Non-parametric shape optimization method for natural vibration design of stiffened shells

• A non-parametric shape optimization method is proposed for natural vibration problems of stiffened thin-walled structures.
• Solutions are given to deal with a specified eigenvalue maximization problem and its reciprocal volume minimization problem.
• Mesh smoothing is implemented simultaneously with shape updating.
• It can be easily implemented in combination with a commercial FEM code.

In this paper, we newly present an effective shape optimization method for natural vibration design of stiffened thin-walled or shell structures. Both the stiffeners and their basic structures are optimized by solving two kinds of optimization problems. The first is a specified eigenvalue maximization problem subject to a volume constraint, and the second is its reciprocal volume minimization problem subject to a specified eigenvalue constraint. The boundary shapes of a thin-walled structure are determined under the condition where the stiffeners and the basic structure are movable in the in-plane direction to their surface. Both problems are formulated as distributed-parameter shape optimization problems, and the shape gradient functions are derived using the material derivative method and the adjoint variable method. The optimal free-boundary shapes are determined by applying the derived shape gradient function to the H1H1 gradient method for shells, which is a parameter-free shape optimization method proposed by one of the authors. Several design examples are presented to validate the proposed method and demonstrate its practical usages.