Quadratically convergent direct calculation of critical points for 3d structures undergoing finite rotations Original Research Article

In this work we present the implementation details of a quadratically converging, Newton-method-based algorithm for direct computation of instability points for 3d structures undergoing finite rotations. The structural model chosen for illustration is the 3d geometrically exact beam. The proposed algorithm makes use of an extended system, where equilibrium equations are supplemented with the loss-of-stability condition which roughly doubles the total number of equations. Nonetheless, the latter requires only an insignificant increase in computational cost due to judicious use of the bordering algorithm for computing the solution. The main thrust of our work is directed towards a careful development of linearized forms of the governing equations employed by Newtonʹs method. The corresponding results are presented both in material and spatial versions. A set of numerical examples is used to illustrate a very satisfying performance of the proposed algorithm.