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    CHAPTER 8BOUNDARY ELEMENT MODELING

    D. W. Herrin, T. W. Wu, and A. F. Seybert

    Department of Mechanical EngineeringUniversity of KentuckyLexington, Kentucky

    1 INTRODUCTION

    Both the boundary element method (BEM) andthe nite element method (FEM) approximate thesolution in a piecewise fashion. The chief differencebetween the two methods is that the BEM solvesthe acoustical quantities on the boundary of theacoustical domain (or air) instead of in the acousticaldomain itself. The solution within the acousticaldomain is then determined based on the boundarysolution. This is accomplished by expressing theacoustical variables within the acoustical domain asa surface integral over the domain boundary. TheBEM has been used to successfully predict (1) thetransmission loss of complicated exhaust components,(2) the sound radiation from engines and compressors,and (3) passenger compartment noise.

    In this chapter, a basic theoretical development of theBEM is presented, and then each step of the process forconducting an analysis is summarized. Three practicalexamples illustrate the reliability and application of themethod to a wide range of real-world problems.

    2 BEM THEORY

    An important class of problems in acoustics is thepropagation of sound waves at a constant frequency . For this case, the sound pressure P at any point uctuates sinusoidally with frequency so that P =pe i t where p is the complex amplitude of the soundpressure uctuation. The complex exponential allowsus to take into account sound pressure magnitude andphase from point-to-point in the medium.

    The governing differential equation for linearacoustics in the frequency domain for p is theHelmholtz equation:

    2p + k2p = 0 (1)

    where k is the wavenumber ( k = /c ). The boundaryconditions for the Helmholtz equation are summarizedin Table 1.

    For exterior problems, the boundary integralequation 1

    C(P)p(P) = S

    pn

    G(r) pG(r)

    ndS ( 2)

    can be developed using the Helmholtz equation[Eq. (1)], Greens second identity, and the Sommerfeld

    Table 1 Boundary Conditions for HelmholtzEquation

    BoundaryCondition

    PhysicalQuantity

    MathematicalRelation

    Dirichlet Sound pressure p = pe( pe )Neumann Normal velocity p

    n = i v n( v n )

    Robin Acoustic impedance p

    n = i

    1

    Z a p

    ( Z a )

    vn and p

    n

    V S

    Q

    P

    r

    Fluid

    Figure 1 Schematic showing the variables for the directboundary element method.

    radiation condition. 1 3 The variables are identi ed inFig. 1. If complex exponential notation is adopted, thekernel in Eq. (2) or the Greens function is

    G(r) = e ikr

    4 r(3)

    where r is the distance between the collocation

    point P and the integration point Q on the surface.Equation (3) is the expression for a point monopolesource in three dimensions. The lead coef cient C(P)in Eq. (2) is a constant that depends on the locationof the collocation point P . For interior problems, thedirect BEM formulation is identical to that shownin Eq. (2) except that the lead coef cient C(P) isreplaced by C 0(P ) , which is de ned differently. 1,2Table 2 shows how both lead coef cients are de ned

    116 Handbook of Noise and Vibration Control . Edited by Malcolm J. Crocker Copyright 2007 John Wiley & Sons, Inc.

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    BOUNDARY ELEMENT MODELING 117

    Table 2 Lead Coefcient Denitions at DifferentLocations

    Location of P C (P ) C0 (P )

    In acoustical domainV

    1 1

    Outside acousticaldomain V

    0 0

    Smooth boundary 1212

    Corners/edges 1 S

    n

    14 r

    dS S

    n

    14 r

    dS

    depending on whether the problem is an interior orexterior one.

    For direct or collocation approaches, 119 the bound-ary must be closed, and the primary variables are thesound pressure and normal velocity on the side of theboundary that is in contact with the uid. The nor-mal velocity ( vn ) can be related to the p/n term inEq. (2) via the momentum equation that is expressedas

    pn

    = i vn (4)

    where is the mean density of the uid.When using the direct BEM, there is a distinction

    between an interior and exterior problem. However,there is no such distinction using indirect BEMapproaches. 2028 Both sides of the boundary areconsidered simultaneously even though only one sideof the boundary may be in contact with the uid. AsFig. 2 indicates, the boundary consists of the inside(S 1) and outside surfaces ( S 2), and both sides areanalyzed at the same time.

