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

• Published in: International journal of non-linear mechanics Vol. 47; no. 2; pp. 240 - 247
• Main Authors: Santos, H.A.F.A., Gao, D.Y.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2012
• Subjects: Approximation; Beams (structural); Canonical dual finite element method; Differential equations; Finite element method; Global optimization; Large deformations; Mathematical analysis; Mathematical models; Maximization; Non-convex variational problem; Non-linear elastic beam; Nonlinearity; Global optimization; Non-convex variational problem; Non-linear elastic beam; Canonical dual finite element method; Large deformations
• ISSN: 0020-7462; 1878-5638
• DOI: 10.1016/j.ijnonlinmec.2011.05.012

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. ► 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. ► Numerical applications are illustrated by several problems with different boundary conditions.