Averaged velocity boundary element method for sound
radiation from vibrating structures : on the use of element
volume velocities for improving computational efficiency
Citation for published version (APA):DeBiesme, F. X. (2009). Averaged velocity boundary element method for sound radiation from vibrating structures : on the use of element volume velocities for improving computational efficiency. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR642570
DOI:
10.6100/IR642570
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Averaged Velocity Boundary Element Method
for
Sound Radiation from Vibrating Structures
On the use of element volume velocities for improving computational efficiency
A catalogue record is available from the Eindhoven University of Technology Library
Averaged Velocity Boundary Element Method for Sound Radiation from Vibrating Structures F.X. DeBiesme - Eindhoven: Technische Universiteit Eindhoven, 2009 - Proefschrift
ISBN: 978-90-386-1762-6
Cover design: Maëlig Pommeret - www.magelidesign.com, France Reproduction: Ipskamp Drukkers B.V., Enschede, The Netherlands
c
Averaged Velocity Boundary Element Method
for
Sound Radiation from Vibrating Structures
On the use of element volume velocities for improving computational efficiency
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een
commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 14 mei 2009 om 16.00 uur
door
Francois-Xavier Marie DeBiesme
prof.dr.ir. J.W. Verheij en
prof.dr. H. Nijmeijer
Copromotor: dr.ir. I. Lopez
Contents
Summary xi
Samenvatting xiii
Notations, symbols and conventions xv
1 Introduction 1
1.1 Noise control by design . . . 1
1.2 Tools for vibro-acoustic computations . . . 2
1.2.1 Analysis tasks . . . 2
1.2.2 Main types of computational methods . . . 2
1.2.3 Acoustic BEM or FEM? . . . 4
1.3 Reducing the number of DOF in acoustic BEM . . . 6
1.3.1 Averaged Velocity BEM as a novel variant . . . 6
1.3.2 Expected advantages of AVBEM . . . 7
1.3.3 Illustration of potential gains . . . 8
1.3.4 Choices for the implementation in this thesis . . . 9
1.3.5 Quantities and errors . . . 9
1.4 Key contributions of the thesis . . . 10
1.5 Outline . . . 10
2 Fundamentals of the Acoustic Boundary Element Method 13 2.1 Introduction . . . 13
2.2 Integral equation for acoustic BEM . . . 13
2.2.1 Wave equation and Helmholtz equation . . . 13
2.2.2 Kirchhoff-Helmholtz integral equation (KHIE) . . . 15
2.3 KHIE discretization and BEM assembly . . . 16
2.3.1 First step: decompose in boundary elements . . . 16
2.3.2 Second step: discretize the acoustic variables . . . 16
2.3.3 Third step: apply the collocation scheme . . . 18
2.3.4 Post processing . . . 18
2.3.6 Geometrical discretization and integration . . . 20
2.3.7 Non-uniqueness . . . 21
2.4 Variants of the Kirchhoff-Helmholtz Integral . . . 22
2.4.1 KHIE variants . . . 22
2.4.2 Discretizing the KHIE variants . . . 22
2.5 Computational and memory costs . . . 23
2.5.1 Number of DOF . . . 23
2.5.2 Assembly cost . . . 23
2.5.3 Solving cost . . . 23
2.5.4 Total computational effort . . . 24
2.5.5 Measure of discretization . . . 25
2.6 Reducing the computational costs . . . 25
2.7 Positioning of Averaged Velocity BEM . . . 29
3 Averaged Velocity BEM 31 3.1 Introduction . . . 31
3.2 Discretization with constant variables . . . 31
3.2.1 Overview . . . 31
3.2.2 Averaged element velocity . . . 33
3.3 Equations of AVBEM . . . 33
3.3.1 First step: decompose in boundary elements . . . 33
3.3.2 Second step: discretize the acoustic variables . . . 33
3.3.3 Third step: apply the collocation scheme . . . 34
3.3.4 Post processing . . . 34
3.3.5 Galerkin and AVBEM . . . 35
3.4 AVBEM in relation to earlier work . . . 35
3.4.1 Discontinuous elements in earlier work . . . 35
3.4.2 Element volume velocities in earlier work . . . 35
3.4.3 AVBEM . . . 38
3.5 Discretization and accuracy of AVBEM . . . 38
3.5.1 Error formula . . . 38
3.5.2 Frequency dependence . . . 39
3.6 Physical considerations for the AVBEM accuracy . . . 39
3.6.1 Accuracy of AVBEM . . . 39
3.6.2 Accuracy of AVBEM compared to SFBEM . . . 40
3.7 Anticipation of the AVBEM error for flexural waves . . . 41
3.7.1 Integral splitting . . . 41
3.7.2 Structures with flexural vibrations . . . 42
Contents vii
4 Assembly for AVBEM 45
4.1 Introduction . . . 45
4.2 Element types and measure of discretization . . . 45
4.3 Treatment of non-singular integrals . . . 47
4.3.1 Gaussian quadrature . . . 47
4.3.2 Expected gains when using AVBEM . . . 48
4.4 Treatment of singular integrals . . . 49
4.4.1 Weak singularity . . . 49 4.4.2 Strong singularity . . . 50 4.4.3 Hypersingularity . . . 50 4.5 Simulations . . . 51 4.5.1 Overview . . . 51 4.5.2 Integration accuracy . . . 52
4.5.3 Simulations on assembly time . . . 54
4.5.4 Discussion . . . 55
4.6 Conclusions . . . 56
5 Accuracy of AVBEM for sound power 57 5.1 Introduction . . . 57
5.2 Description of the plate simulations . . . 57
5.2.1 Introduction . . . 57
5.2.2 General description . . . 58
5.2.3 Computational model of force excited plates . . . 58
5.2.4 Plate properties and modeling details . . . 59
5.2.5 Discretization parameterD . . . 61
5.2.6 Size and number of elements for AVBEM simulations . . . 61
5.2.7 Size and number of elements for QBEM simulations . . . 62
5.3 Simulations for the plate in an infinite baffle . . . 64
5.3.1 Errors as a function ofγ . . . 64
5.3.2 Maximum error as a function of D . . . . 68
5.3.3 Maximum error in three frequency ranges . . . 69
5.4 Results for the plate in a parallelepipedic baffle simulations . . . 73
5.4.1 Description of the baffle . . . 73
5.4.2 Reference solution . . . 74
5.4.3 Error as a function ofγ . . . 74
5.5 Discretization rules for plate-like structures . . . 77
5.5.1 Choosing a discretization rule . . . 77
5.5.2 Using a discretization rule . . . 77
5.6 Discussion . . . 78
5.6.1 DOF reductions . . . 78
6 Accuracy of AVBEM for sound pressure 81
6.1 Introduction . . . 81
6.2 Plates in an infinite flat baffle . . . 82
6.2.1 Description of the simulations . . . 82
6.2.2 Results . . . 82
6.3 Parallelepiped with one face vibrating . . . 84
6.3.1 Description of the simulations . . . 85
6.3.2 Description of the field points . . . 85
6.3.3 Results . . . 86
6.4 Discussion . . . 89
7 Miscellaneous aspects of AVBEM 91 7.1 Introduction . . . 91
7.2 The advantage of reduced assembly time . . . 92
7.2.1 Assembly time versus solving time . . . 92
7.2.2 AVBEM with iterative solver for truck engine noise . . . 92
7.2.3 Results . . . 93
7.2.4 Discussion . . . 93
7.3 Preserving good geometrical modeling with a reduced number of DOF . . . . 94
7.3.1 Description of the simulations . . . 94
7.3.2 Results . . . 95
7.3.3 Discussion . . . 97
7.4 Coarse meshing of a box-like structure with different plate thicknesses . . . . 97
7.4.1 General description . . . 97
7.4.2 Modeling . . . 98
7.4.3 Results . . . 99
7.4.4 Discussion . . . 101
7.5 Summary . . . 102
8 Discussion and Synthesis 105 8.1 Introduction . . . 105
8.2 Discussion of the results . . . 106
8.2.1 Overview . . . 106
8.2.2 Implementation advantages of AVBEM . . . 106
8.2.3 Faster assembly . . . 106
8.2.4 DOF reduction . . . 107
8.2.5 Discretization rules for AVBEM . . . 108
8.2.6 Practical advantages of DOF reductions . . . 111
8.2.7 Theoretical aspects . . . 112
8.3 Prospect of using AVBEM in engineering tools . . . 114
8.3.1 AVBEM with conventional solvers on desktop computers . . . 114
8.3.2 Practical use of AVBEM with conventional iterative solvers . . . 115
Contents ix
8.3.4 Compatibility . . . 119
9 Conclusions and Recommendations 121
9.1 Conclusions . . . 121 9.2 Recommendations . . . 124
Bibliography 127
Aknowledgements 133
Summary
In the field of noise control engineering the numerical computation of the sound radiation from vibrating structures plays an important role in product design. This study is concerned with the Boundary Element Method (BEM), which enables accurate modeling of sound radiation from structures with complex geometry and with complex patterns of vibration. Proper acoustical modeling usually requires large numbers of Degrees Of Freedom (DOF) and conventional BEM algorithms need to solve fully populated linear systems of equations. Therefore, the major practical obstacle for using BEM with conventional solvers is formed by tremendous computational costs and by hardware limitations.
