1 s2.0 S0022460X21002078 main

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Journal of Sound and Vibration 505 (2021) 116135

ContentslistsavailableatScienceDirect

Journal of Sound and Vibration

journalhomepage:www.elsevier.com/locate/jsv

A graph-theory approach to optimisation of an acoustic absorber targeting a speciﬁc noise spectrum that approaches the causal optimum minimum depth

Ian Davis

^a^,^∗

, Andrew McKay

^b

, Gareth J. Bennett

^b

aEﬃcient Energy Transfer ( ηET) Department, Nokia Bell Labs, Blanchardstown Business & Technology Park, Dublin 15, Ireland

bDepartment of Mechanical & Manufacturing Engineering, Trinity College Dublin, Dublin 2, Ireland

a rt i c l e i nf o

Article history:

Received 7 September 2020 Revised 16 March 2021 Accepted 12 April 2021 Available online 14 April 2021 Keywords:

Sound absorption Optimisation Graph theory Metamaterials Numerical modelling

a b s t ra c t

Equivalent circuit analysis is a powerful tool for analysing acoustic systems where a lumped elementmodelisvalid.Theseequivalentcircuits allowanoverallimpedanceof thestructuretobeestimatedwhichfacilitatespredictionsofthereflectivity,transmissibil- ityand/orabsorptivityofthesystem.Complexacousticsystemsarerepresentedbynon- planarequivalentcircuitswhicharechallengingtosimplifytoasingleoverallimpedance valueusingtraditionalKirchoff’sLawsimplifications.Atwo-pointimpedancemethodus- inggraphtheoryallows theimpedanceofacircuitto beestimated withoutsimplifica- tion.Thegraphtheorymethodisappliedtoatypeofacousticabsorberstructurenamed SeMSA (SegmentedMembraneSound Absorber)whichhadpreviouslybeen investigated foratwo-segmentcelldesign.ThismethodallowstheSeMSAanalysistobeexpandedto multi-sectordesignswithawiderparameterspace.Alocaloptimisationroutineisapplied tothegraphtheoryimpedanceestimationtomaximiseacousticabsorptionofSeMSAun- derconsiderationofabsorberdepth,causaloptimalityandthetargetednoisespectra.An- alyticalpredictionsarevalidatedusingnumericalsimulations.Theoptimisedmulti-sector absorberdemonstrates70.5%whitenoiseabsorptioninthe20–4500Hzfrequencyrange withanabsorberdepthof16mmandisjust0.5mmfromthetheoreticalminimumdepth toachievethisabsorptionresponse.

ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Acousticmaterialsforsoundabsorptionhaveevolvedsignificantlyfromtraditionalbulkporous/fibrousabsorberstomod- ernmaterialssuchasmembrane-basedmetamaterials[1],absorbersconsistingofarrangementsofaxially-coupledchannels [2], acousticblack holes that direct acoustic wavesto an absorptivecore[3] andcoiled Helmholtz resonators [4]. These modern materialstypicallytargetsub-wavelengthabsorptioni.e.absorptioncoefficientsofclosetounityareachievedwith materialdepthsoforder

λ

/100,where

λ

^is^the^acousticwavelength.Asummaryofrecentdevelopmentmaybefoundina

∗Corresponding author.

E-mail address: [email protected] (I. Davis).

https://doi.org/10.1016/j.jsv.2021.116135

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Fig. 1. A cylindrical Segmented Membrane Sound Absorber (SeMSA). The cylindrical volume is divided into two air cavity segments by a microperforated plate.

recentreviewpaperbyYangetal.[5].Traditionalbulkabsorberssuchasmelamineorpolyurethaneacousticfoamsgenerally exhibitpoorabsorptionperformancebelow1kHzunlessdepthsoforder100mmareused[6].

While differentmetamaterials utilisedifferentphysicalphenomenasuch asFabry-Perotresonances[7]andelastic cur- vature proﬁlesinside amembrane-plateletsystemthat coupleweaklywithacousticradiation[1],mostmetamaterialsare basedonresonantsystemsthatabsorbeffectivelyinnarrowfrequencybands.Extendingthisfrequencyrangetowiderband- widthsisdesirableinordertocreatebroadbandsoundabsorptioninasmallfootprint.Thekeychallengeforextendingthe absorption bandwidthofan acousticabsorber isatlow frequencieswheretheabsorberdepthissmallerthantheacoustic wavelength.

Anotherkeyconsiderationforthedesignofacousticmaterialsisthesimplicityoftheirconstruction.Inpractical terms theeﬃcacyofanacousticabsorbers(astypicallycharacterisedbyitsabsorptionperunitdepth)mustbeconsideredalong- side thecostandcomplexityofmanufacturingthematerial.Furthermore,ifanacousticabsorberisbeingmanufacturedat scale andacrossa variety ofenvironments andtargetedsoundsources, itisdesirable that theperformance ofthe mate- rialmaybepredicted,tuned andoptimisedforthespeciﬁcapplicationunderconsiderationofavailablevolume,thesound sourcespectrumandotherenvironmentalfactors.

Absorber technologiesthat are made up oflinear combinationsof resistiveandreactive elements canbe represented bylumped-elementmodels.Theselumped-elementmodelsallowtheimpedanceoftheacousticmaterialtoberepresented usingequivalentmechanicalorelectriccircuitmodels [8],andcircuitsimplifications usingKirchoff’sandOhm’sLawscan allowthecircuittobesimplifiedtoasingleoverallimpedancevalue.Oncesimplified,thecircuitcanbeoptimisedtoachieve atargetedabsorptionprofilesuchasmaximalwhitenoiseabsorptioninaprescribedfrequencybandinagivenvolume,as wasdemonstratedusingthe SeMSA(SegmentedMembraneSoundAbsorber)technology [6].Dondaetal.[9]alsopresent an acousticabsorber thatusesanetworkoflumped elementstotargetnarrowband,highabsorptionwithabsorberdepths ofjust

λ

/527.

