Research Article Low-Frequency Acoustic-Structure Analysis

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 583079 8 pageshttpdxdoiorg1011552013583079

Research ArticleLow-Frequency Acoustic-Structure AnalysisUsing Coupled FEM-BEM Method

Jinlong Feng1 Xiaoping Zheng1 Haitao Wang2 HongTao Wang2

Yuanjie Zou3 Yinghua Liu1 and Zhenhan Yao1

1 School of Aerospace Tsinghua University Beijing 100084 China2 Institute of Nuclear and New Energy Technology Tsinghua University Beijing 100084 China3 China Academy of Space Technology Beijing 100094 China

Correspondence should be addressed to Xiaoping Zheng zhengxptsinghuaeducn

Received 19 July 2013 Accepted 5 September 2013

Academic Editor Song Cen

Copyright copy 2013 Jinlong Feng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A numerical algorithmbased on finite elementmethod (FEM) and boundary elementmethod (BEM) is proposed for the analyses ofacoustic-structure coupled problems By this algorithm the structural domain and the acoustic domain are modelled by FEM andBEM respectively which are coupled with each other through the consideration of the appropriate compatibility and equilibriumconditions on the interface of the two domains To improve the computational efficiency the adaptive cross approximation (ACA)approach is incorporated into the proposed algorithm to deal with the nonsymmetric and fully populated matrices resulting fromthe coupling of the FEM and BEMThe validity and the high efficiency of the present approach are demonstrated by two examples

1 Introduction

Acoustic-structure coupling problems can be found in var-ious engineering fields For example in space explorationspacecrafts are subject to heavy acoustic load particularlyduring launching which can impose severe and adverseeffect on the astronauts and the structure of the spacecraftsTherefore it is necessary to develop numerical methods topredict the structural behaviour in the acoustic-structurecoupling problems since these numerical techniques areessential to the improvement of practical engineering design

The finite element method (FEM) and boundary elementmethod (BEM) are both widely used in various engineeringproblems However FEM and BEM are generally appliedin different areas For instance the former is especiallywell suited to the analyses of the problems which involveinhomogeneity or nonlinearity [1 2] whilst the latter is oneof the most powerful and suitable numerical techniques forpredicting the noise around a vibrating system [3] Comparedto FEM [4] BEM reduces the mathematical dimension of theproblem under analysis by one Furthermore in BEM the

Sommerfeld condition is automatically satisfied so that theexternal domain does not need to be bounded As a resultBEM is regarded as the most suitable tool to deal with theproblems in exterior unbounded domains

Therefore for acoustic-structure problems involving sub-merged bodies it should be a natural strategy to combineFEM and BEM in a computational model to make use of therespective advantages of the two methods The combinationof FEM and BEM can be implemented by decomposing theconcerned domain into several subdomains each of whichaccording to its specific physics is modelled by either FEMor BEM [5ndash7] Because of the appropriate compatibility andequilibrium conditions on the respective interface bound-aries the subdomains are coupled to each other

It should be mentioned that the boundary integral equa-tion has a major defect for exterior problems that is it hasnonunique solution at a set of fictitious eigenfrequenciesassociated with the resonant frequencies In the past severalmethods have been proposed to overcome this difficultyThe two most popular methods are the CHIEF methodand Burton-Miller approach The CHIEF method [8] is

2 Mathematical Problems in Engineering

based on the Helmholtz integral equation on the surfaceof the radiating body combined with Helmholtz equationsfor some interior points The resulting system equation isusually solved by the least-square approach The Burton-Miller approach [9] inspired by Panich formulation [10]forms a linear combination of the Helmholtz boundaryintegral equation and its normal derivative providing also avalid solution at any frequency but leading to hypersingularintegrals which can be handled by Guiggiani algorithm [1112] In this study the Burton-Miller approach is adopted

Similar to BEM the coupled FEM-BEM approach leadsto a fully populated system matrix This is thought to bea major drawback of the coupled approach as comparedwith FEM which normally results in a sparse symmetricalsystem matrix If iterative methods like the generalizedminimal residual (GMRES) method [13] in combinationwith suitable preconditioners are used to solve the FEM-BEM system with 119873 degrees of freedom the computationalexpense is in the order of 119874(1198732) If a direct solver isapplied the computational cost is even in the order of119874(1198733)Fortunately over the last few years several fast solutionmethods including fastmultipolemethod (FMM) [14ndash16] andadaptive cross approximation approach [17 18] have beendeveloped These methods reduce the memory requirementsand the computational time significantly when dealing withthe matrix-vector product In this study the ACA approachis adopted since it is much easier to operate compared withFMM which is based on a series expansion of kernel-shapefunction products

In this paper a coupling algorithm combining FEMand BEM is developed for the analysis of acoustic-structureresponse This paper is organized as follows Section 1 is ashort introduction to the background and correspondingresearch status In Section 2 the FE formulations of the struc-ture domain are constructed In Section 3 the application ofthe BEM to Helmholtz problem and its implementation ispresentedThen the coupled equations are built in Section 4In Section 5 the solving algorithms of coupled equationsincluding the GMRES iterative solver and ACA approachare introduced In Section 6 several numerical examples arepresented

