# A general-purpose element-based approach to compute dispersion relations in periodic materials with existing finite element codes

## TL;DR

This paper introduces a versatile elemental approach for computing dispersion relations in periodic materials using standard finite element codes, bypassing the need for access to global matrices.

## Contribution

The authors develop a new elemental-level method to impose Bloch boundary conditions, enabling dispersion relation calculations in any FE code without modifying global matrices.

## Key findings

- Method verified with analytical solutions
- Applicable to various material models and geometries
- Compatible with real and complex algebra solvers

## Abstract

In most of standard Finite Element (FE) codes it is not easy to calculate dispersion relations from periodic materials. Here we propose a new strategy to calculate such dispersion relations with available FE codes using user element subroutines. Typically, the Bloch boundary conditions are applied to the global assembled matrices of the structure through a transformation matrix or row-and-column operations. Such a process is difficult to implement in standard FE codes since the user does not have access to the global matrices. In this work, we apply those Bloch boundary conditions directly at the elemental level. The proposed strategy can be easily implemented in any FE code. This strategy can be used either in real or complex algebra solvers. It is general enough to permit any spatial dimension and physical phenomena involving periodic structures. A detailed process of calculation and assembly of the elemental matrices is shown. We verify our method with available analytical solutions and external numerical results, using different material models and unit cell geometries