Experimental and Numerical Studies on the Hilbert Fractal

Architecture as an Acoustic Metamaterial Gianni Comandini 1 Bristol Composite Institute (BCI), Department of Aerospace Engineering, University of Bristol, UK. Valeska P. Ting2 2 Bristol Composite Institute (BCI), Department of Mechanical Engineering, University of Bristol, UK. Mahdi Azarpeyvand, 3 School of Civil, Aerospace and Mechanical Engineering (CAME) Department of Aerospace Engineering, University of Bristol, UK. Fabrizio Scarpa, 4 Bristol Composite Institute (BCI), Department of Aerospace Engineering, University of Bristol, UK.

ABSTRACT

We evaluate via experiments and numerical methods the transmission loss behaviour of Hilbert Frac- tal Metamaterials (HFMs)s tested in a four-microphone impedance tube. We explore the effect of the fractal order and widths of the cavity slots of HFMs 3D printed in PLA polymer. The COMSOL Finite Element models used here consider both thermoviscous and lossless domains. Tests and simulations have been carried out between 0.2 kHz and 3.0 kHz, with gap widths parametrised between 0.5 mm and 3.0 mm. A broad agreement is observed between the numerical models and the experimental results. The Hilbert fractal with the highest impact in terms of transmission loss is the one represented by the second order, with an experimental peak of almost 50 dB around 1600 Hz. All the HFMs orders show the presence of multiple TL peaks, with the gap width also a critical parameter to tailor the performance of these metamaterials. 1. INTRODUCTION

The Hilbert is perhaps the most evaluated fractal geometrical shape for metamaterial platforms [1]. This peculiar fractal was initially investigated for antenna applications [2]. More recently, this

1 gianni.comandini@bristol.ac.uk

2 v.ting@bristol.ac.uk

3 m.azarpeyvand@bristol.ac.uk

4 f.scarpa@bristol.ac.uk

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space-filling curve has been the subject of studies related to acoustics [3]. In this work, we present a comprehensive numerical and experimental study of Hilbert fractal-based acoustic metamaterials, starting from fractal at order zero, until the fifth order. Samples have been produced using 3D print- ing of polylactic acid (PLA) material. By using PLA rather than other materials, it is possible to re- duce the environmental footprint of the prototypes made by additive manufacturing. All the samples after the experimental campaign were successfully recycled to produce new PLA filaments. 2. DESIGN AND TESTS

The specimens have been designed with an inlet and an outlet on the opposite faces of the samples to allow the acoustic waves to pass through. The test rig used is an impedance tube with four micro- phones and white noise excitation. The test range is between 0.2 kHz and 3.0 kHz. Finite Element simulations have been performed using COMSOL Multiphysics to verify the experimental trends measured, with specific emphasis on the transmission loss (TL) effect provided by the fractal order, plus geometry features like the width of the gaps in the samples. Each fractal sample was produced with six different gap widths. The gap width ranges between 0.5 mm and 3.0 mm, with steps of 0.5 mm. To obtain reliable and statistically robust data, each sample was tested in the impedance tube five times on one side, and another five times by rotating the sample by 180 degrees with respect to the incoming acoustic wave direction.

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Figure 1: Hilbert fractals of (a) order zero, (b) first order (c) and second order. The design process involves a subdivision of the square area in equal portions with the relative centroid. On the left of each sample (a, b, and c) the process to generate the corresponding fractal geometry that starts from a square area.

3. DATA ANALYSIS

The higher peaks of the transmission loss (TL) occur for a gap width of 1.0 mm. The transmission loss increases within the range of frequencies investigated here, for gap values varying from 0.5 mm to 1.0 mm. The values of TL decrease for all the five Hilbert fractal orders investigated, starting from 1.5 mm to 3.0 mm. The TL performance of the fractals is compared with the one of a baseline bulk PLA element with 30% filling volume. The comparison with the baseline underlines a worse insulation effect in all frequencies between 200 Hz and 3000 Hz for a gap width that is not 1.0 mm. In terms of TL, this behaviour happens in both simulations and experiments. According to both experiments and simulations, it appears that open- ings wider than 2.5 mm will perform worse than a simple cube of PLA with the same external dimen- sion (50.8 mm) as the metamaterial one. The peak region shown in Figure 2 is referred to the second- order Hilbert fractal. Between 954Hz and 1788 Hz, a peak with a similar magnitude is also shown by the finite element simulations. The comparison between predictions and experiments follows the same trend, underlining how the peaks broadly occur in the same interval of frequencies. The figures

show that the highest contribution to the TL is the gap width (between 0.5 mm and 1.5 mm). After 2000 Hz, the finite element model provides a slightly different behaviour compared to the experi- mental data. Around 3000 Hz, the results converge around a narrow region near the baseline, as shown in Figure 2a. On the other hand, for the experiment, in Figure 2b, the transmission loss output of the experiments around 300 Hz they do not converge as in the COMSOL model. However, the FEM capture with a good approximation of the TL behaviour between 200 Hz and 2500 Hz.

Figure 2: In figure (a), transmission loss FEM results of the six gaps width between 0.5 mm and 3.0 mm for the second-order Hilbert fractal and comparison with the baseline that is a cube of PLA. Figure (b) shows the experimental results for the same fractal and gap width. 4. FEM

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The finite element model used in this work is based on the use of three separate domains. The first one is represented by a mesh with lossless behaviour. The second one represents the air inside the small passage inside the metamaterial and is modelled considering thermoviscous losses through the resolution of linearized Navier-Stokes equations. In this latter, boundary layers with square elements having a maximum total high of 0.22 mm are used in the mesh. The last contribution considered is the material used to 3D print the samples (PLA). The interaction among the domains is created through coupling [4]. 5. CONCLUSIONS

The main conclusions we can draw from this work are: • The transmission loss of Hilbert Fractal acoustic metamaterials s maximised for a gap width of one millimetre. Other values of the gap width (between 0.5 mm and 3 mm) provide lower ampli- tude TL peaks. • A broad agreement is observed between the numerical model and the experimental results. • The Hilbert fractal metamaterial with the highest impact in terms of transmission loss is repre- sented by the second order, with an experimental peak of almost 50 dB around 1600 Hz. 6. ACKNOWLEDGEMENTS G.C. acknowledges the support of UK EPSRC through the ACCIS Composites Centre for Doctoral Training.

Transmission Loss [dB] 954Hz A788Hz OSAHz, J7eeHz. (a) | Jeaseine ‘Transmission Loss (28) joo 1600 —~2000 25009000 recpseecien bag

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