2 FE Formulation of Structural Domain

A coupling system of acoustic-structure is shown in Figure 1The structure is assumed to be fully submerged in interiorand exterior acoustic fields or one of them In Figure 1the structure domain is denoted by Ω119878 The interior andthe exterior acoustic fields are denoted by Ω

119868 and Ω119864

respectively The inner interface and the outer interface arenamed as 119878119868 and 119878119864 respectively The structure is assumed tobe thin and flexible which is susceptible to acoustic-structureinteraction

FEM is predominantly chosen for simulating linear elas-todynamic systems When the structure Ω119878 is discretized byFEM the resulting system of linear equations reads as

Mu + Cu + Ku = f119904+ T119904 (1)

ΩE

ΩI

ΩS

Structure

Interior acoustic field

Exterior

field

SE

SI

acoustic

Figure 1 Illustration of a coupling system of acoustic-structure

where M C and K denote the global mass matrix thedamping matrix and the stiffness matrix respectively Thevector f

119904is load vector while the vector T

119904resulted from

the tractions on the coupling interface and it will be furtherdiscussed in the following section The global displacementvector is denoted by u

In this paper Rayleigh damping is considered for thedamping matrix which means that

C = 120585M + 120578K (2)

where 120585 and 120578 are the damping parametersIf the load vector f

119904and the acoustic pressure vector T

119904

are both assumed to be harmonic the displacement vector ushould be also harmonic This means that

f119904= f119904119890119894120596119905 T

119904= T119904119890119894120596119905 u = u119890119894120596119905 (3)

where the imaginary unit is denoted by 119894 and 120596 = 2120587119891 is thecircular frequency with the excitation frequency 119891 For therelationship between u and u one obtains

u (119905) = 119894120596u119890119894120596119905 = u (120596) 119890119894120596119905

u (119905) = minus1205962u119890119894120596119905 = 119894120596u (120596) 119890119894120596119905(4)

Substituting (4) into (1) yields

[119894120596M + C minus 119894K120596] u (120596) = f

119904+ T119904 (5)

NASTRAN a finite element package is employed to setup M C K and the right-hand side vector f

119904 Since M C

and K are frequency independent these matrices only haveto be calculated once for a given model

3 The Boundary Element Method forthe Helmholtz Equation

Acoustic problems in frequency domain can be described bythe Helmholtz equation which has a three-dimensional formgiven by

nabla2119901 (x) + 1198962119901 (x) = 0 (6)

Mathematical Problems in Engineering 3

where nabla2 is the Laplace operator 119901(x) is the sound pressureat point x in the fluid 119896 = 120596119888 is the acoustic wavenumberand 119888 is the speed of sound in the fluid

The boundary condition can be expressed as

119901 (x) = 119901 (x) x isin 119878119863

120597119901 (x)120597119899

= 119902 (x) x isin 119878119873(7)

Additionally the three-dimensional Sommerfeld radia-tion condition [19] is

10038161003816100381610038161003816100381610038161003816

120597119901

120597119903minus 119894119896119901

10038161003816100381610038161003816100381610038161003816

le119888

1199032 119903 997888rarr infin (8)

For the exterior problem (8) has to be satisfied to ensure thatthe wave is purely outgoing

A three-dimensional fundamental solution for theHelmholtz equation is given by

119866lowast(x y) = 119890

119894119896119903

4120587119903 (9)

where x is a field point y is a source point and 119903 = |x minus y|is the distance from x to y The key idea is to use Greenrsquossecond identity in combinationwith the property of theDiracdistribution

intΩ119864cupΩ119868

(119901nabla2119866lowastminus 119866lowastnabla2119901) d119881

= int119878119864cup119878119868

(119901120597119866lowast

120597119899minus 119866lowast 120597119901

120597119899) d119878

(10)

By this way the pressure 119901 at an arbitrary point xwithin the acoustic domain Ω119864 cup Ω119868 is given by the integralrepresentation

119901 (x) = minusint119878119864cup119878119868

119866lowast(x y)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878

x isin Ω119864 cup Ω119868

(11)

For scattering problems (11) can be written as

119901 (x) = minusint119878119864cup119878119868

119866lowast(xy)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878 + 119901119868 (x)

x isin Ω119864 cup Ω119868

(12)

Let the point x approach the boundary then the followingconventional boundary integral equation (CBIE) [20] isobtained

119862 (x) 119901 (x) = int119878

(119866lowast 120597119901

120597119899minus 119901

120597119866lowast

120597119899) d119878 (13)

where 119862(x) = 12 if 119878 is smooth around x This CBIE can beemployed to solve the unknown 119901 and 120597119901120597119899 on 119878119864 cup 119878119868

It is well known that the CBIE has a major defect forexterior problems that is it has nonunique solution at a setof fictitious eigenfrequencies associated with the resonantfrequencies [21] A remedy to this problem is to use thenormal derivative BIE in conjunction with the CBIE Takingthe derivative of the integral representation given in (13) withrespect to the normal at a typical point x on 119878119864cup119878119868 and lettingx approach 119878119864 cup 119878119868 one obtains the following hypersingularboundary integral equation (HBIE)