    In short, boundary integral equations like Eq. (2)can be written on both sides of the boundary andthen summed resulting in an indirect boundary integralformulation that can be expressed as

    p(P) = S

    G(r) dp G(r)

    np dS ( 5)

    QP

    Fluid on One or Both Sides

    r

    n

    V n2, p2

    V n1, p1

    V S 2

    S 1

    Figure 2 Schematic showing the variables for theindirect boundary element method.

    In Eq. (5), the primary variables are the single-( dp ) and double-layer ( p ) potentials. The single-layer potential ( dp ) is the difference in the normalgradient of the pressure and can be related to thenormal velocities ( vn1 and vn2), and the double-layer

    potential ( p ) is the difference in acoustic pressure ( p 1and p 2) across the boundary of the BEM model. SinceS 1 is identical to S 2 , the symbol S is used for both inEq. (5) and the normal vector is de ned as pointingaway from the acoustical domain.

    Table 3 summarizes how the single- and double-layer potentials are related to the normal velocity andsound pressure. If a Galerkin discretization is adopted,the boundary element matrices will be symmetric,and the solution of the matrices will be faster thanthe direct method provided a direct solver is used. 21Additionally, the symmetric matrices are preferable forstructural-acoustical coupling. 25 The boundary condi-tions for the indirect BEM are developed by relatingthe acoustic pressure, normal velocity, and normalimpedance to the single- and double-layer potentials.

    More thorough descriptions for the direct and indi-rect BEM are presented by Wu 3 and Vlahopolous, 27respectively. It should be mentioned that the dif-ferences between the so-called direct and indirectapproaches have blurred recently. In fact, high-levelstudies by Wu 29 and Chen et al. 30 ,31 combine bothprocedures into one set of equations. Chen et al. devel-oped a direct scheme using Galerkin discretization,which generated symmetric matrices. However, thesestate-of-the-art approaches are not used in commercialsoftware at the time of this writing.

    3 MESH PREPARATION

    Building the mesh is the rst step in using the BEM tosolve a problem. Figure 3 shows a BEM model usedfor predicting the sound radiation from a gear housing.The geometry of the housing is represented by a BEMmesh, a series of points called nodes on the surface of the body that are connected to form elements of eitherquadrilateral or triangular shape.

    Most commercially available pre- and postprocess-ing programs developed for the FEM may also be usedfor constructing BEM meshes. In many instances, asolid model can be built, and the surface of the solidcan be meshed automatically creating a mesh repre-sentative of the boundary. Alternatively, a wire frameor surface model of the boundary could be createdusing computer-aided design (CAD) software and thenmeshed. Regardless of the way the mesh is prepared,shell elements are typically used in the nite element

    Table 3 Relationship of Single- and Double-LayerPotentials to Boundary Conditions

    Potential SymbolMathematical

    Relation

    Single layer dp p1

    n

    p2 n

    Double layer p p 1 p2

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    118 FUNDAMENTALS OF ACOUSTICS AND NOISE

    Figure 3 Boundary element model of a gear housing.

    preprocessor, and the nodes and elements are trans-ferred to the boundary element software. The materialproperties and thickness of the elements are irrelevantsince the boundary elements only bound the domain.

    Sometimes a structural nite element mesh is usedas a starting point for creating the boundary ele-ment mesh. Sometimes a boundary element meshcan be obtained by simply skinning the structural nite element mesh. However, the structural niteelement mesh is often excessively ne for the sub-sequent acoustical boundary element analyses, lead-ing to excessive CPU (central processing unit) time.Commercially available software packages have beendeveloped to skin and then coarsen structural nite ele-ment meshes. 32 ,33 These packages can automaticallyremove one-dimensional elements like bars and beams,and skin three-dimensional elements like tetrahedronswith two-dimensional boundary elements. Then, theskinned model can be coarsened providing the userwith the desired BEM mesh. An example of a skinned

    and coarsened model is shown in Figure 4.