In this thesis Averaged Velocity BEM (AVBEM) is developed, which is attractive for its sim-plified implementation and its fast assembly. AVBEM is used to find mesh discretization rules that keep errors within a priori chosen value with a minimum number of elements. For AVBEM with a conventional iterative solver the CPU times on a modern desktop PC will re-main within about one minute for model sizes up to 6000 DOF. This CPU time is for a singe frequency line. It is shown that when using the economic discretization rules derived in this study, AVBEM can handle a wide range of noise control engineering problems for such model sizes.
AVBEM has an original combination of three distinctive features. First, the acoustic variables are, at the element level, represented by averaged quantities. For the boundary elements on the vibrating structure, averaged element velocities are prescribed. Second, sub-parametric elements are used, namely elements with a zero order approximation for the acoustic variables and a higher order approximation for the geometry. These higher order polynomial shape functions ensure a proper geometrical modeling without increasing the size of the models. Third, a highly accurate approximation of the input variable (averaged velocity) is used. This is done by modeling the boundary velocity with auxiliary nodes and high order polynomials and integrating it for each element. Most of other BEM variants use isoparametric elements, i.e. elements with the same order of polynomial shape functions for the approximation of acoustic variables and geometry. Such BEM variants are designated as Shape Function BEM or SFBEM.
Numerical simulations are used to derive the rules for economic mesh discretization. The majority of the simulations are for plates with flexural vibration. These plates are placed in different baffles and sound power is computed. The ratio of acoustical and structural wave-lengths is systematically varied and the models are analyzed both with AVBEM and SFBEM.
A few calculations on other structures complete the simulations for sound power. Additional to the plate simulations for sound power, a few similar calculations are performed for sound pressure in the near field and far field of plates. The discretization errors in these simulations remain basically within the error range as for sound power.
A basic summary of the discretization study is that for relatively thin-walled structures with flexural wavelengths smaller than the acoustical wavelength, AVBEM needs significantly less DOF (often 2 to 3 time less) than SFBEM for the same accuracy. On relatively thick-walled structures and on rigid boundaries, only a small DOF reduction is obtained. But for these structures, an already coarse discretization (approximately 3 DOF per acoustical wavelength) ensure an accuracy of 2 dB. These rules are further simplified for situations where the struc-tural wavelength is not known: 3.5 DOF per acoustical wavelength should be used to ensure a 2 dB accuracy. Compared to the industrial practice of using 6 DOF per acoustical wavelength, this coarser discretization reduces the model size with a factor 3. However, this rule can not be applied for very thin-walled structures on which the structural wavelength is less than half the acoustical wavelength. For these structures, it is necessary to discretize with respect to the structural wavelength. Using discretizations with 3 DOF per structural wavelength is assumed to give errors less than 2 dB. This implies more than 6 DOF per acoustical wavelength. The combination of the fast assembly of AVBEM with an iterative solver and with the eco-nomic meshing of the models leads to large CPU time gains for a wide range of engineering applications.
It is recommended to convert Averaged Velocity BEM into Averaged Variable BEM to make it versatile for a broad range of noise control engineering applications. For this purpose only the implementation of two extra Boundary Conditions is needed. Furthermore, application of AVBEM is recommended for so-called coupled problems.
Samenvatting
Numerieke berekening van de geluidafstraling door trillende constructies speelt een belang-rijke rol bij het ontwerpen van geluidarme producten. Deze studie gaat over het gebruik van de akoestische randelementen methode (Boundary Element Method of kortweg BEM) voor dit doel. Deze methode maakt een nauwkeurige berekening mogelijk voor constructies met com-plexe geometrie en trillingsvelden. Goede BEM modellen vereisen doorgaans grote aantallen vrijheidsgraden (DOF) en bij gebruik van conventionele BEM algoritmen moeten lineaire vergelijkingssystemen met volledig gevulde matrices worden opgelost. Daarom vormen in de ingenieurspraktijk de lange rekentijden en hardware beperkingen belangrijke obstakels bij het gebruik van BEM met conventionele solvers.
In dit proefschrift wordt een nieuwe BEM-variant onderzocht, die Averaged Velocity BEM (AVBEM) wordt genoemd. Deze is zeer eenvoudig te implementeren en heeft een snelle matrix assemblage. Met deze AVBEM worden numerieke simulaties gedaan om modellering-regels te vinden, waarmee de discretisatiefouten binnen vooraf gekozen grenzen blijven. Voor AVBEM in combinatie met een conventionele iteratieve solver zullen de rekentijden op een moderne bureaucomputer binnen een minuut blijven voor modellen tot 6000 DOF. Dit zijn de CPU tijden voor één frequentie. Aangetoond wordt dat AVBEM met gebruikmaking van de economische discretisatieregels uit dit proefschrift met zulke modelgroottes een groot scala aan akoestische ontwerpproblemen kan doorrekenen.
AVBEM wordt gekenmerkt door een originele combinatie van drie karakteristieke eigenschap-pen. Ten eerste worden de akoestische variabelen op het niveau van de randelementen be-naderd met gemiddelde grootheden. Voor een randelement op een trillende constructie wordt de gemiddelde snelheid als randvoorwaarde opgelegd. Ten tweede wordt er gebruik gemaakt van subparametrische elementen, te weten randelementen met een nulde-orde benadering voor de akoestische variabelen en met een hogere orde benadering voor de geometrie. Ten derde wordt een nauwkeurige benadering van de gemiddelde elementsnelheden gerealiseerd door gebruik te maken van hulpknooppunten en hogere orde vormfuncties, die de modelgrootte niet beïnvloeden. De meeste BEM-implementaties maken gebruik van isoparametrische ele-menten met dezelfde orde voor de polynoombenaderingen van de akoestische variabelen en van de geometrie. Zulke BEM varianten worden in dit proefschrift aangeduid als Shape Func-tion BEM (SFBEM).
