boundary element simulations for local active noise control using an extended volume

13
Boundary element simulations for local active noise control using an extended volume A. Brancati , M.H. Aliabadi 1 Department of Aeronautics, Imperial College London, South Kensington SW7 2AZ, UK article info Article history: Received 16 August 2010 Accepted 16 June 2011 Available online 6 October 2011 Keywords: Boundary Element Method Local active noise control Control volume Adaptive Cross Approximation Hierarchical matrix format abstract This paper presents a novel local active noise control (ANC) approach formulated using a 3D fast Boundary Element Method (BEM). The proposed method can be easily implemented in a conventional ANC system. The unwanted noise is reduced in a predefined volume, called control volume (CV), by minimising the square modulus of two acoustic quantities, the pressure and one component of the particle velocity. Formulations are presented for one and two control sources. Simulations for both the formulations with various CV sizes, different locations of secondary sources-CV, and a large-scale engineering problem are presented. Practical aspects of the proposed procedure are also described. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The origins of active noise control (ANC) can be traced back to the pioneering work of Paul Lueg [1] and Conover [2]. In the free field, Nelson et al. [3] claimed that a significant global noise reduction (at least 10 dB power attenuation) can be achieved only if the separation distance between the primary monopole source and the control source is less than one-tenth of the wavelength of the disturbance. In the case where the control source is placed at half wavelength from the primary source, no reduction can be accomplished. In an enclosed space Nelson et al. [4] investigated and developed a computer simulation of ANC and verified their models experimentally for harmonic enclosed sound fields. They established that a disturbance can be globally reduced for resonance frequencies and the control source does not require to be separated by less than one half of the wavelength from the primary noise source as for the free field case, even for consider- able number of sources. Ross [5], Hesselm [6] and Berge et al. [7] applied the ANC theory to reduction of noise emanated from transformers. They reported that a 20 dB reduction can be easily achieved even for unsophisticated audio equipment for discrete frequencies of less than 100 Hz, but at higher frequencies the noise attenuation level is not acceptable. It was also established that the level of noise attenuation depends upon the direction of observation, since transformers generate noise from extended surfaces that would require several control sources to obtain a global noise reduction. Early works on the development of local active noise control approach are due to the theoretical and experimental study of Joseph et al. [8] and the numerical work of David and Elliott [9]. It was reported [9] that a 10 dB reduction zone can be obtained for frequencies above the Schroeder frequency and for uniform and diffuse primary noise, and the reduction can be larger, up to one-tenth of the wavelength, if the cancellation point is further from the secondary source. Moreover, the sound pressure level (SPL) away from the cancellation point is almost unaffected. The local ANC approach has been further developed by Garcia-Bonito and Elliott [10]. In their work the primary source is a diffuse enclosed sound field, the secondary source is modelled as a rigid sphere with a vibrating segment and the listener’s head is assumed to be a rigid sphere. Rafaely et al. [11] presented laboratory results for a headrest system. They asserted that a useful performance can be achieved only in the case where the system cancel the pressure at a ‘‘virtual microphone’’ close to the user’s ears that project the quiet area away from the physical microphone. Moreover, they proved that the performance is maintained significant also including the natural movement of the user’s head. In the subsequent work Garcia-Bonito and Elliott [12] demonstrated that the reduction zone can be enlarged by cancelling the pressure and the secondary particle velocity at two different points. In this paper noise in a 3D free field is attenuated by a local ANC approach and simulated using BEM for monotone frequencies. The solution is accelerated by the Adaptive Cross Approximation (ACA) in conjunction with the Hierarchical matrix (H-matrix) format and the GMRES. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements 0955-7997/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enganabound.2011.06.008 Principal corresponding author. Tel.: þ44 7847 113392; fax: þ44 2 07594 1974. E-mail addresses: [email protected] (A. Brancati), [email protected] (M.H. Aliabadi). 1 Tel.: þ44 2 07594 5056; fax: þ44 2 07594 1974. Engineering Analysis with Boundary Elements 36 (2012) 190–202

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Page 1: Boundary element simulations for local active noise control using an extended volume

Engineering Analysis with Boundary Elements 36 (2012) 190–202

Contents lists available at ScienceDirect

Engineering Analysis with Boundary Elements

0955-79

doi:10.1

� Prin

E-m

m.h.alia1 Te

journal homepage: www.elsevier.com/locate/enganabound

Boundary element simulations for local active noise control using anextended volume

A. Brancati �, M.H. Aliabadi 1

Department of Aeronautics, Imperial College London, South Kensington SW7 2AZ, UK

a r t i c l e i n f o

Article history:

Received 16 August 2010

Accepted 16 June 2011Available online 6 October 2011

Keywords:

Boundary Element Method

Local active noise control

Control volume

Adaptive Cross Approximation

Hierarchical matrix format

97/$ - see front matter & 2011 Elsevier Ltd. A

016/j.enganabound.2011.06.008

cipal corresponding author. Tel.: þ44 7847 113

ail addresses: [email protected] (A.

[email protected] (M.H. Aliabadi).

l.: þ44 2 07594 5056; fax: þ44 2 07594 197

a b s t r a c t

This paper presents a novel local active noise control (ANC) approach formulated using a 3D fast

Boundary Element Method (BEM). The proposed method can be easily implemented in a conventional

ANC system. The unwanted noise is reduced in a predefined volume, called control volume (CV), by

minimising the square modulus of two acoustic quantities, the pressure and one component of the

particle velocity. Formulations are presented for one and two control sources. Simulations for both the

formulations with various CV sizes, different locations of secondary sources-CV, and a large-scale

engineering problem are presented. Practical aspects of the proposed procedure are also described.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The origins of active noise control (ANC) can be traced back tothe pioneering work of Paul Lueg [1] and Conover [2].