    It is well known that the BEM can be CPU intensiveif the model has a large number of nodes (i.e., degreesof freedom). The solution time is roughly proportionalto the number of nodes cubed for a BEM analysis,although iterative solvers may reduce the solution

    time. Nevertheless, if solution time is an issue andit normally is, it will be advantageous to minimize thenumber of nodes in a BEM model.

    Unfortunately, the accuracy of the analysis dependson having a suf cient number of nodes in themodel. Thus, most engineers try to straddle theline between having a mesh that will yield accurateresults yet can be solved quickly. The general ruleof thumb is that six linear or three parabolic ele-ments are needed per acoustic wavelength. However,these guidelines depend on the geometry, boundaryconditions, desired accuracy, integration quadrature,and solver algorithm. 34 ,35 Therefore, these guidelinesshould not be treated as strict rules.

    One notable exception to the guidelines is the casewhere the normal velocity or sound pressure on the

    boundary is complicated. Accordingly, the boundarymesh and the interpolation scheme will need to besuf cient to represent the complexity of this boundarycondition. This may require a much ner mesh thanthe guidelines would normally dictate. Regardless of the element size, the shape of the element appears tohave little impact on the accuracy of the analysis, andtriangular boundary elements are nearly as accurate astheir quadrilateral counterparts. 34

    One way to minimize the number of nodes withoutlosing any precision is to utilize symmetry whenappropriate. The common free space Greens function[Eq. (3)] was used for the derivation earlier in thechapter. However, the Greens function can takedifferent forms if it is convenient to do so. Forexample, the half-space Greens function could be

    used for modeling a hemispace radiation problem.

    Coarsened

    FEM Model BEM Model

    Figure 4 Schematic showing a boundary element model that was created using the nite element model as a startingpoint.

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    BOUNDARY ELEMENT MODELING 119

    Similarly, different Greens functions can be usedfor the axisymmetric and two-dimensional cases. 2Symmetry planes may also be used to model rigid oors or walls provided that the surface is in nite orcan be approximated as such.

    The direction of the element normal to the surfaceis another important aspect of mesh preparation.The element normal direction is determined by thesequence of the nodes de ning a particular element. If the sequence is de ned in a counterclockwise fashion,the normal direction will point outward. Figure 5illustrates this for a quadrilateral element. The elementnormal direction should be consistent throughout theboundary element mesh. If the direct BEM is used,the normal direction should point to or away from theacoustical domain depending on the convention usedby the BEM software. In most instances, adjusting thenormal direction is trivial since most commercial BEMsoftware has the built-in smarts to reverse the normaldirection of a mesh or to make the normal directionconsistent.

    4 FLUID PROPERTY SPECIFICATION

    After the mesh is de ned, the uid properties for theacoustical domain can be speci ed. The BEM assumesthat the uid is a homogeneous ideal uid in the linearregime. The uid properties consist of the speed of sound and the mean density.

    In a BEM model, a sound-absorbing material canbe modeled as either locally reacting or bulk reacting.In the local reacting case, the surface impedanceis used as a boundary condition (see Table 1). Inthe bulk-reacting case, a multidomain 36,37 or direct-mixed body BEM 38 analysis should be performed,using bulk-reacting properties to model the absorption.Any homogeneous sound-absorbing material can bedescribed in terms of its bulk properties. Thesebulk properties include both the complex densityand speed of sound for a medium 39 and providean ideal mechanism for modeling the losses of asound-absorbing material. Bulk-reacting properties areespecially important for thick sections of sound-absorbing materials.

    As mentioned previously, the BEM assumes thatthe domain is homogeneous. However, a nonhomoge-neous domain could be divided into several smaller

    i j

    k l

    i l

    k j

    n

    n

    Figure 5 Mannerin which thenormal direction is denedfor a boundary element.

