Thesis_Wave Based Modelling Methods for Steady State Vibro Accoustics

7/30/2019 Thesis_Wave Based Modelling Methods for Steady State Vibro Accoustics

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KATHOLIEKE UNIVERSITEIT LEUVEN

FACULEIT INGENIEURSWETENSCHAPPENDEPARTEMENT WERKTUIGKUNDE

Celestijnenlaan 300B B-3001 Leuven (Heverlee), Belgium

MACHINEBOUW EN AUTOMATISERING

AFDELING PRODUCTIETECHNIEKEN

WAVE BASED MODELLING METHODS

FOR STEADY-STATE VIBRO-ACOUSTICS

Promotoren:Prof. dr. ir. D. VandepitteProf. dr. ir. W. Desmet

Proefschrift voorgedragen tothet behalen van het doctoraatin de ingenieurswetenschappen

door

Bert PLUYMERS

2006D04 Juni 2006


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KATHOLIEKE UNIVERSITEIT LEUVENFACULTEIT INGENIEURSWETENSCHAPPENDEPARTEMENT WERKTUIGKUNDEAFDELING PRODUCTIETECHNIEKEN,MACHINEBOUW EN AUTOMATISERINGCelestijnenlaan 300B, 3001 Heverlee (Leuven), Belgium

WAVE BASED MODELLING METHODS

FOR STEADY-STATE VIBRO-ACOUSTICS

Jury:Prof. dr. ir. G. De Roeck (voorzitter)Prof. dr. ir. D. VandepitteProf. dr. ir. W. DesmetProf. dr. ir. P. SasProf. dr. ir. G. DegrandeProf. dr. W. LauriksProf. R.J. Astley (ISVR Southampton)Prof. dr. ir. S. Vandewalle

Proefschrift voorgedragen tothet behalen van het doctoraatin de ingenieurswetenschappen

door

Bert PLUYMERS

UDC 534:681.3G18 Juni 2006


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c Katholieke Universiteit LeuvenFaculteit Toegepaste WetenschappenArenbergkasteel, B-3001 Heverlee (Leuven), Belgium

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All rights reserved. No part of this publication may be reproduced in any

form, by print, photoprint, microfilm or any other means without writtenpermission from the publisher.

D/2006/7515/30ISBN 90-5682-696-4UDC 534:681.3G18


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Voorwoord

Dont worry about the future;

or worry, but know that worrying is as effectiveas trying to solve an algebra equation by chewing bubblegum.

Baz Luhrmann

Doctoreren is veel meer dan het uitvoeren van je onderzoek en het neerschrij-ven van je resultaten, met die woorden heette Wim mij op 21 augustus 2000welkom op zijn bureau. Hij had overschot van gelijk! Die dag was het beginvan een bijna zes jaar durend avontuur waarvan het wetenschappelijke eindre-sultaat nu in gedrukte vorm voor u ligt. Alvorens aan te vangen met het meertechnische gedeelte, zou ik eerst graag een kort woordje willen richten tot al

de mensen die mij gedurende deze periode op een of ander manier geholpenhebben met bij onderzoek.

Vooraleerst wil ik mijn drie promotoren bedanken omdat ze mij de kansgegeven hebben om dit onderzoek uit te voeren. Vaak, wanneer iemand mijvroeg Wie is uw promotor? en ik antwoordde met Wel, eigenlijk heb ik erdrie? kreeg ik verbaasde gezichten te zien. Ik prijs mij echter gelukkig dat ikmet deze drie mensen heb mogen samenwerken.

Professor Vandepitte is hoofdpromotor van dit doctoraatsonderzoek. Ikdank hem voor de inspirerende samenwerking. Hij heeft, dankzij zijnenthousiasme, zijn opbouwende kritieken en zijn talrijke en steeds zeergrondige paper-reviews voor een groot stuk bijgedragen tot dit onderzoek.Dankzij hem ook waren steeds alle administratieve geplogenheden tot in depuntjes verzorgd. Dirk, bedankt voor dit alles!

Professor Sas wil ik bedanken voor de fijne samenwerking en voor hetvertrouwen dat hij in mij gesteld heeft door mij te laten meewerken aan deorganisatie van ISMA. Verder wil ik hem bedanken dat hij mij wegwijs heeftgemaakt in de wereld der wetenschappelijke conferenties. Ik zal mijn eerstebuitenlandse conferentie, tevens mijn eerste vliegervaring, nooit vergeten. Be-dankt Pol!

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Voorwoord

Professor Desmet wil ik bedanken voor de fijne jarenlange samenwerking.Bij hem kon ik steeds terecht met al mijn technische en minder technische vra-gen. Hij is de man die mij, als begeleider tijdens mijn eindwerk, introduceerdein de wereld van de numerieke akoestiek. Dit deed hij steeds met een niet-aflatend enthousiasme en een ongekend vereenvoudigsvermogen. Zijn enormegedrevenheid en zijn werklust werkten inspirerend. Ik wil hem bedanken voorde vele mogelijkheden die hij mij gegeven heeft om mijn werk aan anderenvoor te stellen en in het bijzonder voor de steun en de vrijheid die ik gekregenheb bij het zoeken van mijn eigen weg. Wim, van harte bedankt voor alles!

Tevens wil ik professor Degrande en professor Lauriks bedanken voor degrondige manier waarop zij, als leden van het leescomite, mijn tekst hebbennagelezen. Ook wil ik professor Vandewalle bedanken dat hij deel wil uitmakenvan mijn jury. I would like to express my gratitude to professor Astley for

reading my text and for coming all the way to Leuven for being a member ofmy jury.

Hierbij dank ik ook het Instituut voor de Aanmoediging van Innovatiedoor Wetenschap en Technologie in Vlaanderen (IWT-Vlaanderen) voor definanciele steun die zij mij heeft verleend door middel van een vierjarige spe-cialisatiebeurs.

Dank aan alle mensen op PMA die iedere dag klaar stonden om te helpen:Lieve, Karin, Carine, Ann, Luc, dank voor jullie hulp; ontelbare keren heb ikjullie lastig gevallen met faxen, onkostennotas, verzekeringsformulieren, pub-licaties, enzovoort. Steeds werd ik met een glimlach geholpen. Een speciaalwoordje van dank richt ik tot Lieve, waarmee ik samen twee ISMA-termijnenlief en leed heb gedeeld. Lieve, merci, het was heel leuk samenwerken! Dankaan de dienst informatica; Jan, merci dat je altijd klaar stond wanneer ik weer

eens een probleem had met mijn laptop, of voor de zoveelste keer langskwammet een nieuwe licentiefile. Ronny, bedankt voor het vele geduld bij het mijleren programmeren en websites bouwen. Paul, dank voor de fijne gesprekkenen voor de onvergetelijke meetcampagnes. Ik ga waarschijnlijk niet vaak meerin de vrieskou op een draaiende maaidorser staan of opgesloten worden in eenof andere afgelegen hangar. Het zijn momenten om te herinneren! Bedanktook Eddy, Viggo en Dirk. Ik heb weliswaar niet vaak professioneel een beroepop jullie gedaan, maar de leuke gesprekken over voetbal, wielrennen en hetleven op PMA blijven mij altijd bij. Jean-Pierre, Polleke, Firmin, Raymond,ook jullie hartelijk dank voor de vele last-minute kleurenprintjes en de velesleutel-koffies. Raymond, dank voor de talloze leuke momenten. Zonder u waser waarschijnlijk nooit sprake geweest van het sociale leven op PMA zoals datnu bestaat. Merci voor het organiseren van de vele PMA-weekends en bbqs.

Bedankt ook voor de vele leuke babbels!Klaas, Maarten, Bert, deze paragraaf is voor jullie! Hartelijk dank dat ik

met mijn vele LaTeX, Linux en C++ vraagjes steeds bij jullie terecht kon.Nog meer bedankt, dat jullie er ook steeds een gepast antwoord op wisten!

I would also like to thank all the foreign Ph.D. students who I had thepleasure of working with: Achim (Hepi hotel) Hepberger, Rogerio (Rocket

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Voorwoord

man) Pirk, Krisztian (Mr. serious scientist) Gulyas, Alessandro (Alec) Za-narini, Ravish (GID) Masti, Silje (Nintzel) Korte, Petra (Packa) Silar, Emil-iano (Emi) Mucchi, Atilla (Forest) Nagy and all the others, thank you all fora very nice time!

Een speciaal bedankje gaat uit naar mijn collega WBMers. Bas, je stondaltijd klaar met raad en daad en je wist elk probleem op te lossen met eengepast Matlab scriptje :-) Ik denk nog met veel plezier terug aan de vaakfilosofische discussies over de WBM. Bedankt voor al je hulp! Ook bedanktvoor de vele leuke momenten op het voet- en voetvolleybalveld. Verder mercivoor het nalezen van delen van mijn tekst. Caroline, van harte dank voor deleuke samenwerking en de fijne tijd op bureau 02.17. Bedankt dat je er altijdwas om de frustraties van het programmeren en het debuggen mee te delenen merci voor het nalezen van delen van dit boekje. Muchas gracias ook voor

de vele leuke momenten naast het werk! Bert, bedankt dat je het onderzoekwil verder zetten en merci voor je hulp deze laatste weken.