1198621015840(x) 119902 (x) = minusint

119878119864cup119878119868

120597119866lowast(x y)120597119899119909

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

1205972119866lowast(x y)

120597119899119909120597119899119910

119901 (y) d119878 x isin 119878119864 cup 119878119868

(14)

where 1198621015840(x) = 12 if 119878119864 cup 119878119868 is smooth around x It shouldbe noted that (14) suffers from the same kind of defect as(13) However if a linear combination of CBIE and HBIE isused the uniqueness of the results can be ensured for exterioracoustic wave problems Therefore

int119878119864cup119878119868

120597119866lowast

120597119899119910

119901 (y) d119878 + 119862 (x) 119901 (x)

+ 120572int119878119864cup119878119868

1205972119866lowast

120597119899 (x) 120597119899 (y)119901 (y) d119878

= int119878119864cup119878119868

119866lowast 120597119901

120597119899d119878

+ 120572 [int119878119864cup119878119868

120597119866lowast

120597119899119909

120597119901

120597119899d119878 minus 1198621015840 (x)

120597119901

120597119899119909

(x)]

(15)

where 120572 is the coupling constant This formulation is calledBurton-Miller formulation [9] for acoustic wave problemsand has been shown to yield unique solutions at all frequen-cies if 120572 is a complex number which for example can bechosen as 120572 = 119894120581 with 119894 = radicminus1 [21] In (15) 120597119901120597119899 = minus119894120596120588vand 120588 and v denote the density of the fluid and the normalvelocity on the interface respectively

When the boundary of the acoustic domain is discretizedinto boundary elements the resulting linear system equationscan be expressed as

[H] [p] = [G] [v] (16)

where the matrices [H] and [G] are obtained by integratingthe fundamental solutions over each boundary element

Equation (16) is the boundary element equation used tosolve the acoustical problem When there is incident sounda free term is needed and (16) becomes

[H] [p] = [G] [v] + [p119868] (17)


4 Coupled FE-BE Formulations

By taking into account the appropriate compatibility andequilibrium conditions at the respective interface bound-aries the fully coupled FE-BE formulations can be derivedHereby the matching grids for the FE and the BE partsare required The compatibility condition at the respectiveinterface boundaries links the acoustic pressure 119901 and thetractions 119879

119899on the structure which can be written in the

following matrix form

119879119899= T sdot n = minus119901 (18)

For an element one has

T119890119904= int119878120590

N119879T dΓ = int119878120590

N119879119879119899n dΓ

= minusint119878120590

N119879n[N119887]119879

d Γp = minusAs119890p

(19)

where As119890= int119878120590

N119879n[N119887]119879dΓSubstituting (19) into (5) and using uniform symbol for

the matrices and vectors yield

119894120596 [M] + [C] minus 119894 [K]120596 [u (120596)] = [f

119904] minus [A119904] [p] (20)

From (20) one obtains

[u (120596)] = 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (21)

Please note that [u(120596)] in (21) incorporates the velocityvector of all of the nodes in the structure and [v] in (17) justdenotes the velocity vector of the nodes on the surface of thestructures The relationship between them reads as

[v] = [R] [u (120596)] (22)

where [R] is a transformationmatrix from the velocity vectorfor all nodes in the structure to the velocity vector for thesurface nodes [22]


[v] = [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (23)

Further more substituting (23) into (17) yields

[H] + [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[A119904] [p]

= [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] + [p

119868]

(24)

Equation (24) is the coupled FEM-BEM equation fromwhich the sound pressure of the nodes on the interfaces canbe obtained

5 Solving Algorithm of the CoupledFEM-BEM Equation

Because of the nonsymmetric and fully populated matricesin the coupled FEM-BEM equations the ACA approachis used in this study to reduce the memory requirementsand the computation cost During iteration process thegeneralized minimal residual (GMRES) method iterativesolver is adopted In the following the ACA and the GMRESalgorithms are introduced briefly More details can be foundin relevant literatures [17 18 23]

51 Adaptive Cross Approximation The ACA algorithm pro-duced by M Bebendorf and S Rjasanow is an effectivetechnique for solving nonsymmetric and fully populatedmatrices The main process is as follows

(1) Index Octree The low-rank approximation of thematrix of FEM-BEM is based on asymptoticallysmooth functions which happens in a well-separateddomain Thus a tree structure for all boundarypoints should be constructed at first For a three-dimensional domain an octree is necessary to be builtto describe the geometry relationship of sections onsurface The information that a tree should recordincludes its father its children its geometry infor-mation and the points contained The least nodesare called leaves whose numbers of boundary pointscontained are less than a given threshold value usedto judge whether a new division is necessary

(2) Partitioning of Matrix Based on the tree the matrixin the FEM-BEM equation can be divided into someblocks At first the relationship between nodes shouldbe established For some node NP in an octree itsneighbour nodes are defined as the nodes which havethe same geometry size and one common point atleast The interaction nodes are those nodes owingthe same geometry size whose fathers are neighbournodes but not for them The relationship is shown asgiven Figure 2 in a two-dimensional domain