    Domain 1Air

    Domain 2Seat

    Figure 6 Passenger compartment modeled as twoseparate acoustical domains.

    0

    80

    70

    60

    50

    40

    30

    20

    10

    0500

    T L ( d B )

    1000 1500 2000Frequency (Hz)

    2500 3000 3500

    ExperimentBEM local reactingBEM bulk reacting

    Figure 7 Comparison of the transmission loss for a linedexpansion chamber using local and bulk reacting models.

    subdomains having different uid properties. Wherethe boundaries are joined, continuity of particle veloc-ity and pressure is enforced. For example, the passen-ger compartment shown in Figure 6 could be modeledas two separate acoustical domains, one for the air andanother for the seat. The seat material properties wouldbe the complex density and speed of sound of the seatmaterial. Another application is muf er analysis witha temperature variation. Since the temperature varia-tions in a muf er are substantial, the speed of soundand density of the air will vary from chamber to cham-ber. Using a multidomain BEM, each chamber can bemodeled as a separate subdomain having different uidproperties.

    The advantage of using a bulk-reacting model isillustrated in Figure 7. BEM transmission loss predic-tions are compared to experimental results for a packedexpansion chamber with 1-inch-thick sound-absorbingmaterial. 38 Both locally and bulk-reacting models wereused to simulate the sound absorption. The resultsusing a bulk-reacting model are superior, correspond-ing closely to the measured transmission loss.

    5 BOUNDARY CONDITIONS

    The boundary conditions for the BEM correspondto the Dirichlet, Neumann, and Robin conditions for

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    120 FUNDAMENTALS OF ACOUSTICS AND NOISE

    Boundary Mesh(2D surface mesh)

    Interior (cavity)

    p s

    z v n

    n ^

    Figure 8 Schematic showing the boundary conditionsfor the direct BEM.

    Helmholtz equation (as shown in Table 1). Figure 8shows a boundary element domain for the direct BEM.The boundary element mesh covers the entire sur-

    face of the acoustical domain. At each node on theboundary, a Dirichlet, Neumann, or Robin boundarycondition should be speci ed. In other words, a soundpressure, normal velocity, or surface impedance shouldbe identi ed for each node. Obtaining and/or select-ing these boundary conditions may be problematic.In many instances, the boundary conditions may beassumed or measured. For example, the normal veloc-ity can be obtained by a FEM structural analysis, andthe surface impedance can be measured using a two-microphone test. 40 Both the magnitude and the phaseof the boundary condition are important. Most com-mercial BEM packages select a default zero normalvelocity boundary condition (which corresponds to arigid boundary) if the user speci es no other condition.

    The normal velocity on the boundary is often

    obtained from a preliminary structural

    nite elementanalysis. The frequency response can be read intoBEM software as a normal velocity boundary condi-tion. It is likely that the nodes in the FEM and BEMmodels are not coincident with one another. However,most commercial BEM packages can interpolate theresults from the nite element mesh onto the boundaryelement mesh.

    For the indirect BEM, the boundary conditionsare the differences in the pressure, normal velocity,and surface impedance across the boundary. Figure 9illustrates the setup for an indirect BEM problem.Boundary conditions are applied to both sides of theelements. Each element has a positive and negativeside that is identi ed by the element normal direction(see Fig. 9). Most dif culties using the indirect BEM

    are a result of not recognizing the rami

    cations of specifying boundary conditions on both sides of theelement.

    To model an opening using the indirect BEM, azero jump in pressure 27 ,28 should be applied to theedges of the opening in the BEM mesh (Fig. 10).Most commercial BEM software has the ability tolocate nodes around an opening so that the user caneasily apply the zero jump in pressure. Additionally,

    Openings

    Side+ Side

    NoiseSource

    Sound-AbsorbingMaterial

    p s z

    v n

    v n

    n ^

    Figure 9 Schematic showing the boundary conditionsfor the indirect BEM.

