Semi-Analytical Algorithm for Metamaterial Parameter Fitting

Table of Links

Abstract and 1. Introduction

1.1 A Polyethylene-based metamaterial for acoustic control

2 Relaxed micromorphic modelling of finite-size metamaterials

2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation)

3 Dispersion curves

4 New considerations on the relaxed micromorphic parameters

4.1 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s bulk material properties

4.2 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s size

4.3 Relaxed micromorphic cut-offs

5 Fitting of the relaxed micromorphic parameters: the particular case of vanishing curvature (without Curl P and Curl P˙)

5.1 Asymptotes

5.2 Fitting

5.3 Discussion

6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P)

6.1 Asymptotes and 6.2 Fitting

6.3 Discussion

7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes

7.2 Fitting

7.3 Discussion

8 Summary of the obtained results

9 Conclusion and perspectives, Acknowledgements, and References

A Most general 4th order tensor belonging to the tetragonal symmetry class

B Coefficients for the dispersion curves without Curl P

C Coefficients for the dispersion curves with P

D Coefficients for the dispersion curves with P◦

8 Summary of the obtained results

We want to emphasize that the main focus of this work is not the result of the fitting of the three different approaches per se, but the semi-analytical fitting algorithm itself and the underlying consistency of the relaxed micromorphic model with respect to the material properties and some of the geometrical characteristic of the metamaterial that it represents. Using the complex but analytically defined expressions of the asymptotes, we can find a numerical fit of all material parameters by only giving the numerical values computed with Comsol Multiphysics® and the apparent mass density ρ of the unit cell. Note that we only use the cut-offs k = 0 and asymptotes k → ∞ for calculating the material parameters while the shape of the curves for intermediate values of k comes automatically.

The routine is completely written with Mathematica allowing us to use symbolic calculations. The essential part of the fitting procedure uses the inbuilt algorithm NMinimize (with the Method RandomSearch) to minimize the mean square error of the asymptotes between the relaxed micromorphic model and the numerical values of the finite element approach in Comsol Multiphysics®. Therefore, in general, if a local minimum is found, it is not guaranteed that it corresponds to a global optimum as well.