A numerical study of elastica using constrained optimization methods

Abstract

Post-buckling behaviour of Elastica is a typical geometric nonlinear problem. Such geometric nonlinear elastic problem provides fundamental model to more complicated problems such as the study of DNA and nano-tubes. In this work, the elastica is discretized into rigid segments linked by elastic rotational springs. By identifying the global or local minima of potential energy of discrete elastica, the configurations of elastica are obtained using the constraints satisfying optmization methods. Genetic algorithm is the mechanism to produce a coarse solution, while sequential quadratic programming is the main numerical tool to generate the final solution. In addition, the post-buckling behaviour of elastica under certain boundary conditions and constraints is explored using the path-following strategy. Extensive numerical studies of elastica under different boundary conditions are carried out. As a result, systematical presentations of post-buckling behaviours of planar elastica are obtained. Instability of pin-pin elastica is examined at the critical point. The comparison of numerical results with analytical solutions shows that the proposed method is plausible. To demonstrate the universality of this energy-based method, both the out-of-plane buckling of spatial elastica and constrained Euler buckling are addressed using the same algorithm. It is demonstrated that the model and method proposed in this work can be easily extended to various applications.