Then for each node of the tree the elements containedare taken as source points and the elements of itsneighbour as field points and the submatrices comingfrom the source points and field points are definedas 119860119899 The submatrices whose entries come from

elements of the node and its interaction nodes aredefined as119860

119891 All of the submatrices119860

119899and119860

119891form

the whole efficient matrix of FEM-BEM equation

(3) Low-Rank Approximation The low-rank approxima-tion in ACA is defined as

A119898times119899 = U119898times119903V119903times119899 =119903

sum

119894=1

u119898times1119894

v1times119899119894 (25)


ND

Neighbour nodes

Interaction nodes

Figure 2 Relationship between nodes

The target of ACA is to achieve

1003817100381710038171003817R119898times1198991003817100381710038171003817 =

10038171003817100381710038171003817A119898times119899 minus A119898times11989910038171003817100381710038171003817 le 120576

1003817100381710038171003817A119898times1198991003817100381710038171003817 (26)

where 120576 is the tolerance of ACA

The ACA algorithm has been applied to the low-rankblocks achieving approximately 119874(119873) for both storage andmatrix-vector multiplication [24] It must be noted thatthe fully pivoted approach is well known to be muchslower than the partially pivoted approach The main reasonbehind this is that the fully pivoted approach requires theknowledge of the full matrix whereas the partially pivotedapproach would only require generation of individual matrixentries

52 Generalized Minimal Residual Method The generalizedminimal residual (GMRES) method is one of the mostpopular iterative solvers for nonsymmetric linear systemsIt was proposed by Saad and Schultz [25] and furtherdeveloped by other researchers [26ndash28] It has the prop-erty of minimizing at every step the norm of the residualvector over a Krylov subspace The algorithm is derivedfrom the Arnoldi process for constructing an orthogonalbasis of Krylov subspace Because of existence of errorthe gained vectors lose orthogonality gradually Thereforepreconditioner is needed In the computer codes for thisstudy the solution of the diagonal blocks is taken as the initialsolution and the maximum number of iteration is set to500

Table 1 Characteristics of the fluid and the sphere

Materials Properties Values

Fluid Acoustic velocity 1524msDensity 1000 kgm3

Sphere

Youngrsquos modulus 207 times 1011 PaPoissonrsquos ratio 03

Density 7669 kgm3

Thickness 015mRadius 5m

Table 2 Characteristics of the fluid and the structure



Structure


Density 4500 kgm3

Thickness 20mmLength 236mWidth 21mHeight 36m

6 Numerical Examples

In this section the proposed coupling approach is appliedto two examples for the simulation of acoustic-structurecoupling problems

61 Benchmark Example The spherical test structure is anelastic sphere shell with a radius of 5m and thickness of015m full of some kind of liquid A point sound sourceis located in the center of the sphere emitting harmonicspherical wave The outer surface of the sphere is free andthe inner surface of the sphere is coupled with the acousticfield The detailed geometrical and material parameters forthe sphere and the fluid are shown in Table 1 In this examplethe damping is ignored

This problem is simulated by using FEM and BEMrespectivelyTheBEMresults are obtained using the proposedalgorithm The FEM results are obtained using NASTRANsoftware The sound pressure amplitudes obtained by thetwo methods are plotted in Figure 3 From the two curveswe can see that the results using BEM agree well with theresults obtained from FEMHowever from Figure 4 in whichcomputational efficiencies of different solving methods arecompared it can be seen that the ACA approach requires theleast computing time

62 Analysis of the Frequency Response of a Rectangular BoxFurthermore the frequency response analysis of a square boxis also presented The rectangular box is a closure chambermade of thin plates Its inner and outer surfaces of thebox are coupled with the interior and the exterior acousticfields respectively In this example the exterior acoustic fieldis assumed to be infinite A point acoustic source located


0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)

Figure 3 Sound pressure curves obtained by BEM and FEM

220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)

Gauss eliminationPreconditioned GMRES iteratorAdaptive cross approximation

Figure 4 Comparison of computing time

outside the box initiates the coupled response of the structureand its surrounding acoustic fields The mesh of the box isshown in Figure 5

Parameters of the box and the fluid are listed in Table 2To model this problem the origin of the Cartesian

coordinate system is set to coincide with the center of thebottom surface of the box while the coordinate axes areparallel with the edges of the box The boxrsquos bottom surfaceis restrained The point acoustic source generating harmonicspherical wave (119901

0= 1Pa) is located at the point (40m

00m 40m) Damping is ignored

X Y

Z

Figure 5 Mesh of the square box

Coupled with both the outer and the inner

0 1 2 3 4 5 6 7 8 9 10

Coupled with the outer acoustic

10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)

Figure 6 Sound pressure under different frequencies

Two cases are considered In the first case the interioracoustic field is neglected whilst in the second case theinterior and the exterior acoustic fields are both consideredIn the two cases the sound pressures at a selected point(118m 00m 2057m) are calculated They are compared inFigure 6

From Figure 6 it can be seen that the trends of the twopressure-frequency curves are similar but the pressures area bit different at some specific frequencies for instance thefrequency is around 20 Figure 7 demonstrates the acousticpressure on the outer surface of the box According to


