a wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave...

Acta Mechanica Sinica (2013) 29(2):219–232DOI 10.1007/s10409-013-0023-4

RESEARCH PAPER

A wideband fast multipole boundary element method forhalf-space/plane-symmetric acoustic wave problems

Chang-Jun Zheng ··· Hai-Bo Chen ··· Lei-Lei Chen

Received: 26 October 2012 / Revised: 18 January 2013 / Accepted: 4 February 2013©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2013

Abstract This paper presents a novel wideband fast mul-tipole boundary element approach to 3D half-space/plane-symmetric acoustic wave problems. The half-space funda-mental solution is employed in the boundary integral equa-tions so that the tree structure required in the fast multipolealgorithm is constructed for the boundary elements in thereal domain only. Moreover, a set of symmetric relationsbetween the multipole expansion coefficients of the real andimage domains are derived, and the half-space fundamentalsolution is modified for the purpose of applying such rela-tions to avoid calculating, translating and saving the multi-pole/local expansion coefficients of the image domain. Thewideband adaptive multilevel fast multipole algorithm asso-ciated with the iterative solver GMRES is employed so thatthe present method is accurate and efficient for both low-and high-frequency acoustic wave problems. As for exterioracoustic problems, the Burton–Miller method is adopted totackle the fictitious eigenfrequency problem involved in theconventional boundary integral equation method. Details onthe implementation of the present method are described, andnumerical examples are given to demonstrate its accuracyand efficiency.

Keywords Helmholtz equation · Boundary elementmethod · Half-space/plane-symmetric problem · Widebandfast multipole method · Noise barrier

The project was supported by the National Natural ScienceFoundation of China (11172291), the National Science Founda-tion for Post-doctoral Scientists of China (2012M510162), andthe Fundamental Research Funds for the Central Universities(KB2090050024).

C.-J. Zheng (�) · H.-B. Chen · L.-L. ChenDepartment of Modern Mechanics,University of Science and Technology of China,230027 Hefei, Chinae-mail: [email protected]

1 Introduction

So far the boundary element method (BEM) has been ap-plied extensively in solving acoustic wave problems, as itinvolves surface discretization only and solves exterior prob-lems (infinite or semi-infinite domain) naturally and exactly.Although the BEM reduces the problem dimensionality byone, the conventional method typically gives rise to fullypopulated and asymmetric coefficient matrices which resultin large storage requirements and prohibitive analysis time.The computational complexity of the conventional method isO(N3) with direct solvers, or O(N2) with appropriate itera-tive solvers, and the storage requirements are O(N2), whereN is the degree of freedom (DOF). This well-known draw-back makes the BEM enormously difficult with large mod-els and thus limited to numerical analyses of small bod-ies at low frequencies. In order to improve the efficiency,various fast approximate techniques, such as the fast multi-pole method (FMM) [1], the fast wavelet transforms [2], theprecorrected fast Fourier transformation (FFT) [3], the H-matrices [4] and the adaptive cross approximation (ACA) [5],have been proposed to accelerate the matrix-vector productsin the BEM. Among them, the FMM seems to be one ofthe most widely accepted techniques in the fast BEM family.Implemented with appropriate iterative solvers, the fast mul-tipole BEM (FMBEM) reduces both the computational com-plexity and storage requirement to O(N) for low-frequencyacoustic problems, for example. Also, the FMM has beenintensively studied in the solution of many problems, suchas acoustics, elastodynamics, electromagnetics, fracture me-chanics [6–27].

In many engineering applications, half-space or plane-symmetric acoustic wave problems are often analyzed, forinstance the acoustic fields around noise barriers standingon an infinite ground. Although the FMBEM approach tofull-space acoustic wave problems has been widely stud-ied in Refs. [10–22], the applications of the FMBEM tohalf-space/plane-symmetric acoustic wave problems are still

