Mixed-variational formulation for phononic band-structure calculation of arbitrary unit cells

TL;DR

This paper introduces an extended mixed-variational formulation for calculating phononic band structures of complex unit cells, offering faster convergence and higher accuracy, applicable to arbitrary 2D and 3D structures.

Contribution

It extends the mixed-variational formulation to handle complex unit cells with arbitrary geometries and properties using numerical integration techniques.

Findings

Faster convergence and greater accuracy than displacement-only variational methods.

Applicable to complex 2D and 3D phononic unit cells.

Demonstrated effectiveness through specific numerical examples.

Abstract

This paper presents phononic band-structure calculation results obtained using a mixed variational formulation for 1-, and 2-dimensional unit cells. The formulation itself is presented in a form which is equally applicable to 3-dimensiomal cases. It has been established that the mixed-variational formulation presented in this paper shows faster convergence with considerably greater accuracy than variational principles based purely on the displacement field, especially for problems involving unit cells with discontinuous constituent properties. However, the application of this formulation has been limited to fairly simple unit cells. In this paper we have extended the scope of the formulation by employing numerical integration techniques making it applicable for the evaluation of the phononic band-structure of unit cells displaying arbitrary complexity in their Bravais structure and in the shape, size, number, and anisotropicity of their micro-constituents. The approach is demonstrated through specific examples