Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters

Abstract

<p style='text-indent:20px;'>In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.</p>

Wave propagation in viscoelastic metamaterials via added-state formulation

“…Hereinafter, the forced responses φ(τ, b) are investigated separately for punctual forces ψ u , ψ v or punctual couples ψ ϕ and different values of the wavevector b, particularizing the stiffness matrix Σ(b), by direct time integrations of Eq. (33). Moreover, the added-state formulation, which by-passes the computational issues related to the presence of convolution integrals in the equations of motion, is exploited to directly assess in time the dynamics of the viscoelastic states.…”

“…According to the latter approach, the optimal solution in the multidimensional space of the design parameters is numerically identified by minimizing or maximizing a suited objective or multi-objective function [28][29][30][31]. Generally, a proper mathematical surrogation of the objective function or a proper dimensionality reduction of the parameter space may help in reducing the computational costs and accelerating the algorithmic convergence [32][33][34]. In the presence of dissipation, the mechanical energy trapped by the propagation-inhibiting local mechanisms is irreversibly absorbed by dampers or -if convenient -extracted by harvesters.…”