220 C.-J. Zheng, et al.

quite few [23–26] and also need further studies. Whendealing with half-space/plane-symmetric problems using theconventional BEM (CBEM), the half-space fundamental so-lution technique can be employed to remove the contribu-tion of the infinite/symmetry plane to the boundary integralequations [28]. With this technique, the discretization of theinfinite/symmetry plane is avoided and only the structuralboundaries need to be discretized. However, this techniquecannot be used directly in the FMBEM, as the fundamen-tal solution in the FMBEM should be expressed in forms ofmultipole expansions. In order to accelerate the solution ofhalf-space/plane-symmetric acoustic wave problems, Yasudaand Sakuma [23], Brunner et al. [24] and Yasuda et al. [25]proposed a set of efficient methods by using the mirror im-age technique, where the half-space fundamental solution isnot utilized, instead, half-space/plane-symmetric problemsare mapped into full-space problems and the original funda-mental solution is employed in the boundary integral equa-tions. Using these methods, the computational complexityand storage requirement can be cut in nearly half for prob-lems with one symmetry plane. However, these methods leadto a bigger tree structure in which it is required to group allboundary elements and their mirror images. The situationgets even worse when the distance between the structure andthe infinite plane increases. Thus, some special techniquesshould be used to improve the efficiency for such cases [23].Bapat et al. [26] presented another half-space FMBEM ap-proach based on the half-space fundamental solution tech-nique, where the tree structure needs to group the boundaryelements of the structure in the real domain only. Althoughtheir method is simple to implement, the local expansion co-efficients of the image domain should be calculated, trans-lated and kept in memory, which actually brings down theefficiency of their method.

Moreover, the FMM approaches to acoustic wave prob-lems in frequency domain consists of the high- and low-frequency methods. The half-space/plane-symmetric ap-proaches proposed by Yasuda and Sakuma [23] and Brun-ner et al. [24] are based on the high-frequency one, andthe approaches presented by Yasuda et al. [25] and Bapatet al. [26] are based on the low-frequency one. The compu-tational complexity is O(NlgN) in the high-frequency FMM,versus O(N) in the low-frequency FMM. However, either ofthem fails in some way outside its preferred frequency re-gion. The high-frequency FMM is known to be unstable forlow-frequency problems, while the low-frequency FMM isfound to be inefficient for high-frequency problems [27]. Ineach case, the difficulty is fundamental and cannot be over-come by simple expedients [29]. Therefore, a wideband ver-sion of half-space FMBEM approach which is accurate andefficient for all frequencies seems quite necessary.

In this study, a novel wideband FMBEM (WFMBEM)approach is presented for 3D half-space/plane-symmetricacoustic wave problems. Different from the approaches pro-posed in Refs. [23–25], the half-space fundamental solution

is utilized in the boundary integral equations so that the treestructure required in the present fast multipole algorithmcould be built for the boundary elements of the structuresin the real domain, instead of in both the real domain andits mirror image. However, different from the method pre-sented by Bapat et al. [26], a set of symmetric relations be-tween the multipole expansion coefficients of the real and theimage domains are derived, and the half-space fundamentalsolution is modified for the purpose of applying such rela-tions to avoid calculating, translating and saving the multi-pole/local expansion coefficients of the image domain. Also,different from all the existed half-space/plane-symmetricFMBEM approaches in Refs. [23–26], the wideband adap-tive multilevel fast multipole algorithm is employed so thatthe present method is accurate, efficient and robust for bothhigh- and low-frequency half-space/plane-symmetric acous-tic problems. Furthermore, as for exterior acoustic problems,the Burton–Miller method [30] is applied to tackle the ficti-tious eigenfrequency problem involved in the conventionalboundary integral equation method.

The remainder of this paper is organized as fol-lows. The boundary integral equations for half-space/plane-symmetric acoustic wave problems are given in Sect. 2. Theformulations for the present wideband half-space FMBEMapproach are outlined in Sect. 3. The wideband half-spaceFMBEM algorithm is presented in Sect. 4. Section 5 givessome numerical examples to demonstrate the accuracy andefficiency of the present method. Section 6 concludes thepaper with further discussions.

2 Boundary integral equations

The Helmholtz equation which is the governing equationin steady-state linear acoustics can be reformulated into aboundary integral equation (BIE) defined on the structuralboundary Γ as follows

c(x)p(x) + −∫Γ

q∗(x, y)p(y) dΓ(y)

=

∫Γ

p∗(x, y)q(y)dΓ(y) + pi(x), (1)

where the coefficient c(x) is 1/2 if Γ is smooth around thesource point x, p(x) is the sound pressure, pi(x) is the in-cident wave, q(y) and q∗(x, y) are the normal derivatives ofp(y) and p∗(x, y), y is the field point, and p∗(x, y) is the fun-damental solution. The symbol −

∫denotes that the integral is

evaluated in the sense of Cauchy principal value. As for 3Dfull-space acoustic wave problems, p∗(x, y) is given as

p∗(x, y) =eikr

4πr, (2)

where k is the wave number and r = |y − x|. In the caseof half-space problems, the half-space fundamental solution,denoted by p∗H(x, y), takes the following form [28]

p∗H(x, y) = p∗(x, y) + βp∗(x, y), (3)

A wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave problems 221

where x is the mirror image of x with respect to the infiniteplane ΓH as shown in Fig. 1, and β is the reflection coeffi-cient which is equal to 1 for a rigid, infinite plane or −1 fora soft, infinite plane. For half-space scattering problems, ifthe incident wave is not parallel to the infinite plane, pi(x) inEq. (1) should be constructed in such a way that it includesthe wave reflected from the infinite plane.

