Widening wave band gaps of periodic plates via shape optimization using spatial Fourier coefficients. (15th January 2021) - Dal Poggetto, Vinícius Fonseca; Arruda, José Roberto de França
Highlights: Two-dimensional Fourier series can be used to describe smooth periodic plate shapes. Shape optimization can be applied using a constrained nonlinear formulation. Resulting shapes yield low-frequency wide band gaps. The same shapes result using Kirchhoff's or Mindlin's theories for small thickness. Resulting shapes are related to which wave modes are separated by the band gap. Abstract: Periodic media have been shown to exhibit wave attenuation in frequency ranges called band gaps. The challenge is designing feasible periodic structures that present total band gaps in the low-frequency range. Plate structures are two-dimensional media that can benefit from the application of these concepts. They can be modeled using either Kirchhoff's or Mindlin's plate theories. Since the periodic properties of plates (geometry and material properties) can be described by a two-dimensional spatial Fourier series, it should be possible to optimize its configuration using the series coefficients. The Fourier series representation is commonly used to compute the dispersion diagrams via the plane wave expansion (PWE) method. In this work, the Fourier series coefficients that describe the spatial distribution of the plate properties are used as optimization variables to obtain solutions that maximize an objective function capable of yielding low-frequency band gaps. In particular, the spatial distribution of the plate thickness is described by a two-dimensional Fourier series. Its coefficients are optimized with constraints on the minimum and maximum values to achieve the widening of low-frequency band gaps. Results show feasible solutions for several values of minimum and maximum thicknesses using Kirchhoff's and Mindlin's plate formulations.