A Lie-Group Shooting Method for Post Buckling Calculations of Elastica
Chein-Shan Liu

doi:10.3970/cmes.2008.030.001
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 30, No. 1, pp. 1-16, 2008
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Keywords Elastica, Post-Buckling, Multiple solutions, Lie-Group shooting method
Abstract In this paper we propose a new numerical integration method of second-order boundary value problems (BVPs) resulting from the elastica of slender rods under different loading conditions and boundary conditions. We construct a compact space shooting method for finding unknown initial conditions. The key point is based on the construction of a one-step Lie group element${\bf G}(T)$ and the establishment of a generalized mid-point Lie group element${\bf G}(r)$ by using the mean value theorem. Then, by imposing${\bf G}(T)={\bf G}(r)$ we can search the missing initial condition through a closed-form solution in terms of the weighting factor$r \in (0,1)$. The Lie-group shooting method is very effective for large deflection problems of elastica even exhibiting multiple solutions.
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