Multiscale Shape Optimization method for Vibration Design of Laminated Porous Shell Structure
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In this study, we propose a concurrent macro and micro shape optimization method for controlling the vibration characteristics of a laminated porous shell structure. The laminated shell has several sub-domains with independent micro-pores layer by layer, and the shapes of macro-shell surface and the micro-pores are optimized by the out-of-plane and by the in-plane shape variations, respectively. The homogenization method is used to bridge the macro-shell and the micro-pores in this multi-scale optimization. The natural frequency and frequency response problems are treated as the vibration design problem, and the solutions to the both problems are presented. A specified natural frequency is maximized in the natural frequency problem or a squared error norm to the target amplitudes at specified frequencies is minimized in the frequency response problem, under the total volume constraint. The state equation of the macro-shell for each problem and the homogenization equations of the unit cells are also used as the constraints. The both shape optimization problem are formulated as a distributed-parameter shape optimization problem, and the shape gradient functions for the macro-shell and the micro-pores are theoretically derived using the adjoint variable method and the material derivative method. The shape gradient functions derived are applied to the vector-type H1 gradient method, a gradient method in the function space, to optimize the shapes of macro-shell and micro-pores, while minimizing the objective function. The validity of the proposed multi-scale shape optimization method is confirmed by several numerical examples for concurrently designing the optimal macro-shell surface and the optimal micro-pore shape in each sub-domain of a laminated shell structure. Furthermore, we confirm that the volume was efficiently distributed among the sub-domains. With the proposed method, the optimal light weight laminated shell structure with smooth macro-shell surface and smooth micro-pore boundary can be created efficiently, while controlling the vibration characteristics.