In the SeMSA technology previously investigated in McKay et al. [6]and shownin Fig. 1, the incident sound causes resonance ofthe membrane-cavity systems which are coupledthrough a microperforatedplate. Thedifferential pressure betweenthetwo cavitiesdrives airthroughthemicroperforateswherethereare largevisco-thermallosses. Soundis also absorbedbydampinginsidethemembranes.SeMSAhadpreviouslybeeninvestigatedforatwo-segmentcylindricaldesign withan equivalentcircuitmodel(see Fig.2(a)) basedonthe impedancesofthetwo membranesplus addedmasses(zm1

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I. Davis, A. McKay and G.J. Bennett Journal of Sound and Vibration 505 (2021) 116135

Fig. 2. Comparison between the impedance estimated by simplifying the equivalent circuit shown in (a) to the two-point impedance model based on Tzeng & Wu’s graph theory approach [13] which analyses the adjacency matrix shown in (b).

andz_m₂),thetwoaircavityvolumes(z_c1 andz_c2)andthemicroperforatedplatedividingthetwoaircavities(zmpp)^.^A^local absorptionroutinewasusedtotunethecircuitimpedances(andhencegeometricparameters)toachieve signiﬁcantwhite noiseabsorptionintherange20–1200Hzatarangeofoveralldepths.

InordertoextendtheanalysisofSeMSAtomorecomplexdesignswithmoresegmentsorsectorsthanthetwo-segment design showninFig.1, amore complexequivalentcircuitwouldbe requiredthan Fig.2(a). Asthe numberofsectorsor segmentsincreasestherewillbemultiplemicroperforatedplatesdividingadjacentaircavitieswhichresultsinanon-planar equivalent circuit. Adding additional parameters to the model allows for additional tuneability ofthe overall impedance which mayextend the bandwidth of the absorber’s performance or allow multiple narrowband frequencyranges to be targeted. However, as the complexity of the circuit increases it becomes more challenging to simplify the circuit using a system of−Y andY− transforms.Examplesimpliﬁcations of threeand four-sector SeMSAcellsmay be found in sectionS1ofthesupplementarymaterial.Itisproposedthatratherthanattemptingtosimplifythesecomplex,non-planar circuits thatan alternativeapproachistakenby applyinga graphtheory methodtocalculatetheimpedancebetweenany two nodesof anycircuit regardless of its complexity(a two-point impedance model).This method doesnot necessitate simpliﬁcationofthecircuit.

Inthispaperthetwo-pointimpedancemodelbasedongraphtheoryisﬁrstvalidatedagainstaKirchoff’sLawsimpliﬁca- tionofatwo-segmentSeMSAimpedancemodel.Thegraphtheorymethodisthenappliedtomulti-sectorcylindricalSeMSA designswithinalocaloptimisationroutineinordertoachievemaximalabsorptioninthefrequencyrange20–4500Hzata rangeofcelldepths.Thismethodmaybeappliedtoanyacousticsystemwherealumpedelementmodelisvalidwhether thegoalissoundabsorptionornotandisparticularlyusefulforanalysingcomplexacousticsystemswithmultipleparallel branchespernode.Acylindricalcelldesignhasbeenselectedtoallowcomparisonwithapreviousstudyofatwo-segment SeMSAdesign[6]andfacilitatefutureexperimentaltestsusingacylindricalimpedancetube.

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2. Two-pointimpedancemodelusinggraphtheory

Graphtheory isthe studyofstructures whichrepresenttherelationsbetweenobjects.Thegraphsthat arestudiedare madeupofmultipleverticesornodesandtheconnectionsbetweenthesenodeswhichareknownasedgesorlinks.Graph theory isapplied in a wide range ofﬁelds frommathematical topology [10] to the analysisof social media interactions [11]andallowsforverycomplexnetworksofverticesandedgestobestudied.

A graphtheory approach maybe applied to solvethe two-point resistances betweenarbitrarynodes ina real-valued resistancenetworkregardlessoftheorderornumberofdimensionsofthecircuit[12].Thissameapproachmaybeextended toRLCcircuitswithcomplex-valuedimpedances[13].Electricalcircuitscanbethoughtofasﬁnitegraphswheretheedges arethecircuitelements(RLCcomponents)andtheverticesarethepointsbetweenthesecircuitelements.

Finitegraphsaremathematicallydescribedbysquarematriceswhichrepresenttheconnectivitybetweennodes/edgesin thenetwork. Theadjacencymatrixisasymmetricmatrixwhichdescribestheadmittances(reciprocaloftheimpedances) betweentheN nodesinthenetwork:

Y=

⎛

⎜ ⎜

⎝

0 y12 ... y1N

y21 0 ... y2N

..

. ... ... ... y_N1 y_N2 ... 0

⎞

⎟ ⎟

⎠

⁽¹⁾

The elementsy_{i j} are theadmittancesbetweennodesi and j.The degreematrixisadiagonal matrixwhichdescribesthe totaladmittanceconnectedtoeachnodeinthenetwork:

D=

⎛

⎜ ⎜

⎝

d1 0 ... 0 0 d₂ ... 0

..

. ... ... ... 0 0 ... d_N

⎞

⎟ ⎟

⎠

⁽²⁾

Eachelementonthemaindiagonalofthedegreematrixd_iisthesumofalladmittancesconnectedtonodei: di=

N j=1

yi j where i=j (3)

ALaplacematrixLisformedfromtheadjacencyanddegreematricesofthenetwork:

L=D−Y (4)

InordertocalculatetheimpedanceofthenetworkrepresentedbytheLaplacematrixLthefollowingeigenvalueequation issolved:

L^†L

ψ

_β=

η

_β

ψ

_β,

η

_β≥0,

β

=0,1,...,N. (5) where ^† denotes the hermitian conjugation,

η

β are the eigenvalues and

ψ

β are the eigenvectors of L^†L. Details on the regularisationandpotentialforsingularitieswhensolvingthisequationwillbeomittedforbrevity,butcanbefoundinthe originalarticle[13].Oncethiseigendecompositionhasbeenperformedtheimpedancebetweenanytwonodesiand jmay becalculatedasfollows:

