Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 1.1 A Polyethylene-based metamaterial for acoustic control 1.1 A Polyethylene-based metamaterial for acoustic control 2 Relaxed micromorphic modelling of finite-size metamaterials 2 Relaxed micromorphic modelling of finite-size metamaterials 2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation) 2.1 Tetragonal Symmetry / Shape of elastic tensors (in Voigt notation) 3 Dispersion curves 3 Dispersion curves 4 New considerations on the relaxed micromorphic parameters 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 4.1 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 4.2 Consistency of the relaxed micromorphic model with respect to a change in the unit cell’s size 4.3 Relaxed micromorphic cut-offs 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 Fitting of the relaxed micromorphic parameters: the particular case of vanishing curvature (without Curl P and Curl P˙) 5.1 Asymptotes 5.1 Asymptotes 5.2 Fitting 5.2 Fitting 5.3 Discussion 5.3 Discussion 6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P) 6 Fitting of the relaxed micromorphic parameters with curvature (with Curl P) 6.1 Asymptotes and 6.2 Fitting 6.1 Asymptotes and 6.2 Fitting 6.3 Discussion 6.3 Discussion 7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes 7 Fitting of the relaxed micromorphic parameters with enhanced kinetic energy (with Curl P˙) and 7.1 Asymptotes 7.2 Fitting 7.2 Fitting 7.3 Discussion 7.3 Discussion 8 Summary of the obtained results 8 Summary of the obtained results 9 Conclusion and perspectives, Acknowledgements, and References 9 Conclusion and perspectives, Acknowledgements, and References A Most general 4th order tensor belonging to the tetragonal symmetry class A Most general 4th order tensor belonging to the tetragonal symmetry class B Coefficients for the dispersion curves without Curl P B Coefficients for the dispersion curves without Curl P C Coefficients for the dispersion curves with P C Coefficients for the dispersion curves with P D 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. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv Authors:
(1) Jendrik Voss, Institute for Structural Mechanics and Dynamics, Technical University Dortmund and a Corresponding Author (jendrik.voss@tu-dortmund.de);
(2) Gianluca Rizzi, Institute for Structural Mechanics and Dynamics, Technical University Dortmund;
(3) Patrizio Neff, Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen;
(4) Angela Madeo, Institute for Structural Mechanics and Dynamics, Technical University Dortmund. Authors: Authors: (1) Jendrik Voss, Institute for Structural Mechanics and Dynamics, Technical University Dortmund and a Corresponding Author (jendrik.voss@tu-dortmund.de); (2) Gianluca Rizzi, Institute for Structural Mechanics and Dynamics, Technical University Dortmund; (3) Patrizio Neff, Chair for Nonlinear Analysis and Modeling, Faculty of Mathematics, University of Duisburg-Essen; (4) Angela Madeo, Institute for Structural Mechanics and Dynamics, Technical University Dortmund.

Part of HackerNoon's growing list of open-source research papers, promoting free access to academic material.

How Symmetric and Skew-Symmetric Tensors Interact

The Science Behind Stronger, Smarter Materials

Can You Reach the End of the Labyrinthine? 

Too Long; Didn't Read

Boost your HackerNoon story @ $159.99! 🚀

Semi-Analytical Algorithm for Metamaterial Parameter Fitting

About Author

Comments

TOPICS

THIS ARTICLE WAS FEATURED IN

Related Stories

Untitled Story

A New Method for Predicting How Waves Move Through Materials

How 3D-Printed Sound Panels Can Absorb Noise More Efficiently

Shhh! The Future of Noise Control Is Twisting Through 3D-Printed Labyrinths

Can 3D Printing Improve Soundproofing? This Study Says Yes

The Science of Soundproofing: A Look at 3D-Printed Acoustic Panels

A New Method for Predicting How Waves Move Through Materials

How 3D-Printed Sound Panels Can Absorb Noise More Efficiently

Shhh! The Future of Noise Control Is Twisting Through 3D-Printed Labyrinths

Can 3D Printing Improve Soundproofing? This Study Says Yes

The Science of Soundproofing: A Look at 3D-Printed Acoustic Panels

Light-Mode

Classic

Newspaper

Dark-Mode

Neon Noir

Minty

HN StartUps