Met behulp van numerieke simulaties worden kwantitatieve regels geformuleerd voor het economisch discretiseren van de randoppervlakken. De meerderheid van deze simulaties is
voor platen met buigtrillingen en voor afgestraald geluidvermogen. De verhouding van de buiggolflengte en de akoestische golflengte wordt hierbij systematisch gevarieerd en discreti-satiefouten worden zowel voor AVBEM als voor SFBEM bepaald. Additioneel worden ook enkele soortgelijke simulaties voor geluiddrukken in het nabije en verre veld uitgevoerd. De discretisatiefouten hierin vallen binnen dezelfde foutenmarge als die voor geluidvermogen. Het onderzoek naar discretisatiefouten is in hoofdlijnen als volgt samen te vatten. Voor relatief dunne constructies waarop buiggolflengten kleiner zijn dan de akoestische golflengte heeft AVBEM significant (vaak twee tot drie keer) minder DOF nodig dan SFBEM voor dezelfde nauwkeurigheid. Op relatief dikwandige constructies en op starre wanden is de DOF reduc-tie voor AVBEM slechts beperkt. Het blijkt echter dat in die situareduc-ties met een vrij grove mesh discretisatie van ongeveer 3 DOF per akoestische golflengte de fouten tot minder dan 2 dB beperkt blijven. Een verdere vereenvoudiging van de discretisatieregels wordt gegeven voor gebruik in die gevallen waarin de buiggolflengten niet voldoende nauwkeurig bekend zijn. In dat geval kan over een groot frequentie bereik een discretisatie van 3.5 DOF per akoestische golflengte worden toegepast met fouten binnen 2 dB. In vergelijking tot de in de industrie veel gebruikte discretisatieregel van 6 DOF per akoestische golflengte, vermindert deze grovere meshing de modelgroottes met een factor 3. Toepassen van discretisatie met 3.5 DOF per akoestische golflengte gaat echter fout voor dunwandige constructies waarop de buiggolflengten kleiner zijn dan de halve akoestische golflengte. Op zulke constructies moet de discretisatie worden bepaald op basis van de buiggolflengte. Er zijn sterke aanwijzingen dat dan bij gebruik van 3 DOF per buiggolflengte de fouten kleiner dan 2 dB blijven. Dit impliceert echter een discretisatie met meer dan 6 DOF per akoestische golflengte.
Een belangrijke conclusie is dat de snelle matrix assemblage van AVBEM in combinatie met het gebruik van een conventionele iteratieve solver en met de DOF reductie door toepassing van de gevonden discretisatieregels, tot forse CPU tijd reducties leidt voor een brede reeks van akoestische toepassingen.
Het proefschrift sluit af met de aanbeveling om AVBEM breder toepasbaar te maken door de omvorming tot Averaged Variable BEM. Hiervoor hoeven alleen twee extra randvoorwaarden te worden geïmplementeerd. Ook wordt aanbevolen om AVBEM te implementeren voor de toepassing op gekoppelde problemen.
Notations, symbols and
conventions
General notations
a, A, α Scalar (italic symbols)
a Vector (bold lowercase symbols)
−
→a Alternative vector notation
A Matrix (bold capital symbols)
¯
a constant value over an element
¯ ¯
a averaged value over an element
Operators
∇f =∂f ∂x−→i + ∂f ∂y−→j + ∂f∂z−→k Gradient in Cartesian coordinates(−→i , −→j, −→k )
∇f = ∂f ∂r−→r + 1 r ∂f ∂φ−→φ + 1 r sin(φ) ∂f ∂θ−→θ
Gradient in spherical coordinates(−→r , −→θ , −→φ )
∇2f = ∂2f ∂x2 +
∂2f ∂y2 +
∂2f
∂z2 Laplacian in Cartesian coordinates
curlΓf (x) = n(x)∧ ∇f(x) Curl operator. n(x) is the normal unit vector to the
surfaceΓ at position x.∧ is the vector cross product.
Roman symbols
a [m] Radius of sphere
a [m] Typical dimension of a structure
c [m s−1] Speed of sound
D = dofadofs Discretization parameter.
dof Number of DOF per wavelength
dofa Number of DOF per acoustical wavelength
dofs Number of DOF per structural wavelength
fmax [Hz] Maximum frequency of interest
fc [Hz] Critical frequency
fmid [Hz] Center frequency of (one-third) octave band
G = e−ikR
4πR Free field Green function
ka [m−1] Acoustic wavenumber
ks [m−1] Structural wavenumber
l [m] Typical length of a BEM element
Lp [dB] Sound pressure levelre 20µPa
LΠ [dB] Sound power levelre 1 pW
N Number or Degrees Of Freedom (DOF)
Nα Shape function used for acoustical quantities for
ele-mentα NG
α Shape function used for geometry for elementα
n Normal unit vector
nr Normal unit vector at position r
ns Normal unit vector at position s
p [Pa] Sound pressure
b
p [Pa] Complex amplitude of sound pressure for harmonic
problems
b
pα [Pa] Complex amplitude of sound pressure for harmonic
problems over elementα ¯
p [Pa] Complex amplitude of sound pressure for harmonic
problems, piecewise constant as for AVBEM
¯
pα [Pa] Complex amplitude of sound pressure for harmonic
problems, constant over elementα
q [m3
s−1] Volume velocity
r [m] Position vector of the receiver point
Re Real part
R [Pa m−3s] Radiation Resistance Matrix (RRM)
R [m] Distance between the source and receiver point
s [m] Position vector of the source point
S [m2] Surface area
u [ms−1] Particle velocity
v = u· n [ms−1] Normal velocity at the boundary of the vibrating structure.
b
v [ms−1] Complex amplitude of normal velocity for harmonic
problems.
b
vα [ms−1] Complex amplitude of normal velocity for harmonic
problems over elementα ¯
v [ms−1] Complex amplitude of velocity for harmonic
prob-lems, piecewise constant as for AVBEM
¯
vα [ms−1] Complex amplitude of normal velocity for harmonic
xvii
¯¯v [ms−1] Complex amplitude of velocity for harmonic
prob-lems, piecewise averaged as for AVBEM
¯¯vα [ms−
1
] Complex amplitude of normal velocity for harmonic problems, averaged over elementα
(x, y, z) Global coordinate system
ZbS [Pa m−
1
s] Specific Acoustic Radiation Impedance Matrix
ZfS [Pa m−
1
s] Specific Acoustic Transfer Impedance Matrix
Zb
A [Pa m−
3
s] Acoustic Radiation Impedance Matrix
Zf
A [Pa m−
3
s] Acoustic Transfer Impedance Matrix
Greek symbols
α and β Element indexes
γ and δ Nodal indexes
γ = ka/ks Wavenumber ratio λa [m] Acoustic wavelength λs [m] Structural wavelength Π [W] Sound power ρ [kg m−3] Density ω [s−1] Angular frequency
(ξ, η) Local coordinates for surface parametrization
Conventions
(2.1) Equation
[2] Literature reference
Acronyms
AV Aeraged Velocity or Averaged Variables.
