Finite difference and finite volume methods for wave-based modelling of room acoustics

356 Pages·2016·24.61 MB·English

by  Brian Hamilton

Finite Difference and Finite Volume Methods for
Wave-based Modelling of Room Acoustics
Brian Hamilton
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF EDINBURGH
2016

Abstract
Wave-basedmodelsofsoundpropagationcanbeusedtopredictandsynthesizesoundsastheywould
beheardnaturallyinroomacousticenvironments. Thenumericalsimulationofsuchmodelswithtradi-
tionaltime-steppinggrid-basedmethodscanbeanexpensiveprocess,duetothesheersizeoflistening
environments(e.g.,auditoriumsandconcerthalls)andduetothetemporalresolutionrequiredbyaudio
rates that resolve frequencies up to the limit of human hearing. Finite difference methods comprise a
simplestartingpointforsuchsimulations,buttheyareknowntosufferfromapproximationerrorsthat
maynecessitateexpensivegridrefinementsinordertoachievesufficientlevelsofaccuracy. Assuch,a
significantamountofresearchhasgoneintodesigningfinitedifferencemethodsthatarehighlyaccurate
whileremainingcomputationallyefficient.
Theproblemofdesigningandusingaccuratefinitedifferenceschemesiscompoundedbythefact
that room acoustics models require complex boundary conditions to model frequency-dependent wall
impedancesovernon-trivialgeometries. Theimplementationofsuchboundaryconditionsinanumeri-
callystablemannerhasbeenachallengeforsometime. Stableboundaryconditionsforfinitedifference
roomacousticssimulationshavebeenformulatedinthepast,butgenerallytheyhaveonlybeenuseful
in modelling trivial geometries (e.g., idealised shoebox halls). Finite volume methods have recently
been shown to be a viable solution to the problem of complex boundary conditions over non-trivial
geometries, andtheyalsoallowfortheuseofenergymethodsfornumericalstabilityanalyses. Finite
volumemethodslendthemselvesnaturallytofullyunstructuredgridsandtheycansimplifytothetypes
ofgridstypicallyusedinfinitedifferencemethods. Thisallowsforroomacousticssimulationmodels
that balance the simplicity of finite difference methods for wave propagation in air with the detail of
finitevolumemethodsforthemodellingofcomplexboundaries.
Thisthesisisanexplorationofthesetwodistinct,yetrelated,approachestowave-basedroomacous-
ticsimulations. Theoverarchingthemeinthisinvestigationisthebalancebetweenaccuracy,computa-
tionalefficiency,andnumericalstability. Higher-orderandoptimisedschemesintwoandthreespatial
dimensionsarederivedandcompared,towardsthegoaloffindingaccurateandefficientfinitedifference
schemes.Numericalstabilityisanalysedusingfrequency-domainanalyses,aswellasenergytechniques
whenever possible, allowing for stable and frequency-dependent boundary conditions appropriate for
room acoustics modelling. Along the way, the use of non-Cartesian grids is investigated, geometric
relationshipsbetweencertainfinitedifferenceandfinitevolumeschemesareexplored,andsomeprob-
lems associated to staircasing effects at boundaries are considered. Also, models of sound absorption
inairareincorporatedintothesenumericalschemes,usingphysicalparametersthatareappropriatefor
roomacousticscenarios.
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Declaration
I declare that that this thesis was composed by myself, that the work herein is my own except where
explicitlystatedotherwiseinthetext,andthatthisworkhasnotbeensubmittedforanyotherdegreeor
professionalqualificationexceptasspecified.
(BrianHamilton)
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Acknowledgements
TherearemanypeoplethatImustthankforhelpandsupportthroughoutthisPhDprogramme.
MydeepestthanksgotomymainPhDsupervisor:Dr. StefanBilbao—forguidancethroughoutall
phasesofthiswork;foryourwisdominallthingsrelatedtoacoustics,signalprocessing,andnumerical
methods;andalsoforyourgoodhumour.
IwouldliketothankProf. LauriSaviojaforco-supervisingthisthesis,andforthemanywelcoming
visitsIhavemadetoAaltoUniversity. ThanksalsotoJonathanBottsandJukkaSaarelmaforcollabo-
rationsandfruitfuldiscussionsregardingallaspectsofnumericalwavesimulation.
Thanks for the support from the friends I have made in Edinburgh over the years. Thanks to my
colleaguesintheAcousticsandAudioGroupforencouragementalongtheway. Inparticular, special
thankstoAlbertoandCraigforcollaborationsandforproofreadingportionsofthisthesis.
Thanks to Prof. Dick Botteldooren, external examiner on this thesis, and Dr. James Hopgood,
internalexaminer,forthoroughreviewsofthisthesisandformanyhelpfulsuggestionsandcorrections.
This work would not have been possible without the financial support of the European Research
Council, undergrantStG-2011-279068-NESS,andoftheNaturalSciencesandEngineeringResearch
CouncilofCanada(NSERC)underthePostgraduateScholarships-Doctoral(PGS-D)Program.
Lastly, thanks to myparents fora lifetimeof encouragement, and sincerethanks to Laurafor per-
sonalsupportthroughoutthisarduousthesis-writingjourney.
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Contents
Abstract i
Declaration iii
Acknowledgements v
1 Introduction 1
