## Date of Award

12-2015

## Document Type

Thesis

## Degree Name

Master of Science (MS)

## Legacy Department

Mechanical Engineering

## Committee Member

Dr. Lonny Thompson, Committee Chair

## Committee Member

Dr. Gang Li

## Committee Member

Dr. Joshua Summers

## Abstract

In this work, the frequency response from two-dimensional polygon and three-dimensional geodesic spheres is numerically simulated using a coupled structural acoustic finite element model. The model is composed of a submerged thin-walled elastic shell structure surrounded by an infinite acoustic air domain. Infinite elements are used to simulate the far-field acoustic radiation condition. Results for the faceted polygon and geodesic sphere are compared to their canonical counterpart viz. a circle/ring and a spherical shell. A unique feature of this study is to compare results as the number of facets in the polygon or geodesic is increased, such that the surface area converges in the limit of a large number of facet sides approaching the geometry of a circle or sphere. In this work the ratio of acoustic wavelength to the local geometric parameter of edge length in 2-D, and facet area in 3-D is proposed and varied to quantify the comparison between the faceted shapes with that of the corresponding reference circle or sphere. A threshold ratio is proposed, up to which scattering response of a polygonal/geodesic spherical scatterer matches the scattering response of a circle/sphere which has the same diameter as the circumscribing circle/sphere of the polygon/geodesic sphere. This ratio is an approximation and can be considered as a guide rule for design. Conversely, this ratio can be used for the inverse scattering problem, where from a known scattering response, the faceted geometry can be predicted without prior knowledge. The geodesic sphere was invented by Buckminster Fuller in the early 1950’s, has been of interest in architecture due to the larger open interior spaces which can be constructed. Of particular interest in this work is the hierarchical geometric structure of the geodesic sphere which increasingly approximates a spherical surface as the hierarchy (degree) increases. The geodesic sphere has been modelled by taking an icosahedron and projecting the triangular faces onto a surface of the sphere using vector geometry. The scattering response of elastic structures in the mid-frequency resonance band depends strongly on the total mass. For comparisons, the natural frequency of the hierarchical geometries generated in 2-D and 3-D, are designed to have the same total mass. Using this approach, differences in natural frequency and scattering response are driven primarily by changes in overall stiffness and stiffness distribution, and to a lesser degree, by changes in mass distribution. To give a wide range of frequency response, natural vibration frequencies for the different elastic shells have been extracted up to 3000 Hz corresponding to the nondimensional frequency ka = 55, where k is the wavenumber defined by the circular frequency over the acoustic wave speed (speed of sound in air), and a is the diameter of the circle/shell which circumscribes the scatterer. Convergence with the natural frequencies of ring/sphere is observed as the hierarchy in polygons (number of sides) and the geodesic sphere (degree) increases. The target strength is calculated at the important front and back locations on the surface of the elastic scatterer subject to an incoming plane acoustic wave along the major axis aligned with the geometry. More frequency data points near the natural frequencies are used to provide increased resolution needed to capture the peak amplitudes in the response at resonance. Target strength at the same location, calculated for the circle/spherical scatterer is compared and quantified by the ratio of wavelength to facet dimension. Scattering from rigid bodies has been studied to validate the elastic scattering response in air.

## Recommended Citation

Kumar, Ankit, "Finite Element Analysis of the Effect of Acoustic Wavelength to Hierarchical Side Length and Facet Area for Elastic Scattering from Polygonal Rings and Geodesic Spheres" (2015). *All Theses*. 2511.

https://open.clemson.edu/all_theses/2511