Dank aan alle sportievelingen: de donderdag/dinsdag-middag shotters, demaandag-middag squashers, de woensdag-avond fietsers, de dinsdag-middaglopers, het waren steeds momenten om naar uit te zien. De talloze IBTs,citadel-ascension-days, tours en rondes, .. het zijn stuk voor stuk mooie herin-neringen.

Merci ook aan de leden van het BAD. Heren, wie had ooit kunnen denkendat een mopje over Judas om 3 uur s nachts ergens in een godverlaten gatin de Walen1 zon gevolgen kon hebben! Niet toevallig zijn jullie net diemensen die ervoor zorgden dat de wekelijkse Happy Hours vaak doorliepen totvroeg in de morgen en dat het woord kroegentocht een totaal andere dimensiekreeg. En dan met zon kanjer van een voorzitter, de sleutel eigenlijk voor

een gegarandeerd succes! Bedank allemaal voor de vele leuke activiteiten, detalrijke geanimeerde gesprekken2 en de maandelijkse gatherings, dat er nogvele mogen volgen!

Een speciaal woordje van dank ook aan de drie musketiers. Dank aan Gus,mijn Madrileense mental-coach. Hij leerde me bergen te beklimmen. Dankaan Bram, mijn Beselaarse physical-coach. Hij was altijd op de juiste momentop de juiste plaats met het juiste middel3. Dank aan Wim, mijn Antwaaarpsepartner-in-meerdere-crimes. Gast, merci voor de talrijke leuke momenten, desportieve prestaties, de fijne gesprekken, de frisse duvels en de liters koffie.Heren, het was een feesje dat nog lang mag duren!

Verder dank aan iedereen die mee verantwoordelijk is voor wat bekendstaat als het SET. Omdat het onbegonnen werk is om alle leuke momentenvan de voorbije jaren op te sommen, hieronder een kleine bloemlezing van

de hoogtepunten die mij nog lang gaan bijblijven: De koning te rijk op eenstoeltje in de Geckos met nen Hoegaarden Grand-Cru in de hand / Nachtelijke

1Bij nader onderzoek blijkt het Jalhay te zijn.2Gaande van discussies over tandwielingrijpfrequenties tot boeiende uiteenzettingen over

regenbogen.3Loperamide Hydrochloride

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Voorwoord

Rielse eitjes met ne goeie whiskey / Ne side-by-side in de Footloose / Metde Jean in de Weirdos .. Greg, zijn da nie ons studenten? / De zon zienondergaan in Santa-Barbara of ne Leffe drinken in San-Fransisco, het is eenmoeilijke keuze / Den Alp in 4 uur, in 2 uur of int wit, hoe dan ook hijblijft de moeite / Despeekes in de Giraf en Xmas bier in het cafe der engelen/ de schuur / One tok tok tok tok tok tok tok One .. met ne squivel-stepderaan vast / Pol? .. Ja, Pol?? .. tzennekik .. / Effe die pannenkoekengaan wegsmijten / Een paars-groene kerst gewenst / Tres absolut con lemonpor favor / Chauffeuren in Reservoir Dogs stijl / Patience .. ik had mijnmomenten / Goesting in een zakske chips? / Was het nu 5-3 of 5-2 ? /Vrijdagmorgen 7u ontbijt, daar horen zachtgekookte eitjes bij / Peanuts in deCantop Lounge / stressballetjes / . . .

Tot slot nog een speciaal woordje van dank aan mijn ouders, mijn broer

en mijn zus voor hun jarenlange steun en motivatie. Mama, papa, merci voorde mogelijkheden die jullie mij geboden hebben. Ik heb een hele weg afgelegd,maar ik had het nooit kunnen doen zonder jullie!

BertJuni, 2006

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Abstract

This dissertation considers the development of wave based prediction meth-ods for the analysis of steady-state vibro-acoustic problems. Conventionalelement based prediction methods, such as the finite element method (FEM),are commonly used, but are restricted to low-frequency applications. Thewave based method (WBM) is an alternative deterministic technique whichis based on the indirect Trefftz approach. The WBM is computationally veryefficient, allowing the analysis of problems at higher frequencies. This disser-tation reports on an extension of the WBM to multi-domain acoustic problemsand problems involving unbounded acoustic fluid domains, such as transmis-sion, scattering and radiation problems. The efficiency of the WBM is mostpronounced for problems of moderate geometrical complexity. A hybrid fi-nite element-wave based method combines the strengths of the two methods,namely, the high computational efficiency of the WBM and the ability of theFEM to model problems of arbitrary geometrical complexity. Numerical vali-dation examples show the enhanced computational efficiency of the WBM forproblems of moderate geometrical complexity and of the hybrid method forreal-life engineering problems.

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Beknopte samenvatting

Dit proefschrift beschrijft de ontwikkeling van golfgebaseerde modellerings-

technieken voor het analyseren van stationaire vibro-akoestische proble-men. Vaak worden klassieke elementgebaseerde methoden, zoals de eindige-elementenmethode (EEM) en de randelementenmethode (REM), toegepastvoor de analyse van dit type van problemen. De toepasbaarheid van de ele-mentgebaseerde methoden is echter beperkt tot het laagfrequente gebied. Degolfgebaseerde methode (GBM) is een alternatieve methode die gebaseerd isop een indirecte Trefftz benadering. De GBM is rekenkundig zeer efficient,waardoor de methode ook toepasbaar is voor het analyseren van problemenbij hogere frequenties. Dit proefschrift rapporteert over de ontwikkeling vaneen uitbreiding van de GBM voor het behandelen van akoestische proble-men bestaande uit meerdere subdomeinen en voor het analyseren van prob-lemen in oneindige fluda. De efficientie van de GBM is het meest uitge-sproken voor problemen met een bescheiden geometrische complexiteit. Eenhybride eindige-elementen-golfgebaseerde methode combineert de sterke pun-ten van de twee technieken, namelijk de hoge rekenkundige efficientie van deGBM en de toepasbaarheid van de EEM voor problemen met een willekeurigegeometrische complexiteit. Numerieke validatievoorbeelden tonen het ver-hoogde prestatievermogen aan van zowel de uitgebreide GBM voor problemenmet een bescheiden geometrische complexiteit, als van de hybride methodevoor levensechte ingenieurstoepassingen.

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List of symbols

Abbreviations

1D : one-dimensional

2D : two-dimensional

3D : three-dimensional

ABC : absorbent boundary condition

AMLS : automated multi-level substructuring

API : application programming interface

BC : boundary condition

BE : boundary element

BEM : boundary element method

BiCG-Stab : biconjugate gradient stabilized

BKM : boundary knot method

CAD : computer aided designCAE : computer aided engineering

CMS : component mode synthesis

CPU : central processing unit

CVE : complex envelope vectorization

DEM : discontinuous enrichment method

DGM : discontinuous Galerkin method

dof : degree of freedom

dofs : degrees of freedom

DRMFS : dual reciprocity MFS

DtN : Dirichlet-to-Neumann

EFGM : element-free Galerkin method

ESSM : energy source simulation method

FE : finite element

FEM : finite element method

FETI : finite element tearing and interconnecting

FETI H : FETI Helmholtz

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List of symbols

FETI 2LM : FETI 2 Lagrangian multipliersFFE : full-field equations method

FFE1 : full-field equations method of the first kind

FFE2 : full-field equations method of the second kind

flops : floating point operations

FRF : frequency response function

GFEM : generalized FEM

GLS : Galerkin least squares

GP : Gauss point

GLS : Galerkin gradient least squaresGMRES : generalized minimal residual

HFBEM : high frequency BEM

IE : infinite element

IEM : infinite element method

L-MAG : local mesh-dependent augmented Galerkin

L2M : L2 method

LHS : left-hand side

LPBEM : lumped parameter BEM

LST : least squares Trefftz

MFS : method of fundamental solutions

MPBEM : multi-pole BEM

MPI : message passing interface

NRBC : non-reflecting boundary condition

NFE : null-field equations methodPML : perfectly matched layer

PUFEM : partition-of-unity FEM

PWM : plane wave method

QMR : quasi-minimal residual

QS-FEM : quasi stabilized FEM

RBF : radial basis function

RFB : residual-free bubble

RHS : right-hand side

SEA : statistical energy analysis

UWVF : ultra-weak variational formulation

VA : vibro-acoustic

VCTR : variational theory of complex raysWB : wave based

WBEM : wave boundary elements

WBM : wave based method

WBS : wave based substructuring

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List of symbols

Arabic symbols

A : plane wave amplitude [Pa]

A : WB system matrix

A : WB system submatrix resulting from residual formu-lation of condition

b : WB loading vector

B : WB matrix with wave function gradients

Bfe : FE matrix with shape function gradients

c : speed of sound [m/s]

c : FE-WB coupling vector

C : constantcen : n-th order even angular Mathieu function of the first

kind

C : WB coupling matrix

Cfe : FE damping matrix

D : plate bending stiffness [Nm]

E : Youngs modulus [Pa]

f : elliptical semi-focal distance [m]

f : frequency [Hz ]

fint : frequency limitation due to FE interpola-tion error

[Hz ]

fpol : frequency limitation due to FE pollutionerror

[Hz ]

fxw, fyw, fzw : wave function scaling factors

f : WB loading vector

f() : function of f : WB loading subvector resulting from residual formu-

lation of condition ffe : FE loading vector

F : line force [N/m]

g() : function of G : Greens kernel function

h : FE mesh size [m]

h : dimensionless FE mesh sizeH(2)n : n-th order Hankel function of the second kind