In the free field, Nelson et al. [3] claimed that a significantglobal noise reduction (at least 10 dB power attenuation) can beachieved only if the separation distance between the primarymonopole source and the control source is less than one-tenth ofthe wavelength of the disturbance. In the case where the controlsource is placed at half wavelength from the primary source, noreduction can be accomplished. In an enclosed space Nelson et al.[4] investigated and developed a computer simulation of ANC andverified their models experimentally for harmonic enclosed soundfields. They established that a disturbance can be globally reducedfor resonance frequencies and the control source does not requireto be separated by less than one half of the wavelength from theprimary noise source as for the free field case, even for consider-able number of sources. Ross [5], Hesselm [6] and Berge et al. [7]applied the ANC theory to reduction of noise emanated fromtransformers. They reported that a 20 dB reduction can be easilyachieved even for unsophisticated audio equipment for discretefrequencies of less than 100 Hz, but at higher frequencies thenoise attenuation level is not acceptable. It was also establishedthat the level of noise attenuation depends upon the direction ofobservation, since transformers generate noise from extended

ll rights reserved.

392; fax: þ44 2 07594 1974.

Brancati),

4.

surfaces that would require several control sources to obtain aglobal noise reduction.

Early works on the development of local active noise controlapproach are due to the theoretical and experimental study ofJoseph et al. [8] and the numerical work of David and Elliott [9]. Itwas reported [9] that a 10 dB reduction zone can be obtained forfrequencies above the Schroeder frequency and for uniform anddiffuse primary noise, and the reduction can be larger, up toone-tenth of the wavelength, if the cancellation point is furtherfrom the secondary source. Moreover, the sound pressure level(SPL) away from the cancellation point is almost unaffected. Thelocal ANC approach has been further developed by Garcia-Bonitoand Elliott [10]. In their work the primary source is a diffuseenclosed sound field, the secondary source is modelled as a rigidsphere with a vibrating segment and the listener’s head isassumed to be a rigid sphere. Rafaely et al. [11] presentedlaboratory results for a headrest system. They asserted that auseful performance can be achieved only in the case where thesystem cancel the pressure at a ‘‘virtual microphone’’ close to theuser’s ears that project the quiet area away from the physicalmicrophone. Moreover, they proved that the performance ismaintained significant also including the natural movement ofthe user’s head. In the subsequent work Garcia-Bonito and Elliott[12] demonstrated that the reduction zone can be enlarged bycancelling the pressure and the secondary particle velocity at twodifferent points.

In this paper noise in a 3D free field is attenuated by a local ANCapproach and simulated using BEM for monotone frequencies. Thesolution is accelerated by the Adaptive Cross Approximation (ACA) inconjunction with the Hierarchical matrix (H-matrix) format andthe GMRES.

Page 2: Boundary element simulations for local active noise control using an extended volume

Table 1Pressure for a pulsating sphere under prescribed uniform flux for five wave

numbers (k¼1, 2, 3, 4, 5).

Wave num.

k

Analytical solution Standard BEM ACA H-matrix

GMRES

Real

part

Imag.

part

Real

part

Imag.

part

Real

part

Imag.

part

1 0.5000 0.5000 0.4994 0.5000 0.4996 0.5003

2 0.8000 0.4000 0.7994 0.4001 0.7997 0.4006

3 0.9000 0.3000 0.8969 0.2958 0.8965 0.3001

4 0.9412 0.2353 0.9412 0.2368 0.9419 0.2361

5 0.9615 0.1923 0.9606 0.1932 0.9615 0.1930

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 191

The BEM is a general and effective numerical technique for 3DHelmholtz acoustic simulations [13,14]. In the last two decadesthe ANC has been studied using the BEM by many authors, forexample: Cunefore and Koopmann [15], Guang-Hann [16], Yangand Tseng [17], and Bai and Chang [18]. These studies are focusedon attenuating the offending noise in a global sense. In the study[15] the secondary sources are not monopoles or point sources asin previous works, but extended vibrating surfaces in free space.The BEM is used to find the total radiated power from a pulsatingsphere and from vibrating surfaces within a box and minimisedby using the secondary source vibration surface velocity. TheGuang-Hann [16] work is focused on the reduction of noisegenerated in the airport. The method of the images is used tocreate the ground and a new fundamental solution is calculatedby including the impedance of such a surface. The secondarysources have finite dimensions, fixed locations and sizes. Theeffort of Yang and Tseng [17] is mainly focused on the optimalposition of loudspeakers in 2D and 3D cases. The indirect BEM isused to simulate the sound propagation while the sequentialquadratic programming (SQP) was selected as optimiser. Bai andChang [18] performed the ANC of a noise radiated in enclosureswith known normal specific acoustic impedance. The total timeaverage acoustic potential energy is selected as the cost functionto be minimised and used to optimise the positions and theamplitudes of the secondary sources.

