Global optimal solutions to general sensor network localization problem

The dynamic contact of two nonlinear Gao beams that are connected with a joint is modeled, analyzed, and numerically simulated. Contact is modeled with either (i) the normal compliance condition, or (ii) the unilateral Signorini condition. The model is in the form of a variational equality in case (i) and a variational inequality in case (ii). The existence of the unique variational solution is established for the problem with normal compliance and the existence of a weak solution is proved in case (ii). The solution in the second case is obtained as a limit of the solutions of the first case when the normal compliance stiffness tends to infinity. A numerical algorithm for the problem is constructed using finite elements and a mixed time discretization. Simulation results, based on the implementation of the algorithm, of the two cases when the horizontal traction vanishes or when it is sufficiently large to cause buckling, are presented. The spectrum of the vibrations, using the FFT, shows a rather noisy system.

Dynamic contact of two GAO beams

The post-buckling problem of a large deformed beam is analyzed using the canonical dual finite element method (CD-FEM). The feature of this method is to choose correctly the canonical dual stress so that the original non-convex potential energy functional is reformulated in a mixed complementary energy form with both displacement and stress fields, and a pure complementary energy is explicitly formulated in finite dimensional space. Based on the canonical duality theory and the associated triality theorem, a primal–dual algorithm is proposed, which can be used to find all possible solutions of this non-convex post-buckling problem. Numerical results show that the global maximum of the pure-complementary energy leads to a stable buckled configuration of the beam, while the local extrema of the pure-complementary energy present unstable deformation states. We discovered that the unstable buckled state is very sensitive to the number of total elements and the external loads. Theoretical results are verified ...

Post-buckling solutions of hyper-elastic beam by canonical dual finite element method

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. Zalinescu and his coworker are addressed. The paper is ended with some open problems and future works in global optimization and nonconvex mechanics.

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

Boundary-value problems in solid mechanics are often addressed, from both theoretical and numerical points of view, by resorting to displacement/rotation-based variational formulations. For conservative problems, such formulations may be constructed on the basis of the Principle of Stationary Total Potential Energy. Small deformation problems have a unique solution and, as a consequence, their corresponding total potential energies are globally convex. In this case, under the so-called Legendre transform, the total potential energy can be transformed into a globally concave total complementary energy only expressed in terms of stress variables. However, large deformation problems have, in general, for the same boundary conditions, multiple solutions. As a result, their associated total potential energies are globally non-convex. Notwithstanding, the Principle of Stationary Total Potential Energy can still be regarded as a minimum principle, only involving displacement/rotation fields. The existence of a maximum complementary energy principle defined in a truly dual form has been subject of discussion since the first contribution made by Hellinger in 1914. This paper provides a survey of the complementary energy principles and also accounts for the evolution of the complementary-energy based finite element models for geometrically non-linear solid/structural models proposed in the literature over the last 60 years, giving special emphasis to the complementary-energy based methods developed within the framework of the geometrically exact Reissner-Simo beam theory for the analysis of structural frames.

Complementary-Energy Methods for Geometrically Non-linear Structural Models: An Overview and Recent Developments in the Analysis of Frames

This paper presents a canonical dual mixed finite element method for the post-buckling analysis of planar beams with large elastic deformations. The mathematical beam model employed in the present work was introduced by Gao in 1996, and is governed by a fourth-order non-linear differential equation. The total potential energy associated with this model is a non-convex functional and can be used to study both the pre- and the post-buckling responses of the beams. Using the so-called canonical duality theory, this non-convex primal variational problem is transformed into a dual problem. In a proper feasible space, the dual variational problem corresponds to a globally concave maximization problem. A mixed finite element method involving both the transverse displacement field and the stress field as approximate element functions is derived from the dual variational problem and used to compute global optimal solutions. Numerical applications are illustrated by several problems with different boundary conditions.

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space This formulation employs as approximate functions the exact statically admissible force fields, ie , those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space

H.A.F.A. Santos

Papers

Canonical dual finite element method for solving post-buckling problems of a large deformation elastic beam

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables