ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) 444 –450 DOI 10.3813/AAA.919074 Acoustic Performance of aBarrier Embedded With Helmholtz ResonatorsUsing a Quasi-Periodic BoundaryElement Technique Samaneh M. B. Fard1),Herwig Peters1),Steffen Marburg2),Nicole Kessissoglou1) 1) School of Mechanical and Manufacturing Engineering, The University of NewSouth Wales, Sydney, NSW 2052, Australia. [email protected] 2) Institute of Vibroacoustics of Vehicles and Machines, Faculty of Mechanical Engineering, Tecnical University of Munich, Boltzmann Str.15, 85748 Garching, München, Germany Summary Barriers are widely used to attenuate environmental noise from vehicles to residential areas. This paper numer- ically explores the acoustic performance of abarrier for tailored lowfrequencynoise reduction using aquasi- periodic boundary element method, whereby the barrier is represented by afinite number of periodic sections along its length. Results for the insertion loss of abarrier with single and multiple Helmholtz resonators em- bedded along the top edge of each periodic barrier section and tuned to different frequencies are compared to results for an equivalent straight barrier in the absence of the Helmholtz resonators. The acoustic performance of the noise barrier with and without the embedded Helmholtz resonators is also examined for elevated trafficnoise sources typically corresponding to engine and exhaust noise of light and heavy vehicles. Diffraction overthe top edge of the barrier and reflections from the ground on both the source and receiversides of the barrier are taken into account. ©2017 The Author(s).Published by S. Hirzel Verlag · EAA. This is an open access article under the terms of the Creative Commons Attribution (CCBY4.0)license (https://creativecommons.org/licenses/by/4.0/). PACS no. 43.50.Gf 1. Introduction random edge profile barrier does not perform as well as a conventional noise barrier at lowfrequencies. Other stud- Barriers are commonly used as acontrol measure to re- ies investigated the acoustic performance of curved noise duce outdoor sound propagation from road trafficnoise barriers [10]. Incorporating cavities or resonators in barrier affecting nearby residential areas. The amount of reduc- designs is another approach to attenuate noise at particular tion largely depends on the barrier geometry and material frequencies [7, 11, 12, 13]. High levels of attenuation were properties. The simplest method to increase the efficiency presented for awave-trapping barrier by Pan et al. [7]. The of anoise barrier is generally to increase its height, but wave-trapping barrier comprised of multiple wedges with such increases are not always practical due to aesthetics, perforated surfaces, aback cavity and an internal lining, cost, maintenance and safety reasons. Forexample, barri- and wasshown to significantly reduce the sound pressure ers subject to wind loads have height constraints, thereby levelatlow frequencies. reducing their effectiveness. Theoretical methods for estimating the sound attenu- There are anumber of different designs to the top of ation produced by anoise barrier are well established, abarrier in an attempt to modify the diffracted wavesand for example, see [14, 15, 16]. The finite element method improve barrier performance compared to astraight barrier (FEM)has been used to compute the insertion loss of of similar height [1, 2, 3, 4, 5, 6, 7, 8]. Areviewofprofiled anoise barrier [17, 18]. Tyurina et al. [17] developed noise barriers including T-shaped, Y-shaped and arrow- an FEM model for a4mhigh noise barrier and vali- shaped barriers revealed that in most cases, aT-shaped bar- dated their numerical results with experimental tests. An rier performs better than the other noise barrier designs [3]. FEM acoustic topology optimisation model wasdevel- Shao et al. [9] showed that abarrier with arandom edge oped to design anoise barrier with minimum Zwickers profile along the top improvedbarrier performance com- loudness [18]. Anumerical approach that can satisfy the pared to astraight barrier.However,itwas shown that the Sommerfeld radiation condition is the boundary element method (BEM), which has been widely used to predict the acoustic performance of arange of barrier designs Received2November 2015, [1, 2, 10, 19, 20, 21, 22]. Boundary integral equations were accepted 23 March 2017. solved using the BEM and reasonable agreement between ©2017 The Author(s). Published by S. Hirzel Verlag · EAA. 444 This is an open access article under the terms of the CC BY 4.0 license. Fard et al.:Barrier embedded with Helmholtz resonators ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) predicted and measured insertion loss values wasobtained [1, 2, 19, 20]. Lam [21] modified the BEM formulation to predict the