    Zero Jump Condition Junction

    Figure 10 Special boundary conditions that may beused with the indirect BEM.

    special treatment is important when modeling threeor more surfaces that intersect (also illustrated inFig. 10). Nodes must be duplicated along the edge andcompatibility conditions must be applied. 27 ,28 Thoughthis seems complicated, commercial BEM softwarecan easily detect and create these junctions applyingthe appropriate compatibility conditions.

    Many muf ers utilize perforated panels as attenua-tion mechanisms, and these panels may be modeled byspecifying the transfer impedance of the perforate. 41,42The assumption is that the particle velocity is con-tinuous on both sides of the perforated plate but thesound pressure is not. For example, a perforated plateis shown in Fig. 11. A transfer impedance boundarycondition can be de ned at the perforated panel andexpressed as

    Z tr = p 1 p 2

    vn(6)

    Perforated Plate

    P 2P 1

    v n

    Figure 11 Schematic showing the variables used todene the transfer impedance of a perforate.

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    BOUNDARY ELEMENT MODELING 121

    0

    454035302520

    T L ( d B )

    1510

    50

    1000 2000 3000Frequency (Hz)

    4000 5000

    Measured

    Perforated Tube

    BEM

    Figure 12 Transmission loss for a concentric tube reso-nator with a perforate.

    where Z tr is the transfer impedance, p 1 and p 2 arethe sound pressures on each side of the plate, and vnis the particle velocity. The transfer impedance canbe measured or estimated using empirical formulas.In these empirical formulas, the transfer impedance isrelated to factors like the porosity, thickness, and holediameter of a perforated plate. 43 ,44 Figure 12 showsthe transmission loss results computed using the BEMresults for an expansion chamber with a perforatedtube.

    Another useful capability is the ability to specifyacoustic point sources in a BEM model. Noisesources can be modeled as a point source if they areacoustically small (i.e., the dimensions of a sourceare small compared to an acoustic wavelength) andomnidirectional. Both the magnitude and the phase of the point source should be speci ed.

    6 SPECIAL HANDLING OF ACOUSTICRADIATION PROBLEMS

    The BEM is sometimes preferred to the FEM foracoustic radiation problems because of the ease inmeshing. However, there are some solution dif cultieswith the BEM for acoustic radiation problems. Boththe direct and indirect methods have dif culties thatare similar but not identical. With the direct BEM,the exterior boundary integral equation does nothave a unique solution at certain frequencies. Thesefrequencies correspond to the resonance frequenciesof the airspace interior to the boundary (with Dirichletboundary conditions). Though the direct BEM resultswill be accurate at most frequencies, the soundpressure results will be incorrect at these characteristicfrequencies.

    The most common approach to overcome the

    nonuniqueness dif

    culty is to use the combinedHelmholtz integral equation formulation, or CHIEF,method. 11 A few overdetermination or CHIEF pointsare placed inside the boundary, and CHIEF equationsare written that force the sound pressure to be equal tozero at each of these points. Several CHIEF pointsshould be identi ed inside the boundary because aCHIEF point that falls on or near the interior nodalsurface of a particular eigenfrequency will not provide

    a strong constraint since the pressure on that interiornodal surface is also zero for the interior problem. Asthe frequency increases, the problem is compoundedby the fact that the eigenfrequencies and the nodal sur-faces become more closely spaced. Therefore, analysts

    normally add CHIEF points liberally if higher fre-quencies are considered. Although the CHIEF methodis very effective at low and intermediate frequen-cies, a more theoretically robust way to overcomethe nonuniqueness dif culty is the Burton and Millermethod. 5

    Similarly, for an indirect BEM analysis, there is anonexistence dif culty associated with exterior radi-ation problems. Since there is no distinction betweenthe interior and exterior analysis, the primary variablesof the indirect BEM solution capture information onboth sides of the boundary. 27 At the resonance fre-quencies for the interior, the solution for points on theexterior is contaminated by large differences in pres-sure between the exterior and interior surfaces of theboundary. The nonexistence dif culty can be solved

    by adding absorptive planes inside or by specifying animpedance boundary condition on the interior surfaceof the boundary. 27

    The lesson to be learned is that exterior radiationproblems should be approached carefully. However,excellent acoustical predictions can be made using theBEM, provided appropriate precautions are taken.