AVBEM Vibro-acoustic BEM using averaged element velocities for the approximation of the boundary condition on the vibrating structure.
AVBEM UP AVBEM using the reduced integration scheme defined in 4.4.
BEM Boundary Element Method.
BEM<3> The number between <.>, following a BEM method, indicates the level of Gaus-sian integration used for the element integrals.
DOF Degrees Of Freedom.
FEM Finite Element Method.
FMM Fast Multipole Method
IBEM Indirect BEM. BEM formulation in which pressure and velocity do not appear explicitly, but as potential layers. These potential layers are also referred to as pressure jump and velocity jump over the boundary.
KHIE Kirchhoff-Helmholtz Integral Equation.
LBEM BEM using linear shape functions for the approximation of the acoustic quanti-ties. In this thesis, the linear isoparametric elements can be 4 node quadrilateral or 3 node triangular.
QBEM BEM using serendipity quadratic shape functions for the approximation of the acoustic quantities. In this thesis, the serendipity quadratic isoparametric ele-ments can be 8 node quadrilateral or 6 node triangular.
RRM Radiation Resistance Matrix. It denotes the real part of ZbA.
SF Shape Function.
Chapter 1
Introduction
1.1
Noise control by design
Most people in modern society daily experience noise in the meaning of unwanted sound. Whether one lives close to a highway, undergoes a MRI scan in a hospital or uses household appliances, people are aware of the annoying, the disturbing and the fatiguing effects that sound can have.
In the last half-century noise control engineering has become a fast growing and widespread field of activities. About forty years ago major activities were related to the building acous-tics area, to the development of air- and spacecraft and to the development of naval ships with low underwater noise signatures. However, since then the scope has been significantly widened. New legislation with respect to environmental noise has increased the noise con-trol engineering efforts in areas such as transportation noise (road and railway traffic) and industrial noise. Legislation with respect to occupational noise stimulates the development of quieter production machines and manufacturing processes. But in addition to legislation driven activities, new consumer demands have also increased the noise control efforts drasti-cally. Sound quality has become an important criterion of choice to consumers and, contrarily to the legislation, consumers do not reject loud noise only, but often also more subtle sounds. For example, the proud owner of a new digital photo camera will be embarrassed to loose the candor of a picture, because of the clearly audible zooming of the camera. Therefore, in the development of many products, noise control activities are not primarily driven by legislation, but by consumer demands on comfort and sound quality.
Noise control by design implies that a newly developed product is given certain intrinsic prop-erties with respect to the noise emission source strength or the sound quality. It is distinguished from another common engineering practice, in which noise control is not part of the product design process, but is added afterward and is situation dependent. This thesis is concerned with a further development of one of the analysis tools, which are nowadays available for en-gineers involved in noise control by design. The many available tools may be divided in two basic categories, namely those for experimental analysis and those for computational analy-sis. The use of such tools is often complementary, but their combination has lead to powerful analysis methods as well (e.g. modal analysis; microphone array technology).
In this thesis a contribution is made to the further development of the acoustic Boundary Ele-ment Method (BEM). This computational method can be applied, for example, to predict the sound emission from vibrating structures, like engines and also the sound transmission losses due to complex shaped and partially open enclosures around such vibro-acoustic sources. In our study we investigate a novel variant of this method, which may as this author thinks -fundamentally contribute to diminishing computational costs and computer memory require-ments, which hamper a widespread use of BEM in low noise design practice. However, before our research goal is stated more in detail, a brief orientation with respect to computational methods for noise control by design is given.
1.2
Tools for vibro-acoustic computations
1.2.1
Analysis tasks
For controlling the noise generated by structural vibrations, two complementary strategies exist. One is by changing the dynamical properties of the structures in such a way that the vi-bration amplitudes of the structure are reduced or that the sound radiation efficiency is dimin-ished. Reduction of radiation efficiency means that the same structural vibration amplitudes cause lower sound levels. The other approach is to improve the attenuation of the emitted sound when it propagates into the surroundings of the source. In product design both strate-gies are often combined, as is illustrated with the following automotive example. Making truck engines quieter is the responsibility of diesel engine developers. Their task is to control the internal mechanical and combustion forces and the vibration and sound radiation proper-ties of the engine structure. However, obeying the legal requirements for pass-by noise is the responsibility of the vehicle designers. They have to design additional noise control measures on the vehicle, which need to be effective at minimum cost. This is a complex task, because of many conflicting design requirements (cooling, weight reduction et cetera).
Both types of design activities include the use of extensive experimental testing. But the great costs for experimental testing of design variants and the need to shorten product development cycles has formed an incentive for developing and using powerful computational tools in all phases of vehicle design. The same is true for the design of many other products.
1.2.2
Main types of computational methods
Vibro-acoustics is, loosely speaking, the field of knowledge which describes the interaction between the structural vibration and the sound waves in the adjacent gas or liquid. This includes sound radiation by vibrating structures, sound induced vibration of structures and acoustical scattering on flexible structures.
The choice of a computational method will depend, first of all, on physical considerations. For example, the acoustic vibrations may be linear or non-linear. The acoustic medium may be homogeneous or inhomogeneous. The sound may be stationary or transient. It may contain one or many frequency components, etc. Besides physical considerations also economical and engineering considerations play a role when choosing a convenient method. With respect
1.2 Tools for vibro-acoustic computations 3
to computational costs, besides the complexity of certain physical phenomena, also the large spatial dimensions of vibrating structures and adjacent acoustic domains may lead to very ex-pensive computational problems. Of course, the affordability of the analysis depends on the type of the product involved. In the aerospace industry the criteria for the affordable computer infrastructure and computational efforts are different from those in typical machinery indus-try. But in both design environments it is easy to be confronted with vibro-acoustic problems, which exceed the capacity for applying certain analysis methods in an economical way. With respect to engineering considerations, among the factors, which govern the choice of a com-putational method, are the required accuracy of the output data, the required amount of spatial or temporal details of the output data and the available type and accuracy of input data. The discussion of these aspects in more detail is outside the scope of this introduction.
The study of this thesis is concerned with one particular type of method, namely the Boundary Element Method (BEM) and the primary interest in this study is the modeling of sound radia-tion from vibrating structures. Therefore, at this place in the introducradia-tion, the main objective is to position BEM in rather loose terms relative to other methods, which are in use for this type of application.
BEM and FEM
The boundary element method is a numerical solution method for partial differential equa-tions and its application in the field of vibro-acoustics is just one of many. The term element indicates that BEM is a solution method in which the spatial domain is subdivided in discrete elements and the term boundary indicates that these elements only describes the boundary of the domain.
A more general numerical solution method for solving partial differential equations, is the Finite Element Method (FEM). In vibro-acoustic analysis it is applied for computations both of structural vibration and of acoustic fields. The most visible distinction of acoustic FEM compared to acoustic BEM is that in FEM the discrete elements are not just on the boundary, but fill up the volume of the acoustic domain. So in BEM, element meshes are 2D, whereas in FEM they are 3D. However, at this place in the discussion it is more relevant to mention a property, which these methods share. Both with BEM and FEM detailed spatial and temporal information can be obtained on field variables, such as sound pressure, particle velocity and others. The elements have nodes and in a BEM or FEM model the discrete values of the field variables at the nodes are obtained by solving linear systems of equations.
