Tailoring static deformation of frame structures based on a non-parametric shape–size optimization method

Abstract

In this study, a non-parametric shape–size optimization method is developed for tailoring the static deformation of large-scale frame structures. This deformation control design is one of the important problems in the stiffness design of frame structures, and enables us to create a smart or a high performance structure for a specific ability of deformation. As the objective functional, we introduce the sum of squared error norms for achieving the desired displacements on specified members, and assume that each frame member varies in the off-axis direction with changing cross sections. The shape gradient function, the size gradient function, and the optimality conditions for this problem are theoretically derived with the Lagrange multiplier method, the material derivative method, and the adjoint variable method. The optimal shape–size variations that minimize the objective functional are determined by using the H1 gradient method for frame structures. With the proposed method, the optimal arbitrarily formed frame structures with the optimal cross sections can be obtained without any shape and size parameterization while maintaining their smoothness. The validity and practical utility of this method for tailoring the static deformation of frame structures are verified through design examples.