This paper presents a novel and general local ANC approachthat can be directly applied to a conventional digital signalprocessing (DSP) system without significant changes. The mainidea is to minimise the square modulus of the pressure and thesquare modulus of the total particle velocity in one direction in apredefined volume rather than at discrete points. This techniqueaims to extend the noise reduction volume into a larger zone thanfor a standard point cancellation procedure. A BEM formulationfor 3D Helmholtz equation is utilised to solve the problem andrun simulations. A first approach to the problem, using a singlecontrol source, has been compared with the conventional strategyin the literature for a diffuse primary noise in an infinite domain.Furthermore, this analysis is focused in the vicinity of thesecondary source. The results demonstrate the efficiency of thenew technique using three control volumes with different sizes.The formulation is extended for using two control sources andresults are presented. Certain aspects on practical applicationof the proposed strategy are described. Finally, a large-scaleengineering problem of noise reduction inside an aircraft cabinis presented.

2. Boundary element method

By considering a boundary (G) of a domain (O), the problem issolved in terms of the pressure pðxÞ and the particle velocity qðxÞ(with respect to the normal of the boundary surface at theconsidered node). The boundary integral equation for Helmholtzproblem can be written as [13]

Cðx0Þpjðx0Þþ

ZG

qnðx0,xÞpðxÞ dGðxÞ ¼ZG

pnðx0,xÞqðxÞ dGðxÞ

þ

ZO

pnðx0,XsÞ

1

c2bðXsÞ dOðXs

Þ

ð2:1Þ

where pnðx0,xÞ and qnðx0,xÞ are the pressure and particle velocityfundamental solutions, respectively, and Cðx0Þ depends on thelocation of the point x0 (see [13]). The last term refers to thepresence of sources within the domain O with strength bðXs

Þ=c2

and c is the sound velocity.

The integral equation in (2.1) is discretised into N constantelements and the resulting system of equations can be repre-sented in matrix form as

Hp¼GqþsðXsÞ ð2:2Þ

where H and G are coefficient matrices corresponding to integrals ofthe product of the Jacobian of transformation with boundary particlevelocity and pressure fundamental solutions, respectively, p and q arethe boundary pressure and particle velocity vectors, respectively.Finally, the last integral in Eq. (2.1) produces the vector s, created byNP sources within the domain O, such as monopoles and planewaves.

Including the boundary conditions yields a linear system ofequations of the form

AY¼ Fþs ð2:3Þ

where Y is the vector containing the unknown boundarypressures and particle velocities, A is a coefficient matrix and Fis obtained by multiplying the prescribed BCs with the corre-sponding columns of the G and H matrices.

The main drawbacks of the BEM consist of the fact that thematrix A is densely populated and non-symmetric, and thestorage requirement is of OðN2Þ, which slow down the solutiontime and waste the computational advantage of discretising onlythe geometry boundaries. In the recent past, various techniqueshave been explored to overcome these difficulties which includeblock-based solvers [19], lumping techniques [20], iterative sol-vers [21,22] and fast multipole method [23,24].

In this contribution a purely algebraic technique has beenadopted, i.e., the Adaptive Cross Approximation [25]. It has beendemonstrated [26] that this technique in conjunction with theH-matrix format and the GMRES (RABEM code) decreases the CPUtime significantly for 3D Helmholtz simulations. In order to assess theaccuracy and efficiency of the proposed approach, a simple bench-mark problem, whose analytical solution is well-known [27], hasbeen utilised, i.e., a uniform pulsating sphere. To determine thepressure on the surface of the sphere, the radius, distance, normaluniform radial vibrating velocity and acoustic impedance are allconsidered equal to unity. Moreover, a standard BEM code has alsobeen used to compare the results. The mesh utilised is composed of1622 nodes and 3240 elements. Table 1 compares the pressureobtained at five wave numbers, i.e., k¼1, 2, 3, 4, 5. As evident theBEM solutions are in close agreement with all the analytical values.The proposed approach results to be between 7.5 and 10.3 timesfaster than the standard method. It should be noted that the finer themesh, the higher the speed up ratio.

Further details on this technique are give in the Appendix.

2.1. Pressure and particle velocity at selected internal points

The pressure PðXÞ at number of selected internal points X canbe obtained from the boundary solutions of pressure and particle

Page 3: Boundary element simulations for local active noise control using an extended volume

Table 2Values of the terms in Eq. (3.5).

a b c d f g

RDP2

sR dDR

DP2sI dD

RDPpRPsR dD

RDPpIPsI dD

RDPpRPsI dD

RDPpIPsR dD

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202192

velocity as

PðXÞ ¼�HpþGqþSðXsÞ ð2:4Þ

where H and G are matrices similar to H and G, but evaluated atinternal points, SðXs

Þ refers to the presence of extra sources at theinternal points and the vector P0ðXÞ collecting the internal particlevelocity along an axis is calculated by

P0ðXÞ ¼�H0pþG

0qþS0ðXs

Þ ð2:5Þ

where superscript ‘‘0’’ denotes derivative with respect to one ofthe Cartesian coordinates (x1, x2, x3); S0ðXs

Þ is a vector containingthe contribution to the particle velocity generated by the sourceswithin the domain O, and H

0and G

0contain the integrals of the

derivatives of the fundamental solutions [13,14]. In this contribu-tion the ACA is also utilised to evaluate the matrices H, G, H

0and

G0

and the H-matrix format to store them.Some basic subroutines utilised in our BEM code have been

taken from [14,26] where the code has been benchmarked.