X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)

Figure 7 (a) Sound pressure distribution for the coupled system without considering the inner field (b) Sound pressure distribution for thecoupled system considering the inner field

the graphs it is found that when the frequency varies from20 to 100 the pressure distributions on the outer surface ofthe box are almost the same for the two cases Therefore itcan be concluded that for this example the neglection of theinner acoustic field in the modelling does not lead to ruinousresult

7 Conclusions

In this study an algorithm based on the coupling of FEMand BEM is developed for the analysis of acoustic-structureresponse The FEM is employed to model the structure partwhile the BEM is used to discretize the acoustic domainThe two domains are coupled with each other through theconsideration of the appropriate compatibility and equilib-rium conditions on the interface of the two domains To

improve the computational efficiency the ACA approachis incorporated into the proposed algorithm to deal withthe nonsymmetric and fully populated matrices resultingfrom the coupling of the FEM and BEM By the proposedalgorithm the fictitious frequency problem can also beavoided The validity and the high accuracy of the presentalgorithm are demonstrated by a benchmark example Themodeling for the frequency response of a rectangular boxis also presented The numerical examples show that thecoupling method has great potential to deal with large-scalecomplex acoustic-structure problems

Acknowledgment

Thiswork is jointly supported by theNationalNatural ScienceFoundation of China (nos 11272182 and 11072128)


References

[1] K J Bathe Finite Element Procedures Prentice Hall EnglewoodCliffs NJ USA 1996

[2] O C Zienkiewicz and R L TaylorThe Finite Element MethodButterworth-Heinensann Oxford UK 5th edition 2000

[3] C A Brebbia and J Dominguez Boundary Elements AnIntroductory Course McGraw-Hill London UK 1989

[4] O C Zienkiewicz and R L TaylorThe Finite Element Methodvol 1-2 McGraw-Hill London UK 4th edition 1991

[5] O C Zienkiewicz D W Kelly and P Bettess ldquoThe coupling ofthe finite element method and boundary solution proceduresrdquoInternational Journal for Numerical Methods in Engineering vol11 no 2 pp 355ndash375 1977

[6] C C Spyrakos and D E Beskos ldquoDynamic response offlexible strip-foundations by boundary and finite elementsrdquo SoilDynamics and Earthquake Engineering vol 5 no 2 pp 84ndash961986

[7] O von Estorff ldquoCoupling of bem and fem in the time domainsome remarks on its applicability and efficiencyrdquoComputers andStructures vol 44 no 1-2 pp 325ndash337 1992

[8] H A Schenck ldquoImproved integral formulation for acousticradiation problemsrdquo Journal of theAcoustical Society of Americavol 44 pp 41ndash58 1968

[9] A J Burton and G F Miller ldquoThe application of integralequation methods to the numerical solution of some exteriorboundary-value problemsrdquo Proceedings of the Royal Society Avol 323 pp 201ndash210 1971

[10] I O Panich ldquoOn the question of solvability of the externalboundary value problem for the wave equation and Maxwellrsquosequationrdquo Russian Mathematical Surveys vol 20 no 1 pp 221ndash226 1965

[11] M Guiggiani G Krishnasamy T J Rudolphi and F J Rizzo ldquoAgeneral algorithm for the numerical solution of hypersingularboundary integral equationsrdquo Journal of AppliedMechanics vol59 no 3 pp 604ndash614 1992

[12] J J Rgo Silva H Power and L C Wrobel ldquoA numericalimplementation of a hypersingular boundary element methodapplied to 3D time-harmonic acoustic radiation problemsrdquo inBoundary Elements XIV Computational Mechanics Publica-tions pp 271ndash287 Southampton and Elsevier London UK1992

[13] Y Saad Iterative Methods for Sparse Linear Systems SIAMPhiladelphia Pa USA 2nd edition 2003

[14] V Rokhlin ldquoRapid solution of integral equations of classicalpotential theoryrdquo Journal of Computational Physics vol 60 no2 pp 187ndash207 1985

[15] L Greengard and V Rokhlin ldquoA fast algorithm for particlesimulationsrdquo Journal of Computational Physics vol 73 no 2 pp325ndash348 1987

[16] A P Peirce and J A L Napier ldquoA spectral multipole methodfor efficient solution of large-scale boundary element modelsin elastostaticsrdquo International Journal for Numerical Methods inEngineering vol 38 no 23 pp 4009ndash4034 1995

[17] M Bebendorf ldquoApproximation of boundary element matricesrdquoNumerische Mathematik vol 86 no 4 pp 565ndash589 2000

[18] M Bebendorf and S Rjasanow ldquoAdaptive low-rank approxima-tion of collocation matricesrdquo Computing vol 70 no 1 pp 1ndash242003

[19] A Sommerfeld Partial Differential Equations in Physics Aca-demic Press New York NY USA 1949

[20] A F Seybert B Soenarko F J Rizzo and D J Shippy ldquoAnadvanced computational method for radiation and scattering ofacoustic waves in three dimensionsrdquo Journal of the AcousticalSociety of America vol 77 no 2 pp 362ndash368 1985

