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 multiobjective
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.