3. Noise attenuation using a single control source

This section presents a novel approach for attenuating anunwanted noise in a prescribed volume. The approach consistsof reducing the noise in an area of interest, here called control

volume (CV) and denoted as D.To reduce the noise level within D, the integral of the square

modulus of the total pressure PðXDÞ at the points in the volume D

is used as the cost function to be minimised, that is

fcðXDÞ ¼

ZDjPðXDÞj

2 dD ð3:1Þ

with XDAD and where PðXDÞ is given by Eq. (2.4).As evident, for a given primary noise distribution, the above

cost function is minimised by the secondary source field thatis generated by a 3D object with hard boundary conditions(i.e., q¼0) everywhere except for a vibrating portion of thesurface (i.e., qa0) (see Fig. 1).

The optimum vibration velocity of the secondary sourcesurface qsðxvÞ can be evaluated using a constant a that relates theoptimum secondary field solution to a solution obtained by anyvelocity of secondary source vibrating surface qsðxvÞ as follows:

qsðxvÞ ¼ aqsðxvÞ ð3:2Þ

where xv refers to the points of the secondary vibrating surface.

Fig. 1. The problem configuration including vibrating surface, secondary source

and control volume.

The pressure P and particle velocity P0 at any point of thedomain (including the boundary points) is generated by both theprimary and the secondary fields. Due to the linearity of the waveequation, each of these fields can be evaluated separately, andsummed together as follows:

P¼ PpþaPs ¼ ðPpRþ iPpIÞþðaRþ iaIÞðPsRþ iPsIÞ

P0 ¼ P0pþaP0s ¼ ðP0pRþ iP0pIÞþðaRþ iaIÞðP

0sRþ iP0sIÞ ð3:3Þ

where the subscripts p and s refer to the primary and secondaryquantities, respectively, and the subscripts R and I refer to the realand imaginary parts, respectively.

Substituting relations (3.3) into the cost function (3.1) yieldsthe following expression:

fcðXDÞ ¼

ZD½ðPpRþaRPsR�aIPsIÞ

2þðPpIþaIPsRþaRPsIÞ

2� dD ð3:4Þ

The complex constant a is determined by minimising the costfunction (i.e., setting the cost function derivative with the respectto aR and aI equal to zero). Hence, a system with two equations isobtained where the solution a can be written as follows:

a¼ �cþd

aþb,

f�g

aþb

!ð3:5Þ

where the constants values in the above expression are shown inTable 2.

In order to obtain the value of a, the system of equations (2.3)is solved first for the primary field and again for the secondarysource field with any prescribed values of the boundary condi-tions at the vibrating surface. Therefore, each term in Eq. (3.5) isevaluated by Eq. (2.4) applied to the primary and secondary fields,separately.

The solution (3.5) represents the optimum noise attenuationthat can be achieved since the second derivative of fc with respectto a is always positive, hence

@2

@a2R

ZDjPðXDÞj

2 dD¼@2

@a2I

ZDjPðXDÞj

2 dD¼ 2

ZDðP2

sRþP2sIÞ dD ð3:6Þ

4. Results for a single control source

In this section ANC is simulated by the BEM for a singlesecondary source. It is shown that our approach can yield a moreextended noise attenuation area compared to the conventionalpoint cancellation strategy [10,11] for local ANC simulations.

Next, the mathematical model of the primary noise and thesecondary source is presented. In the second subsection the plotsof the novel strategy are presented.

All the figures are plotted using a grid of 101�101 points andthree different values of reduction: 6, 10, 20 dB.

4.1. Primary noise and secondary source model

The primary offending noise is considered to be diffused and itis generated by plane waves with homogeneous direction ofpropagation [10]. The primary noise pressure at each point of

Page 4: Boundary element simulations for local active noise control using an extended volume

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 193

coordinate X is given by equation [10,12]

PpðXÞ ¼Xjmax

j ¼ 1

Xlmax

l ¼ 1

ðajl�ibjlÞeikek�X ð4:1Þ

where i is the imaginary unit, k is the wave-number, ek is theplane wave versor and � is the operator of the scalar product. Thevalues of jmax and lmax are 6 and 12, respectively, used to generate72 plane waves distributed in all directions of the space whichcreate a diffuse primary noise. The values of ajl and bjl arerandomly chosen from a uniform distribution.

The secondary source field is created according to [10,12] by asphere of radius 0.08 m with a constant active segment of 1201.

Fig. 2. Mathematical model of the problem: GS vibrating surface, Gz case of the

speaker, D control volume, primary noise plane waves.

Fig. 3. Average noise reduction area generated by a rigid sphere with an active segme

Therefore for a free space simulation the boundary (G) is dividedby the secondary (Gs) vibrating surfaces and the residual 2401segment surface has been considered to be infinitely hard(Neumann BCs with q¼0) as shown in Fig. 2.

The sphere that models the loudspeaker is discretised by 325nodes and 646 triangular super-parametric (linear geometry andconstant unknowns) elements. The vibrating segment has thex1x3-plane and the x1x2-plane as symmetry planes (see Fig. 2).

The noise attenuation is obtained from the average result of 20samples. Such simulations are quite time consuming since calcu-lating a for 20 samples needs the knowledge of the primary andthe secondary field solutions as well as the value of the pressureat the internal points that constitute the CV. However, it shouldbe noted that the solving matrices of Eq. (2.2) and the internalpoint matrices (2.4) are evaluated only once, since they dependupon the geometry of the problem that is never modified.

