The isogeometric paradigm is aimed at unifying the geometric and analysis descriptions of engineering problems. This unification is brought about by employing the same basis functions describing the geometry to approximate the physical response. Non-uniform rational B-splines (NURBS) are commonly used for this purpose and are adopted in the present work for the design of elastic arches. Design for optimal shape and stiffness distribution is considered. Manufacturing constraints are imposed on shape and sizing variables. Shape changes are represented by altering spatial location of the control points and the associated weights. Sizing variables, that control the stiffness distribution, are defined at the control points and interpolated using the same spline basis functions. Since analysis, sizing, and shape design share the same underlying description, consistent discrete sensitivities can be easily evaluated analytically, greatly improving the performance of the optimisation process. While sizing should reflect the influence of local stress states, shape design is preferably performed at a global level. Thus, a multilevel approach is utilised, where shape design is carried out at a coarser level. Projecting the shape design sensitivities bridges the gap between the different levels. A variational formulation of essential manufacturing constraints for sizing and shape optimal design is introduced. The design framework is applied to fundamental frequency maximisation problems.
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