I : active acoustic intensity vector [W/m2]

j : imaginary unit =1

k : acoustic wave number [m1]

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List of symbols

k : dimensionless acoustic wave numberkb : structural bending wave number [m1]

kxw, kyw, kzw : acoustic wave number components [m1]

Kfe : FE stiffness matrix

L : length [m]

Lx, Ly, Lz : dimensions of bounding box [m]

M cn : n-th order even radial Mathieu-Hankel function ofthe fourth kind

Mfe : FE mass matrix

M sn : n-th order odd radial Mathieu-Hankel function of thefourth kind

n : normal direction

n : normal vector [nx ny nz]T

nav : number of frequencies

ne : number of FE

n : number of response points

nfe : number of FE nodes

ns : normal vector on structural component

nw : number of wave functions

nW : number of wave functions

N : wave function truncation factor

N : number of dofs

Nfe : FE shape function

N : frame functionNV : number of WB subdomains

Nfe : vector of FE shape functions

N : vector of frame functions

p : acoustic pressure [Pa]

p : FE shape function interpolation order

p : WB weighting function

pa : WB weighting function contribution factor [Pa]

pfe : FE nodal pressure value [Pa]

pfe : vector of FE nodal pressures

p : frame pressure [Pa]

p : frame function contribution factor [Pa]

p : vector of frame function contribution factorsp : frame weighting function

pq : particular pressure solution function for a pointsource in free field conditions

pw : wave function contribution factor [Pa]

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List of symbols

pw : vector of wave function contribution factorspw : vector of WB weighting function contribution factors

q : acoustic volume velocity point source (perunit volume)

[s1]

q : elliptical dimensionless parameter

qfe : FE loading vector

Q : acoustic volume velocity point source(source strength)

[m3/s]

Q : acoustic volume acceleration point source(source strength)

[m3/s2]

Q : FE-WB coupling matrix

r : position vector

r : polar radial coordinate [m]

R : distance [m]

R : circle radius [m]

R : residual error associated with acoustic quantity sen : n-th order odd angular Mathieu function of the first

kind

S : surface [m2]

t : time [s]

t : thickness [m]

v : acoustic velocity vector [m/s]

vfe : FE loading vector

v : frame normal velocity [m/s]v : frame function contribution factor [m/s]

v : frame function weighting function

v : vector of frame function contribution factors

vn : acoustic normal velocity [m/s]

V : 3D acoustic domain

wi : wave function series truncation number

wfe : FE loading vector

wns : structural normal displacement [m]

W : active acoustic power [W]

Wfe : FE weighting function

x,y,z : cartesian coordinates [m]

Z, Zn, Zint : acoustic (normal) impedance [Pas/m]Zfe : FE dynamic stiffness matrix

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List of symbols

Greek symbols

: Dirac-delta function

av : average absolute prediction error [dB re 2.105Pa]

p : absolute prediction error [dB re 2.105Pa]

: elliptical radial coordinate [m]

: average relative prediction error

p : relative prediction error

: elliptical angular coordinate [rad]

: loss factor

: boundary curve

a : boundary curve collecting p, v and Z

I : interface curve between two volume domains (WBmethodology)

p : boundary curve on which pressure boundary condi-tions are applied

s : boundary curve which coincides with a structuralcomponent

v : boundary curve on which normal velocity boundaryconditions are applied

Z : boundary curve on which normal impedance bound-ary conditions are applied

: boundary curve at infinity at which the Sommerfeldradiation condition is applied

: wavelength [m]

: frame variable, i.e. Lagrangian multiplier

: vector of frame variables

: double layer potential

: Poisson coefficient

w : acoustic wave function

w : vector of acoustic wave functions

0 : ambient fluid density [kg/m3]

s : structural density [kg/m3]

: single layer potential

: FE interpolation error

: polar angular coordinate [rad] : Rayleigh method acoustic source formulation

: circular frequency [rad/s]

n : eigenfrequency [rad/s]

: 2D acoustic domain

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List of symbols

: boundary surfacea : boundary surface collecting p, v and Zi : interface surface between two volume domains (FE

methodology)

I : interface surface between two volume domains (WBmethodology)

Ip : I on which pressure continuity conditions apply

Iv : I on which normal velocity continuity conditionsapply

H : hybrid interface surface between two volume domains

p : boundary surface on which pressure boundary con-ditions are applied

s : boundary surface which coincides with a structuralcomponent

v : boundary surface on which normal velocity boundaryconditions are applied

Z : boundary surface on which normal impedanceboundary conditions are applied

: boundary surface at infinity at which the Sommerfeldradiation condition is applied

: position coordinate [m]

: position vector

: boolean number

Miscellaneous symbols

a : vector (lower case, bold character)

A : matrix (upper case, bold character)

: predefined value of : approximation of() : associated with WB subdomain e : associated with FE e+, : evaluated on the positive or negative side of a sur-

face or curve, as compared to the normal direction

| | : absolute value of : L2-norm of[1, 2] : closed integer interval from 1 up to 2() : real part of () : imaginary part of

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List of symbols

CEIL() : ceiling function returning the integer value just largerthan the double precision argument

T : transpose of : complex conjugate of : the empty setV : the boundary of volume V

: partial differential operator

d : differential operator

: gradient vector

2 : Laplacian operatorLv : normal velocity operator

Leq : equivalent normal velocity operator

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Table of contents

Voorwoord I

Abstract V

Beknopte samenvatting VII

List of symbols IX

Table of contents XVII

I Introduction and state-of-the-art 1

1 Introduction 3

1.1 The need for numerical tools in vibro-acoustic engineering 31.2 CAE for vibro-acoustics: strengths and limitations . . . . 41.3 The dual research goal . . . . . . . . . . . . . . . . . . . . 7

1.3.1 New developments to the wave based method . . . 71.3.2 Development of a hybrid finite element - wave

based method . . . . . . . . . . . . . . . . . . . . . 81.4 Outline of the dissertation . . . . . . . . . . . . . . . . . . 9

2 State-of-the-art in deterministic modelling of steady-state vibro-acoustic problems 112.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Definition of steady-state vibro-acoustic problems . . . . . 122.3 The Finite Element Method . . . . . . . . . . . . . . . . . 15

2.3.1 Basic formulations . . . . . . . . . . . . . . . . . . 152.3.2 Treatment of unbounded domains . . . . . . . . . 182.3.3 Errors involved with the use of the FEM . . . . . . 222.3.4 FEM properties . . . . . . . . . . . . . . . . . . . . 23

2.4 Extensions to the FEM . . . . . . . . . . . . . . . . . . . 252.4.1 FE process optimization . . . . . . . . . . . . . . . 26

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Table of contents

2.4.2 Domain decomposition methods . . . . . . . . . . 282.4.3 Stabilized methods . . . . . . . . . . . . . . . . . . 292.4.4 Generalized methods . . . . . . . . . . . . . . . . . 302.4.5 Multi-scale methods . . . . . . . . . . . . . . . . . 322.4.6 Special purpose transformation methods . . . . . . 34

2.5 The Boundary Element Method . . . . . . . . . . . . . . . 352.5.1 Direct BEM . . . . . . . . . . . . . . . . . . . . . . 352.5.2 Indirect BEM . . . . . . . . . . . . . . . . . . . . . 362.5.3 BEM properties . . . . . . . . . . . . . . . . . . . 38

2.6 Extensions to the BEM . . . . . . . . . . . . . . . . . . . 392.6.1 Rayleigh based methods . . . . . . . . . . . . . . . 402.6.2 Fast multi-pole methods . . . . . . . . . . . . . . . 412.6.3 Generalized methods . . . . . . . . . . . . . . . . . 42

2.7 Trefftz and hybrid Trefftz methods . . . . . . . . . . . . . 442.7.1 Equivalent source methods . . . . . . . . . . . . . 452.7.2 Hybrid Trefftz methods . . . . . . . . . . . . . . . 492.7.3 Least-squares T-elements . . . . . . . . . . . . . . 502.7.4 Ultra weak variational formulation . . . . . . . . . 502.7.5 Wave Based Method . . . . . . . . . . . . . . . . . 51

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 53

II The wave based method 55

3 The wave based method: concepts and processes 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Wave based modelling . . . . . . . . . . . . . . . . . . . . 573.2.1 Multi-domain wave based modelling . . . . . . . . 58

3.2.1.1 Partitioning into convex subdomains . . . 583.2.1.2 Mathematical description of the continu-

ity conditions . . . . . . . . . . . . . . . . 603.2.2 Wave function selection . . . . . . . . . . . . . . . 61

3.2.2.1 Homogeneous solution . . . . . . . . . . . 613.2.2.2 Wave function truncation . . . . . . . . . 643.2.2.3 Wave function scaling . . . . . . . . . . . 643.2.2.4 Wave function linear dependency . . . . . 653.2.2.5 Particular solution . . . . . . . . . . . . . 66

3.2.3 Wave based model construction . . . . . . . . . . . 663.2.3.1 Formulation of residuals . . . . . . . . . . 66

3.2.3.2 Construction of the system matrices . . . 673.2.4 Solution of the wave based model . . . . . . . . . . 713.2.5 Postprocessing . . . . . . . . . . . . . . . . . . . . 71

