Parameter-free shape identification method for frame structure with desired static deformation

Abstract

In this paper, a free form optimization method is proposed for achieving a desired static deformation of a frame structure. Deformation control is one of the important problems in stiffness design of frame structures, and it also enables to give a function to frame structures. As an objective functional, we introduce the sum of squared error norms to the desired displacements on specified members, and the assumption that each frame member varies in the off-axis direction. The shape gradient function and the optimality conditions for this problem are theoretically derived with the Lagrange multiplier method and the material derivative method. The optimal shape variation that minimizes the objective functional is determined as the displacement-field by applying the negative shape gradient function as fictitious external forces to the frame members. With the proposed method, the optimal arbitrarily formed frame structure can be obtained without any shape parameterization while maintaining the smoothness. The validity and practical utilities of this method for the static deformation control of frame structures are verified through design examples.