insertion loss of abarrier incorporating at- tenuation due to atmospheric turbulence. Gasparoni et al. [22] investigated fork shaped barriers using an openBEM Figure 1. Illustration of the quasi-periodic boundary. software. BEM simulations of barriers were implemented into noise mapping software whereby the acoustic perfor- mance of several complexbarrier designs were compared with astraight barrier of equal height [10]. To save computational cost, periodic and quasi-periodic structures have been used to solvemodels of three-dimen- sional (3D) infinite structures. Hashimoto et al. [23] tested an optimization analysis for periodic sound barriers com- posed of elastic scatterers and subject to plane wave exci- tation. Lam [24] numerically studied plane wave incidence on the surface of aperiodic structure. 3D plane wave scat- tering in doubly periodic structures wasinvestigated us- ing Floquet theory [25]. In recent work, Ziegelwanger et al. [26] coupled the fast multiple method with aquasi- periodic boundary element method to simulate the sound field scattered around anoise barrier.Karimi et al. [27] compared the insertion loss for asonic crystal barrier using Figure 2. Configuration of asingle periodic barrier section show- aperiodic BEM technique, whereby aunit cell comprised ing monopole sources. arow of cylindrical scatterers. In this work, a3Dquasi-periodic BEM technique re- cently developed by the authors [28] is implemented to The acoustic pressure is assumed to be periodic, such investigate the acoustic performance of abarrier with em- that the acoustic pressure at each collocation point on a bedded Helmholtz resonators tuned to asingle frequency typical boundary section is equal to the acoustic pressure or multiple frequencies. Only asmall section of the bar- at the corresponding point on the other sections. Hence, rier is considered and periodically repeated in one direc- only the acoustic pressures at the collocation points of the tion, thus significantly reducing the simulation run times centre boundary Γ0 need to be determined. and storage requirements compared to afull 3D model. The barrier wasmodelled using four-node super-para- Diffraction overthe top edge of the barrier and reflections metric discontinuous linear boundary elements. Aconver- from the ground on both the source and receiversides of gence analysis to identify the required number of elements the barrier are taken into account. Results are compared per wavelength wasconducted. Atruncation analysis was to those for an equivalent straight barrier in the absence also carried out to identify the minimum number of bar- of the Helmholtz resonators and for elevated trafficnoise rier sections for the quasi-periodic model to represent an sources. infinitely long barrier.The full description of the quasi- periodic BEM technique is givenin[28]. 2. Numerical methods 2.2. Finite element model 2.1. Quasi-periodic boundary element method Three-dimensional barrier models were also developed us- ing commercial finite element software COMSOL Multi- In the quasi-periodic boundary element method, the boun- physics (v4.3b)tovalidate the convergence analysis re- dary of the barrier Γ is divided into small sections Γ as n sults for the quasi-periodic BEM. Comparisons were only shown in Figure 1. The boundary Γ is then represented by presented for abarrier with single Helmholtz resonator due summation of the boundary sections Γ as n to limitations in computational resources for the finite el- ement model. Aperfectly matched layer (PML)was used Γ=Γ−M∪Γ−M+1∪···∪Γ−1 ∪Γ0 ∪Γ1 ∪... to approximate radiation into the infinite exterior acous- ···∪Γ ∪···∪Γ ∪Γ . (1) n N−1 N tic domain. The thickness of the PML wasscaled with N and M are the number of boundaries on each side of the frequencysuch that its thickness wasequal to one wave- length. The acoustic domain in the FE model wasdis- initial boundary Γ0.From the boundary condition givenby equation (1),the total length of the barrier is givenby cretized using quadratic Lagrangian elements. To achieve an accurate prediction of pressure overafinite element, LT =(N+M+1)xp. (2) the dimensions of the elements were chosen to be less than one fifth of an acoustic wavelength. Aperiodic boundary where xp is the length of one section of the periodic barrier, condition wasapplied at the lateral edges, thus extending as shown in Figure 2. the length of the barrier to infinity in one dimension. 445 ACTA ACUSTICA UNITED WITH ACUSTICA Fard et al.:Barrier embedded with Helmholtz resonators Vol. 103 (2017) 3. Barrier models Aperiodic section of arectangular barrier with athick- ness of 0.5 mand aheight of 3m wasinitially modelled. The length of each periodic section was1m. Asingle Helmholtz resonator wasthen embedded along the top part of each barrier section.