3.3 Wave based model properties . . . . . . . . . . . . . . . . 713.3.1 Degrees of freedom . . . . . . . . . . . . . . . . . . 733.3.2 Problem discretization & approximation functions 73

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3.3.3 Accuracy of derived variables . . . . . . . . . . . . 733.3.4 System matrix properties . . . . . . . . . . . . . . 733.3.5 Problem geometric complexity . . . . . . . . . . . 743.3.6 Computational performance & frequency range

applicability . . . . . . . . . . . . . . . . . . . . . . 743.4 Numerical implementation . . . . . . . . . . . . . . . . . . 75

3.4.1 Input data . . . . . . . . . . . . . . . . . . . . . . 763.4.2 WBM implementation . . . . . . . . . . . . . . . . 76

3.4.2.1 Phase 1: Analysis of the input data . . . 773.4.2.2 Phase 2: Creation of the topology map . 773.4.2.3 Phase 3: Construction and solution of

the WB model . . . . . . . . . . . . . . . 773.4.2.4 Phase 4: Outputting results . . . . . . . 81

3.4.3 Output data . . . . . . . . . . . . . . . . . . . . . 813.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 WBM performance evaluation in the full audio range 834.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 Wave function truncation rules . . . . . . . . . . . . . . . 83

4.2.1 A priori defined fixed number of wave functions . . 844.2.2 Frequency dependent number of wave functions . . 85

4.3 Numerical validation: 2D car-like cavity . . . . . . . . . . 864.3.1 Problem description . . . . . . . . . . . . . . . . . 864.3.2 Model descriptions . . . . . . . . . . . . . . . . . . 87

4.3.2.1 Finite element models . . . . . . . . . . . 884.3.2.2 Wave based models . . . . . . . . . . . . 89

4.3.3 Convergence analysis . . . . . . . . . . . . . . . . . 894.3.4 Prediction accuracy analysis at high frequencies . . 92

4.3.4.1 A priori defined fixed number of wavefunctions . . . . . . . . . . . . . . . . . . 93

4.3.4.2 Frequency dependent number of wavefunctions . . . . . . . . . . . . . . . . . . 101

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 Multi-domain wave based modelling 1095.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 Impedance coupling methodology . . . . . . . . . . . . . . 109

5.2.1 Mathematical description of the impedance conti-nuity conditions . . . . . . . . . . . . . . . . . . . 110

5.2.2 Motivation for the use of the impedance continuityconditions . . . . . . . . . . . . . . . . . . . . . . . 111

5.2.3 Introduction of the impedance coupling in the WBformulations . . . . . . . . . . . . . . . . . . . . . 113

5.3 Numerical validation: simply shaped 3D cavity . . . . . . 1155.3.1 Problem description . . . . . . . . . . . . . . . . . 115

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5.3.2 Model descriptions . . . . . . . . . . . . . . . . . . 1175.3.2.1 Finite element models . . . . . . . . . . . 1175.3.2.2 Wave based models . . . . . . . . . . . . 119

5.3.3 Selection of the impedance coupling factor Zint . . 1255.3.3.1 Undamped case . . . . . . . . . . . . . . 1265.3.3.2 Damped case . . . . . . . . . . . . . . . . 1295.3.3.3 Conclusions . . . . . . . . . . . . . . . . . 130

5.3.4 Response fields . . . . . . . . . . . . . . . . . . . . 1305.3.4.1 Undamped case . . . . . . . . . . . . . . 1315.3.4.2 Damped case . . . . . . . . . . . . . . . . 132

5.3.5 Convergence analysis . . . . . . . . . . . . . . . . . 1335.3.5.1 Undamped case . . . . . . . . . . . . . . 1345.3.5.2 Damped case . . . . . . . . . . . . . . . . 136

5.3.5.3 Conclusions . . . . . . . . . . . . . . . . . 1375.3.6 Frequency response analysis . . . . . . . . . . . . . 139

5.3.6.1 Undamped case . . . . . . . . . . . . . . 1395.3.6.2 Damped case . . . . . . . . . . . . . . . . 1445.3.6.3 Conclusions . . . . . . . . . . . . . . . . . 145

5.4 Experimental validation: 3D car-like cavity . . . . . . . . 1475.4.1 Problem description . . . . . . . . . . . . . . . . . 1475.4.2 Model descriptions . . . . . . . . . . . . . . . . . . 148

5.4.2.1 Finite element models . . . . . . . . . . . 1485.4.2.2 Wave based models . . . . . . . . . . . . 149

5.4.3 Response fields . . . . . . . . . . . . . . . . . . . . 1515.4.4 Convergence analysis . . . . . . . . . . . . . . . . . 1525.4.5 Experimental validation: frequency response spectra 155

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 158

6 The WBM: application to unbounded problems 1596.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1596.2 The WBM for unbounded problems . . . . . . . . . . . . 160

6.2.1 Subdomain partitioning . . . . . . . . . . . . . . . 1606.2.2 Wave function selection . . . . . . . . . . . . . . . 1616.2.3 Wave based model . . . . . . . . . . . . . . . . . . 164

6.3 Numerical validation: cavity backed plate . . . . . . . . . 1646.3.1 Problem definition . . . . . . . . . . . . . . . . . . 1656.3.2 Model descriptions . . . . . . . . . . . . . . . . . . 165

6.3.2.1 Finite element and boundary elementmodels . . . . . . . . . . . . . . . . . . . 165

6.3.2.2 Wave based models . . . . . . . . . . . . 1676.3.3 Response fields . . . . . . . . . . . . . . . . . . . . 1746.3.4 Convergence analysis . . . . . . . . . . . . . . . . . 1746.3.5 Rectangular truncation curve . . . . . . . . . . . . 177

6.3.5.1 Enclosing rectangle . . . . . . . . . . . . 1776.3.5.2 Coinciding rectangle . . . . . . . . . . . . 179

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6.3.5.3 Comparison of various truncation curves 1826.4 Numerical validation: bass-reflex loudspeaker . . . . . . . 184

6.4.1 Problem definition . . . . . . . . . . . . . . . . . . 1846.4.2 Model descriptions . . . . . . . . . . . . . . . . . . 185

6.4.2.1 Finite element and boundary elementmodels . . . . . . . . . . . . . . . . . . . 185

6.4.2.2 Wave based models . . . . . . . . . . . . 1856.4.3 Response fields . . . . . . . . . . . . . . . . . . . . 1866.4.4 Convergence analysis . . . . . . . . . . . . . . . . . 188

6.5 The WBM for transmission calculations . . . . . . . . . . 1896.5.1 Problem definition . . . . . . . . . . . . . . . . . . 1896.5.2 Introduction of a baffle plane . . . . . . . . . . . . 1896.5.3 Wave function selection . . . . . . . . . . . . . . . 190

6.5.4 Wave based model . . . . . . . . . . . . . . . . . . 1916.6 Numerical validations . . . . . . . . . . . . . . . . . . . . 191

6.6.1 An aperture in a baffle plane . . . . . . . . . . . . 1916.6.2 A flexible plate mounted in a baffle plane . . . . . 194

6.7 Conclusions and future work . . . . . . . . . . . . . . . . 1956.7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . 1956.7.2 Future work . . . . . . . . . . . . . . . . . . . . . . 195

III The hybrid finite element - wave basedmethod 199

7 The hybrid finite element - wave based method: concepts

and processes 2017.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2017.2 Strengths and limitations of the FEM and the WBM . . . 2017.3 Direct coupling approach . . . . . . . . . . . . . . . . . . 203

7.3.1 Different coupling conditions . . . . . . . . . . . . 2037.3.1.1 Direct pressure and velocity coupling . . 2047.3.1.2 Direct impedance coupling . . . . . . . . 2067.3.1.3 Direct mixed impedance-pressure coupling 207

7.3.2 Solution of the coupled model . . . . . . . . . . . . 2097.3.3 Discussion of the direct coupling approaches . . . . 210

7.4 Indirect coupling approach . . . . . . . . . . . . . . . . . . 2117.4.1 Different frame definitions . . . . . . . . . . . . . . 212

7.4.1.1 Indirect pressure coupling . . . . . . . . . 212

7.4.1.2 Indirect velocity coupling . . . . . . . . . 2147.4.1.3 Indirect impedance coupling . . . . . . . 2167.4.1.4 Indirect mixed impedance-pressure cou-

pling . . . . . . . . . . . . . . . . . . . . 2177.4.2 Solution of coupled model . . . . . . . . . . . . . . 2197.4.3 Discussion of the indirect coupling approaches . . 221

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7.4.4 Selection of frame functions . . . . . . . . . . . . . 2217.4.4.1 Frame function selection criteria . . . . . 2227.4.4.2 Frame function definitions . . . . . . . . 225

7.5 Direct vs. indirect hybrid couplings . . . . . . . . . . . . . 2297.6 Numerical implementation . . . . . . . . . . . . . . . . . . 231

7.6.1 Input data . . . . . . . . . . . . . . . . . . . . . . 2327.6.2 Hybrid FE-WB method implementation . . . . . . 233

7.6.2.1 Phase 1: Analysis of the input data . . . 2347.6.2.2 Phase 2: Creation of the topology map . 2347.6.2.3 Phase 3: Construction and solution of

the hybrid model . . . . . . . . . . . . . . 2347.6.2.4 Phase 4: Outputting results . . . . . . . 237

7.6.3 Output data . . . . . . . . . . . . . . . . . . . . . 2377.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 237