Zi j= N β=2

1

η

β

( ψ

βⁱ−

ψ

β^j

)

² ⁽⁶⁾

The impedanceofan acousticsystemsuch asa soundabsorberwhich maybedescribedby a lumped-elementmodel may be represented by an electrical circuit analogy, assuming that flow effects and non-linear effects maybe assumed to be negligible. Theimpedance betweenanytwo nodesin thisequivalentcircuit maythereforebe calculated usingthe graph theory method described above. In order to validate this approach, the impedance calculated for a two-segment cylindricalSeMSAabsorberisfoundbysimplifyingthecircuit(seeFig.2(a))toasingleimpedancevalueusingKirchoff’sLaw simplifications (seetheoriginal article[6]fordetail) andcomparedwiththeimpedancevaluecalculated usingEq.6.The two impedancevaluesarecomparedforarangeoffrequenciesinFig.2(c).Thetwoimpedancevaluesareidenticalwhich demonstrates the efficacy ofthe graph theory approach.This graph theory method isapplied to more complex acoustic systemsinthefollowingsections.

Foranelectroacousticcircuitanalogy,thereﬂectioncoeﬃcientofthematerialundernormal-incidencecanbecalculated fromthetwo-pointimpedancecalculatedfromthegraphtheorymethod:

R1N =

(

^z^ˆ1N−zˆ0

)

/

(

^z^ˆ1N+zˆ0

)

⁽⁷⁾

where zˆ₀ isthe characteristicspecific impedanceofthefluid medium.Notethe (^ˆ·) ^symbol^denotes ^that ^thisîs â^specific impedanceinunitsofRaylsorkg/m²s.Theother impedanceelementsdiscussedhereinwithouta (^ˆ·)^symbol âreⁱⁿûnits

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Fig. 3. Isometric view of the cylindrical sound absorbers under investigation: a microperforated plate backed by a sealed air cavity, a two-segment SeMSA design and a multi-sector SeMSA design. The acoustic impedances are labelled for each example.

of kg/m⁴s andmust be convertedto a specificimpedance value beforeestimating thereflectioncoefficient. The normal- incidenceacousticabsorptioncoefficientmaybecalculatedfromthereflectioncoefficient:

α

=1−

|

^R1N

|

² ⁽⁸⁾

Note that the acoustic impedances, reflection coefficients andabsorption coefficientsare dependent on the frequency f whichhasbeenomittedforbrevity.

Thekeybenefitofthegraphtheorymethoddiscussedhereinisitssimplicity.Alternativemethodsofanalysingacoustic networkssuch asthe generalisedtwo-portmobility-matrixformalismintroduced byGlav and ˚Abom[14]alsoaccount for flow effects, makingit a veryusefulmodel forestimatingimpedances inflow ductsforexample. Theacoustic absorbers analysedhereinaregeometricallycompactwithrespecttotheacousticwavelengthandthereforealumpedelementmodel isconsideredvalid.

3. SeMSAwithmultiplesectors 3.1. Lumpedelementimpedances

Theproposed soundabsorberdesignisan evolutionofthetwo-segmentSeMSAdesigntoamorecomplexmulti-sector design.Theabsorbergeometryisstillcylindrical,butinsteadofbeingsubdividedintotwosegmentsthecylindricalvolume issubdividedintoNsectors. Acylindricalgeometryisusedsothatthemulti-sectorSeMSAmaybeexperimentallyinvesti- gated usingacylindricalimpedancetube(in thesamemannerasthetwo-segmentdesign [6]),although thestudyherein islimitedto analyticalpredictionswithnumericalvalidation.Inordertoanalyticallycomparethe morecomplexsectored SeMSAdesignswithexistingdesigns,themicroperforatedplatesoundabsorberbackedbyasealedaircavityexaminedby Maa [15]as well as the two-segment SeMSA design are used asbenchmarks. The threeabsorber typesinvestigated are shownschematicallyinFig.3.

Inalumpedelementsystemtheindividuallumpedimpedancescanbethoughtofasboththebuildingblocksandtune- ableelementsoftheoverallsystem.Forexample,awoodwindinstrumentcanbeanalysedusinglumpedelementstopredict thetonalityandtimbreofits responsewithdifferentﬁngerings[16].Foran absorberweseektotunetheelementslikea musicalinstrumenttoachieveadesiredabsorptionresponse.Therearethreebuildingblocksweuseinthemicroperforated plateabsorberandSeMSAdesigns(eachoftheseacousticimpedancetermshaveunitskg/m⁴s):

z_mi=

ζ

⁺ ^j

ω

0m_i

S²_i (9)

zci= kci

j

ω

0

where kci=c²₀

ρ

⁰

SiD (10)

zmppi=−ˆz0

jk0

σ

ⁱ

J0

di 2

−jω0

ν⁰

J2

di 2

−jω0

ν0

^[^t⁺⁰^.⁸⁵^dⁱ^·

( σ

ⁱ

)

^]

Smpp (11)

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Table 1

Deﬁnitions of symbols representing the physical parameters listed in Table 2 for the acoustic absorbers shown in Fig. 3 .

Symbol Physical parameter Units Minimum value Maximum value

N Number of subdivisions in multi-sector cell 3 7

R SeMSA cell radius mm 20 40

D SeMSA cell depth mm 10 50

σi Porosity of plate i 0 0.25

t i Thickness of plate i mm 0.1 1

θi Angle between plate i and plate (i −1) rads 2 π/N0 . 25(2 π/N) 2 π/N+ 0 . 25(2 π/N)

d i Diameter of holes in plate i mm 0.2 1

m i Mass of membrane & added mass at sector i g 0.02 1 ζ Damping inside membrane kg/m ⁴s 2 . 78 ×10 ⁵ 2 . 78 ×10 ⁵

Table 2

Parameter space for the sound absorbers under study (see Fig. 3 ) for a ﬁxed cell radius R : a microperforated plate backed by a sealed air cavity, a two-segment SeMSA design and an N-sector SeMSA design where N > 2 .