Fig. 1 Nomenclature for a half-space problem

The boundary conditions on Γ (Γ = Γp + Γq + Γz) aregiven as

p(x) = p(x), on Γp, (4)

q(x) = iρωv(x), on Γq, (5)

p(x) = zv(x), on Γz, (6)

where i denotes the imaginary unit, ρ is the medium density,ω is the angular frequency, v(x) is the normal velocity, and zis the acoustic impedance. The quantities with overbars in-dicate given values on the boundary. As for exterior acousticwave problems, it is necessary to introduce the Sommerfeldradiation condition [31] at infinity which ensures that all ra-diated and scattered waves are outgoing.

Equation (1), usually referred to as the conventionalBIE (CBIE), can be employed to calculate the unknownboundary values. However, the BEM based on it fails to yieldunique solutions for exterior acoustic problems at the eigen-frequencies of the associated interior problems [32]. Theseeigenfrequencies are usually called fictitious eigenfrequen-cies because they do not have any physical significance, butjust arise from the drawback of the boundary integral rep-resentation for solving exterior wave propagation problems.In order to tackle this difficulty, two main methods, appro-priate for practical applications, have been proposed overthe last several decades, i.e., the combined Helmholtz in-tegral equation formulation (CHIEF) [32] and the Burton–Miller method [30]. In this study, we employ the Burton–Miller method which is more rigorous to conquer the ficti-

tious eigenfrequency problem than the CHIEF, especially inthe high frequency range [33].

The Burton–Miller formulation which is a linear com-bination of the CBIE and its normal derivative can be writtenas

c(x)p(x) + αc(x)q(x) + −∫Γ

q∗(x, y)p(y) dΓ(y)

+α=

∫Γ

q∗(x, y)p(y) dΓ(y) =∫Γ

p∗(x, y)q(y) dΓ(y)

+α−∫Γ

p∗(x, y)q(y) dΓ(y) + pi(x) + αpi(x), (7)

where α is the coupling constant that can be chosen asi/k [34], the symbol =

∫denotes that the integration is carried

out in the sense of Hadamard finite part of the divergent in-tegral, and f (x) = ∂ f (x)/∂n(x).

Discretizing Γ using N (e.g., piecewise constant) sur-face elements leads to the following equation (e.g., foracoustic radiation problems)

12

pi +

N∑j=1

hi j p j =

N∑j=1

gi jq j, (8)

where

hi j =

∫Γ j

[q∗(x, y) + βq∗(x, y)]dΓ(y), (9)

gi j =

∫Γ j

[p∗(x, y) + βp∗(x, y)]dΓ(y), (10)

for the CBIE (i.e., Eq. (1)), or

hi j =

∫Γ j

[q∗(x, y) + βq∗(x, y) + αq∗(x, y)

+αβq∗(x, y)]dΓ(y), (11)

gi j =

∫Γ j

[p∗(x, y) + βp∗(x, y) + αp∗(x, y)

+αβp∗(x, y)]dΓ(y), (12)

for the Burton–Miller formulation (i.e., Eq. (7)). In the aboveequations, Γ j represents the piecewise constant element j asan example. Collecting Eq. (8) for all boundary source pointsand expressing them in matrix form result in the followinglinear algebraic equations

hhhppp = gggqqq. (13)

Equation (13) can be rearranged into the following form byapplying the boundary conditions

AAAψψψ = BBBφφφ, (14)

where ψψψ and φφφ are the unknown and known vectors, respec-tively; AAA and BBB are the coefficient matrices corresponding tothem. Equation (14) can now be solved and all the unknownboundary values are then obtained. Once this has been done,one can calculate the sound pressure at any internal point byusing the discretized form of Eq. (1) with c(x) = 1.


Strongly singular and hypersingular boundary integralsare found in Eqs. (1) and (7). In order to evaluate them accu-rately, various singularity subtraction techniques have beenproposed in Refs. [35–39]. However, most of these tech-niques are still cumbersome and sometimes time-consuming,especially for FMBEM. For instance, when the formulasare regularized using the fundamental solution of Laplace’sequation, multipole expansion formulas and other translationformulas have to be implemented not only for the fundamen-tal solution and its derivatives of the Helmholtz equation butalso for those of Laplace’s equation. In contrast, when piece-wise constant elements are employed to discretize the bound-ary surface, the strongly singular and hypersingular bound-ary integrals can be evaluated directly and efficiently. More-over, the piecewise constant element discretization makesthe implementation of the FMBEM easier. According toour previous work [40–42], we have a non-singular versionof Eqs. (1) and (7) for the piecewise constant element dis-cretization as follows12

p(x) +∫Γ\Γx

q∗(x, y)p(y)dΓ(y)

