Isogeometric sizing and shape optimization of 3D beams and lattice structures at large deformations

A computational method for optimizing the shape of the centerline curve and the spatial variation of geometric and material sizing parameters of the cross-sections of elastic, 3-dimensional beams and beam structures subject to large deformations is presented in this work. The approach is based on the concept of isogeometric analysis, i.e., the representation of geometry and the discretization of the numerical solution using spline functions. Here, mixed isogeometric collocation methods are used to discretize the geometrically exact 3D beam model. These spline representations are extended to the parameterization of the design variables, which are the initial centerline curves of the beams, as well as cross-sectional sizing properties, which may be varying along the beam axis and can be functionally graded through the cross-sections. To tailor the mechanical deformation behavior of a beam or beam structure, a nonlinear optimization problem is formulated and solved using gradient-based methods. For this purpose, all required gradients and sensitivities are derived analytically. The potential of this holistic design optimization approach is demonstrated in application to tailoring of elastic metamaterials and beam lattice structures, as well as 4D printing of multi-material laminate beams.