8 The hybrid finite element - wave based method: numer-ical validations 239

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 2398.2 Numerical validation: simply shaped 3D cavity . . . . . . 239

8.2.1 Problem description . . . . . . . . . . . . . . . . . 2408.2.2 Model descriptions . . . . . . . . . . . . . . . . . . 241

8.2.2.1 Finite element models . . . . . . . . . . . 2418.2.2.2 Wave based models . . . . . . . . . . . . 2418.2.2.3 Hybrid FE-WB models . . . . . . . . . . 241

8.2.3 Convergence analysis . . . . . . . . . . . . . . . . . 2528.2.3.1 200Hz predictions . . . . . . . . . . . . . 2528.2.3.2 400Hz predictions . . . . . . . . . . . . . 2568.2.3.3 700Hz predictions . . . . . . . . . . . . . 2588.2.3.4 Conclusions . . . . . . . . . . . . . . . . . 258

8.2.4 Frequency response analysis . . . . . . . . . . . . . 2608.2.4.1 The H441N1 models . . . . . . . . . . . 2618.2.4.2 The H2873N1 models . . . . . . . . . . 2678.2.4.3 The H9025N1 models . . . . . . . . . . 2728.2.4.4 Conclusions . . . . . . . . . . . . . . . . . 272

8.3 Numerical validations: real-life car cavities . . . . . . . . . 2748.3.1 Coupe car cavity . . . . . . . . . . . . . . . . . . . 275

8.3.1.1 Problem description . . . . . . . . . . . . 2758.3.1.2 Numerical results . . . . . . . . . . . . . 275

8.3.2 Mono-volume car cavity . . . . . . . . . . . . . . . 2798.3.2.1 Problem description . . . . . . . . . . . . 2798.3.2.2 Numerical results . . . . . . . . . . . . . 280

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 287

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IV Conclusions 289

9 Conclusions and future developments 291

9.1 Main achievements . . . . . . . . . . . . . . . . . . . . . . 2919.2 Main conclusions . . . . . . . . . . . . . . . . . . . . . . . 292

9.2.1 WBM performance analysis . . . . . . . . . . . . . 2929.2.2 Multi-domain WB modelling . . . . . . . . . . . . 2939.2.3 The WBM for unbounded domains . . . . . . . . . 2939.2.4 The hybrid FE-WB method: development and

validations . . . . . . . . . . . . . . . . . . . . . . 2949.3 Future developments . . . . . . . . . . . . . . . . . . . . . 295

Bibliography 298

V Addenda 317

A The WBM for 2D unbounded vibro-acoustics 319A.1 Problem definition . . . . . . . . . . . . . . . . . . . . . . 319A.2 The wave based method . . . . . . . . . . . . . . . . . . . 322

A.2.1 Partitioning into convex subdomains . . . . . . . . 322A.2.2 Mathematical description of the subdomain con-

tinuity conditions . . . . . . . . . . . . . . . . . . . 324A.2.3 Acoustic pressure expansion . . . . . . . . . . . . . 324A.2.4 Structural displacement expansion . . . . . . . . . 327A.2.5 Coupled vibro-acoustic wave model . . . . . . . . . 328

A.2.5.1 Structural boundary conditions . . . . . . 328A.2.5.2 Acoustic boundary and continuity con-

ditions . . . . . . . . . . . . . . . . . . . 329A.2.5.3 Assembly of the wave model . . . . . . . 333

Curriculum Vitae 335

List of Publications 337

Nederlandse samenvatting I

1 Inleiding en doelstelling van het onderzoek . . . . . . . . . I1.1 Het gebruik van numerieke voorspellingsmetho-

den voor vibro-akoestische problemen . . . . . . . I

1.2 Computer ondersteunde modelleringstechniekenvoor vibro-akoestiek: sterktes en beperkingen . . . II1.3 De dubbele doelstelling van het onderzoek . . . . . III

2 Definitie van een vibro-akoestisch probleem . . . . . . . . IV3 Uitbreiding van de golfgebaseerde methode . . . . . . . . VI

3.1 De golfgebaseerde methode . . . . . . . . . . . . . VI

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3.2 Prestatie-evaluatie in het audio-gebied . . . . . . . IX3.3 Ontwikkeling van een stabiele multi-domein kop-

pelingsmethode . . . . . . . . . . . . . . . . . . . . XI3.4 De GBM voor problemen in oneindige fluda . . . XV

4 Hybride eindige elementen-golfgebaseerde methode . . . . XX4.1 Hybride modelleringsstrategie . . . . . . . . . . . . XX4.2 Directe koppeling . . . . . . . . . . . . . . . . . . . XXII4.3 Indirecte koppeling . . . . . . . . . . . . . . . . . . XXIII4.4 Vergelijking tussen directe en indirecte koppelingen XXV4.5 Numerieke validatie . . . . . . . . . . . . . . . . . XXV

5 Besluiten . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIX

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Part I

Introduction and

state-of-the-art

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Chapter 1

Introduction

1.1 The need for numerical tools in vibro-

acoustic engineering

Ever more restrictive legal regulations regarding noise and vibration emissionsand immissions as well as the growing customer demand for noise and vibra-tion comfort, force design engineers to take into account the vibro-acousticbehaviour of products and processes in the design optimization. As time ismoney and less is more, Computer Aided Engineering (CAE) tools have be-come an essential part in the product development process. CAE involves

the aid of numerical tools for the functional performance optimization of aproduct. Typical performance attributes include mechanical strength andstiffness, noise and vibration levels, durability, etcetera. The possibility ofevaluating these attributes on virtual prototypes at almost any stage of thedesign process is reducing the need for very expensive and time consumingphysical prototype testing.

Over the last decades, Computer Aided Design (CAD) techniques haveevolved into mature and widely used tools to support the geometrical designprocess to allow for digital mock-up creation and to steer the computer aidedmanufacturing of components and parts. Despite their steady improvementsand extensions, CAE techniques are still primarily used by analysis special-ists. In order to turn them into easy-to-use, versatile tools that are also easilyaccessible for designers, several bottlenecks have to be resolved. The latter

include the lack of efficient numerical techniques for solving system-level func-tional performance models, the lack of sufficiently large computer power andthe lack of transparency and compatibility between geometry driven CADdesign tools and numerical grid driven CAE analysis tools.

In 1965, Intel co-founder Gordon Moore stated that the number of tran-sistors on a chip would double about every two years. His prophecy became

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1 Introduction

reality and the still ongoing quest for smaller and faster transistors lifts com-puter power to new heights every day. The availability of vast amounts ofcomputational resources at affordable prices is turning CAE into a vital toolin the industrial product design process.

This dissertation discusses the development of novel CAE tools, dedicatedfor the efficient analysis of steady-state vibro-acoustic problems.

1.2 CAE for vibro-acoustics: strengths and

limitations

Ideally, modelling and analysis tools for vibro-acoustics should be applica-

ble over the whole frequency range of interest, which is typically the fullaudio-frequency range from 20Hz up to 20kHz. In practice, however, specificmethods are applicable only in a limited frequency range.

At low frequencies the response of the system is usually described in termsof modes, which are relatively few in number. These modes are typically calcu-lated using deterministic numerical methods, and in particular finite element(FE) (Zienkiewicz et al., 2005) and boundary element (BE) (Von Estorff, 2000)methods, which are both well developed and well established. The use of com-mercial and in-house FE and BE packages in industry is widespread. Thereare difficulties, however, particularly as frequency increases. The size of thenumerical models increases rapidly, so that the computational cost becomesvery expensive. Increasing computer power and applications of various specialtechniques, such as for instance component mode synthesis (Craig Jr., 1981)

and multi-level substructuring (Bennighof et al., 2000), can ameliorate the sit-uation somewhat, but the basic problem still remains. Full FE models of cars,for example, can be developed for frequencies up to a few hundred Hz, but thefrequency range of interest spans some kH z (Freymann, 2000). In summary,deterministic methods, such as the FE method (FEM) and the BE method(BEM), encounter severe difficulties as frequency increases (Zienkiewicz, 2000;Marburg, 2002).

At higher frequencies there are very many modes, and the vibrations canbe better described in terms of waves which propagate through the system.Energy methods come to the fore. The system is regarded as being built upfrom interconnected component subsystems, each of which has many modesand the response is given in terms of subsystem energy and the flow of energythrough the structure. Foremost among such methods is Statistical Energy

Analysis (SEA) (Lyon and DeJong, 1995). However, SEA runs into prob-lems at lower frequencies, when the theory, assumptions and approximationsbehind SEA break down. In summary, SEA encounters severe difficulties asfrequency decreases.

As a result, there is a mid-frequency gap in the modelling capabilities:too high for deterministic methods, too low for statistical methods. Unfor-

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1.2 CAE for vibro-acoustics: strengths and limitations

tunately, this is typically the frequency range of highest human sensitivityto noise, and one that typically dominates subjective perception. Currentresearch is focussed on mid-frequency issues. Broadly, approaches can begrouped into three various types, although there is significant overlap in meth-ods and philosophies between the types.