SEA
In vibro-acoustics the strength of BEM and FEM, namely their capacity for detailed modeling, may form a weakness as well, when it comes to daily engineering applications. To be accurate, the size of the elements needs to be smaller than the structural and acoustic wavelengths in-volved. A consequence is that when length dimensions of structures and acoustic domains are much larger than the structural and acoustic wavelengths, the number of elements becomes very large. Although the capacity of computer hard- and software grows continuously, this still hampers the economic application of these numerical methods for large structures, like
ships, airplanes and buildings. However, often model size does not form just a technical or a economical limitation. When structural and acoustic wavelengths become relatively small, not only the amount of detailed information often grows prohibitively for engineering purposes, but also its accuracy often deteriorates severely. This has to do with uncontrollable uncertain-ties in system properuncertain-ties, inaccuracies in input data or just lack of sufficiently detailed input data.
Therefore, parallel to the development of the element methods, other methods have been de-veloped and play their role in vibro-acoustic engineering design. A widely used method is Statistical Energy Analysis (SEA), see e.g. [31]. In this computational method the struc-ture and the acoustic domain are subdivided in subsystems with dimensions larger than the involved wavelengths. The interaction of the subsystems and their responses are described by energy type quantities, which represent average values over the positions of subsystems and over frequency bands. Again systems of linear equations need to be solved, but these are much smaller than for FEM and BEM models. They are economically affordable even for very large structures such as ships and complete buildings. However, given the requirement that subsystems have to be bigger than the structural and acoustic wavelengths, the applica-bility of SEA is often doubtful or even impossible in the lower end of the frequency range of interest. Moreover, structures are often composed of various parts, some which are suited for the use of SEA, whereas other parts of the system are better suited for analysis with an ele-ment method. Therefore, research efforts are being made to develop methods, which combine both modeling principles, see e.g. [57]. This makes element methods still worth considering even in somewhat higher frequency ranges.
Because the research in this thesis is concerned with acoustic BEM, now a brief discussion will be presented on the positioning of BEM compared to FEM and on specific aspects in relation to low noise design problems in vibro-acoustics.
1.2.3
Acoustic BEM or FEM?
When using an element method for modeling structural vibration, the more generally applica-ble is FEM. However, for the acoustic domain both FEM and BEM are candidates and means to couple the structural and the acoustic domains are available for both methods. Here we limit the discussion to those vibro-acoustic applications for which the acoustics is linear, the medium is homogeneous and the response has to be computed either in the frequency domain or in the time domain. In those situations there seems no a priori physical reason to prefer one of the methods.
For the sake of clarity, the discussion will be illustrated with figure 1.1 (from Meier [37]). In a BEM model such as shown in figure 1.1 a, the proper number of elements depends on the type of elements, the surface area, the geometrical shape, the shortest acoustical wavelength involved and the required accuracy. When the detailed data of the surface vibrations are known, then - for example - the radiated sound power of the engine and of its individual components can be calculated. BEM is also very suitable for calculating the sound pressures anywhere in the infinite acoustic domain. For calculating the same outputs FEM can be used as well. In that case a certain volume around the engine has to be filled with acoustic finite
1.2 Tools for vibro-acoustic computations 5
(a) BEM (b) FEM
Figure 1.1:BEM and FEM models for free field radiation. The illustrations are from Meier [37].
elements. In the free field case, this may look like the mesh in figure 1.1 b. Figures 1.1 a and 1.1 b illustrate therefore characteristic differences between both methods:
1. The first difference is the 2D nature of the acoustic BEM mesh and the 3D nature of the FEM mesh. The larger number of FEM elements needed, compared to BEM elements, is often mentioned as a distinct advantage of BEM. However, this difference is not al-ways practically relevant, because the FEM matrices composing the system of linear equations to be solved are sparse, whereas they are fully populated in BEM. This prop-erty may cause the solution of big FEM models to be faster than that of BEM models with much less elements, or better with a much smaller number of degrees of freedom. Also differences in computer storage requirements need to be considered.
2. The second difference is the necessary truncation of the domain when using FEM. By using in the BEM model the appropriate free field Green’s function, the acoustical boundary condition for the engine is correctly modeled. In the acoustic FEM model of figure 1b a finite domain (a sphere) is filled with elements. The truncation of the do-main at a certain radius as in the figure may cause serious errors in the acoustic dodo-main responses, which may affect the accuracy of the calculated sound power significantly. Ameliorating this error by using a large volume is economically unwanted and phys-ically sometimes still doubtful. To avoid artificially induced acoustical reflections on the outer mesh boundary, the boundary conditions at these outer nodes should be the same as for the infinite domain. Among the different solutions which have been
devel-oped for this problem, the so-called infinite element method (IFEM) [5, 3] have proved successful and are nowadays available from several software vendors. When apply-ing this method, both the acoustic responses inside the FEM volume and in the outer IFEM region are correctly modeled and the advantage of the sparse matrix structure is preserved.
From a more fundamental viewpoint one might say that FEM is the preferred method when non-linearity or inhomogeneity (e.g. large temperature gradients) needs to be taken into ac-count. But given the situation of linear acoustics and homogeneous media, BEM becomes more attractive when the ratio of the domain boundary area to the domain volume decreases. This severely reduces the number of DOF for BEM as compared to FEM. Given this gen-eral situation, it is not surprising that on the one hand the developments in FEM focus on decreasing the number of DOF, for example by applying the above mentioned IFEM. On the other hand current developments in BEM focus on overcoming the severe practical limitations caused by the size and the density of the BEM matrices by the use of fast solvers. Compared to conventional BEM, Fast Multipole Methods (FMM) and multigrid methods show enormous reductions in computer memory storage and computation times. Recently FMM codes have become commercially available, see e.g. [1]. Even on PC-type computers they can handle BEM models, which are much larger than can be handled with conventional solvers. Still the computational times for these large models remain relatively moderate. These methods change the feasibility of acoustic BEM analysis drastically. In section 2.6 a brief characteriza-tion of these various methods will be given. However, the purpose of this thesis is to explore a complementary path. An alternative way of handling the acoustical variables on the element level is studied, which simplifies BEM implementations and reduces the assembly time and the number of DOF.
1.3
Reducing the number of DOF in acoustic BEM
1.3.1
Averaged Velocity BEM as a novel variant
BE methods traditionally use a polynomial approximation of the acoustical variables, like in FE methods. However, the mathematical requirements for the approximation of the variables are different for BEM and FEM and there is more freedom in the choice of the approximation for BEM. This study focuses on an approximation method for the acoustic variables, which is as accurate as using a polynomial approximation but with a reduced number of DOF. There are very few publications on the use of different approximations of the variables for vibro-acoustic methods based on the Kirchhoff-Helmholtz integral equation. These include the methods of Marburg and Schneider [51] and Fahnline and Koopmann [15]. Marburg et al. suggested the use of discontinuous polynomial approximation for BEM, which successfully reduced the number of elements. However, in their work the number of DOF per element increases, leading to a negligible total DOF reduction. Fahnline et al. use a lumped parameter approximation over the elements for which the variables are volume velocity and averaged pressure. Their theoretical study of the use of lumped parameters for vibro-acoustic radiation
1.3 Reducing the number of DOF in acoustic BEM 7
problems forms a firm foundation for the use of lumped parameters for the computation of radiated sound power. Although they do not apply lumped parameters in a commonly used BEM-type algorithm and little is said about the accuracy of their method, some implications of their work did give significant inspiration for developing the novel Averaged Velocity BEM (AVBEM) variant in this thesis. Another significant source of inspiration was the theoretical and experimental work of Holland and Fahy [24]. Their work underlines the crucial impor-tance of the accurate approximation of element volume velocities in their sound radiation analysis of flexural vibration of plates.