[21] R Kress ldquoMinimizing the condition number of boundaryintegral operators in acoustic and electromagnetic scatteringrdquoThe Quarterly Journal of Mechanics and Applied Mathematicsvol 38 no 2 pp 323ndash341 1985

[22] M J Allen and N Vlahopoulos ldquoIntegration of finite elementand boundary element methods for calculating the radiatedsound from a randomly excited structurerdquo Computers andStructures vol 77 no 2 pp 155ndash169 2000

[23] O von Estorff S Rjasanow M Stolper and O ZaleskildquoTwo efficient methods for a multifrequency solution of theHelmholtz equationrdquo Computing and Visualization in Sciencevol 8 no 3-4 pp 159ndash167 2005

[24] I Benedetti M H Aliabadi and G Davı ldquoA fast 3D dualboundary element method based on hierarchical matricesrdquoInternational Journal of Solids and Structures vol 45 no 7-8pp 2355ndash2376 2008

[25] Y Saad and M H Schultz ldquoGMRES a generalized minimalresidual algorithm for solving nonsymmetric linear systemsrdquoSIAM Journal on Scientific and Statistical Computing vol 7 no3 pp 856ndash869 1986

[26] C Y Leung and S P Walker ldquoIterative solution of largethree-dimensional BEM elastostatic analyses using the GMREStechniquerdquo International Journal for Numerical Methods inEngineering vol 40 no 12 pp 2227ndash2236 1997

[27] M Merkel V Bulgakov R Bialecki and G Kuhn ldquoIterativesolution of large-scale 3D-BEM industrial problemsrdquo Engineer-ing Analysis with Boundary Elements vol 22 no 3 pp 183ndash1971998

[28] S Amini and N D Maines ldquoPreconditioned Krylov subspacemethods for boundary element solution of the Helmholtzequationrdquo International Journal for Numerical Methods in Engi-neering vol 41 no 5 pp 875ndash898 1998

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


based on the Helmholtz integral equation on the surfaceof the radiating body combined with Helmholtz equationsfor some interior points The resulting system equation isusually solved by the least-square approach The Burton-Miller approach [9] inspired by Panich formulation [10]forms a linear combination of the Helmholtz boundaryintegral equation and its normal derivative providing also avalid solution at any frequency but leading to hypersingularintegrals which can be handled by Guiggiani algorithm [1112] In this study the Burton-Miller approach is adopted

Similar to BEM the coupled FEM-BEM approach leadsto a fully populated system matrix This is thought to bea major drawback of the coupled approach as comparedwith FEM which normally results in a sparse symmetricalsystem matrix If iterative methods like the generalizedminimal residual (GMRES) method [13] in combinationwith suitable preconditioners are used to solve the FEM-BEM system with 119873 degrees of freedom the computationalexpense is in the order of 119874(1198732) If a direct solver isapplied the computational cost is even in the order of119874(1198733)Fortunately over the last few years several fast solutionmethods including fastmultipolemethod (FMM) [14ndash16] andadaptive cross approximation approach [17 18] have beendeveloped These methods reduce the memory requirementsand the computational time significantly when dealing withthe matrix-vector product In this study the ACA approachis adopted since it is much easier to operate compared withFMM which is based on a series expansion of kernel-shapefunction products

In this paper a coupling algorithm combining FEMand BEM is developed for the analysis of acoustic-structureresponse This paper is organized as follows Section 1 is ashort introduction to the background and correspondingresearch status In Section 2 the FE formulations of the struc-ture domain are constructed In Section 3 the application ofthe BEM to Helmholtz problem and its implementation ispresentedThen the coupled equations are built in Section 4In Section 5 the solving algorithms of coupled equationsincluding the GMRES iterative solver and ACA approachare introduced In Section 6 several numerical examples arepresented

2 FE Formulation of Structural Domain

A coupling system of acoustic-structure is shown in Figure 1The structure is assumed to be fully submerged in interiorand exterior acoustic fields or one of them In Figure 1the structure domain is denoted by Ω119878 The interior andthe exterior acoustic fields are denoted by Ω

119868 and Ω119864

respectively The inner interface and the outer interface arenamed as 119878119868 and 119878119864 respectively The structure is assumed tobe thin and flexible which is susceptible to acoustic-structureinteraction

FEM is predominantly chosen for simulating linear elas-todynamic systems When the structure Ω119878 is discretized byFEM the resulting system of linear equations reads as

Mu + Cu + Ku = f119904+ T119904 (1)

ΩE

ΩI

ΩS

Structure

Interior acoustic field

Exterior

field

SE

SI

acoustic

Figure 1 Illustration of a coupling system of acoustic-structure

where M C and K denote the global mass matrix thedamping matrix and the stiffness matrix respectively Thevector f

119904is load vector while the vector T

119904resulted from

the tractions on the coupling interface and it will be furtherdiscussed in the following section The global displacementvector is denoted by u

In this paper Rayleigh damping is considered for thedamping matrix which means that

