On SDP Method for Solving Canonical Dual Problem in Post Buckling of Large Deformed Elastic Beam

TL;DR

This paper introduces a novel primal-dual semi-definite programming algorithm based on canonical duality to solve nonconvex post-buckling problems of large elastic beams, enabling the identification of multiple stable and unstable solutions.

Contribution

It develops a new canonical dual finite element method and PD-SDP algorithm for accurately solving all possible post-buckled solutions in large deformed elastic beams.

Findings

Global minimum corresponds to stable buckled configuration.

Local maximum corresponds to unbuckled state.

Local minimum indicates an unstable buckled state sensitive to parameters.

Abstract

This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.