The AVBEM developed and studied in this thesis solves the BEM equation system, by using single values for the element velocities and pressures. As input variables, the averaged ele-ment velocities are used. AVBEM therefore uses a single DOF per eleele-ment. In the collocation method, which is used in this thesis, a single node is defined at the element centroids. How-ever, AVBEM uses auxiliary polynomial shape functions and nodes to obtain accurate approx-imations of the averaged velocity over the elements in a pre-calculation. This pre-calculation does not influence the number of DOF of the BEM model. In contrast to the conventional ap-proximation in BEM of the acoustic variables with polynomial shape functions and nodes, this averaged variables approximation permits a reduction of DOF without sacrificing too much accuracy.
In BEM, additionally to the acoustical variables, the geometry of the elements needs to be approximated as well. Polynomial shape functions are used for that purpose in AVBEM, in the same way as commonly done in BEM. This preserves the accuracy of geometrical modeling on curved surfaces and for coarse meshes, but it does again not affect the number of DOF and thus the computational cost of solving the BEM equation system.
Summarizing, the characteristics of AVBEM are its use of subparametric elements and aux-iliary shape functions. The subparametric elements use constants for the acoustic variables and a higher order polynomial approximation of the geometry. The auxiliary shape functions are higher order polynomials used for accurate approximation of the averaged velocities in a pre-processing step.
1.3.2
Expected advantages of AVBEM
The use of constant variables in AVBEM is expected to lead to a number of advantages, which are separated under two main categories.
1. Simplified assembly: Assembly is, loosely speaking, a group of computations which have to be done for each element and for each node belonging to that element. For AVBEM they only have to be performed for each element. The assembly is therefore faster. Additionally, because the elements of AVBEM are smaller for the same amount of DOF, one expects that a lower order numerical integration can be used to handle non-singular integrals. The handling of non-singularities is also much easier when the acoustical variables are considered constant over one element. Other advantages include the disap-pearance of free term coefficient computations, because the single node at the center of the AV elements is always on a smooth boundary. Moreover, since one element
repre-sents one node and therefore one DOF, the management of the data is much simplified. All these small advantages lead to a significantly faster assembly.
2. DOF reduction: AVBEM is expected to allow for coarser discretizations than SFBEM (i.e. BEM using shape functions for the approximation of the acoustic variables), while retaining similar accuracy, see sections 3.5.2 and 3.6. All stages of the BEM compu-tation -i.e. assembly, solving and post processing- will profit from such a reduction. Besides a shorter computation time, computer memory usage will also be reduced.
1.3.3
Illustration of potential gains
To illustrate the potential gains in computational time due to faster assembly and DOF reduc-tion, an example will be considered already in this general introduction. Figure 1.2 shows the BEM mesh of a rail segment used for acoustic studies. The details of these acoustic stud-ies are unimportant for this section. Concerning the computational aspect, the computational efforts when the same geometrical mesh is modeled with AV elements and with commonly used 8 node quadrilateral quadratic elements are compared. For QBEM (i.e. SFBEM using quadratic elements), the mesh size is 12700 DOF, whereas for AVBEM it is 3843 DOF. This corresponds with a DOF reduction factor of 3.3. For both simulations, a Gaussian solver was used and the assembly and solving times are shown in table 1.1.
Figure 1.2:A rail segment used for illustrating the computational gains due to DOF reduction
when using AVBEM.
It is seen that for QBEM, the sum of assembly and solving times is about 1.5 hour, whereas for AVBEM it is reduced to about 3.3 minutes. In the particular simulations shown here, accuracy of radiated sound power was not affected. This example illustrates how for a con-ventional BEM formulation and a widely used type of solver, large gains are obtained by using AVBEM. Both assembly and solving time are considerably reduced, but solving time reduc-tion is predominant. An important topic of research in this thesis is to study such potential gains in more detail. Especially, potential gains for iterative solvers, which are much faster than Gaussian solvers, will be discussed.
1.3 Reducing the number of DOF in acoustic BEM 9
Method used QBEM AVBEM Reduction DOF 12700 3848 3.3 Assembly time [h:m:s] 00:04:18 00:00:23 11.2 Solving time [h:m:s] 01:27:04 00:02:58 29.3 Sum [h:m:s] 01:31:22 00:03:21 27.2
Table 1.1:Assembly and solving times for QBEM and AVBEM for the same geometrical mesh
on a rail segment.
1.3.4
Choices for the implementation in this thesis
There is a wide range of BEM formulations available nowadays. The implementation in this thesis is limited to a so-called direct formulation and a collocation scheme, see chapters 2 and 3. However, as will be discussed in section 2.6, this is not considered as a severe limitation. The replacement of shape functions by averaged acoustic variables seems to be complemen-tary and not conflicting with most others developments for reduction of computational costs. The implementations are realized in the commercial code NEO [42].
In this thesis, only uncoupled problems -as opposite to coupled problems- are studied. When a vibrating structure is radiating sound, then it acts upon the acoustic medium and the acoustic medium acts upon the structure. When the medium is light (like air) and the structure is not too light or too lightly damped, the acoustic loading does not affect the structural vibrations. These remain unchanged compared to the in vacuo situation. However, for very lightweight structures in air or for plate and shell type structures in water the acoustic load may severely affect structural vibrations. In those cases a coupled FEM/BEM analysis is required when using BEM for sound radiation calculations, see e.g. [25]. Although coupled problems are not studied, they will be discussed in section 9.2. Further research on using AVBEM for coupled problems is considered as very promising.
1.3.5
Quantities and errors
Sound power and sound pressure
With acoustic BEM, many acoustic quantities can be calculated such as sound pressure, par-ticle velocity, sound intensity and sound power. In this study the performance of AVBEM is primarily concentrated on sound power radiated from vibrating structures. Radiated sound pressures are also studied to a lesser extent. Sound power is a quantity which is often used for evaluating the source strength of equipment. It is hardly influenced by the surroundings in which a source is installed and therefore suitable to characterize intrinsic source strength properties for low noise design purposes. On the other hand, when sound has to be reduced at prescribed locations, spatial field information is needed and then sound pressures at field points have to be calculated.
Errors
When studying the performance of different BEM formulations, a major practical issue is the required accuracy. In noise control engineering applications, the answer on the accuracy question depends on the type and the goal of the analysis. One general aspect is the uncertainty in input data (i.e. boundary conditions, geometry, etc.). Therefore, BEM model accuracies better than 1 dB are usually practically irrelevant. However, apart from inaccurate input data, economical considerations play an important role as well. Increased computational speed at the cost of some accuracy may be attractive as long as the size of the errors can be controlled. In an early design stage, the problems to be tackled are, for example, comparison of alternative noise control packages, ranking of engine enclosures with apertures at different locations and ranking of partial sources contributions, of transfer paths and of certain frequency bands of the sound spectrum. In this stage less accuracy is often accepted in exchange for increased speed. In situations, where expensive measures are designed to realize incremental gains, like, for example, in some automotive applications, much more accuracy is needed. Some increase of computational costs has then to be accepted. But in many vibro-acoustic radiation simulations errors of 2-3 dB will be acceptable and in some cases even up to 5 dB. Users of SEA for higher frequency modeling in the same noise control engineering applications, have also to deal with inaccuracies typically in the range 1-5 dB.