=

∫Γ\Γx

p∗(x, y)q(y)dΓ(y)

+i

2k

(1 −

∫ 2π

0

eikR

2πdθ

)q(x) + pi(x), (15)

12

p(x) +α

2q(x) +

∫Γ\Γx

q∗(x, y)p(y)dΓ(y)

+α

∫Γ\Γx

q∗(x, y)p(y)dΓ(y) =∫Γ\Γx

p∗(x, y)q(y)dΓ(y)

+α

∫Γ\Γx

p∗(x, y)q(y)dΓ(y) + pi(x) + αpi(x)

+i

2k

(1 −

∫ 2π

0

eikR

2πdθ

)q(x)

−α(

ik2−

∫ 2π

0

eikR

4πRdθ

)p(x), (16)

where Γ\Γx denotes the boundary Γ excluding the elementΓx in which the source point x exists, and R = R(θ) is thedistance from x to the peripheral of the elment as shown inFig. 2.

Furthermore, it is well-known that Eq. (14) is very time-consuming to obtain and solve in a conventional way evenwith appropriate iterative solvers. In order to acceleratethe solution process and reduce the storage requirement, theFMM approach is employed in this study. The FMM ap-proach to acoustic wave problems in frequency domain con-sists of the low- and high-frequency types, and either ofthem fails in some way outside its preferred frequency re-gion. The high-frequency FMM is known to be unstablefor low-frequency problems, while the low-frequency FMMis found to be inefficient for high-frequency problems [27].Hence, it needs to construct a wideband version which is ac-

curate and efficient for all frequencies. Several WFMBEMapproaches have already been proposed for full-space acous-tic wave problems in Refs. [17–21]. But in this study, theFMM approaches for both low and high frequencies are uni-fied to first obtain an efficient WFMBEM approach for half-space/plane-symmetric acoustic wave problems.

Fig. 2 A piecewise constant element Γx on which the source pointx is placed

3 WFMBEM formulations

In order to apply the FMM approach, the fundamental so-lution of the Helmholtz equation should be expanded into asuitable form. The WFMBEM employed in this study usesa series expansion formula of the fundamental solution inlow-frequency region and a plane wave expansion formulain high-frequency region. As either of these two expan-sion formulas meets some difficulties outside the preferredfrequency domain, a wideband version which is a combina-tion of both the low- and high-frequency FMM approachesis constructed in this section. In the low-frequency region,the full-space fundamental solution (i.e., Eq. (2)) is expandedinto the following series

p∗(x, y) =ik4π

∞∑n=0

n∑m=−n

(2n + 1)Imn (k,−→Oy)Om

n (k,−→Ox), (17)

where O is an expansion point near y as shown in Fig. 3, Imn

and Omn are defined as

Imn (k,aaa) = jn(kr)Ym

n (θ, φ), (18)

Omn (k,aaa) = h(1)

n (kr)Ymn (θ, φ), (19)

and Imn are the complex conjugates of Im

n , jn and h(1)n are the

n-th order spherical Bessel and Hankel functions of the firstkind, Ym

n is the spherical harmonics defined as

Ymn (θ, φ) = cm

n Pmn (cos θ)eimφ, (20)

where cmn =

√(n − m)!/(n + m)!, Pm

n denote the associatedLegendre functions; r, θ and φ represent the three spherical

coordinates of some vector aaa, such as−→Ox or

−→Oy, for instance.


Fig. 3 Multipole expansion points and boundary nodes in half-space FMM

The plane wave expansion formula of the full-spacefundamental solution (i.e., Eq. (2)) is written as follows

p∗(x, y) =ik

16π2

∫S

eikk·−−→x′x T (k, kkk,−−→Ox′) e−ikk·−→OydS . (21)

Here x′ is an expansion point near x as depicted in Fig. 3, theintegration is taken over the unit sphere S , kkk denotes the out-ward unit vector on S , and the diagonal translation functionT is given as

T (k, kkk,aaa) =∞∑

n=0

in(2n + 1)h(1)n (ka)Pn(kkk · aaa), (22)

where a = |aaa|, aaa = aaa/a, and Pn stand for the Legendre poly-nomials.