Absorber type z mParameters z cParameters z mppParameters No. of parameters

Microperforated Plate – D σ^;^t;^d ⁴

Two-Segment SeMSA m 1, m 2; ζ D ;θ σ;t ;d 8

Three-Sector SeMSA m 1, m 2, m 3; ζ D ;θ1, θ2 σ1, σ2, σ3;t 1, t 2, t 3; d 1, d 2, d 3 16 Four-Sector SeMSA m 1, m 2, m 3, m 4; ζ D ;θ1, θ2, θ3 σ1, σ2, σ3, σ4; t 1, t 2, t 3, t 4; d 1, d 2, d 3, d 4 21 . .

. . . . . . . . . . . . .

N-Sector SeMSA m 1, m 2, . . . , m N; ζ D ; θ1, . . . , θN−1 σ1, . . . , σN; t 1, . . . , t N; d 1, . . . , d N 5 N + 1

where

ω

0 intheangularfrequency,zˆ₀isthecharacteristicimpedanceoftheﬂuidmedium,

ν

0isthekinematicviscosityof the ﬂuidmedium, k₀ isthewavenumber,

ρ

0 is thedensity of the ﬂuidmedium, j=√

−1andJ_n are thenthorder Bessel functions. TheFok function(

σ

i)^accounts^forhole-to-holeinteractions[17].Manyalternativeestimations ofthe acoustic impedanceofamicroperforatedplateimpedanceareavailable;Eq.(11)wasselectedasitmatchestheimplementationused by the COMSOL’sinteriorperforatedplateboundarycondition[18]whicheases comparisons betweentheanalysisherein andanumericalmodel.Thismicroperforatedplateimpedancemodelassumesthatthermaleffectsarenegligiblecompared toviscouseffects,platevibrationsarenegligibleandnon-lineareffectsareignored.Theequationsdescribingimpedanceszm

andzcmaybefoundinMerhaut[8].

ThecavityareatermS_ivariesdependingonthetypeofabsorberbeinginvestigated:

Si=

⎧ ⎪

⎨

⎪ ⎩

S_cell, foramicroperforatedplateabsorber,wherei=1. 0.5

( θ

⁻^sin

( θ ))

^R², foratwosegmentSeMSAdesign,wherei=1. S_cell−0.5

( θ

⁻^sin

( θ ))

^R², foratwosegmentSeMSAdesign,wherei=2. Scell

( θ

i/2

π )

^for^allmulti-sectorSeMSAdesigns.

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whereS_cell=

π

^R²^.^Definitions^forâllôther^terms^may^be^foundⁱⁿ^Table¹^.Âll^parametersâre^geometricând^therefore^freely

tuneablewiththeexceptionof

ζ

,thedampingfactorinsidethemembrane.Thisparametermustbeestimatedempirically.In thisstudythedampingparameterfor0.2mmlatexmembraneasdeterminedbyMcKayetal.[6]isapplied.Theparameter spaceincreasesasmoresectorsareaddedaspertable2.Themicroperforatedplatethicknesscanbeaslowas0.1mmwhich wouldmaketheassumptionofnegligibleplatevibrationsinaccurate,howeverinapracticalimplementationathickerplate maybe usedwhoseplatethicknessisreducedlocallytothemicroperforationsby counter-boringlarger diameterholes in serieswiththemicroperforatedholes.FormoredetailsonthismanufacturingprocessseeMcKayetal.[6].

3.2. EffectofcellsizeontheimpedanceofaSeMSAabsorber

For thecaseof acircularcell, the area oftheMPPs isproportional to the cellradius butthe area ofthe membranes andprojectedareaofthecavitiesarebothproportionaltothesquareoftheradius.Eqs.(9)–(11)thenshowthatz_mi∝1/r⁴ (or z_mi∝1/r² ifthearealdensity,m_i/S_i, iskeptconstant),z_ci∝1/r² andz_mppi∝1/r.Thismeans that asthe cell radius is changed,theimpedancesofthecell’scomponentsdonotallchangeatthesameratewhichleadstoachangeinabsorption responsewithcellsize.

Formostother SeMSAarrangements a similar argumentholds:the sizeof thecell cannot beincreasedin awaythat causesallthemembraneareasandMPPareastogrowatthesamerate.Onecounterexampleisthatofarectangulartwo- segment SeMSAwhichisallowed togrowalong onedimensionto whichtheMPPisalignedandthearealdensityofthe membraneiskeptconstant.

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Fig. 4. Adjacency matrices and circuit layouts for multi-sector SeMSA designs. The vertical connections between node 1 and nodes ( 2 , . . . , N + 1 ) have acoustic impedances z mi. The vertical connections between node N + 2 and nodes ( 2 , 3 , . . . , N + 1 ) have acoustic impedances z ci. The horizontal connections joining nodes ( 2 , . . . , N + 1 ) have acoustic impedances z mppi.

3.3. Graphtheorymodel

TheLaplacematrixrepresentationoftheimpedancesoftheequivalentcircuitsfortheabsorberdesignsunderinvestiga- tionwillhaveanincreasingnumberofnodesN asthenumberofdesignsparametersincreases:

N =

2, foramicroperforatedplateabsorber.

N+2 forallSeMSAdesigns. (13)

The adjacency matricesfor arange ofmulti-sector SeMSA designs are shownin Fig. 4.This demonstrates that thenon- planarequivalentcircuitsformulti-sectorSeMSAdesignshaveabipyramidstructure.NotethattheLaplacematrixapproach toestimatingtheimpedancesisnotlimitedtotheSeMSAdesignsbutcouldbeappliedtoanyacousticsystemregardlessof theshapeandlayoutofitsequivalentcircuit,assumingalumped-elementmodelisvalid.

