Non-parametric free-form optimal design of frame structures in natural frequency problem

Optimal design of structures with respect to their mechanical behavior is essential and basically required in structural engineering. In this study, we propose a non-parametric free-form optimization method based on the variational method to design frame structures composed of arbitrarily curved linear elastic members. The natural frequency maximization problem of frame structures is formulated as a non-parametric shape optimization problem under the volume constraint. Under the assumption that each member varies in the out-of-plane direction to its centroidal axis, the shape gradient functions and the optimality conditions are theoretically derived by the Lagrange multiplier method and the formulae of the material derivative. Then, the derived shape gradient functions are applied to a gradient method in the Hilbert space with a P.D.E (Partial Differential Equation) smoother, which is referred as the H1 gradient method for frame structures. Moreover, a simple switching technique of the objective functional is presented for overcoming the discontinuity problem of repeated eigenvalues, which often appears in natural frequency maximization problem. With this combination of the three techniques, the optimal free-form frame structures owning smoothly curved members can be obtained without any preliminary shape parameterization, and the effectiveness and validity of the proposed method are verified through three design examples.