Equilibrium Path Analysis Including Bifurcations with an Arc-Length Method Avoiding A Priori Perturbations

Wrinkling or pattern formation of thin (floating) membranes is a phenomenon governed by buckling instabilities of the membrane. For (post-) buckling analysis, arc-length or continuation methods are often used with a priori applied perturbations in order to avoid passing bifurcation points when traversing the equilibrium paths. The shape and magnitude of the perturbations, however, should not affect the post-buckling response and hence should be chosen with care. In this paper, our primary focus is to develop a robust arc-length method that is able to traverse equilibrium paths and post-bifurcation branches without the need for a priori applied perturbations. We do this by combining existing methods for continuation, solution methods for complex roots in the constraint equation, as well as methods for bifurcation point indication and branch switching. The method has been benchmarked on the post-buckling behaviour of a column, using geometrically non-linear isogeometric Kirchhoff-Love shell element formulations. Excellent results have been obtained in comparison to the reference results, from both bifurcation point and equilibrium path perspective.