Multi-objective optimal design of mechanical metafilters based on principal component analysis

In this paper, an advanced computational method is proposed, whose aim is to obtain an approximately optimal design of a particular class of acoustic metamaterials, by means of a novel combination of multi-objective optimization and dimensionality reduction. Metamaterials are modeled as beam lattices with internal local resonators coupled with the microstructure through a viscoelastic phase. The dynamics is governed by a set of integro-differential equations, that are transformed into the Z-Laplace space in order to derive an eigenproblem whose solution provides the dispersion relation of the free in-plane propagating Bloch waves. A multi-objective optimization problem is stated, whose aim is to achieve the largest multiplicative trade-off between the bandwidth of the first stop band and the one of the successive pass band in the metamaterial frequency spectrum. Motivated by the multi-dimensionality of the design parameters space, the goal above is achieved by integrating numerical optimization with machine learning. Specifically, the problem is solved by combining a sequential linear programming algorithm with principal component analysis, exploited as a data dimensionality reduction technique and applied to a properly sampled field of gradient directions, with the aim to perform an optimized sensitivity analysis. This represents an original way of applying principal component analysis in connection with multi-objective optimization. Successful performances of the proposed optimization method and its computational savings are demonstrated.