4. Causaloptimality

Theacousticcausalityconstraint[7]tellsusthetheoreticalminimumdepththatcanachieveagivenabsorptionspectrum

α

(

λ

),wherewhere

α

^is^thenormal-incidenceabsorptioncoeﬃcientand

λ

^is^the^acousticwavelength:

D≥ 1 4

π

²

B_eff B₀

^∞

0

ln[1−

α ( λ )

^]^d

λ

⁼^D^min ⁽¹⁴⁾

whereB_effrepresentsthebulkmodulusofthesoundabsorberinthestaticlimit.ForthemicroperforatedplateandSeMSA designs underinvestigationthisstaticlimitisequaltothebulkmodulusofairB₀ becausethemembranephysicsisaccu- rately represented asa limp mass(under the assumptionof nopre-tension andnegligiblebending stiffness).If thelimp membranewastensionedorstiff thenB_effwouldbeequaltoB₀plustheadditionalstiffeningeffect.AccordingtoEq.(14)it isnotpossibletoachievebroadbandperfectabsorptioninaﬁnitethicknessbutitispossibletohavepeakswheretheab- sorptionisperfectatasinglepoint.Theequationalsoshowsusthattheminimumthicknessofanabsorbertoachievehigh absorptioninawidefrequencybandisdominatedbytheabsorptionresponseatlowfrequencies/largewavelengths.

In reference to the absorption coeﬃcient spectra for the the acoustic absorbers underinvestigation, it is prudent to comparetheactualdepthoftheabsorberDusedtoachieveagivenabsorptionspectrum

α

(^f)^with^thetheoreticalminimum depthachievableascalculatedbyEq.(14):

^D=D_min

( α )

−D (15)

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As ^D^tends^towards^zero^theâbsorption^spectrumîs^closer^to^the^causal^limit^thatîsâchievableât^the^prescribed ^depth.

For a singleabsorber design we can calculate themean offset fromthe actual absorber depth andthe causally optimal minimumdepth:

^Dm= 1 ND

_N

D i=1

(

^Dmin

( α

i

)

−D_i

)

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whereiistheindexofthedepthoftheabsorber,

α

iistheabsorptionspectrumgeneratedbytheabsorberunderinvestiga- tionatdepthiandandN_Disthenumberofabsorberdepthsinvestigated.Intheresultsdiscussedbelow21absorberdepths aretestedfrom10mmto50mmat2mmincrements.

5. Objectivefunctionforoptimisation

The performance ofanacoustic absorbermaybe quantified bylinearaveraging oftheabsorption coefficientspectrum inadefinedfrequencyrangetogiveanoverallabsorptioncoefficient,

α

¯^.^Thisâpproach^was^previously âpplied^to^the^two- segmentSeMSAdesigninthefrequencyrange20–1200Hz.Inordertoextendthisanalysis,theoverallabsorptioncoefficient maybeconsideredalongsidethedepthoftheabsorber(D)andhowcloselyitreachesthecausallimit(^D⁾^to^further^quan- tifyits effectiveness.Furthermore,simply averagingthe absorptioncoefficient spectrumis onlysuitable forcharacterising the effectiveness of the absorber forspectrally-flat (white noise)absorption. If the targetedsound source forabsorption has a non-white noise spectrumit is sensibleto consider the powerspectrum of the soundsource when estimatingan absorber’seffectiveness.

Considerasourcesoundwhosesoundspectrumisdescribedbythediscreteauto-spectraldensityfunctionGxx[k]where k isthefrequencybinindex. Thesoundpowerreductionthat asoundabsorber witha discreteabsorption spectrum

α

^[^k^]

wouldachieveinDecibelsisgivenby:

C_α=10log₁₀

(

¹−c_α

)

⁽¹⁷⁾

wherec_α istheoverallabsorptioncoeﬃcentweightedbythesoundspectrumofthetargetednoisesource:

c_α=

k2

k=k1

α

^[^k^]^G^xx^[^k^]

k2

k=k1

Gxx[k]

⁽¹⁸⁾

wherek₁ isthefrequencybinindexofthelowestfrequencyandk₂isthefrequencybinindexofthehighestfrequencyin thebandwidthofinterest.

ThesoundpowerreductionfactorC_α providesasuitablecostfunctionthatcanbeminimisedinanoptimisationroutine inordertogeneratetheparametersetforasuitableN-SeMSAdesign.Thestepsforthisoptimisationareasfollows:

1. ThecelldepthD,cellradiusRandnumberofsubdivisionsNareprescribed/ﬁxed.

2. AninitialparametersetisselectedasthemeanvaluesoftheremainingfreeparametersinTable2.

3. An optimisation routine (interior-point stochastic gradient descent algorithm) minimises the cost function C_α in Eq.(17)by modifying thefree parameters within the boundsspeciﬁedin Table1.The absorption spectrum

α

^is^esti-

matedfromtheacousticimpedancez₁_N calculatedfromEq.(6).

4. Theoptimalabsorptionspectrumisreturnedbytheoptimisationroutineaswellastheparametersetthatachievedthis result.

ThisprocedureensuresthattheoptimisationroutinegeneratestheidealSeMSAgeometricparametersforthetargetsound spectrum.

6. Results

6.1. Analyticalsimulations

Fig. 5 showsthe predicted absorption responses of the three absorber types tested: a microperforated plate, a two- segment SeMSAdesign anda rangeofmulti-sector SeMSAdesigns with(^N=3,...,6)^.^The ^absorber ^size^was^ﬁxed ^with R=40mm.ThecostfunctioninEq.(17)hasbeenminimisedinordermaximisewhitenoiseabsorptioninthe20–4500Hz frequencyrange(Gxx[k]=1,forallk).Thisoptimisationprocedurewasrepeatedforarangeofabsorberdepths.Themini- mumdepth(10mm)caseisshowninblue,themaximumdepth(50mm)caseisshowninredandtheintermediatedepths aregradedbetweenblueandred.Astheabsorberdepthisincreasedtheabsorptionbandwidthextendstolowerfrequencies foreachdesign,asispredictedbyEq.(14).

