Complete Solutions to Post-Buckling Problems of Large Deformed Elastic Beam
David Gao (Univ Ballarat/ANU)
Instability in Solids and Structures
Mon 10:45 - 12:15
Barus-Holley 190
Post-buckling of large deformed beam provides one of fundamental challenging problems in engineering sciences. Mathematically speaking, the post-bifurcation is mainly due to the nonconvexity of total potential energy. Numerical discretization of any nonconvex variational problem leads to a global optimization problem, which could possess many local extrema. It is known in computational science that most of nonconvex optimization problems are considered as NP-hard [1] and can't be solved by traditional direct approaches. Unfortunately, this well-known fact in computer science and global optimization is not fully recognized in computational mechanics. In this talk, the speaker will first show that the well-known von Karman equation is actually linear in one-dimensional problems and can’t be used for study post-buckling phenomenon. Based on the real large deformation beam model proposed in 1996, the speaker will show that if the axial load is big than the Euler buckling load, the total potential of this model is a nonconvex (double-well) functional which can be used for modeling post-buckling phenomenon. By using mixed finite element method, the associated nonconvex variational problem is discretized as a global optimization problem, which can be solved completely by the canonical duality theory developed by the speaker over the past 10 years. The speaker will show that the global minimizer and local maximizer are stable, while the local minimizer is very sensitive to both numerical discretization and external loads. References: [1] D.Y. Gao, and H.D. Sherali, 2008. Canonical duality: Connection between nonconvex mechanics and global optimization, in Advances in Appl. Mathematics and Global Optimization, pp.249-316, Springer, 2009. [2] K. Cai, D.Y. Gao, Q.H. Qin (2013) Post-buckling Solutions of Hyper-elastic Beam by Canonical Dual Finite Element Method, http://arxiv.org/abs/1302.4136