Phononic band structure of honeycomb lattice with defects, using spectrally formulated finite element method

TL;DR

This paper introduces a spectral finite element method to accurately compute phononic band structures of honeycomb lattices with defects, using Bloch formulation and supercell analysis to handle high frequencies and defect effects.

Contribution

It develops a spectral FEM approach with Timoshenko beam elements for phononic band analysis, including supercell techniques to model defects and band structure reconstruction.

Findings

Spectral FEM accurately analyzes high-frequency phononic bands.

Supercell analysis reveals defect-induced band splitting and folding.

Primitive and supercell band structures are reconciled.

Abstract

A spectrally formulated finite element analysis based methodology has been proposed to calculate phononic band structure of reticulated honeycomb lattices having translationally invariant repetitive elements called unit cells. Bloch formulation captures dynamics of infinite structure through that of a unit cell. While conventional FEM is generally used for analyzing dynamics of such unit systems, here, constituent structural members are treated as 1D waveguide and modeled as Timoshenko beam frame element, enabling application of spectral FEM, suitable for accurately analyzing the dynamics, particularly efficient at very high frequencies. Using exact solutions as shape functions spares dense meshing. Resulting eigenvalue problem is solved by Wittrick-Williams method, an iterative scheme. Subsequently, band structures are obtained for supercells- units comprising multiple elemental unit cells; compared and reconciled with those obtained using elemental cell (termed primitive unit cell to distinguish from supercell). Primitive cell band structures are reconstructed from Supercell band structures. Supercell band structures show some spurious bands, which are explained in terms of band folding in the primitive cell band structure. Supercell allows treatment of defects as a periodic feature with certain defect density. Of particular observation in such band structures is the separation of bands, known as degeneracy breaking.