Sensitivity analysis to geometrical imperfections in shell buckling via a mixed generalized path-following method

An efficient continuation method is proposed for a direct sensitivity analysis of the critical point to geometrical imperfections, e.g. expressed as combination of given shapes, for thin-walled structures prone to buckling. After a finite element discretization, the critical point is defined by a system of nonlinear algebraic equations imposing equilibrium and critical condition according to the null vector method. An arc-length equation is added to follow a path of critical points in the imperfection space. Remarkable novelties are achieved. Firstly, the Jacobian of the extended system is entirely computed analytically by means of a solid-shell model and a strain-based modeling of the geometrical deviation. Moreover, a mixed scheme with independent element-wise stress variables is devised for a more efficient and robust iterative solution compared to the standard null vector method. The mixed algorithm speeds up the sensitivity analysis allowing larger imperfection steps and much fewer factorizations of the condensed stiffness matrix, only one per imperfection in its modified version with constant matrix over the step. Finally, the critical point derivatives with respect to the imperfection parameters are also obtained analytically and can be used to generate a gradient-based critical path for a quick search of the worst-case imperfection.