Due to the diffuse distribution of the primary field of Eq. (4.1),at each point of the field the reduction in decibels (dB) is providedby dividing the average square modules of the controlled field bythe average squared modulus of the diffuse primary field andtaking the log 10 of this ratio multiplied by 10, as follows:

ReductionðXÞ ¼ 10 log10

P20j ¼ 1 jPðXÞj

2jP20

j ¼ 1 jPpðXÞj2j

ð4:2Þ

4.2. Results for different CV dimensions

As evident, the proposed strategy can be fully realised only if boththe primary and secondary fields are perfectly known at every CVpoint. This means to place several reference microphones close at theCV points to predict the noise level. Nevertheless, such a system is

nt of 1201 and by the point cancellation strategy at the point (19.4, 0.0, 0.0) cm.

Page 5: Boundary element simulations for local active noise control using an extended volume

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202194

unfeasible for a practical application and a different approach is thusrequired.

A possibility to overcome this drawback consists of hypothe-sising that the primary field is constant at the control volumepoints. This hypothesis is supported by considering a CV with anopportune size such that the characteristic wavelength of theunwanted noise is larger than its dimensions. On the contrary, thesecondary source can be considered to have its effective response.

In substance, under this hypothesis an adaptive feed-forward ANCsystem requires a single reference microphone that, in case of singlecontrol source, could be located behind the loudspeaker, and a singleerror microphone to be placed close to the silent zone [28,29]. Thesecondary field generated by the loudspeaker at the CV points(i.e., polar diagram) can be evaluated for a standard configurationby a common acquisition system for acoustic measurement andwithout any primary disturbance. Finally, the error microphone canadaptively adjust the loudspeaker response by a single attainment.

The integrals over the CV to evaluate a, Eq. (3.5) and Table 2,are simplified since the primary quantities can be located outsidethe integrals and the Legendre-Gauss quadrature rule using eightnode brick elements with seven sampling points has beenadopted for their evaluations.

Table 3Characteristics of the CVs.

CV

classification

Dimens.

(cm)

Location of diagonal

points

Number

nodes

Number

elements

x1 x2 x3 x1 x2 x3 x1 x2 x3

S, small 5 8 4 17 �4 �2 22 4 2 24 8

M, medium 10 16 8 15 �8 �4 25 8 4 140 72

L, large 15 24 12 13 �12 �6 28 12 6 270 160

Fig. 4. Average noise reduction are

To test the proposed strategy, three simulation resultsobtained from three CVs with different dimensions are comparedwith results from the conventional point cancellation strategy.The primary noise is evaluated at the centre of each CV.

The results obtained by the point cancellation strategy for fourfrequencies (109, 273, 546, 1092 Hz) are shown in Fig. 3. It can benoted that the results shown in this figure are very similar to theresults obtained by Garcia-Bonito and Elliott [12].

The first CV (small) has dimensions of 5�8�4 cm along thex1,x2,x3-axis, respectively. The two diagonal points are located at(17,�4,�2) and (22,4,2) from the centre of the sphere. The meshused for the CV is composed of 24 nodes and 8 elements. Thesecond CV (medium) has a volume eight times the first. The twodiagonal points are located at (15,�8,�4) and (25,8,4) from thecentre of the sphere. The mesh is composed of 140 nodes and 72elements. The last CV (large) has a volume 27 times the first. Thetwo diagonal points are located at (13,�12,�6) and (25,12,6)from the centre of the sphere. The CV is meshed with 270 nodesand 160 elements. Table 3 presents the characteristics of the CVs.

Figs. 4–6 show the average total pressure field in thex1,x2-plane when the noise attenuation is achieved by using thesmall, the medium and the large CVs, respectively. All of them arecompared with the case of a single point cancellation that isshown in Fig. 3. From Fig. 3, it can be seen that a morehomogeneous distribution of the noise attenuation is possibleby using the formulation introduced here.

5. Noise attenuation using two control sources

In the previous sections the noise has been attenuated usinga single secondary source. Herein the addition of a second

a generated with the small CV.

Page 6: Boundary element simulations for local active noise control using an extended volume

Fig. 5. Average noise reduction area generated with the medium CV.

Fig. 6. Average noise reduction area generated with the large CV.

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 195

Page 7: Boundary element simulations for local active noise control using an extended volume

Table 4Values of the terms in Eqs. (5.4)–(5.7).