Forthe microperforatedplatedesign,the bandwidthof theabsorber doesnot extendastheabsorber depth increases butthefrequencyresponseshiftstolowerfrequencies,bringingmoreabsorptionintothefrequencyrangeunderstudy.For

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Fig. 5. Absorption spectra for a range of absorber depths from 10 mm (dark/blue) to 50 mm (light/yellow). Target spectrum = white noise. Frequency range = 20–4500 Hz. R = 40 mm. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 6. Overall absorption coeﬃcient and causal optimality of a microperforated plate (MPP), two-segment SeMSA (2SS) and multi-sector SeMSA ( NSS, where N > 2 ) absorber. Target spectrum = white noise. Frequency range = 20–4500 Hz. Each curve is the absorption response at a different depth from 10 mm to 50 mm in 2 mm increments.

all theSeMSAdesignsthebandwidthextendsconsiderablyastheabsorber depthincreases.Itisdiﬃculttodiscern which SeMSA designperforms best intermsof overallabsorption by observingthe absorptioncurves,so the overallabsorption weighted by theinput sound spectrum (c_α) isshown asa function ofabsorber depth in Fig.6(a). Forwhite noisec_α is equal to thelinear averageof theabsorption curvesshownin Fig.5. Theoverall absorption increasesmarkedly between themicroperforatedplateabsorber(MPP)andallSeMSAdesigns.BetweenallSeMSAdesignstheoverallabsorptionisquite similar, butpeaks fora multi-sectordesign withN=4atthe maximumdepth. Atsmaller absorber depths closerto the minimumtheabsorptionresponse ishigherwhenmoresectorsareadded.Theoverallabsorption increasesmonotonically withabsorberdepthineachcase.

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Fig. 7. Absorption spectra for a range of absorber depths from 10 mm (dark/blue) to 50 mm (light/yellow). Target spectrum = fan noise. Frequency range

= 20–4500 Hz. R = 40 mm. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

TheoptimisationroutinewasrepeatedforasmallerabsorberdesignwithR=20mmwhichdemonstratesverydifferent behaviour to the R=40mm absorbers, seeFig. 6(b).In thisexamplethe two-segment SeMSA (2SS) clearlyoutperforms the other designsin termsofoverall absorptionatall depths. TheSeMSAabsorption performance decreases considerably as more sectors are added andactually performs worse than the MPP for N>3. The individual sectors are constrained to smallervolumesasthe numberofsectorsincreases whichlimitsthe abilityofthe optimisationroutine toextend the absorptionresponseofthemulti-sectorabsorberdesignswithinthissmalleroverallabsorbervolume.Notethatthemicrop- erforatedplate(MPP) absorber responseis identicalfortheR=20 mmandR=40mm casessince theimpedanceofan MPPabsorberisunaffectedbythecross-sectionaldimensionsoftheabsorber.

Fig.6(a)and(b)quantifytheabsorptionperformanceoftheabsorberdesignsanddemonstratethebenefitsandpotential limitationstoaddingmoresectorstoaSeMSAdesign.However,itisunclearbyanalysingtheabsorptioncoefficientsalone whetherwehaveachievedclosetothetheoreticallimitsofabsorptionperformanceateachabsorberdepth.Inordertoasses this, Eq.(16)wascalculatedusingeachoftheabsorptionspectrainFig.5asinputs.Thisquantifiesthedifferencebetween theactualabsorber depthandtheminimumabsorberdepththatcantheoreticallygeneratethesameabsorptionresponse, see Fig. 6(c) and (d). For the MPP absorber we see that the absorber is, on average, 18 mm from the causal optimum minimumdepthtoachievethesameabsorptionresponse.Thisshowsagreatdealofroomforimprovement,whichexplains howSeMSAisabletoachievesignificantlyhigherabsorptionresponseswithinthesamedepth.FortheSeMSAdesignsthe ^Dmvalue remainsbelow1mmforall designsforbothabsorbersizes. Interestingly,thisholds trueevenformulti-sector SeMSAdesignswhichshowloweroverallabsorptionresponsesforR=20mm.Thisisduetothefactthatthemulti-sector SeMSAabsorbersgenerateabsorptionpeaksatrelativelylowfrequencieswhichareclosetowhatistheoreticallypossibleat eachdepth,howeverthebandwidthoftheseabsorptionpeaksissmallandleadstolowerc_α values.Theabsorptionspectra fortheR=20mmabsorberdesignsmaybefoundinsectionS2ofthesupplementarymaterial.

In real acoustic mitigation scenarios the target soundsource(s) for absorption may not be spectrally flat asanalysed above. The optimisationroutine isrepeatedasabove withthecost function(see Eq.(17)) modifiedby a changeofinput spectrum Gxx[k].Inthiscasethe auto-spectraldensityfunctionGxx[k] estimatedfora compactaxialfan (SanyoDenkiSan Ace,120mmdiameter)frommicrophonemeasurementstakeninsideananechoicchamberisusedtoweightc_α inEq.(18). Theanalysedfrequencyrangeismaintainedat20Hzto4500Hz.TheabsorptionspectraforarangeofoptimisedR=40mm absorbers are shown in Fig. 7 that target the acoustic signature of this axial fan. The normalised auto-spectral density functionofthefanisalsoshownineachsub-figurewithblack-dashedlines.

The absorptionspectraforthefan noisecasedemonstrateverydifferentbehaviourwhencomparedtothewhitenoise example.Fannoiseshowsstrongtonalpeaksattheblade-passfrequencyofthefanandits harmonicswhichisat683Hz forthisfandesignatthisrotationalspeed.Atlowfrequenciesclosetotheblade-passfrequency(500–750Hz)thereisalso considerablebroadbandnoisegeneration.Theoptimisationroutineproducesabsorptionspectrawhichtradethewideband- widthabsorptionobservedforthewhitenoisecaseforanarrowbandabsorption responsecentredclosetotheblade-pass frequency.Thistrade-off isillustratedinthecausalityconstraintinEq.(14).As moresectorsareaddedtothemulti-sector designstheabsorptionresponsemaybefurthertunedtoexhibitpeaksofabsorptionattheharmonicsofthefanblade-pass frequency. Theseadditionalabsorptionpeaksallowthe multi-sectordesignto achievean overallabsorption responsethat

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Fig. 8. Colormap of overall absorption coeﬃcent c α(left). Delta between the causally optimal minimum distance and the SeMSA cell depth for the absorption spectra shown in the previous section (right). Target spectrum = axial fan. Frequency range = 20–4500 Hz. R = 40 mm.

increasesasmoresectorsareadded,seeFig.8.Interestinglythetwo-segmentSeMSAdesigndeliverstheworstabsorption response at low depths as the optimisationroutine struggles to locate signiﬁcant absorption close tothe fan blade-pass frequency. As the absorber depth increases the superior bandwidth ofthe two-segment absorber leads to higheroverall absorptionthanfortheMPPdesign.