Efficient alternative deterministic methods

Energy methods

Hybrid approaches, combining deterministic and energy methods

Efficient alternative deterministic methodsAmong the alternative deterministic techniques, some remove the polynomialapproximation common in element based methods, by imposing direct, wave-like solutions to the equations of motion within the domain. Examples of theseelement-free methods include the Trefftz method (Trefftz, 1926), the varia-tional theory of complex rays (Riou et al., 2004) and the partition-of-unitymethods (Melenk and Babuska, 1996). Other techniques instead approximatethe equation of motion itself by smoothing the short-wavelength response insome frequency band. This allows for a coarse mesh to be used in a FE-likemanner to predict the long-wavelength amplitude variation of the response.Complex envelope vectorization (Carcaterra and Sestieri, 2004) is one suchapproach.

Energy methods

SEA provides no detail about the variation of the average response withfrequency or with position in a subsystem. There are also further de-ficiencies, for example it may predict the wrong average. Current re-search is addressing these issues. Recent work at Cambridge University(Langley and Cotoni, 2004) has provided estimates of the variance of SEApredictions based on asymptotic models of the subsystem modal statistics.The conditions under which these asymptotic conditions hold, i.e. how highmust the frequency be, are unknown and form a current research objective.Other work aims to remove or relax the assumptions of SEA. Statistical modalEnergy distribution Analysis (SmEdA) (Maxit, 1998) allows for the fact thatthere may not be equipartition of modal energy and this may result in sub-stantial inaccuracies in response prediction. The energy distribution method(Mace, 2003) predicts the distribution of energy within the structure from

modal descriptions of the individual subsystems, typically generated by con-ventional FE analysis. This is akin to an inverse SEA model, and gives anSEA-like description.

Hybrid approachesOne mid-frequency issue of real practical importance is the case where stiff,

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1 Introduction

low mode-count components are coupled to flexible, high mode-count com-ponents. One example is a car body. There are very many panel bendingmodes and these are coupled to stiff substructures such as the body framesor the in-plane motion of the floor pan. The difficulty is that the two typesof component are well modelled by either FEM or SEA respectively, but thestructure as a whole is not well modelled by one single approach. The sizeof a full FE model is prohibitively large, while some components have sofew modes that the SEA assumptions do not apply. This has motivated thesearch for hybrid techniques which combine FE and SEA models of differ-ent regions of the structure. Such situations are sometimes referred to asfuzzy structures and modelled using such theories. A promising approachis the hybrid FE/SEA method developed recently at Cambridge University(Shorter and Langley, 2005a,b). The philosophy behind the method is to

model the stochastic, high mode-count subsystem by direct and reverberantwave fields, which are coupled to the deterministic, for instance FE, regionsof the structure.

While full FE analysis is impossible in the mid-frequency region, small,or local, FE models can be used to predict various parameters or the lo-cal behaviour. For example, small FE models can be used to predict thecharacteristics of wave propagation in the structures, together with reflectionand transmission at joints. Such information can then be used to determinethe response of the structure as a whole. The computational cost is a smallfraction of a global FE analysis, since calculating the propagation over largedistance becomes a trivial task. One such approach is the spectral FEM(Peplow and Finnveden, 2004), which involves the development of special el-ements by making simple polynomial shape function approximations to the

deformation perpendicular to the direction of wave propagation. In the morerecent Wave/FE approach (Mencik and Ichchou, 2006) conventional FE mod-els are postprocessed using an eigenanalysis of the transfer matrix. A finalapplication of small FE models is the calculation of coupling loss factors us-ing energy distribution methods. Subsystem properties can be randomised togive robust parameter estimates.

Concluding remarksIn each of the three approaches to bridge or at least narrow the mid-frequencygap, the availability of efficient deterministic modelling tools is very essential.In the first approach, they constitute the core technology, while in the twoother approaches, they serve either as a pre- and postprocessing tool for energybased methods or as an essential component in a hybrid coupling.

A unique feature of deterministic methods is that they provide detailed,spatially distributed information at any position and frequency of interest foran individual realisation of the considered vibro-acoustic system. The twoother approaches, however, involve always some sort of averaging in spaceand/or time and/or frequency and provide response information that is rep-resentative for the ensemble average of various realisations of the considered

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1.3 The dual research goal

system. The latter assumes implicitly that a sufficient degree of randomvariability and uncertainty exists between the various realisations within theensemble. This assumption is definitely met at high frequencies, but its justi-fication starts to degrade when moving towards the mid-frequency range. Inmany applications only a small to moderate degree of variability is at stake,such that a deterministic approach, even in the mid-frequency range, can pro-vide information that is mandatory in the design and analysis phase and thatis of complementary added value to the averaged information provided byenergy based probabilistic methods.

These two observations constitute the major motivation for the develop-ment of an alternative, generally applicable, efficient, deterministic predictiontechnique for mid-frequency applications.

1.3 The dual research goal

Section 1.2 illustrates the need for a deterministic prediction technique whichis applicable in the mid-frequency range. The scope of this doctoral research isto develop a technique that complies with the required features: deterministic,efficient, accurate, reliable and generally applicable. In order to obtain this,a dual goal is pursued.

1. Further development of the Wave Based Method (WBM) (Desmet,1998), which has shown to be a highly efficient deterministic predictiontechnique for vibro-acoustic problems involving a moderate geometricalcomplexity.

2. Extension of the work ofVan Hal (2004), who laid down the fundamentalprinciples for a hybrid finite element-wave based coupling approach.

1.3.1 New developments to the wave based method

The WBM (Desmet, 1998) is a deterministic prediction technique which isbased on an indirect Trefftz approach. Unlike other conventional determinis-tic methods, the WBM applies approximation functions which exactly satisfythe underlying mathematical equations, and in this way incorporate a prioriknown information about the solution into the numerical scheme. As a result,the method is shown to be computationally more efficient than other deter-ministic methods, making it applicable also for problems in the mid-frequencyrange. However, in order to fully benefit from the methods efficiency, the

problem geometry has to be of moderate complexity.The method so far has been validated for several bounded two-dimensional

(2D) vibro-acoustic problems and some three-dimensional (3D) problems.However, questions arise regarding the performance evolution of the methodover frequency and regarding the stability of the coupling conditions appliedwhen assembling two WB subdomains together. Also, unbounded problems,

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1 Introduction

i.e. problems involving infinite fluid domains, have not been analyzed withthe WBM yet.

The reported doctoral research has developed several novel features of theWBM:

The WBM is applied for problems in the low-, mid- and high-frequencyrange, in order to identify its computational characteristics in the entireaudio frequency range.

A novel impedance based coupling is introduced for coupling WB sub-domains. Due to the introduction of artificial damping as a result ofthe coupling impedance, the novel coupling is more stable than theconventional coupling, which is based on pressure and normal velocitycontinuity.

The wave based (WB) methodology is extended towards the explicitmodelling of unbounded domains. Selection of appropriate wave func-tions in unbounded fluid domains, allows the WBM to tackle transmis-sion, radiation and scattering problems.

In view of an extensive numerical validation programme and in order to fullyexploit the efficiency of the WBM, a high-performance implementation of themethod has been developed.

1.3.2 Development of a hybrid finite element - wavebased method

To fully benefit from the computational efficiency of the WBM, the consid-ered problem domain should have a moderate geometrical complexity. Real-life engineering problems, in general, do not meet this condition. Therefore,Van Hal (2004) proposed a hybrid coupling between the FEM and the WBMin order to combine the strengths of the two methods, i.e. the ability of theFEM to model any problem, regardless of its geometrical complexity, andthe high computational efficiency of the WBM for problems with moderatelycomplex geometries. The problem domain is divided into large, homogeneoussubdomains which are tackled with the WBM, while the small, geometricallymore complex regions are tackled with the FEM. At the resulting interfacesbetween the FE and the WB subdomains, interface conditions are appliedvia Lagrangian multipliers. The performance of novel hybrid FE-WB methodwas illustrated for 2D vibro-acoustic problems.

The reported doctoral research extends the use of the hybrid FE-WBmethod towards 3D vibro-acoustic problems. Furthermore, novel indirectcoupling strategies are proposed, enforcing other, more stable, continuity con-ditions using the Lagrangian multipliers. Also direct coupling approaches aredeveloped. In this way, the Lagrangian multipliers are omitted from themodel, yielding smaller numerical models.

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1.4 Outline of the dissertation

In order to apply the hybrid FE-WB method for general 3D applications,a high-performance implementation has been developed.

1.4 Outline of the dissertation

The dissertation consists of 5 parts and is organized as follows.

Part I: Introduction and state-of-the-artPart I comprises an introductory chapter and an extensive overview of thestate-of-the-art in deterministic numerical modelling techniques. In order toposition the wave based methods developed in the framework of this Ph.D.,conventional deterministic methods are discussed regarding their basic prin-ciples, their strengths and limitations and their practical applicability.

Part II: The wave based methodPart II involves the WBM developments. Chapter 3 reviews the current stateof the WBM, listing the basic principles and discussing a high-performanceimplementation of the method. Chapter 4 investigates the computationalefficiency of the WBM in the low-, mid- and high-frequency range. Chapter 5discusses a novel coupling approach for multi-domain WB problems. Chapter6 discusses a new development to the WBM in order to make the methodapplicable for problems involving unbounded acoustic fluid domains.

Part III: The hybrid finite element - wave based methodThe development and validation of several new direct and indirect couplingstrategies to couple FE with WB models are discussed in chapter 7. Numerical

validation examples are discussed in chapter 8.

Part IV: ConclusionsPart IV concludes the dissertation, stating the main conclusions of the re-search, the main achievements and some recommendations for future research.

