A Comprehensive Framework for Design and Optimization of Graded Metamaterials Subjected to Concentrated Bandwidth Excitations

Abstract This paper presents a comprehensive framework for the efficient design and optimization of graded mechanical metamaterials subjected to finite-bandwidth stochastic excitations. This task presents two significant challenges: the high computational cost of high-fidelity finite-element (FE) analyses and the high dimensionality of the design space for finite structures. The proposed framework overcomes these challenges by integrating three key components. First, a surrogate reduced-order model—based on a multiregion, Craig-Bampton component mode synthesis and a principal component analysis (PCA)/kriging predictor—is employed to rapidly and accurately predict the substructural reduced-order matrices. Second, a state-space stochastic excitation model is used, enabling the system’s performance (response variance) to be solved efficiently via the Lyapunov equation. Third, graded metamaterial parameterizations are used to reduce the high-dimensional geometry problem to a low-dimensional design vector. Bayesian optimization (BO) is then used to find the optimal grading parameters that minimize the vibration amplification. The framework is demonstrated on a two-dimensional (2D) metamaterial, where the surrogate model accuracy in predicting the stochastic performance is validated. The results reveal a critical trade-off between performance and computational cost. The optimal design parameterization is shown to be highly dependent on the excitation bandwidth. The framework is proven to be an effective tool for selecting the best design and optimization strategy under realistic stochastic conditions.