Giventhewidebandwidthofsoundbeingtargetedforabsorptioninthisstudy,theacousticabsorptionperformance of a bulkporousabsorberisalsoconsideredforcomparison.Absorptionspectraforamelamineacousticfoamwasextracted froma manufacturer’s datasheet[19]andc_α wascalculated forboth thewhitenoise andfan noisecases.For a25mm thick melaminefoamsamplec_α wasestimatedas0.62forwhitenoiseabsorptionand0.48forfannoiseabsorptionwhich isconsiderablelowerthantheabsorptionvaluesobservedinFigs.6and8.

6.2. Numericalsimulations

Theanalyticalpredictionsoftheabsorptionresponses ofboththeMPPandSeMSAabsorbersdescribedabovearebased onthegraphtheoryapproachdescribedearlierinthisarticle.Inordertovalidatethattheseanalyticalpredictionsareinfact accurate predictionsof therealabsorptionresponses, theanalytical predictionsare comparedwithnumericalsimulations using COMSOLMultiphysics.However, the membranemodelused withinthe simulationaccountsfor dampinginsidethe membranesusingadifferentphysicalmodeltoan acousticimpedancevalueaslistedinTable1.Toamendthis,theeffect ofmembrane dampingvaluewasnot factoredintothe simulationsi.e.anyacousticabsorption occursintheholesof the microperforatedplates.Inordertoaligntheanalyticalpredictionswiththisundampedmodel,theoptimisationroutinewas runwith

ζ

=0andallother parametersinTable1wereleftfreeintheoptimisation.Thetargetspectrumtominimisethe costfunctioninEq.(17)istheaxialfanspectrumasdiscussedabove.

Fig. 9(a) showsthegeometryofthe threesector SeMSAdesign that hasbeengenerated bythe optimisationroutine.

Thedetails ofthismodelmaybe foundintheMethodssectionbelow.Fig.9(b)comparestheabsorptionspectrumforthis 10mmdeepabsorberaspredictedusingthegraphtheorymethod(analytical)andascalculatedbytheCOMSOLmodel(numerical).Thetwospectrashowexcellentagreement.Fig.9(c)showstheintegrateddissipationinthethreemicroperforated plates(MPPs)insidetheSeMSAcell,andtheacousticpressureamplitudesaveragedovereachsectorareshowninFig.9(d).

TheannotationsinFig.9(a)indicatethelocationsofMPPsandsectors1–3.Eachoftheabsorptionpeaksisassociatedwith a spikeinpowerdissipationinthemicroperforatedplateswhichisdriven byhighacousticvelocitiesthoughthe perforations drivenbythemotionofthemembranes.Eachsector’sresonancealsocorrespondstoasharpincreaseintheacoustic pressure inall sectors, demonstratingthecoupling oftheacoustic systembetweensectors.The highestacousticpressure amplitudeisobservedinadifferentsectorateachresonancefrequency.Fig.9(e)–(g)showtheacousticpressureﬁeldsinside the SeMSAcell atthethreeresonancefrequencies.Theseimagesillustratetheuniformity oftheacousticpressureﬁeld in thethreesectors. Thisuniformitydemonstratesthevalidityofthelumped-elementmodelforanalysingthephysics ofthe SeMSAcellandexplainsthecloseagreementintheabsorptionspectrainFig.9(b).

7. Discussion

Thepresentedgraphtheorymethodforcalculatingthetwo-pointimpedanceofacircuitmaybeappliedtoanyacoustic systeme.g.waveguide,absorberorbarrierwherealumped-elementmodelisvalid.Themethodhasbeenappliedtomodel theresponseofanoptimisedacousticabsorberunderconsiderationofabsorbersizeandthespectrumofthetargetedsound source forabsorption,generatingthegeometric parameters tomaximiseoverallabsorption. Thismethodensures thatthe mosteffectiveabsorberpossibleperunitvolumeisgeneratedforeachabsorberdesigninvestigated,asevidentbythehigh overallabsorptionvaluesandcloseproximitytocausaloptimaldepthsforthegeneratedabsorptionspectra.Fundamentally thecausaloptimalityconstraintensuresthatnoabsorbercanexhibitperfectbroadbandresponseinaﬁnitevolume.

This study has demonstrated that adding sectors can improve the absorption response of SeMSA, howeverthe ideal SeMSA design isgenerally dependent on thecross-sectional size of theabsorber. Forwhite noise absorption witha R= 20mmabsorberthehighestabsorptionresponseisobservedwithatwo-segmentdesign.However,theabsorptionresponse ofa larger(R=40mm)SeMSAdesignpeaksfora four-sectordesignathigherabsorber depthanda seven-sectordesign

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Fig. 9. Numerical simulation results for a three sector SeMSA cell, R = 20 mm, D = 10 mm. Target spectrum = axial fan. Frequency range = 20–2500 Hz.

Analytical: C = −2 . 91 dB. Numerical: C = −2 . 95 dB.

atlower absorberdepths. Thereisnosingleabsorber whichdeliversthebestabsorption performanceacross allscenarios investigated; forpractical implementation, the correct choice must be madeunder consideration of available space and the characteristics of the soundsource. Furthermore, the manufacturing complexity of the absorber must be takeninto considerationaswellastheabsorptionperformance.SpectralweightingfunctionssuchasA-weightingmayalsobefactored into the cost function to account forpsychoacoustic effects ofsounds atdifferent frequencies. Adding moresectors may increase the absorption bandwidthwhich isespecially beneﬁcial forthe noiseabsorption ofbroadband sources, however manufacturingtheseabsorbersatscalemaybechallengewithoutadditivemanufacturing.