As shown in Fig. 1, the half-space fundamental solution(i.e., Eq. (3)) represents the interaction between point x andy, and the interaction between the mirror-image point x andy. In the FMBEM approach, the half-space fundamental so-lution should also be expanded into a suitable form. Bapat etal. [26] utilized the series expansion formula (i.e., Eq. (17))to obtain the expansion form of Eq. (3). But in this study, inorder to construct an efficient half-space FMBEM approach,we first modify the half-space fundamental solution into thefollowing form

p∗H(x, y) = p∗(x, y) + βp∗(x, y), (23)

and then reformulate Eqs. (9) and (10) as follows

hi j =

∫Γ j

[q∗(x, y) + βq∗(x, y)

]dΓ(y), (24)

gi j =

∫Γ j

[p∗(x, y) + βp∗(x, y)

]dΓ(y). (25)

So that, using Eqs. (17) and (21), we can express the bound-ary integrals over a boundary element Γ j which is far awayfrom the source point x as follows

hi j p j

gi jq j

⎫⎪⎪⎬⎪⎪⎭ =ik4π

∞∑n=0

n∑m=−n

(2n + 1)[Mm

n (k,−−→Oyj)Om

n (k,−→Ox)

+Mmn (k,−−→Oyj)Om

n (k,−→Ox)

], (26)

and

hi j p j

gi jq j

⎫⎪⎪⎬⎪⎪⎭ =ik

16π2

∫S

eikk·−−→x′x[F(k, kkk,

−−→Oyj)T (k, kkk,

−−→Ox′)

+F(k, kkk,−−→Oyj)T (k, kkk,

−−→Ox′)

]dS , (27)

where y j is a field point on Γ j; Mmn (k,−−→Oyj) and F(k, kkk,

−−→Oyj),

defined in the following, are multipole moments of low- andhigh-frequency FMM approaches in the real domain, respec-tively.

Mmn (k,−−→Oyj) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

∫Γ j

∂Imn (k,−→Oy)

∂n(y)pjdΓ(y), for hi j p j,

∫Γ j

Imn (k,−→Oy)qjdΓ(y), for gi jq j,

(28)

and

F(k, kkk,−−→Oyj)

=

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∫Γ j

−ikkkk · n(y)e−ikk·−→Oy p jdΓ(y), for hi j p j,

∫Γ j

e−ikkkk·−→Oyq jdΓ(y), for gi jq j.

(29)

Moreover, Mmn (k,−−→Oyj) and F(k, kkk,

−−→Oyj) which are mul-

tipole moments in the image domain are also required inEqs. (26) and (27). Since we can obtain them directly from

Mmn (k,−−→Oyj) and F(k, kkk,

−−→Oyj) using the relations derived in the

Appendix, we do not need to consider them until the M2Land F2H procedures in the following.

Multipole moments of a group of � boundary elementsthat are close to O, exactly speaking, all elements in the sameleaf cell defined in the next section and centered at O, can beadded up together to form the moments of the leaf cell

Mmn (k,O) =

�∑j=1

Mmn (k,−−→Oyj), (30)

F(k, kkk,O) =�∑

j=1

F(k, kkk,−−→Oyj). (31)

In the low-frequency FMM approach, the moment cen-ter can be shifted from O to O′ according to the follow-ing M2M (moments to moments) translation formula, if

|−−→O′x| > |−−→O′y|.

Mmn (k,O′) =

∞∑n′=0

n′∑m′=−n′

∑l∈L(n,n′ ,m,m′)

(2n′ + 1)(−1)m′Wn,n′ ,m,m′,l


×I−m−m′l (k,

−−−→O′O)M−m′

n′ (k,O), (32)

where Wn,n′ ,m,m′,l is given by

Wn,n′,m,m′ ,l = (2l + 1)in′−n+l

⎛⎜⎜⎜⎜⎜⎝n n′ l

0 0 0

⎞⎟⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎜⎝

n n′ l

m m′ −m − m′

⎞⎟⎟⎟⎟⎟⎠ , (33)

and( · · ·· · ·

)stands for the Wigner 3j symbol [43], and the set

L is defined as

L(n, n′,m,m′) = {l| l ∈ Z, n + n′ − l : even,

max{|m + m′|, |n − n′|} � l � n + n′}. (34)

The M2L (moments to local expansion coefficients)translation formula where the contribution from the infi-nite/symmetry plane is taken into account can be expressedas

Lmn (k, x′) =

∞∑n′=0

n′∑m′=−n′

∑l∈L(n,n′ ,m,m′)

(2n′ + 1)(−1)m+m′Wn′,n,m′ ,m,l

×[Om+m′

l (k,−−−→O′x′)Mm′

n′ (k,O′)

+Om+m′l (k,

−−−→O′x′)Mm′

n′ (k,O′)]. (35)

Then, the local expansion center can be shifted from x′ tox′′ according to the following L2L (local expansion to localexpansion) translation formula

