Isogeometric sizing and shape optimisation of beam structures

Based on the recently proposed framework of isogeometric analysis, a study of structural sizing and shape optimisation of curved beam structures is presented. The idea of the isogeometric approach is to analyse the structure using its intrinsic and hence exact geometric definition. The structural response is approximated using the same basis functions describing the geometry. The presented isogeometric framework employs non-uniform rational B-splines. Shape changes can be represented by altering both spatial location of control points and the corresponding weights towards the optimal design. Adaptation of identical geometric representations for both design and analysis resolves the problem of repetitive conversion between the computer-aided and facet based geometries used in conventional finite element analysis. Sizing variables defined at the control points are interpolated using the same basis functions. Due to the fact that both analysis, sizing and shape design share the same underlying description, direct sensitivities can be easily evaluated analytically which improves the performance of the optimisation process. While sizing should reflect local stress states, shape design is preferably performed on a global level. Thus, a multilevel design approach is proposed, where shape design is carried on a coarser level. Projecting the shape design sensitivities bridges the gap between the different levels. The proposed design formulation is applied to compliance minimisation problems. Numerical results indicate the applicability of the method.