In this thesis, the possible trading of accuracy against speed will be studied. The performance in this respect of AVBEM will be compared with that of conventional SFBEM. With the noise control engineering applications which have been mentioned above in mind, errors of 1-5 dB will be covered in these comparisons. This type of information was not available in existing literature up to now.
1.4
Key contributions of the thesis
The purpose of this thesis is to study AVBEM and its validity and applicability for vibro-acoustic simulations for noise control by design. Qualitative and quantitative results are pre-sented for the computational gains due to a faster assembly and due to DOF reductions, while preserving the accuracy.
The first contribution of the thesis is a novel BEM variant. The novelty lies in an original combination of several features. These are the combination of constant approximation for the acoustic variable with higher order polynomial approximations of the geometry and the retained accuracy of the averaged velocity approximation for coarser meshes. The second original contribution is the systematic study of discretization errors as a combined function of acoustical and structural wavelengths for sound radiation from plates.
1.5
Outline
The structure of the thesis is as follows:
1.5 Outline 11
goal is to present the basic background information, which is needed to develop the novel AVBEM formulation in the next chapter and to position it in comparison with other recent developments.
• Chapter 3 presents the use of averaged variables as approximations of the acoustic
vari-ables on the boundary elements and their use in an otherwise conventional BEM formu-lation. Moreover, a brief discussion is presented on other uses of averaged variables in vibro-acoustic sound radiation studies. The chapter includes brief physical justifications for the validity of AVBEM in vibro-acoustic analysis and ends with the formulation of the research topics of this thesis.
• Chapter 4 studies the consequences of using averaged variables for the assembly process
in BEM. Numerical integration aspects for AVBEM are studied in comparison with those for common types of elements in SFBEM. Simulations are presented on accuracy and speed for different implementations.
• Chapter 5 presents simulations, which are designed to investigate the potential DOF
reductions for AVBEM compared to SFBEM, while preserving the same accuracy. The simulations are for sound power. Results are presented for force excited plates in both infinite and parallelepipedic baffles. The simulations are used for an extensive study on discretization errors, both for AVBEM and for BEM with quadratic shape functions. Novel results are presented on the combined influence of the structural and acoustical wavelength on discretization errors. Additionally these results are used for deriving discretizations rules.
• Chapter 6 shows similar simulations as in chapter 5, but now for sound pressures
rela-tively close to the vibrating structures. The number of simulations is much smaller than in chapter 5, because the main purpose here is to demonstrate that AVBEM gives not only acceptable accuracy for sound power or far field pressures, but also for near field pressures.
• Chapter 7 presents some examples to illustrate practical advantages of AVBEM. These
examples concern the faster assembly, the improved geometrical shape modeling and the coarser uniform meshes on strongly inhomogeneous and complex vibrating struc-tures.
• In Chapter 8, a discussion and synthesis of the results takes place.
• Chapter 9 summarizes the main conclusions and presents a few recommendations for
further implementations and also for further study of discretization errors both for AVBEM and SFBEM.
Chapter 2
Fundamentals of the Acoustic
Boundary Element Method
2.1
Introduction
The goal of this chapter is to present some of the basics of the acoustical Boundary Ele-ment Method as a necessary background for the developEle-ment of the Averaged Velocity BEM (AVBEM) in the following chapters of this thesis. A more complete introduction can be found in a number of specialized books [7, 60, 63, 18]. For the sake of simplicity, the discussion in this chapter is mainly limited to exterior problems using a direct BEM formulation and a collocation scheme, although in section 2.3 the direct BEM formulation using a Galerkin scheme as well as the indirect formulation using a Galerkin scheme are briefly mentioned. In section 2.2 the Kirchhoff-Helmholtz Integral Equation (KHIE) is introduced. It forms the basic integral equation for BEM. The assembly of the BEM matrix system obtained from the discretization of the KHIE is discussed in section 2.3. In section 2.4, variants of the KHIE commonly used with numerical methods are mentioned. In section 2.5, computational effort and memory usage of BEM are considered. In section 2.6 a brief survey of recent develop-ments aiming at reducing the computational cost is presented. Finally in section 2.7 AVBEM is briefly positioned compared to the aforementioned approaches.
2.2
Integral equation for acoustic BEM
2.2.1
Wave equation and Helmholtz equation
Let us consider a vibrating body, radiating sound into a surrounding acoustic medium as in figure 2.1. Acoustic perturbations propagate as waves through a compressible fluid. The BEM studies addressed in this thesis deal with sound propagation in non-moving air, which may be considered as homogeneous and inviscous. The mathematical description of this propaga-tion is given by the linearized wave equapropaga-tion in (2.1), with the sound pressurep(r, t) as the
r
s
Receiver
Source
R
Figure 2.1: A sound radiating body. The position vectors of the receiver and source points are shown in a Cartesian coordinate system.
dependent field variable. r represents the position vector andt the time. ∇2p(r, t)− 1
c2 ∂2
∂t2p(r, t) = 0, (2.1)
wherec is the (phase) propagation speed. The derivation of this wave equation can be found
in many acoustic textbooks, e.g. Kinsler et al. [27] or Pierce [44]. The acoustical particle velocity u(r, t) is another field variable for which the same wave equation holds. It is related
to the sound pressure by the inviscid Euler equation
ρ0
∂u(r, t)
∂t + ∇p(r, t) = 0. (2.2)
Given the primary concern of the work in this thesis, i.e. low noise design of linearly vibrating structures, the mathematical analysis is preferably performed in the frequency domain. There-fore, solutions of the wave equation with harmonic time dependence (i.e. single frequencies) are studied. The field sound pressure is then expressed asp(r, t) = Rep(r)eb iωt and simi-larly the particle velocity as u(r, t) = Reu(r)eb iωt . Introducing the time harmonic pressure in (2.1) yields the homogeneous Helmholtz equation
∇2p(r) + kb 2p(r) = 0,b (2.3)
wherek = ω/c is the wave number and ω = 2πf the radian frequency. Then the Euler
equation can be written as
iρ0ωbu+ ∇bp = 0. (2.4)
Additionally, when a source of unit strength is present at a position s, the solution of the inhomogeneous Helmholtz equation (∇2p(r) + kb 2
b
p(r) =−δ(r − s)) is the Green’s function.
In a 3D free space, it reads
G(r, s) = e−ik|r−s| 4π|r − s| =
e−ikR
2.2 Integral equation for acoustic BEM 15
The fundamental equation of BEM as studied in this thesis, is an integral version of the Helmholtz equation, which is called the Kirchhoff-Helmholtz integral equation. This equation is briefly discussed in the next subsection.
2.2.2
Kirchhoff-Helmholtz integral equation (KHIE)
There are several methods for deriving the Kirchhoff-Helmholtz Integral Equation1from the
Helmholtz equation. See e.g. [44, 63] for methods based on the second Green’s identity and e.g. [7, 18] for methods based on a weighted residual variant of the homogeneous Helmholtz equation. All lead to the following integral over Sa, the surface surrounding the acoustic
volume Z
Sa[∇G(r, s)b
p(s)− G(r, s)∇bp(s)]· nsdS = C(r)bp(r). (2.6) ns is the normal unit vector toSa at position s. Equation (2.6) is commonly used in three situations: interior radiation problems, exterior radiation problems and scattering problems. For an interior problem,Sais the cavity wall surface. For an exterior problem,Sa = S + Sinf , where S represents the surface in contact with the radiating body and Sinf represents that part of the surface, which moves in the limit to infinity. For a scattering problem, the situation is the same as for an exterior problem, except that a free termpbinc(r) must be added on the right hand side for the incident wave.