    7 BEM SOLUTIONEven though BEM matrices are based on a surfacemesh, the BEM is often computationally and memoryintensive. Both the indirect and direct proceduresproduce dense matrices that are not sparse, as is typicalof nite element matrices. For realistic models, the sizeof the matrix could easily be on the order of tens of thousands. The memory storage of an N N matrixis on the order of N 2 , while the solution time using adirect solver is on the order of N 3 . As the BEM modelgrows, the method sometimes becomes impractical dueto computer limitations.

    One way to overcome the solution time dif cultyis to use an iterative solver 45 with some appropriatepreconditioning. 46 ,47 Iterative solvers are much fasterthan conventional direct solvers for large problems. 48Also, there is no need to keep the matrix in memory,although the solution is slower in that case. 49 Addition-ally, BEM researchers have been working on differentvariations of the so-called fast multipole expansionmethod based on the original idea by Rokhlin 5053 inapplied physics.

    8 POSTPROCESSING

    Boundary element results can be viewed and assessedin a number of different ways. The BEM matrixsolution only computes the acoustical quantities on thesurface of the boundary element mesh. Thus, only thesound pressure and/or normal velocity is computed onthe boundary using the direct method, and only thesingle- and/or double-layer potentials are computedusing the indirect BEM. Following this, the acoustical

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    122 FUNDAMENTALS OF ACOUSTICS AND NOISE

    quantities at points in the eld can be determinedfrom the boundary solution by integrating the surfaceacoustical quantities over the boundary, a processrequiring minimal computer resources. As a result,once an acoustical BEM analysis has been completed,

    results can be examined at any number of eld pointsin a matter of minutes. This is a clear advantage of using numerical approaches like the BEM over thetime-intensive nature of experimental work. However,the numerical results in the eld are only as reliable asthe calculated acoustical quantities on the boundary,and the results should be carefully examined to assurethey make good engineering sense.

    To help evaluate the results, commercial softwareincludes convenient postprocessing capabilities todetermine and then plot the sound pressure resultson standard geometric shapes like planes, spheres,or hemispheres in the sound eld. These shapes donot have to be de ned beforehand, making it veryconvenient to examine results at various locations of interest in the sound eld. Furthermore, the user can

    more closely inspect the solution at strategic positions.For example, Fig. 13 shows a sound pressure contourfor the sound radiated by an engine cover. A contourplot of the surface vibration is shown under theengine cover proper, and the sound pressure resultsare displayed on a eld point mesh above the coverand give a good indication of the directivity of thesound at that particular frequency.

    Additionally, the sound power can be computedafter the matrix solution is completed. One advantageof the direct BEM is that the sound power and radiationef ciency can be determined from the boundarysolution directly. This is a direct result of only oneside of the boundary being considered for the solution.However, determining the sound power using theindirect BEM is a little more problematic. Normally,

    the user de

    nes a sphere or some other geometricshape that encloses the sound radiator. After the soundpressure and particle velocity are computed on the

    Surface Vibration Contour

    Sound Pressure Contour

    Figure 13 Contour plot showing the sound pressurevariation on a eld point plane located above an enginecover.

    geometric shape, the sound power can be determinedby integrating the sound intensity over the area of theshape. Results are normally better if the eld pointsare located in the far eld.

    Another possible use of BEM technology can be to

    identify the panels that contribute most to the sound ata point or to the sound eld as a whole. For instance, aBEM mesh was painted onto a diesel engine and thenvibration measurements were made at each node on theengine surface. The measured vibrations were used asthe input velocity boundary condition for a subsequentBEM calculation. The sound power contributions (indecibels) from the oil pan and the front cover of a diesel engine are shown in Fig. 14. As the gureindicates, the front cover is the prime culprit at 240 Hz.This example illustrates how the BEM can be used asa diagnostic tool even after a prototype is developed.

