Shape optimization of space frame structures for maximizing a natural frequency

Abstract

This paper presents a non-parametric, or a node-based, shape optimization method for designing the optimal geometry of a space frame structure for a natural frequency problem. For a frame structure composed of arbitrarily curved linear elastic members, a natural frequency maximization problem for a specified mode subjected to a volume constraint is formulated as a distributed-parameter shape optimization problem. With the eigenvalue of a specified mode as the objective functional and the assumption that each frame member varies in the out-of-plane direction to the centoroidal axis, the shape gradient function and the optimality conditions are theoretically derived by the Lagrange multiplier method and the material derivative method. An optimal geometry is determined by applying the negative shape gradient function as fictitious external forces to the frame members and analyzing an optimal variation that minimizes the objective functional. This methodology was developed by the authors as the free-form optimization method for frame structures, which is a gradient method in a Hilbert space. For achieving the eigenvalue of a specified mode to be maximized, multiple root problem is avoided by tracking the specified mode through the optimization process using Modal Assurance Criterion (MAC). The effectiveness of the proposed method were verified through several design problems.