Focusing on the exterior radiation problems, the KHIE is
Z
S+Sinf[∇G(r, s)b
p(s)− G(r, s)∇bp(s)]· nsdS = C(r)bp(r). (2.7) Sinf obeys the Sommerfeld condition of radiation and the integral overSinf leads to a zero value. The value of the free term coefficientC(r) depends on the position of the receiver point.
If the receiver point belongs to the volume considered, i.e. located outside of the structure for an exterior radiation problem, then its value is one. If it belongs to another volume, it equals zero. If the source point lies on the (smooth) boundaryS then its value is 1/2. However, for
positions on a non-smooth boundary (for which the normal is not uniquely defined), a small computation has to be performed to determineC, see e.g. [7] p63. The KHIE is written in its
final form with the help of Euler’s equation (2.4) as
Z S ∇G(r, s)bp(s)· nsdS + Z S ikρ0c0G(r, s)bu(s)· nsdS = C(r)bp(r). (2.8) In order to keep the notations simple, the normal projections inside the integrals are rewritten
as ∇n sG = ∇G· ns b v = bu· ns . (2.9)
1The term Kirchhoff-Helmholtz integral equation refers to a constant frequency equation. A time domain version
Equation (2.8) then becomes Z S∇ n sG(r, s)bp(s) dS + Z S ikρ0c0G(r, s)bv(s) dS = C(r)bp(r). (2.10) In words, the KHIE assumes a known analytical functionG(r, s) inside the volume and only b
p(s) and bv(s) on the boundary surface are (partly) unknown. Therefore the solution anywhere
can be computed when the field values on the boundary surface have been computed. The normal velocity and the pressure on the boundary can be considered simply as weighting factors of the known solutionG and its normal derivative. Thus the main task of BEM is to
compute the complete field on the boundary surface.
2.3
KHIE discretization and BEM assembly
This section will focus on discretizing the Kirchhoff-Helmholtz integral equation according to the collocation method. Attention will be given to defining terms and concepts commonly used in BEM, such as shape functions and Gaussian quadrature. The notation used is quite similar to that in Migeot et al. [38]. For more details the reader is referred to e.g. Ciskowski et al. [7], Gaul et al.[18] or Wu et al. [63].
2.3.1
First step: decompose in boundary elements
For a complex surface, the integrals in (2.10) have to be computed numerically, using quadra-ture formulas. The first step is then to divide the surface of the strucquadra-ture into elements of simple form, in order to facilitate the implementation of the quadrature formulas. Applying the discretization of the surface in elements within the KHIE leads to
C(r)bp(r) = ET X α=1 Z Sα∇ n sG(r, s)bpα(s) dSα+ jkρ0c0 ET X α=1 Z SαG(r, s)bv α(s) dSα. (2.11)
2.3.2
Second step: discretize the acoustic variables
The spatial distributions of pressure and normal velocity are usually not known analytically, but only at a certain number of discrete points. They are therefore expressed for each element as a linear combination of shape functions with nodal values. Over an element, the pressure and the normal velocity are expressed as
( b pα(s) =PDγ=1Nαγ(s) bpγ b vα(s) =PDγ=1Nαγ(s) bvγ (2.12)
γ is the nodal index and D is the number of nodes per element. Nα(s) denote the shape functions. The substitution of these approximations of the pressure field and of the surface
2.3 KHIE discretization and BEM assembly 17
normal velocity modifies both integrals in (2.11) to
C(r)bp(r) = ET X α=1 Z Sα∇ n sG(r, s) D X γ=1 Nαγ(s)bpγdSα +jkρ0c0 ET X α=1 Z Sα G(r, s) D X γ=1 Nαγ(s)bvγdSα. (2.13)
In order to take the nodal values out of the integrals, (2.13) is rewritten as
C(r)bp(r) = ET X α=1 D X γ=1 b pγ Z Sα∇ nG(r, s)N αγ(s) dSα +jkρ0c0 ET X α=1 D X γ=1 b vγ Z Sα G(r, s)Nαγ(s) dSα. (2.14)
As explained in [38] p. 20, the sum overγ and the sum over α can be permuted. The double
sum can be seen as a loop over all the elements including a loop over the nodes belonging to
each element or alternatively, as a loop over all the nodes including a loop over the elements in contact with that node. Using this property, one can take the nodal valuespbγ andbvγ out of the first sum.
C(r)bp(r) = DT X γ=1 b pγ E X α=1 Z Sα∇ n sG(r, s)Nαγ(s) dSα +jkρ0c0 DT X γ=1 b vγ E X α=1 Z Sα G(r, s)Nαγ(s) dSα. (2.15)
Notice that the sums are now over the total number of nodesDT and the number of elements
E in contact with a node. Equation (2.15) can be written in a compact form by defining ( aγ(r) =PEα=1 R Sα∇nsG(r, s)Nαγ(s) dSα bγ(r) =−jkρ0c0PEα=1RSαG(r, s)Nαγ(s) dSα (2.16)
reducing the equation to
DT X γ=1 b pγaγ(r)− DT X γ=1 b vγbγ(r) = C(r)bp(r). (2.17)
Assuming that r is constrained to a finite number of discrete points, (2.17) can be written in matrix notation as
where the vectors p and v contains the nodal pressures and normal velocities on the boundary, whereas pf may represent the pressure anywhere in the field.
The matrices Af and Bf are of sizeF × DT, withDT the total number of acoustical nodes andF the number of field points. The elements of Af and Bfare defined in (2.16). Cf is the diagonal matrix of sizeF× F containing the free term coefficients. The subscript f denotes
receiver points belonging to the field. In the subsequent text, the construction of matrices A,
B and C is referred to as assembly.
2.3.3
Third step: apply the collocation scheme
As presented in (2.18), the system is not solvable because there are three unknowns in a single equation. Since pf may represent the values of sound pressure anywhere in the field, it can also represent the boundary. Therefore, r is placed successively at each nodal position on the boundary. This step is referred to as collocation scheme (see [63] p55) and the nodal positions are called collocation points. By doing so we select that solution which satisfies the KHIE at the collocation points. This leads to pf = p and the BEM matrix system becomes
(A− C)p − Bv = 0. (2.19)
Notice that the subscript f has been removed from the matrices because the receiver point belongs now to the boundary. In this thesis the investigations are limited to cases where a velocity boundary condition is known (Neumann BC). Equation (2.19) is therefore solved for p.
2.3.4
Post processing
If field pressure is of interest, after solving the BEM matrix system (2.19), p and v can be substituted in (2.18) and the pressure anywhere in the field can be calculated. If sound power is of interest, it can be computed directly from the boundary variables by integrating pressure and normal velocity over the radiator’s surface and averaging over a periodT = 2π/ω, Π = R
S 1 T
RT
0 pu·n dt dS. Using the complex definition of pressure and velocity from the previous sections, the radiated sound power can be written as
Π = 1 2 Z S Re{bpbv∗} dS = 12 ET X α=1 Z Sα Re{bpαvb∗α} dSα. (2.20)
Replacingpbαandvbαby their expression in (2.12), (2.20) is written in a matrix notation
Π = 1
2Re
pTMv∗ (2.21)
with the elements of M being
mγδ =PEα=1 Z
Sα