    Boundary element method postprocessing is notalways a turnkey operation. The user should carefullyexamine the results rst to judge whether con denceis warranted in the analysis. Furthermore, unlike

    measurement results, raw BEM results are alwayson a narrow-band basis. Obtaining the overall orA-weighted sound pressure or sound power mayrequire additional postprocessing depending on thecommercial software used. Also, the transmission lossfor a muf er or a plenum system cannot be exporteddirectly using many BEM software packages. Thisrequires additional postprocessing using a spreadsheetor mathematical software.

    9 EXAMPLE 1: CONSTRUCTION CABA construction cab is an example of an interior acous-tics problem. The construction cab under considera-tion is 1 .9 1.5 0.9 m3 . Due to the thickness of the walls, and the high damping, the boundary wasassumed to be rigid. A loudspeaker and tube were

    attached to the construction cab, and the sound pres-sure was measured using a microphone where the tubeconnects to the cab. All analyses were conducted atlow enough frequencies so that plane waves could beassumed inside the tube. Medium-density foam wasplaced on the oor of the cab.

    First, a solid model of the acoustical domain wasprepared, and the boundary was meshed using shellelements. A commercial preprocessor was used to pre-pare the mesh, which was then transferred into BEMsoftware. In accordance with the normal convention forthe commercial BEM software in use, the element nor-mal direction was checked for consistency and chosento point toward the acoustical domain. For the indi-rect BEM, the normal direction must be consistent,pointing toward the inside or outside.

    In this case, both the direct and indirect BEMapproaches were used. For the indirect BEM, theboundary conditions are placed on the inner surface,and the outer surface is assumed to be rigid (normalvelocity of zero). For both approaches, the measuredsound pressure at the tube inlet was used as a boundarycondition, and a surface impedance was applied to the oor to model the foam. (The surface impedance of the foam was measured in an impedance tube. 40 ) All

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    BOUNDARY ELEMENT MODELING 123

    Figure 14 BEM predicted sound power contributions from the oil pan and front cover of a diesel engine.

    Figure 15 Schematic showing the BEM mesh andboundary conditions for the passenger compartment of aconstruction cab.

    other surfaces aside from the oor were assumed to be

    rigid. The boundary conditions are shown in Fig. 15.Since the passenger compartment airspace ismodally dense, a ne frequency resolution of 5 Hzwas used. The sound pressure results are compared at apoint in the interior to measured results in Fig. 16. Theresults demonstrate the limits of the BEM. Althoughthe boundary element results do not exactly match themeasured results, the trends are predicted well and theoverall sound pressure level is quite close. Determining

    the pressure at a single point is arguably the most chal-lenging test for a boundary element analysis. The BEMfares better when the sound power is predicted sincethe sound pressure results are used in an overall sense.

    10 EXAMPLE 2: ENGINE COVER IN A PARTIAL ENCLOSURE

    The sound radiation from an aluminum engine coverin a partial enclosure was predicted using the indirectBEM. 54 The experimental setup is shown in Fig. 17The engine cover was bolted down at 15 locations tothree steel plates bolted together ( 34 inches thick each).The steel plates were rigid and massive comparedto the engine cover and were thus considered rigidfor modeling purposes. A shaker was attached to theengine cover by positioning the stinger through ahole drilled through the steel plates, and high-densityparticleboard was placed around the periphery of thesteel plates. The experiment was designed so that theengine cover could be assumed to lie on a rigid half space. The engine cover was excited using white-noiseexcitation inside a hemianechoic chamber.

    To complicate the experiment, a partial enclosure

    was placed around the engine cover. The plywoodpartial enclosure was 0.4 m in height and was linedwith glass ber on each wall. Although the addedenclosure is a simple experimental change, it hada signi cant impact on the sound radiation and theway in which the acoustical system is modeled. Thisproblem is no longer strictly exterior or interior sincethe enclosure is open, making the model unsuitable forthe direct BEM; the indirect BEM was used.

