deformation analysis, thin-walled structure can buckle even before the stress reaches its elastic limit [31][32][33][34][35][36][37]. Mathematically speaking, many fundamentally difficult problems in engineering and the sciences are mainly due to the nonconvexity of their modeling.…”

Section: Nonconvex Analysis/mechanics and Difficultiesmentioning

confidence: 99%

RETRACTED: Canonical duality–triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex systems

Gao

Ruan

Latorre

2015

Mathematics and Mechanics of Solids

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Canonical duality-triality is a breakthrough methodological theory, which can be used not only for modeling complex systems within a unified framework, but also for solving a wide class of challenging problems from real-world applications. This paper presents a brief review on this theory, its philosophical origin, physics foundation, and mathematical statements in both finite and infinite dimensional spaces. Particular emphasis is placed on its role for bridging the gap between nonconvex analysis/mechanics and global optimization. Special attentions are paid on unified understanding the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization, as well as the theorems, methods, and algorithms for solving these challenging problems. Misunderstandings and confusion on some basic concepts, such as objectivity, nonlinearity, Lagrangian, and generalized convexities are discussed and classified. Breakthrough from recent challenges and conceptual mistakes by M. Voisei, C. Zȃlinescu and his co-worker are addressed. Some open problems and future works in global optimization and nonconvex mechanics are proposed.

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“…This general subdifferential equation governs many finite-deformation nonsmooth systems (see [11][12][13][14][15][16] the Green strain tensor e is defined by the quadratic differential operator A:…”

Section: • A*ow(au) + Of(u)mentioning

confidence: 99%

“…Based on this general theory, a series of complementary variational principles for finite deformation nonsmooth mechanics have been established (see [11][12][13][14][15][16]). Theoretical analysis shows that for finite-deformation systems governed by the linear physical laws (i.e.…”

Section: • A*ow(au) + Of(u)mentioning

confidence: 99%