Theproximityoftheabsorberdepthtothecausaloptimalminimumdepth(^D⁾îs^notâ^sufficient^predictorôfâ^high- performing absorber in terms ofoverall absorption. However, thisvalue does indicate that the absorber’s low-frequency response isclosetomaximumfortheprescribed absorberdepth. Inthisanalysisthe backwallsoftheabsorbershasnot been factored into the absorber depths D.In accordance withthe mass law, theideal wall material should have a high density in order to keep the overall absorber volume ascompact aspossible while ensuring minimal sound leakage. It would be possible to extend the graph theory analysis to include the effects of this soundleakage with a given set of materialpropertiesandtooptimisetheabsorberwallthicknessesforafinitevolume.

The overallimpedanceoftheabsorbershasbeenthe focusofthisstudy,howeveritwouldalsobepossibleto analyse theimpedancebetweenanypairofnodeswithin theequivalentcircuit.Thiscouldbeused forexampletoquantifylosses throughspeciﬁcelementssuchasindividualmicroperforatedplatesandmembranes.Thisstudyhasalsofocusedonnormal-

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Fig. 10. Adjacency matrix (left) for a 3 ×3 rectangular grid form of SeMSA (right).

incidenceabsorptionbutcouldbeextendedtograzingincidencesoundbyapplyingadifferentpotentialatdifferentpoints oftheequivalentcircuit.Thisanalysiscouldbeappliedtoacousticliners.

The soundabsorbersanalysed inthisstudyhaveallbeencylindricalinordertoaidfutureexperimental investigations usinga cylindricalimpedancetube.An additionalexampleofarectangulargridSeMSAabsorber isshowninFig.10.This geometryhastheadvantagethatitcantessellate.Thecentralrectangularsub-volumeinthegridhasfourmicroperforated platesjoiningneighbouringsub-volumeswhichwillfurthercomplicatethe equivalentcircuit.Thegraphtheory methodis thereforeidealforanalysingsuchacomplexdesign,whichrepresentsjustoneofawidearrayofpotentialfuturepossibilities forstudyusingthemethodologyoutlinedinthisarticle.

8. Methods

8.1. Analyticaloptimisationroutine

The analytical optimisation routinewas performedinMATLAB using thefmincon function forincreasing depths, D.In ordertoprevent errorsintheminimisationofthecostfunctionwherethesolutionconvergesonalocalminimawhichis notgloballyoptimal,anumberofcheckswereinplace.If, attheendoftheoptimisationroutineatagivenDthesolution doesnotexceedthefollowingthresholds:

• c_α>0.1,ascalculatedbyEq.(18),

• ^Dm>−1 mm,ascalculatedbyEq.(15).

the optimisation routine at the currentdepth is re-initialisedwith a random combinationof thefree parameters inthe rangesdeﬁnedbyTable1.

8.2. Numericalsimulations

ThenumericalmodelsinvestigatedinthisstudywereanalysedinCOMSOLMultiphysicsinthefrequencydomain.Thetop facesofthemodelweremodelledasalinearelasticmembranewiththematerialpropertiesoflatexrubber.Themembrane isﬁxedalongtheperimeterofeachsector.Additionalmasswasaddedtothemembranescoveringeachsectortomatchthe prescribed m_i termsgivenbytheanalyticaloptimisation routine.Anormally-incident,uniformacousticpressureof0.1Pa wasappliedtoeachmembraneforeveryfrequencytested.

Each sector cavityinthe cell wasmodelled usingpressure acoustics domains(no thermoviscouslossesare included).

The microperforations weremodelled usingthermoacousticsdomains inorderto capturethethermoviscouslossesinside the perforations with a boundary layer mesh. Twenty mesh nodeswere placed inside the viscous penetration depth at thehighestfrequencytestedinordertoaccuratelycapturetheviscousboundarylayer.Thethermoacousticsdomainswere extended 0.5mm eithersideof theendsof theperforations inorderto captureendeffects.Theabsorptivelossesinside theSeMSAcellwere quantiﬁedbyintegratingthetotalthermoviscouslossesinsidethethermoacousticdomains.Allother boundariesinthesimulationweremodelledashardwallsi.e.haveinﬁniteacousticimpedance.

Additionalinformation

The authorsdeclarethat theresearch wasconductedintheabsenceofanyrelationshipsthat could beconstruedasa potentialconﬂictofinterest.

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DeclarationofCompetingInterest

The authors declare that they have no knowncompeting ﬁnancial interests or personal relationshipsthat could have appearedtoinﬂuencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

Ian Davis:Conceptualization, Methodology, Software,Visualization,Writing - originaldraft. Andrew McKay:Investiga- tion,Software,Datacuration,Writing-review&editing.GarethJ.Bennett:Supervision,Writing-review&editing.

Acknowledgements

This research was partly funded under the Irish Research Council Enterprise Partnership Scheme (Postdoctoral) EP- SPD/2017/123withﬁnancialcontributionsfromTrinityCollegeDublin,NokiaBellLabsandtheIrishResearchCouncil Supplementarymaterial

Thesupplementarymaterialdocumentincludesthefollowing:

• Examplesof theKirchoff’sLaw simpliﬁcationsofthe biypramidequivalentcircuits fora three-andfour-sector SeMSA design.

• AdditionalabsorptionspectraforawidecombinationofparameterssuchasSeMSAcellradius,absorberdepth,targeted frequencyrangeandtargetedsoundspectrum.

• Colormapsof(^cα)^and^plots^of^Dmforalltheabove.

• An additionalnumericalsimulation forafour-sector SeMSAdesign,further validatingtheeﬃcacyofthe graphtheory method.

Supplementarymaterialassociatedwiththisarticlecanbefound,intheonlineversion,at10.1016/j.jsv.2021.116135 References

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