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    124 FUNDAMENTALS OF ACOUSTICS AND NOISE

    Figure 16 Sound pressure level comparison at a point inside the construction cab. (The overall A-weighted soundpressure levels predicted by BEM and measured were 99.7 dB and 97.7 respective.)

    Figure 17 Schematic showing the experimental setup ofan engine cover located inside a partial enclosure.

    A structural nite element model of the cover wascreated from a solid model of the engine cover. Thesolid model was automatically meshed using parabolictetrahedral nite elements, and a frequency responseanalysis was performed. The results of the niteelement analysis were used as a boundary conditionfor the acoustical analysis that followed.

    Using the same solid model as a starting point,the boundary element mesh was created by meshingthe outer surface of the solid with linear quadrilateralelements. The boundary element mesh is simpler andcoarser than the structural nite element mesh. Since

    features like the small ribs have dimensions much lessthan an acoustic wavelength, they have a negligibleeffect on the acoustics even though they are signi cantstructurally. Those features were removed from thesolid model before meshing so that the mesh wascoarser and could be analyzed in a timely manner.The boundary condition for the engine cover is thevibration on the cover (i.e., the particle velocity). Thecommercial BEM software used was able to interpolate

    the vibration results from the structural nite elementmodel onto the surface of the boundary element mesh.

    A symmetry plane was placed at the base of theengine cover to close the mesh. Since this is anacoustic radiation problem, precautions were taken toavoid errors in the solution due to the nonexistencedif culty for the indirect BEM discussed earlier.Two rectangular planes of boundary elements werepositioned at right angles to one another in the spacebetween the engine cover boundary and the symmetryplane (Fig. 18). An impedance boundary condition wasapplied to each side of the planes. Since the edgesof each plane are free, a zero jump in pressure wasapplied along the edges.

    Symmetry Plane

    AcousticImpedancePlanes

    Local AcousticImpedanceZero Jump in

    Sound Pressure

    Engine CoverVibration

    Figure 18 Schematic showing the boundary conditionsthat were assumed for a vibrating engine cover inside apartial enclosure.

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    BOUNDARY ELEMENT MODELING 125

    Figure 19 Comparison of the sound power from the partial enclosure. Indirect BEM results are compared with thoseobtained by measurement. (The overall A-weighted sound power levels predicted by BEM and obtained by measurementwere both 97.6 dB.)

    The thickness of the partial enclosure was neglectedsince the enclosure is thin in the acoustical sense(i.e., the combined thickness of the wood and theabsorptive lining is small compared to an acousticwavelength). A surface impedance boundary conditionwas applied on the inside surface of the elements,and the outside surface was assumed to be rigid (zerovelocity boundary condition). As indicated in Fig. 18,a zero jump in pressure was applied to the nodes onthe top edge.

    As Fig. 19 shows, the BEM results comparedreasonably well with the experimental results. Theclosely matched A-weighted sound power resultsare largely a result of predicting the value of thehighest peak accurately. The differences at the otherpeaks can be attributed to errors in measuring thedamping of the engine cover. A small change in thedamping will have a large effect on the structural FEManalysis and a corresponding effect on any acousticcomputational analysis that follows. Measuring thestructural damping accurately is tedious due to datacollection and experimental setup issues involved.

    11 CONCLUSION

    The objective of this chapter was to introduce theBEM, noting some of the more important develop-ments as well as the practical application of the methodto a wide variety of acoustic problems. The BEM isa tool that can provide quick answers provided that asuitable model and realistic boundary conditions canbe applied. However, when the BEM is looked atobjectively, many practitioners nd that it is not quite

    what they had hoped for. Today, many problems arestill intractable using numerical tools in a purely pre-dictive fashion. For example, forces inside machinery(i.e., engines and compressors) are dif cult to quan-tify. Without realistic input forces and damping inthe structural FEM model, numerical results obtainedby a subsequent BEM analysis should be consideredcritically. Certainly, the BEM may still be useful fordetermining the possible merits of one design overanother. Nevertheless, it is hard to escape the suspi-cion that many models may not resemble reality asmuch as we would like.

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