C = 120585M + 120578K (2)

where 120585 and 120578 are the damping parametersIf the load vector f

119904and the acoustic pressure vector T

119904

are both assumed to be harmonic the displacement vector ushould be also harmonic This means that

f119904= f119904119890119894120596119905 T

119904= T119904119890119894120596119905 u = u119890119894120596119905 (3)

where the imaginary unit is denoted by 119894 and 120596 = 2120587119891 is thecircular frequency with the excitation frequency 119891 For therelationship between u and u one obtains

u (119905) = 119894120596u119890119894120596119905 = u (120596) 119890119894120596119905

u (119905) = minus1205962u119890119894120596119905 = 119894120596u (120596) 119890119894120596119905(4)


[119894120596M + C minus 119894K120596] u (120596) = f

119904+ T119904 (5)

NASTRAN a finite element package is employed to setup M C K and the right-hand side vector f

119904 Since M C

and K are frequency independent these matrices only haveto be calculated once for a given model

3 The Boundary Element Method forthe Helmholtz Equation

Acoustic problems in frequency domain can be described bythe Helmholtz equation which has a three-dimensional formgiven by

nabla2119901 (x) + 1198962119901 (x) = 0 (6)




119901 (x) = 119901 (x) x isin 119878119863

120597119901 (x)120597119899

= 119902 (x) x isin 119878119873(7)


10038161003816100381610038161003816100381610038161003816

120597119901

120597119903minus 119894119896119901

10038161003816100381610038161003816100381610038161003816

le119888

1199032 119903 997888rarr infin (8)



119866lowast(x y) = 119890

119894119896119903

4120587119903 (9)




= int119878119864cup119878119868

(119901120597119866lowast

120597119899minus 119866lowast 120597119901

120597119899) d119878

(10)


119901 (x) = minusint119878119864cup119878119868

119866lowast(x y)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878

x isin Ω119864 cup Ω119868

(11)


119901 (x) = minusint119878119864cup119878119868

119866lowast(xy)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878 + 119901119868 (x)

x isin Ω119864 cup Ω119868

(12)


119862 (x) 119901 (x) = int119878

(119866lowast 120597119901

120597119899minus 119901

120597119866lowast

120597119899) d119878 (13)



1198621015840(x) 119902 (x) = minusint

119878119864cup119878119868

120597119866lowast(x y)120597119899119909

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

1205972119866lowast(x y)

120597119899119909120597119899119910

119901 (y) d119878 x isin 119878119864 cup 119878119868

(14)


int119878119864cup119878119868

120597119866lowast

120597119899119910

119901 (y) d119878 + 119862 (x) 119901 (x)

+ 120572int119878119864cup119878119868

1205972119866lowast

120597119899 (x) 120597119899 (y)119901 (y) d119878

= int119878119864cup119878119868

119866lowast 120597119901

120597119899d119878

+ 120572 [int119878119864cup119878119868

120597119866lowast

120597119899119909

120597119901

120597119899d119878 minus 1198621015840 (x)

120597119901

120597119899119909

(x)]

(15)



[H] [p] = [G] [v] (16)



[H] [p] = [G] [v] + [p119868] (17)






119879119899= T sdot n = minus119901 (18)


T119890119904= int119878120590

N119879T dΓ = int119878120590

N119879119879119899n dΓ

= minusint119878120590

N119879n[N119887]119879


(19)

where As119890= int119878120590



119894120596 [M] + [C] minus 119894 [K]120596 [u (120596)] = [f

119904] minus [A119904] [p] (20)


[u (120596)] = 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (21)


[v] = [R] [u (120596)] (22)



[v] = [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (23)


[H] + [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[A119904] [p]

= [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] + [p

119868]

(24)










119899and119860

119891form




sum

119894=1

u119898times1119894

v1times119899119894 (25)


ND

Neighbour nodes

Interaction nodes



1003817100381710038171003817R119898times1198991003817100381710038171003817 =


1003817100381710038171003817A119898times1198991003817100381710038171003817 (26)







Sphere


Density 7669 kgm3





Structure


Density 4500 kgm3








0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)


220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)








X Y

Z



0 1 2 3 4 5 6 7 8 9 10


10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)





X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119901 (x) = 119901 (x) x isin 119878119863

120597119901 (x)120597119899

= 119902 (x) x isin 119878119873(7)


10038161003816100381610038161003816100381610038161003816

120597119901

120597119903minus 119894119896119901

10038161003816100381610038161003816100381610038161003816

le119888

1199032 119903 997888rarr infin (8)



119866lowast(x y) = 119890

119894119896119903

4120587119903 (9)




= int119878119864cup119878119868

(119901120597119866lowast

120597119899minus 119866lowast 120597119901

120597119899) d119878

(10)


119901 (x) = minusint119878119864cup119878119868

119866lowast(x y)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878

x isin Ω119864 cup Ω119868

(11)


119901 (x) = minusint119878119864cup119878119868

119866lowast(xy)

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

120597119866lowast(x y)120597119899119910

119901 (y) d119878 + 119901119868 (x)

x isin Ω119864 cup Ω119868

(12)


119862 (x) 119901 (x) = int119878

(119866lowast 120597119901

120597119899minus 119901

120597119866lowast

120597119899) d119878 (13)



