Free-form optimization for controlling time-dependent responses of shell structures

In this study, we propose a parameter-free free-form optimization method to control the time-dependent response of shell structures. The design objective is to minimize the dynamic compliance or to control the dynamic responses at arbitrary domains and times for an arbitrary time-dependent loading under volume constraint. The unsteady optimum design problems are formulated as a distributed-parameter shape optimization problem based on the variational method, and the shape gradient functions with respect to the shape variation in the normal direction to the surface are derived based on the material derivative and the adjoint variable methods. The shape gradient functions theoretically derived are applied to the H¹ gradient method for shells, a gradient method in the function space. With the proposed approach, the optimum free form of the shell structures for the time-dependent response problems can be obtained while minimizing the objective functional and maintaining the smoothness of the form. Several shape design examples under a continuous dynamic force or an impulse force are demonstrated to show the effectiveness of the proposed method, and the results are discussed.