A general isogeometric polar approach for the optimisation of variable stiffness composites: Application to eigenvalue buckling problems

This study presents a general approach for the multi-scale design of variable stiffness composites (VSCs). The first-level problem of the multi-scale two-level optimisation strategy (MS2LOS) is solved to determine the optimal distribution of the VSC stiffness properties at the macroscopic scale satisfying the requirements of the problem at hand. In this phase, the VSC laminate is modelled as an equivalent homogeneous anisotropic plate whose behaviour is described in terms of polar parameters (PPs), which vary locally over the structure. The First order Shear Deformation Theory is used to take into account the influence of the transverse shear stiffness on the mechanical response of the VSC and Basis Spline (B-Spline) surfaces are employed to represent the PPs fields. In this background, the expression of the gradient of the buckling factor is determined analytically by exploiting the properties of the polar formalism and of the B-Spline surfaces. Moreover, the effect of the discrete variables, involved in the definition of the B-Spline surfaces, on the performances of the optimised solution is investigated. The effectiveness of the approach is proven on two benchmark problems dealing with the maximisation of the first buckling load of a VSC laminate, subject to feasibility and geometric requirements, taken from the literature. The results obtained by means of the MS2LOS based on the polar formalism outperform those reported in the literature, which are obtained through an optimisation strategy based on lamination parameters.