Lmn (k, x′′) =

∞∑n′=0

n′∑m′=−n′

∑l∈L(n,n′ ,m,m′)

(2n′ + 1)(−1)mWn′ ,n,m′,−m,l

×Im−m′l (k,

−−−→x′x′′)Lm′

n′ (k, x′). (36)

Also, the M2M, M2L and L2L translation formulas of thehigh-frequency FMM for half-space/plane-symmetric acous-tic problems, denoted as F2F, F2H and H2H for distinguish-ing them in this paper, are given as

F(k, kkk,O′) = eikk·−−−→O′OF(k, kkk,O), (37)

H(k, kkk, x′) = T (k, kkk,−−−→O′x′)F(k, kkk,O′)

+T (k, kkk,−−−→O′x′)F(k, kkk,O′), (38)

H(k, kkk, x′′) = eikk·−−−→x′x′′H(k, kkk, x′). (39)

Finally, for the group of � boundary elements that are close to

O and far away from the source point x,�∑

j=1

hi j p j or�∑

j=1

gi jq j

can be expressed in the form of expansion using the localexpansion coefficients as

�∑j=1

hi j p j or�∑

j=1

gi jq j

=ik4π

∞∑n=0

n∑m=−n

(2n + 1)Lmn (k, x′′)Im

n (k,−−→x′′x), (40)

in the low-frequency FMM, or

�∑j=1

hi j p j or�∑

j=1

gi jq j =ik

16π2

∫S

eikk·−−→x′′xH(k, kkk, x′′)dS , (41)

in the high-frequency FMM.In the wideband FMM, we use the following M2F for-

mula to convert the low-frequency moments to the high-frequency moments

F(k, kkk,O) =∞∑

n=0

n∑m=−n

(2n + 1) i−nYmn (kkk)Mm

n (k,O). (42)

The local expansion coefficients of the high-frequency FMMcan also be converted to those of the low-frequency FMM byusing the following H2L formula

Lmn (k, x′) =

in

4π

∫S

Ymn (kkk)H(k, kkk, x′)dS . (43)

As for the Burton–Miller formulation, since it is a linearcombination of the CBIE and its normal derivative, we needonly to modify Eqs. (40) and (41) as follows

�∑j=1

hi j p j or�∑

j=1

gi jq j =ik4π

∞∑n=0

n∑m=−n

(2n + 1)Lmn (k, x′′)

×⎡⎢⎢⎢⎢⎢⎢⎣Im

n (k,−−→x′′x) + α

∂Imn (k,−−→x′′x)

∂n(x)

⎤⎥⎥⎥⎥⎥⎥⎦ , (44)

�∑j=1

hi j p j or�∑

j=1

gi jq j =ik

16π2

∫S

[1 + αikkkk · n(x)

]eikk·−−→x′′x

×H(k, kkk, x′′)dS . (45)

4 WFMBEM algorithm

With all the formulations given above, we are able to con-struct the present WFMBEM algorithm for half-space/plane-symmetric acoustic wave problems. In this study, we useboth the low- and high-frequency FMM approaches andswitches between them depending on the level in the treestructure as shown in Fig. 4.

Step 1. Discretization

Discretize the structural boundary Γ as in the routineCBEM processes, for instance using the piecewise constanttriangular element discretization employed in the numericalexamples of this paper.

Step 2. Construction of the tree structure

Consider a cube enclosing Γ as the cell of level 0. Thendivide this cell (a parent cell) into eight equal cubes and callany of them a cell (a child cell) of level 1 if there are bound-ary elements in it. Keep dividing a cell in this way until theelement number in it is less than a specified number, and callthis childless cell a leaf cell.


Fig. 4 Wideband FMM

To make the following descriptions clearer, we definesome terms here. Two cells at the same level are said to beadjacent if they share at least one vertex. If two cells are notadjacent but their parent cells are adjacent, they are said tobe well-separated. The list of all well-separated cells of cellC forms the interaction list of C. Cells whose parent cells arenot adjacent to the parent cell of C are called far cells of C.The switching level s between the low- and high-frequencyFMM is the level satisfying ds+1 < D and ds � D (where ds

and ds+1 are the sizes of cells at level s and s + 1, and D thethreshold between the low- and high-frequency FMM).

Step 3. Upward pass

Calculate the multipole moments of all cells down fromlevel 2. As for a leaf cell C at level l, the multipole mo-ments are the summation of moments from all boundaryelements included in C to the center of C. If l > s, thelow-frequency moments (Eq. (28)) are needed, otherwise thehigh-frequency moments (Eq. (29)) are needed. As for anon-leaf cell C at level l, shift the multipole moments fromthe center of its child cells to its center by the M2M trans-lation (Eq. (32)) if l � s, or the F2F translation (Eq. (37)) ifl < s, then add up all the translated moments together to formthe multipole moments of C. However, if l = s, the M2F con-version (Eq. (42)) should be used after the M2M translationto convert the low-frequency moments to the high-frequencymoments.

