Concurrent multiscale optimization method for natural vibration design of porous structures

Abstract

This paper proposes a concurrent multiscale optimization method of macrostructure topology and microstructure shapes for porous structures, aiming at maximizing the vibration eigenvalue. The multi-material distribution of the macrostructure and the shape of the microstructures are optimized by topology and shape optimization respectively. The homogenized properties of the porous materials are calculated using the homogenization method, and the homogenized elastic tensor and density are applied to the macrostructure. The optimum distribution of the porous material in the macrostructure is determined by a multi-material topology optimization using the GSIMP method. The KS function is introduced to solve the repeated-root problem that lies in the maximization of vibrational eigenvalues. The area-constrained multiscale optimization problem is formulated as a distributed parameter optimization problem, and the sensitivity functions are derived using the Lagrange multiplier method and the adjoint variable method. Based on the obtained sensitivity functions, the design variables are updated using the H1 gradient method. The effectiveness of the proposed method for maximizing vibration eigenvalues is confirmed through numerical examples.