1.1 Introductoryremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesisobjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Outlineofthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Maincontributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 RelatedPublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Background 11
2.1 Roomacousticsfundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Linearandlosslesssoundpropagationinair . . . . . . . . . . . . . . . . . . . 11
2.1.2 Soundsourcesinfreespace . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Boundaryconditionsmodellinglocallyreactivewalls . . . . . . . . . . . . . . 17
2.1.4 Soundabsorptioninair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.5 Roomimpulseresponses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2 Computationalmethodsforroomacousticsinthetime-domain . . . . . . . . . . . . . 33
2.2.1 Geometricmethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.2 Finitedifferencemethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.3 Finiteelementandfinitevolumemethods . . . . . . . . . . . . . . . . . . . . 40
2.2.4 Boundaryelementmethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.2.5 Fouriermethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3 Roomacousticsmodellingin3-DwiththesimpleCartesianscheme 45
3.1 Thebasic3-Dscheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Anequivalentstaggeredgridformulation . . . . . . . . . . . . . . . . . . . . 49
3.1.3 Energystabilityanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.4 Numericaldispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 Boundaryconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
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3.2.2 Frequency-independentboundaryconditions . . . . . . . . . . . . . . . . . . 65
3.2.3 Boundaryconditionsforthestaggeredscheme . . . . . . . . . . . . . . . . . 75
3.2.4 Frequency-dependentboundaryconditions . . . . . . . . . . . . . . . . . . . 76
3.3 Incorporatingviscothermallosses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3.1 Energyanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.2 Astaggeredfirst-orderviscousscheme . . . . . . . . . . . . . . . . . . . . . 85
3.3.3 Boundaryconditionsfortheviscothermalscheme . . . . . . . . . . . . . . . . 86
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Finitedifferenceschemesforthewaveequationinfreespace 91
4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.1 SamplingconsiderationsonCartesiangrids . . . . . . . . . . . . . . . . . . . 92
4.1.2 Explicittwo-stepfinitedifferenceschemesforthewaveequation. . . . . . . . 96
4.2 Finitedifferenceschemesforthe1-Dwaveequation . . . . . . . . . . . . . . . . . . 102
4.2.1 Thesimplestscheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.2 Parametrisedapproximationstothe1-DLaplacian . . . . . . . . . . . . . . . 104
4.2.3 Schemeswithhigh-orderspatialdifferencingin1-D . . . . . . . . . . . . . . 106
4.2.4 High-orderaccurateschemesusingmodifiedequations . . . . . . . . . . . . . 109
4.3 Cartesianschemesforthe2-Dwaveequation . . . . . . . . . . . . . . . . . . . . . . 113
4.3.1 Thesimplestschemein2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3.2 Parametrisedapproximationstothe2-DLaplacian . . . . . . . . . . . . . . . 115
4.3.3 Schemeswithhigh-orderspatialdifferencesin2-D . . . . . . . . . . . . . . . 118
4.3.4 Compact(K+1)-pointschemesin2-DwithK 8 . . . . . . . . . . . . . . 121
≤
4.3.5 High-order(K+1)-pointschemesin2-DwithK 24 . . . . . . . . . . . . 126
≤
4.3.6 Optimised(K+1)-pointschemesin2-DwithK 24. . . . . . . . . . . . . 129
≤
4.3.7 Comparingschemesforwidebandaccuracy . . . . . . . . . . . . . . . . . . . 130
4.3.8 Numericalexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.4 Hexagonalschemesforthe2-Dwaveequation. . . . . . . . . . . . . . . . . . . . . . 142
4.4.1 Thehexagonalgridandsamplingconsiderations . . . . . . . . . . . . . . . . 142
4.4.2 ParametrisedapproximationstotheLaplacian . . . . . . . . . . . . . . . . . . 145
4.4.3 Thesimpleseven-pointhexagonalscheme . . . . . . . . . . . . . . . . . . . . 147
4.4.4 Afamilyof(K+1)-pointcompactschemeswithK 12 . . . . . . . . . . . 148
≤
4.4.5 High-order(K+1)-pointschemeswithK 24 . . . . . . . . . . . . . . . . 150
≤
4.4.6 Numericalexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.5 Cartesianschemesforthe3-Dwaveequation . . . . . . . . . . . . . . . . . . . . . . 156
4.5.1 Parametrisedapproximationstothe3-DLaplacian . . . . . . . . . . . . . . . 156
4.5.2 Schemeswithhigh-orderspatialdifferencingin3-D . . . . . . . . . . . . . . 159
4.5.3 Compact(K+1)-pointschemesin3-DwithK 26 . . . . . . . . . . . . . 162
≤
4.5.4 High-orderandoptimised(K+1)-pointschemeswithK 62 . . . . . . . . 170
≤
4.5.5 Comparingschemesforwidebandaccuracy . . . . . . . . . . . . . . . . . . . 173
4.5.6 Numericalexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
4.6 Non-Cartesianschemesforthe3-Dwaveequation. . . . . . . . . . . . . . . . . . . . 181
4.6.1 Non-Cartesiangridsandsamplingconsiderations . . . . . . . . . . . . . . . . 181
4.6.2 DiscreteLaplaciansonFCCandBCCgrids . . . . . . . . . . . . . . . . . . . 186
4.6.3 Thesimplestnon-CartesianschemesonFCCandBCCgrids . . . . . . . . . . 188
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