1198621015840(x) 119902 (x) = minusint

119878119864cup119878119868

120597119866lowast(x y)120597119899119909

120597119901

120597119899(y) d119878

+ int119878119864cup119878119868

1205972119866lowast(x y)

120597119899119909120597119899119910

119901 (y) d119878 x isin 119878119864 cup 119878119868

(14)


int119878119864cup119878119868

120597119866lowast

120597119899119910

119901 (y) d119878 + 119862 (x) 119901 (x)

+ 120572int119878119864cup119878119868

1205972119866lowast

120597119899 (x) 120597119899 (y)119901 (y) d119878

= int119878119864cup119878119868

119866lowast 120597119901

120597119899d119878

+ 120572 [int119878119864cup119878119868

120597119866lowast

120597119899119909

120597119901

120597119899d119878 minus 1198621015840 (x)

120597119901

120597119899119909

(x)]

(15)



[H] [p] = [G] [v] (16)



[H] [p] = [G] [v] + [p119868] (17)






119879119899= T sdot n = minus119901 (18)


T119890119904= int119878120590

N119879T dΓ = int119878120590

N119879119879119899n dΓ

= minusint119878120590

N119879n[N119887]119879


(19)

where As119890= int119878120590



119894120596 [M] + [C] minus 119894 [K]120596 [u (120596)] = [f

119904] minus [A119904] [p] (20)


[u (120596)] = 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (21)


[v] = [R] [u (120596)] (22)



[v] = [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (23)


[H] + [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[A119904] [p]

= [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] + [p

119868]

(24)










119899and119860

119891form




sum

119894=1

u119898times1119894

v1times119899119894 (25)


ND

Neighbour nodes

Interaction nodes



1003817100381710038171003817R119898times1198991003817100381710038171003817 =


1003817100381710038171003817A119898times1198991003817100381710038171003817 (26)







Sphere


Density 7669 kgm3





Structure


Density 4500 kgm3








0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)


220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)








X Y

Z



0 1 2 3 4 5 6 7 8 9 10


10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)





X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119879119899= T sdot n = minus119901 (18)


T119890119904= int119878120590

N119879T dΓ = int119878120590

N119879119879119899n dΓ

= minusint119878120590

N119879n[N119887]119879


(19)

where As119890= int119878120590



119894120596 [M] + [C] minus 119894 [K]120596 [u (120596)] = [f

119904] minus [A119904] [p] (20)


[u (120596)] = 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (21)


[v] = [R] [u (120596)] (22)



[v] = [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] minus [A119904] [p] (23)


[H] + [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[A119904] [p]

= [G] [R] 119894120596 [M] + [C] minus 119894 [K]120596

minus1

[f119904] + [p

119868]

(24)










119899and119860

119891form




sum

119894=1

u119898times1119894

v1times119899119894 (25)


ND

Neighbour nodes

Interaction nodes



1003817100381710038171003817R119898times1198991003817100381710038171003817 =


1003817100381710038171003817A119898times1198991003817100381710038171003817 (26)







Sphere


Density 7669 kgm3





Structure


Density 4500 kgm3








0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)


220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)








X Y

Z



0 1 2 3 4 5 6 7 8 9 10


10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)





X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










ND

Neighbour nodes

Interaction nodes



1003817100381710038171003817R119898times1198991003817100381710038171003817 =


1003817100381710038171003817A119898times1198991003817100381710038171003817 (26)







Sphere


Density 7669 kgm3





Structure


Density 4500 kgm3








0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)


220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)








X Y

Z



0 1 2 3 4 5 6 7 8 9 10


10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)





X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20 30 40 50 60 70 80 90 100Frequency (Hz)

BEMFEM

10minus2

10minus1

100

101

Soun

d pr

essu

re (P

a)


220

200

180

160

140

120

100

80

60

40

20

0 20 40 60 80 100Frequency (Hz)

Tim

e (s)








X Y

Z



0 1 2 3 4 5 6 7 8 9 10


10minus3

10minus2

10minus1

100

Soun

d pr

essu

re (P

a)

Frequency (Hz)





X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X Y

Z10 Hz

0220201801601401201008006004002

X Y

Z20 Hz

065060550504504035030250201501005

X Y

Z30 Hz

03027024021018015012009006003

X Y

Z40 Hz

01301201101009008007006005004003002001

X Y

Z50 Hz

01101009008007006005004003002001

X Y

Z60 Hz

0201801601401201008006004002

X Y

Z70 Hz

181614121

060402

08

X Y

Z80 Hz

0807060504030201

X Y

Z90 Hz

0280260240220201801601401201008006004002

X Y

Z100 Hz

04035030250201501005

(a)

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

X Y

Z

10 Hz 20 Hz 30 Hz 40 Hz 50 Hz

60 Hz 70 Hz 80 Hz 90 Hz 100 Hz

0260240220201801601401201008006004002

181614121

060402

08

0201801601401201008006004

01009008007006005004003002

015013011009007005003001

0201801601401201008006004002

181614121

060402

08

01201101009008007006005004003002001

01601401201008006004002

0280260240220201801601401201008006004002

(b)



7 Conclusions



Acknowledgment



References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Low-Frequency Acoustic-Structure Analysis