Micropore shape optimization for natural vibration design of porous structures

In this paper, we present a shape optimization method for periodic microstructures to maximize a specified vibration eigenvalue of a porous macrostructure. The homogenized elastic moduli calculated by the homogenization method are applied to the macrostructure to connect the microstructures with the macro structure. The KS function is introduced to solve the repeated eigenvalue problem hidden in vibration eigenvalue optimization. The shape optimization problem subject to the volume constraint considering the microstructures is formulated as a distributed-parameter optimization problem, and the shape gradient function is derived by the Lagrange multiplier method and the adjoint variable method. The shape gradient function is applied as a distributed force to update the design boundaries of the unit cells of the microstructures by the H1 gradient method. The smooth boundary shapes obtained by the H1 gradient method are suitable for manufacturing with a 3D printer. In the numerical examples, the eigenvalues and the optimum shapes were compared changing the number of the domains of the microstructures in the macrostructure. As a result, the effectiveness of shape optimization method for microstructures aimed at maximizing the vibration eigenvalue of a macrostructure was confirmed.