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

Abstract

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.