Step 4. Downward pass

Calculate the local expansion coefficients of all cells atall levels with l � 2. The local expansion coefficients of cellC at level l consist of two parts. One is obtained from allcells in the interaction list of C by using the M2L transla-tion (Eq. (35)) if l > s or the F2H translation (Eq. (38)) ifl � s; and the other is obtained from all the far cells of Cby using the L2L translation (Eq. (36)) if l > s or the H2Htranslation (Eq. (39)) if l � s. However, as to a cell at level 2,

the coefficients are obtained by the M2L or F2H translationonly. In the downward pass, when l = s, the H2L conver-sion (Eq. (43)) should be used to convert the local expansioncoefficients of the high-frequency FMM to those of the low-frequency FMM.

Step 5. Evaluation of boundary integrals

Now, evaluate the boundary integrals of all sourcepoints contained in a leaf cell C at level l. Firstly, calculatethe contributions from all boundary elements in both C andits adjacent cells directly as in the CBEM. Then, calculate thecontributions from all the other cells (i.e., cells in the inter-action list and far cells of C) by shifting the local expansioncoefficients from the center of C to the source points. If l > s,the low-frequency formula (Eq. (40)) should be used, other-wise if l � s, the high-frequency formula (Eq. (41)) shouldbe employed. The summation of all these contributions givesthe final value of boundary integrals.

Using the above procedures, we can calculate the ma-trix and known vector product on the right-handside ofEq. (14). However, when calculating the matrix and un-known vector product on the left-handside of Eq. (14), weneed to update the unknown vector in iterative solvers, suchas the GMRES method [44] employed in this paper, and thencontinue at Step 2 for the matrix-vector product until the so-lution converges within a specified tolerance. In addition,the above procedures show the normal WFMBEM algorithmfor half-space/plane-symmetric acoustic wave problems. Inorder to further improve the efficiency, the adaptive algo-rithm [13] and the L2 modification technique [14] are alsoemployed in this study.

5 Numerical examples

Numerical examples are employed in this section to demon-strate the accuracy and efficiency of the present widebandhalf-space FMBEM approach to large-scale acoustic waveproblems. The present approach has been implemented in acode using Fortran 95, which can be applied to solve threedimensional half-space and also plane-symmetric acousticwave problems.

In all the examples presented below, the media ofacoustic fields are assumed to be air with the density ofρ = 1.20 kg/m3 and the sound speed of C = 340.0 m/s.Piecewise constant triangular elements are employed to dis-cretize the boundary surfaces. As for the FMM, the maxi-mum number of elements contained in a leaf cell is set to100. The number of truncation terms is determined by thefollowing semi-empirical formula [6, 27]

p = kdl + c ln(kdl + π), (46)

where dl is the size of cells at level l, and c the coefficient de-pending upon the precision of the arithmetics. The thresholdD between the low- and high-frequency FMM approachesis set to 0.2λ according to Ref. [9], where λ is the acousticwavelength. All boundary integrals are evaluated by using


the Gaussian quadrature formula with 10 integration points.All numerical results given below were obtained on a desk-top PC with an Intel 2.93 GHz Core CPU and 3.45 GB RAM.

5.1 Interior acoustic field of a rectangular box

The rectangular box, as depicted in Fig. 5, is selected as aninterior acoustic wave example in this section. The soundpressures on the left and right side surfaces are given as pL

and pR, respectively. The other four surfaces are assumed tobe rigid walls, i.e., normal velocities are specified as zero onthem. Thus, the analytical sound pressure at an internal pointx (x1, x2, x3) is given as

p(x) = pLsin(kl − kx1)

sin(kl)+ pR

sin(kx1)sin(kl)

, 0 � x1 � l. (47)

Fig. 5 The rectangular box model

In the numerical analysis, l = w = h = 1.0 m, pL = 0and pR = 100.0 Pa. The coefficient c in Eq. (46) is set to 5 forsingle precision. The iterative solver GMRES employed inthis study terminates iterations when the residual is below thetolerance of 10−3. Since the problem is an interior one, andthe model and boundary conditions are plane-symmetric, weare able to utilize the present wideband half-space FMBEMapproach based on the CBIE formulation.