RDðP��P��P��P��Þ dD 1 2 3 4

a� P0s1IPs1IUs2I P0s1RPs1IPs2I P0s1IPs1RPs2I P0s1RPs1RPs2I

b� P0s2IPs1IPs2I P0s2RPs1IPs2I P0s2IPs1RPs2I P0s2RPs1RPs2I

c� P0s1IP0s2IPs1I P0s1RP0s2RPs1I P0s1RP0s2IPs1I P0s1IP

0s2RPs1I

d� P0s1IP0s2IPs2I P0s1RP0s2RPs2I P0s1RP0s2IPs2I P0s1IP

0s2RPs2I

f� P0s2IP2s1I P0s2RP2

s1I P0s2IP2s1R P0s2RP2

s1R

g� P0s1IP2s2I P0s1RP2

s2I P0s1IP2s2R P0s1RP2

s2R

h� P02s1IPs2I P02s1RPs2I P02s1IPs2R P02s1RPs2R

l� P02s2IPs1I P02s2RPs1I P02s2IPs1R P02s2RPs1R

m� P02s2IP2s1I P02s2RP2

s1I P02s2IP2s1R P02s2RP2

s1R

bc� P0s1IP0s2I P0s1RP0s2R P0s1RP0s2I P0s1IP

0s2R

Ps1IPs2I Ps1IPs2I Ps1RPs2I Ps1RPs2IRDðP��P��P��P��Þ dD 5 6 7 8

a� P0s1IPs1IPs2R P0s1RPs1IPs2R P0s1IPs1RPs2R P0s1RPs1RPs2R

b� P0s2IPs1IPs2R P0s2RPs1IPs2R P0s2IPs1RPs2R P0s2RPs1RPs2R

c� P0s1RP0s2IPs1R P0s1IP0s2RPs1R P0s1IP

0s2IPs1R P0s1RP0s2RPs1R

d� P0s1IP0s2IPs2R P0s1RP0s2RPs2R P0s1RP0s2IPs2R P0s1IP

0s2RPs2R

f� – – – –

g� P0s2IP2s2I P0s2RP2

s2I P0s2IP2s2R P0s2RP2

s2R

h� – – – –

l� P02s2IPs2I P02s2RPs2I P02s2IPs2R P02s2RPs2R

m� P02s1IP2s2I P02s1RP2

s2I P02s1IP2s2R P02s1RP2

s2R

bc� P0s1RP0s2I P0s1IP0s2R P0s1IP

0s2I P0s1RP0s2R

Ps1IPs2R Ps1IPs2R Ps1RPs2R Ps1RPs2R

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202196

secondary source proves that a higher noise attenuation level canbe achieved.

The importance of the two speaker locations on the overallnoise reduction is highlighted with three different configurationsin the next section.

5.1. Mathematical model

The addition of the second control source requires another costfunction that should be minimised contemporaneously withEq. (3.1). Such a cost function can be written as follows:

fc2ðXDÞ ¼

ZDjP01ðXDÞj

2 dD ð5:1Þ

where P01 is the component along the x1-direction of the particlevelocity.

The second loudspeaker is also modelled as a hard 3D objectwith a vibrating surface. Its optimal vibration velocity qs2ðxvÞ isevaluated by a second constant b as for the previous case

qs2ðxvÞ ¼ bqs2ðxvÞ ð5:2Þ

The total pressure and the total component of the total particlevelocity along one direction at a generic point ðXÞ are evaluated,respectively, as in expression (3.3) by adding an extra term thatrefers to the second control source as follows:

P¼ PpþaPs1þbPs2

P0 ¼ P0pþaP0s1þbP0s2 ð5:3Þ

where the subscripts s1 and s2 refer to the first and the secondcontrol source, respectively.

Expressions in (5.3) are substituted into the cost functions(3.1) and (5.1) that are minimised by a and b (i.e., the costfunction derivatives with the respect to aR, aI , bR and bI are setequal to zero). A system with four equations and four unknowns(aR, aI , bR and bI) is obtained whose solution is displayed below.The solution of such system can be simplified, as shown below, byconsidering the primary field as constant at the CV points [12].

Each term of the a and b requires solution of Eq. (2.3) andapplication of Eqs. (2.4) and (2.5) three times, first for the primaryfield, by considering both the secondary sources as rigid surfaces,and again twice for the two secondary sources, acting indepen-dently, for any initial value of the vibrating velocity (qs1 and qs2)and by considering the other secondary source as a rigid sphere.

The solution for the standard procedure in terms of aR, aI , bR

and bI can be written as follows:

aR ¼�PpIðl1þ l2�d1�d2þd7�d8ÞþPpRðl3þ l4�d3þd4�d5�d6Þ

m1þm2þm3þm4þm5þm6þm7þm8

þP01pIðb4�b1þg1�b6�b7þg3ÞþP01pRðg2�b2�b3þb5�b8þg4Þ

þ2ð�bc1�bc2�bc3þbc4þbc5�bc6�bc7�bc8Þ

ð5:4Þ

aI ¼�PpIðl3þ l4�d3þd4�d5�d6Þ�PpRðl1þ l2�d1�d2þd7�d8Þ

m1þm2þm3þm4þm5þm6þm7þm8

þP01pIðg2�b2�b3þb5�b8þg4Þ�P01pRðb4�b1þg1�b6�b7þg3Þ

þ2ð�bc1�bc2�bc3þbc4þbc5�bc6�bc7�bc8Þ

ð5:5Þ

bR ¼�PpIðc6�c1�c2�c5þh1þh2ÞþPpRðc3�c4�c7�c8þh3þh4Þ

m1þm2þm3þm4þm5þm6þm7þm8

þP01pIðf1þ f3�a1�a4þa6�a7ÞþP01pRðf2þ f4�a2þa3�a5�a8Þ

þ2ð�bc1�bc2�bc3þbc4þbc5�bc6�bc7�bc8Þ

ð5:6Þ

bI ¼�PpIðc3�c4�c7�c8þh3þh4Þ�PpRðc6�c1�c2�c5þh1þh2Þ

m1þm2þm3þm4þm5þm6þm7þm8

þP01pIðf2þ f4�a2þa3�a5�a8Þ�P01pRðf1þ f3�a1�a4þa6�a7Þ

þ2ð�bc1�bc2�bc3þbc4þbc5�bc6�bc7�bc8Þ

ð5:7Þ

where P01pI and P01pR are the real and imaginary parts of componentalong the x1-direction of the primary particle velocity, respec-tively. All the other quantities in the above solutions are reportedin Table 4.

6. Results for two control sources

In this section the adoption of a system with two controlsources is investigated. The importance of the control sourcelocations is described next. A simplified procedure is alsopresented. As for the single source, results are obtained fromthe average results of 20 samples and for four frequencies (i.e.,109, 273, 546, 1092 Hz).

6.1. Standard procedure

In this subsection the control sources and CV are located atdifferent places and three different configurations are tested (seeFig. 7).

