  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.