Part V: AddendaPart V comprises the authors publication list and curriculum vitae. Further-more, appendix A discusses some additional WBM formulations.

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Chapter 2

State-of-the-art indeterministic modelling of

steady-state vibro-acoustic

problems

2.1 Introduction

This chapter discusses the state-of-the-art in deterministic numerical mod-elling techniques for the analysis of steady-state vibro-acoustic problems.

The chapter starts with a mathematical description of vibro-acoustic prob-lems in section 2.2. The remaining sections give an overview of existing de-terministic numerical methods, applicable for solving this type of problems.

Section 2.3 discusses the widely used finite element method (FEM). Section2.4 lists some of the recent developments to optimize the FEM in order to makeit applicable in a wider frequency range. The direct and the indirect boundary

element method (BEM) and some extensions to these methods are describedin sections 2.5 and 2.6, respectively. A third class of deterministic methodsdiscussed are the so-called Trefftz and hybrid Trefftz methods. This classcomprises the wave based method (see section 2.7.5), which is the baselinemethodology of this dissertation. Section 2.8 closes the chapter with someconcluding remarks.

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2 State-of-the-art in deterministic modelling of steady-state vibro-acoustic

problems

2.2 Definition of steady-state vibro-acousticproblems

This section provides a mathematical description of vibro-acoustic problems.In view of problem classification, the following definitions are introduced.

bounded versus unbounded problems: Acoustic problems which takeplace in a bounded environment are denoted as bounded problems. Un-bounded problems involve transmission, radiation and scattering prob-lems in an unbounded acoustic medium.

interior versus exterior problems: In order to make this distinction,a closed problem boundary is required. If the problem boundary com-

pletely encloses the considered acoustic fluid and no information outsidethe boundary is needed, the problem is called an interior problem. Incase the problem domain of interest is restricted to the outer part, theproblem is denoted as an exterior problem.

coupled versus uncoupled: Whenever a vibrating structure is surroundedby a fluid, there is a fluid-structure or vibro-acoustic interaction. Thevibrating structure causes the surrounding fluid particles to vibrate, re-sulting in pressure waves. On the other hand, the acoustic pressurewaves in the fluid act as an external loading onto the structure. In re-ality a problem is always coupled. However, sometimes it is justified topartly neglect or even completely discard the coupling. Such problemsare referred to as uncoupled problems. The noise radiated from a ro-

tating engine block, for instance, may be considered as an uncoupledproblem, i.e. the vibrations of the engine generate sound waves, butacoustic waves impinging on the solid engine block do not influence thevibrational behaviour significantly. A large class of problems in buildingacoustics are problems where the coupling may be completely discarded.For instance, the acoustic pressure field inside a room does not interactwith the solid, rigid walls.

Consider an uncoupled, unbounded, exterior, vibro-acoustic problem, asshown in figure 2.1. The acoustic pressure loading on the structural compo-nents is neglected and the presence of the structural parts is modelled as anacoustic normal velocity boundary condition onto the fluid parts which are indirect contact with the structure. In this way, only the equations describing

the behaviour of the fluid and the associated boundary conditions define themathematical problem.

Figure 2.1 shows a closed boundary surrounded by a fluid, which is char-acterised by its speed of sound c and its ambient fluid density 0. Thefluid domain V is excited by an acoustic volume velocity point source qat circular frequency . The time-harmonic pressure response is given by

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2.2 Definition of steady-state vibro-acoustic problems

V

yx

zn

p

q

r

rq

WsWv

WZWp

W8

Figure 2.1: An uncoupled, unbounded, exterior, vibro-acoustic problem

p(r, t) = p(r, )ejt with r =

x y zT

the position vector, T the trans-pose operator, j the imaginary unit

1 and with t denoting the time. Fromhere onwards, the steady-state solution p(r, ) is abbreviated as p(r).

Assuming that the system is linear, the fluid is inviscid and the process isadiabatic, the steady-state acoustic pressurep(r) is governed by the Helmholtzequation (Morse and Ingard, 1968).

2p(r) + k2p(r) = j0(r, rq)q (2.1)

with

2

=

2x2

+2y 2

+2z 2

the Laplacian operator, k = /c the acoustic

wave number and with a Dirac-delta function.The boundary of the considered acoustic problem domain V is denoted as

V = and consists of three parts: = a s . a (= v Z p) represents the part of the boundary at which

one of the three following acoustic boundary conditions applies:

acoustic normal velocity boundary conditions at v, expressed as

r v : Lv(p(r)) = vn(r) (2.2)with Lv() the velocity operator

Lv() = j0

n

(2.3)

withn

= nT the derivative in the normal direction, n =nx ny nz

Tthe vector normal to the fluid domain V and

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problems

= x

y

zT. vn(r) is the predefined normal veloc-

ity.

acoustic normal impedance boundary conditions at Z, expressedas

r Z : Lv(p(r)) = p(r)/Zn(r) (2.4)with Zn(r) the predefined normal impedance.

acoustic pressure boundary conditions at p, expressed as

r p : p(r) = p(r) (2.5)with p(r) the predefined pressure.

s corresponds to that part of the boundary that coincides with a vi-brating structural component. The vibrations of the structure act asan acoustic normal velocity boundary condition on the fluid domain V.These boundary conditions are expressed as

r s : Lv(p(r)) = nsTnjwns(r) (2.6)with ns the normal vector of the structural component and with wns(r)the steady-state normal displacement of the structure.

At the Sommerfeld radiation condition for outgoing waves is ap-plied (Colton and Kress, 1998). This condition ensures that no acousticenergy is reflected at infinity and is expressed as

lim|r|

|r|p(r)|r| +jkp(r)

= 0 (2.7)The Helmholtz equation (2.1) together with the associated boundary con-

ditions (2.2), (2.4), (2.5), (2.6) and (2.7), defines a unique pressure field p(r).Once the pressure field p(r) is determined, other acoustic quantities can

be derived (Morse and Ingard, 1968).

The acoustic particle velocity vector v(r) is determined proportional tothe gradient of the pressure field

v(r) =j

0p(r) (2.8)

The active acoustic intensity vector I(r) representing the flow of acous-tic energy is

I(r) =1

2(p(r) v(r)) (2.9)

with () the real part and with the complex conjugate of .

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2.3 The Finite Element Method

The active acoustic power W through a surface S is determined as theintegral of the active acoustic intensity through the considered surfaceand is calculated as

W =

S

I(r) n dS (2.10)

with n the normal vector on S.

These formulations yield a mathematical description for a general uncou-pled, unbounded, exterior, vibro-acoustic problem. For any interior vibro-acoustic problem the same formulations can be applied. However, in this case(2.7) is irrelevant since no effects at infinity are included in the model. Incase of s = , one obtains the formulations of a purely acoustic (interior orexterior) problem.

For fully coupled vibro-acoustic problems, consisting of an acoustic partand a structural part, these formulations only describe the acoustic part. Fora description of the pressure loading of the acoustic fluid on to the structuralcomponents and the description of the dynamic behaviour of the structuralcomponents themselves, other formulations are required. Since they are out ofthe scope of this dissertation, the reader is referred to literature (Fahy, 1985;Leissa, 1993; Junger and Feit, 1993) for more detailed information. AppendixA discusses the formulations for 2D fully-coupled problems.

Several numerical methods have been developed for solving the mathe-matical descriptions of the vibro-acoustic problems introduced in this section.The following sections give an overview of the state-of-the art in deterministicnumerical modelling methods.


The FEM (Zienkiewicz et al., 2005) is a deterministic prediction techniquewhich discretizes the problem domain into a large but finite number of smallelements. Within these elements, the dynamic response variables, being for in-stance the structural displacements or the acoustic pressures, are described interms of simple, polynomial shape functions. However, since these shape func-tions are no exact solutions of the governing differential equations, a (very)fine discretization is required to obtain reasonable prediction accuracy.

2.3.1 Basic formulations

In order to apply the FEM for uncoupled bounded vibro-acoustic prob-

lems, the considered acoustic cavity V is subdivided into a number of

non-overlapping FE domains Ve,

V =

nee=1

Ve with Vi Vj = , i = j

, as

shown in figure 2.2. In each element, a number of nefe nodes is defined atsome particular locations in the element. The total number of nodes in the

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problems

discretization is nfe =

ne

e=1

nefe . The boundary e = Ve of each element Ve

is composed of five non-overlapping parts

e = ep ev eZ es ei

,namely those parts, which are intersections of the element boundary e withthe problem boundary

ep =

e p , ev = e v, es = e s,eZ =

e Z), and the common interface ei between two adjacent ele-ments.

W Wp vU U WZ U Ws

V = VU ee=1

ne

Ve

W W* *= U We e

W W W W W= U U U Up v Z iWse e ee ee

V

Figure 2.2: A bounded vibro-acoustic problem and its FE discretization

Within each element Ve, the exact solution of the Helmholtz equation (2.1)is approximated by a linear combination of nefe simple (polynomial) elementshape functions

p(r) p(r) =nefe

fe=1pefeN

efe (r) r Ve (2.11)

Based on the element shape functions Nefe , which are locally defined inone element Ve, some global shape functions Nfe may be constructed, whichare defined in the entire acoustic cavity V. In each element Ve to which nodef e belongs, the global shape function Nfe is identical to the correspondingelement shape function Nefe , while it is zero in all other elements. In this way,a global pressure expansion may be defined as

p(r) p(r) =nfefe=1

pfeNfe (r) = Nfe(r)pfe r V (2.12)

The contribution factors pfe , stored in the (nfe 1) column vector pfe,are the unknown nodal degrees of freedom (dofs). In general, the unknowndofs represent nodal pressure values. The associated shape functions Nfe arestored in the (1 nfe ) row vector Nfe.