In the first configuration (Fig. 7a) the vibrating segments ofboth sources are directed towards the x1-axis and the CV is placedbetween them. In the second configuration (Fig. 7b) the vibratingsegments are directed towards the centre of the CV and one ofthem is placed at a larger distance from this point. Finally, the lastconfiguration (Fig. 7c) is similar to the second, but both of themare placed at the same distance from the centre of the CV. Thepoint at which the value of the primary field (pressure andparticle velocity) has been calculated and maintained constantis at 19.4 cm from the origin in all of the three configurations.

Figs. 8–10 show the average total pressure field in thex1x2-plane and x1x3-plane, ((a) and (b) in each figure, respec-tively) due to a primary diffuse field and two secondary fieldsgenerated by two rigid spheres with an active segment of 1201

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Fig. 7. The three configurations tested: (a) configuration 1, (b) configuration 2, (c) configuration 3.

Fig. 8. Average noise reduction area for the first configuration.

Fig. 9. Average noise reduction area for the second configuration.

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 197

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Fig. 10. Average noise reduction area for the third configuration.

Fig. 11. Average noise reduction area of the simplified procedure.

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202198

and radius of 8 cm when the noise attenuation is achieved in themedium size CV.

It can be noted that the best noise attenuation level is reachedwith the second configuration for the initial two lowest frequen-cies. As the frequency increases the third configuration provides ahigher noise attenuation level. Moreover, the first configurationshould be avoided since a considerable noise reduction is reachedonly in the symmetry plane between the two sources. This effectoccurs because the two sources create two parallel acoustic fieldsso that they interact reciprocally and the noise is essentiallycreated by them. This configuration highlights that the location ofthe secondary sources is a basic design consideration.

6.2. Simplified procedure

This subsection presents the results of a simplified procedureemploying two loudspeakers.

The hypothesis of considering constant the primary field at theCV points can be extended by considering the primary particle

velocity equal to zero at the same points. Thus, the solutions(5.4)–(5.7) are simplified such that the terms P01pI and P01pR are setto zero and the related terms are eliminated, resulting also in areduction of the computational effort.

The average total pressure field in the x1x2-plane andx1x3-plane, when the pressure and only the secondary compo-nents along the x1-direction of the particle velocity are minimisedin a CV, is shown in Fig. 11. The second configuration of Fig. 7 isutilised.

Comparing Figs. 11 and 9 highlights a greater noise reductionarea which is now attached to the closer source. This suggeststhat minimising the secondary source particle velocity compo-nents instead of the total makes the proposed strategy feasible ina practical application without reducing the extension of thequiet zone.

For a practical application, only a single further error microphoneis required, with respect to the case of a single control source, forevaluating the secondary particle velocity along a single directiononly and to adapt the response of the control source.

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Table 5Average noise reduction level reached inside the CV.

Frequency (Hz) fc without ANC fc with ANC Noise reduction (dB)

82 6.90�10�10 1.98�10�12 25.43

110 1.31�10�10 5.88�10�12 13.48

142 4.87�10�12 1.80�10�12 4.31

182 1.40�10�9 3.56�10�11 15.96

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 199

7. Results for a large-scale engineering problem

This section presents the results obtained by the proposedstrategy when applied to a large-scale engineering problem, i.e.,the ANC in an aircraft cabin.

The model of the cabin represents a portion of fuselage limitedby a rear and a front panel and it is included in a cuboid ofdimensions 2�1.9�2.3 (see Fig. 12). Inside, the cabin is com-posed of two lines of three seats. Each line is 0.5�1.35 m withheight 1.054 m. Due to the geometrical symmetry, only half cabinis utilised. The cabin mesh is composed of 3914 nodes and 7699constant elements in order to deal with up to 250 Hz (i.e., 10elements per wavelength are guaranteed). The headrest is 0.46 mlong (along the x2-axis) and the CV is a 0.2�0.32�0.2 m cuboid(so that it covers most of the headrest extension) and it is locatedin the front line at the seat close to the edge of the fuselage (seeFig. 13). The integration in the CV is performed by subdividing itinto 160 cuboid linear elements.

Fig. 12. Aircraft cabin geometry.

Fig. 13. Zoom on the aircraft cabin geometry.

The boundary conditions (BCs) were set to represent a possiblereal circumstance. The front and rear panel are modelled with softBCs (p¼0), whereas the symmetry plane as a hard panel (q¼0). Inthe model the seats, the floor and the ceiling have soundabsorption properties with absorbing coefficients varying linearlyat different frequencies. In particular, at 70 Hz the absorbingcoefficients of seats, floor and ceiling have absorbing coefficientequal to 0.14, 0.10 and 0.06, respectively, and at 200 Hz equal to0.53, 0.30 and 0.20.

The primary disturbance is generated by a monopole locatedat the point (1.87, 0.25, 1.90) close to the symmetry panel and it isevaluated in the centre of the CV and maintained constant. Thecontrol source is the same of the previous examples, but its radiusis now only 4 cm, and its centre is located at the point(0.760,1.525,1.460) close to the ceiling above the CV. Its vibrationportion is headed in the negative x3 direction.

Four frequencies are analysed, i.e., 82, 110, 142, 182 Hz, beingclose to the resonance frequencies of the cabin at a typical jetnoise frequency range (50–200 Hz) [30]. Table 5 shows the costfunction without and with the ANC, and the noise reduction level(in dB) obtained inside the CV for each analysed frequency.

Figs. 14 and 15 compare the SPL (in dB) inside the cabin at twofrequencies, i.e., 82 and 182 Hz, without (a) and with (b) theproposed ANC technique.

