Constrained isogeometric design optimization of lattice structures on curved surfaces: computation of design velocity field

A constrained isogeometric design optimization method is presented for lattice structures located on a specified curved surface. Lattices on curved surfaces have been utilized in various engineering applications like medical stents, non-pneumatic wheel frames, and so on. When it comes to design problems, however, the lattice needs to be located on a specified surface for its manufacturability as well as performance, which results in nonlinear constraints in configuration design. We define lattice structures and their design variables on planar rectangular surfaces, and utilize the concept of free-form deformation (FFD) and the global curve interpolation to obtain the analytical expressions for the control net of lattice structure on curved surfaces. The material derivative of the analytical expressions eventually leads to precise design velocity field. The analytical configuration design sensitivity for a spatial Timoshenko beam is derived. In numerical examples, we verify the derived configuration design sensitivity and the developed computation scheme of design velocity field by comparison with finite differences, and several configuration design optimization examples are demonstrated.