Assume that the global pressure expansion (2.12) satisfies a priori boththe pressure boundary conditions along p and the pressure continuity be-tween two adjacent elements along their common interface ei (conforming

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elements). The pressure approximation violates the governing Helmholtzequation (2.1) within the acoustic cavity V, the normal velocity boundarycondition along v, the impedance boundary condition along Z, the nor-mal velocity boundary conditions from the structural components along sand the inter-element velocity continuity between two adjacent elements onei . The involved residuals on these relations are forced to zero in an integralsense by application of a weighted residual formulation. Contributions of theenforcement of the inter-element velocity continuity cancel out each other inthis formulation, yielding the following weighted residual expression

V

Wfe(r)2 p(r) + k2 p(r) +j0(r, rq)q d V

v j0Wfe (r) (Lv(p(r)) vn(r)) ds

j0Wfe (r)Lv(p(r)) nsTnjwns(r)d

Z

j0Wfe (r)

Lv(p(r)) p(r)

Zn(r)

d = 0

(2.13)

where Wfe represents a weighting function.By application of integration by parts and the divergence theorem, (2.13)

may be rewritten in its weak form

2V

1

c2Wfe (r) p(r)d V +j

Z

0Zn(r)

Wfe(r) p(r)d

+

VWfe (r) p(r)d V j V

0Wfe(r)(r, rq)qd V (2.14)

+jv

0Wfe (r)vn(r)d +js

0Wfe (r)nsTnjwns(r)d = 0

For the sake of brevity, the position dependency of the matrices and thevectors will be omitted in their notation.

Adopting a Galerkin approach in that the weighting functions Wfe areexpanded as a linear combination of the same shape functions Nfe and sub-stituting expansion (2.12) in the weak form (2.14), yield a square matrixequation 2Mfe +jCfe + Kfepfe = qfe vfe wfe (2.15)with Mfe the (nfe nfe) acoustic mass matrix

Mfe = V

1c2

NfeT Nfed V (2.16)

with Cfe the (nfe nfe ) acoustic damping matrix

Cfe =

Z

0Zn

NfeT Nfed (2.17)

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problems

with Kfe the (nfe nfe ) acoustic stiffness matrix

Kfe =

V

BfeT Bfe d V (2.18)

with Bfe defined as

Bfe =Nfe (2.19)

and with qfe , vfe and wfe the (nfe 1) acoustic loading vectors

qfe = j

V

0NfeT (r, rq)qd V (2.20)

vfe = j

v

0NfeT vnd (2.21)

wfe = j

s

0NfeT ns

Tnjwnsd (2.22)

A more compact way of writing the FE matrix equation (2.15) is

Zfepfe = ffe withZfe = 2Mfe +jCfe + Kfeffe = (qfe vfe wfe) (2.23)

The vector pfe is the solution vector, collecting all unknown dofs, beingthe nodal pressure values. The pressure b oundary conditions along p aretaken into account by assigning the prescribed values directly to the nodaldofs, such that these dofs are no longer unknowns of the FE model and canbe condensed out of the FE system matrix Zfe . Zfe is sometimes referredto as the dynamic FE stiffness matrix.

2.3.2 Treatment of unbounded domains

A difficulty in application of the FEM for unbounded vibro-acoustic prob-lems, lies in the effective treatment of unbounded domains. Because theFEM is based on a discretization of the problem domain into small ele-

ments, it cannot directly handle unbounded problems. An artificial bound-ary is needed to truncate the unbounded problem into a bounded problem(Wolf and Song, 1996). Special techniques are then required to reduce spuri-ous reflection of waves at the truncation boundary. The region between theproblem boundary and the truncation boundary is modelled using conven-tional FE. Three strategies are applied in reducing the spurious reflections at

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the artificial boundary:

1. absorbing boundary conditions

2. infinite elements

3. absorbing layers

The next sections briefly discuss the three approaches. For a more elab-orated overview, the reader is referred to Thompson and Pinsky (2004);Harari (2006) and Thompson (2006).

Absorbing boundary conditionsAbsorbing boundary conditions (ABCs), also called non-reflecting bound-ary conditions (NRBCs), prevent spurious acoustic reflections at the trunca-

tion surface by application of specific boundary conditions at the truncationboundary, see figure 2.3.

absorbing boundary

conditions

Figure 2.3: Absorbing boundary conditions at the truncation boundary

A distinction is made between local and global NRBCs. Local schemesretain the sparsity and banded structure of the FE system matrices. Ingeneral, the order of the NRBC determines its accuracy in preventing spu-rious reflections. In case of higher-order local NRBCs, auxiliary variables,i.e. dofs, are introduced into the system. From a FE point of view,this is seen as a mesh refinement with an associated introduction of ad-ditional dofs. Local NRBCs tend to lose accuracy for higher frequencies(Shirron and Babuska, 1998). Low-order conditions are easy to implementand applicable to more general shapes of the truncation surface. However,care must be taken with low-order boundary conditions, in that the trunca-tion surface must be placed sufficiently far from the radiating or scatteringobject. If not, large spurious reflections can pollute the entire numerical so-lution. This, however, increases the (large) bounded domain to be modelled

with FE. High-order NRBCs are more accurate, but are more difficult to im-plement due to the presence of high-order derivatives. Givoli (2004) reviewsdifferent forms of local absorbing boundary conditions.

Global NRBCs relate all the degrees of freedom on the artificial truncationsurface to each other such that the sparsity of the system matrices is signifi-cantly reduced which results in a significant increase in computational load.

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problems

Furthermore, special procedures for implementation and parallellization arerequired. The Dirichlet-to-Neumann mapping (DtN) falls in this category ofapproaches (Keller and Givoli, 1989; Harari et al., 1998; Nicholls and Nigam,2004). The order of the series expansion of the DtN boundary condition de-termines the accuracy of the procedure. Despite the theoretical correctnessof the approach, the fact of being a global method degenerating the sparsestructure of the FE system matrices is a major practical disadvantage.

Infinite elementsAn alternative to absorbing boundary conditions at the truncation bound-ary is explicitly modelling the exterior region with infinite elements (IE)(Bettess, 1992). Infinite elements are similar to finite elements, in that theydiscretize the problem domain. However, with infinite elements one face of

the element is located at the truncation surface, while the opposite elementfaces extend to infinity, see figure 2.4.

infinite elements

Figure 2.4: Infinite elements at the truncation boundary

Within the IE, shape functions model the acoustic field variables. Thesefunctions combine a suitable amplitude decay and a wave-like radial variationfor modelling outgoing travelling waves. By increasing the radial order of theshape functions, the accuracy increases, but so do the involved computationalefforts.

Two leading approaches have emerged within the infinite element world(Gerdes, 2000).

A first group are the so-called unconjugated Burnett elements(Burnett and Holford, 1998a,b). These elements are based on a formula-tion applying the same functions as both shape functions and weighting

functions.

A second group of elements are the Astley-Leis elements, also calledas the wave envelope elements (Astley et al., 1994, 1998). For theseelements, the weighting functions are conjugates of the shape functionsand a geometric weighting factor is introduced.

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The unconjugated Burnett formulation is considered to provide the bestnear-field accuracy, whereas the Astley-Leis formulation is most effective inthe far-field (Shirron and Babuska, 1998). The conjugated IE leads to fre-quency independent system matrices and involves a fairly simple integrationof polynomial functions, but the resulting matrices are not symmetric. Theunconjugated formulation yields matrices that are symmetric but frequencydependent. Moreover, matrix coefficients result from more tedious numericalintegrations. Ill-conditioning with the conjugated formulation can be avoidedby selection of a suitable radial basis. In case of the unconjugated Burnettformulation, the condition number increases rapidly irrespective of the radialbasis (Astley and Coyette, 2001a). The performance of both conjugated andunconjugated formulations deteriorates at higher frequencies and stretchedinterfaces (Astley and Coyette, 2001b).

Absorbing layersThe perfectly matched layer (PML) concept, originally introduced byBerenger (1994) for electromagnetic waves, is another option for modellingthe exterior acoustic field. The idea is to introduce an exterior layer of finitethickness at the truncation boundary, see figure 2.5, such that the outgo-ing waves are absorbed in the exterior layer before reaching the end of thatlayer. By splitting a scalar field and by proper selection of the PML coef-ficients, plane wave reflection for an arbitrary angle of incidence is theoret-ically eliminated. The PML is usually formulated in cartesian coordinates.Collino and Monk (1998) have generalized the PML concept to curvilinearcoordinates. The PML converges to perfect absorbance if the layer thickness

is increased. Practical implementation of the PML involves several numeri-cal parameters, namely the width of the layer, the number of divisions andthe variation of the PML coefficients and their maximal values. Researchregarding the optimization of these parameters is still ongoing.

absorbing layer

Figure 2.5: Absorbing layer at the truncation boundary

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problems

2.3.3 Errors involved with the use of the FEMAccuracy of FE approxi

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