8. Conclusions and future work

In this paper a novel approach to the local ANC was investi-gated. The proposed strategy consists of utilising an enclosed areawhere the modulus of the total pressure and the component ofthe total particle velocity are minimised rather than cancelled atdiscrete points. Results demonstrated that a more homogeneousnoise attenuation level is achieved in the predefined volume,called control volume (CV). A BEM formulation accelerated by theACA in conjunction with the H-matrix format and the GMRESsolver was utilised to solve the Helmholtz equation in the 3Dfield. Two ANC formulations, using a single and two controlsources, respectively, were presented.

The proposed strategy is appealing as its application to aconventional DSP can be achieved since only the related transferfunction (to include the CV parameters) needs to be modified.This is due to the fact that the primary disturbance is requiredonly at a single point, and the extension of the noise reductionarea depends only upon the secondary source response.

It was also demonstrated that the use of an additionalsecondary source can provide a greater noise reduction. Further-more, the secondary particle velocity can be used instead of thetotal particle velocity to make the strategy practical and toaccelerate the execution time of the DSP system without affectingthe noise attenuation level.

An optimised control source location can increase the perfor-mance of the proposed strategy in enclosed space.

The effectiveness of the proposed CV approach would need tobe tested experimentally in the future.

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Fig. 14. SPL inside an aircraft cabin at 82 Hz: (a) without ANC, and (b) with ANC.

Fig. 15. SPL inside an aircraft cabin at 182 Hz: (a) without ANC, and (b) with ANC.

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202200

Acknowledgements

This work was carried out as part of an European researchproject (SEAT: Smart tEchnologies for stress free Air Travel) AST5-CT-2006-030958 coordinated by Imperial College London.

The authors are especially grateful to Dr. Vincenzo Mallardoand Dr. Vincent Marant for our many and always fruitfulconversations.

Appendix A. Adaptive Cross Approximation

As afore-mentioned it is well-known that one of the maindrawbacks of the BEM is that the governing matrix is denselypopulated and non-symmetric, slowing down the solution timeof the procedure. To overcome this disadvantage, the ACA inconjunction with the H-matrix and the GMRES are adopted toachieve a faster solution without reducing the accuracy level.

The assembly time of the linear system of equations (2.3) isaccelerated by calculating only a few entries of the originalmatrix. The basic idea of the ACA is to divide the whole matrixinto two rank (low and full rank) blocks based on size and distancebetween a group of collocation points and a group of boundaryelements. A full rank block is represented entirely, whereas a low-rank block permits a particular approximation where only a few

entries of the original block are required to represent the entireblock (see Fig. A1):

CCCk ¼~A � ~B

T¼Xk

i ¼ 1

~a i �~b

T

i ðA:1Þ

where ~A is of order m� k and ~B is of order n� k, with k5m andk5n being the rank of the new representation. The approximat-ing block Ck satisfies the relation JC�CkJF reJCJF , where J � JF

represents the Frobenius norm and e is the prescribed accuracy. Itshould be noted that the level of approximation is set in advancethrough e and a higher accuracy level is adaptively reached by ahigher value of k.

The process leading to the subdivision in sub-blocks and totheir further classification (full rank and low rank) is based on apreliminary hierarchical partition of the matrix index set aimed atgrouping subsets of indices corresponding to contiguous nodesand elements. The idea behind this technique is based on theconsideration that the integrals of contiguous elements due to asingle collocation point are almost identical, especially for highdensity meshes. The same consideration is valid for the integralsof a single element due to a number of collocation points. Thesolving matrix is thus divided into sub-blocks till a block isclassified admissible (it allows the approximated representation)or its size is above a certain value, i.e., the cardinality.

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Fig. A2. Block-wise representation of two ACA generated matrices. Sphere composed by (a) 646 constant triangular elements; (b) 1126 constant triangular elements.

Fig. A1. Low rank approximation: a few entries of the original block represents the entire block.

A. Brancati, M.H. Aliabadi / Engineering Analysis with Boundary Elements 36 (2012) 190–202 201

The admissibility condition can be written as

minðdiam Or ,diam OcÞrZ distðOr ,OcÞ ðA:2Þ

where Or denote the cluster of elements containing the discreti-sation nodes corresponding to the row indices of the consideredblock and Oc is the set of elements over which the integration iscarried out to compute the coefficient corresponding to thecolumn indices; diam and dist indicate the diameter and thedistance of the two clusters, respectively, and Z40 is a parameterinfluencing the number of admissible blocks on one hand and theconvergence speed of the adaptive approximation of low-rankblocks on the other hand [31].

The cardinality and Z assume a predominant role on speedingup the CPU time and the optimised values depend upon the sizeand the frequency range of the problem [26]. The structure of theH-matrices reduces the memory storage requirements and accel-erates the building process and the matrix-vector multiplication[32]. The proposed procedure is of O(N) for both storage andmatrix-vector multiplication [33,34].

Fig. A2 shows two solving matrix representations after theapplication of the ACA (R¼ 50) to a sphere composed by 646 and1126 triangular elements, respectively. The dark grey and thelight grey block represent the full rank and the low-rank blocks,respectively. In particular, the more the light grey, the lower thevalue of the rank k of the blocks. As evident as the degrees offreedom increases the matrix blocks approximated by theACA increase and a great part of the matrix is represented in alow-rank format.

It should be noted that using the ACA does not require tomodify or rewrite the routines for the boundary integration inpreviously developed codes.

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