First, as depicted in Fig. 5, the sound pressures at nineinternal points on the central line of the box are calculatedand compared with the analytical solutions for wave numberk = 1.0 m−1. The number of elements is 9 600 for the entiremodel, versus 4 800 for the half model. It is observed fromFig. 6 that the numerical results obtained by the CBEM withthe GMRES solver and the WFMBEM for both the entireand half models are very close to each other, and agree wellwith the analytical solutions.

Next, we study the effects of increasing number of el-ements on the efficiency of different BEM approaches. Thetotal number of elements varies from 1 536 to 60 000 for theentire model, versus from 768 to 30 000 for the half model.The CPU times and memory requireed by different BEM ap-proaches to solve the problem are compared in Figs. 7 and8, where “Number of elements” indicates that of the en-

tire model and the number of elements for the half modelis just half of that of the entire model. Moreover, we re-fer to “CBEM-LU” and “CBEM-GMRES” as the results ob-tained by the CBEM with the LU-decomposition and GM-RES solvers, respectively. Figures 7 and 8 clearly demon-strate the efficiency of the present WFMBEM approaches incomparison with the CBEM approaches. Also, the presenthalf-space WFMBEM is found to be more efficient thanthe full-space WFMBEM for plane-symmetric problems, forinstance, for the mesh with 6 144 elements for the entiremodel, the full-space WFMBEM uses 42.33 s of the CPUtime and 160 MB of the memory, but the present half-spaceWFMBEM only uses 23.19 s of the CPU time and 52 MB ofthe memory.

Fig. 6 Sound pressure on the central line of the box model(k = 1.0 m−1)

Fig. 7 CPU times used to solve the box model

The efficiency of the present half-space WFMBEM isfurther compared with that of the full-space WFMBEM inTable 1, where “Number of elements” also denotes that ofthe entire model, “CPU” and “RAM” represent the CPUtimes and memory requirements, respectively. “Ratios” inTable 1 are the ratios of the CPU times and memory require-ments between the half-space and full-space WFMBEM ap-proaches. It is demonstrated in Table 1 that the CPU times


of the present half-space WFMBEM are about 1/2 of thoseof the full-space WFMBEM, and the memory requirementsof the present half-space WFMBEM are only about 1/3 ofthose of the full-space WFMBEM for the rectangular boxexample.

5.2 Exterior acoustic field of a pulsating sphere

A sphere of radius a pulsating with a uniform normal veloc-ity vn in an infinite domain is chosen as an exterior acousticwave example in this section to verify the present widebandhalf-space FMBEM approach. The analytical solution to thesound pressure at a distance r from the center of the sphereis given as

p(r) = vniρcka2

r(1 − ika)eik(r−a). (48)

Fig. 8 Memory required to solve the box model

Table 1 Result comparisons for the box example

NumberHalf-space WFMBEM Full-space WFMBEM Ratios

CPU/s RAM/MB CPU/s RAM/MB CPU RAM

9 600 47.39 117 80.45 373 0.589 0.314

21 600 98.84 280 169.88 1 022 0.582 0.274

38 400 196.09 589 338.14 1 688 0.580 0.349

60 000 298.22 832 519.80 2 148 0.574 0.387

In the numerical analysis, the radius a = 1.0 m and thenormal velocity vn = 1.0 m/s. The coefficient c in Eq. (46)is also set to 5 for single precision, and the tolerance forconvergence of the iterative solver GMRES is set to 10−3.Since this infinite problem is plane-symmetric, the presentwideband half-space FMBEM can be applied. Sound pres-sures at an internal point with r = 6.0 m are calculated andcompared with the analytical solutions in Fig. 9 for a wavenumber range from 0 to 10.0 m−1, where the element num-ber is 10 800 for the entire model, versus 5 400 for the halfmodel. It is shown in Fig. 9 that the results obtained by thepresent wideband half-space FMBEM with the CBIE formu-lation (Half-space WFMBEM (CBIE)) follow the analytical

Fig. 9 Sound pressure at an internal point with r = 6a

solutions closely except in the vicinity of fictitious eigen-frequencies (i.e., k = π, 2π, 3π, · · · ) where the predicted re-sults become completely erroneous. However, if the Burton–Miller formulation is employed to tackle the fictitious eigen-frequency problem, the numerical results of both the half-space CBEM and WFMBEM approaches are very close andaccurate over the whole wave number range.

The relative errors of the results of different BEM ap-proaches are compared in Fig. 10 for k = 1.0 m−1, where“Number of elements” indicates that of the entire model andthe number of elements for the half model is just half of thatof the entire model. It is demonstrated that the relative er-rors of CBEM-LU, CBEM-GMRES and FMBEM results are

Fig. 10 Relative errors of the solutions for the pulsating sphereexample


very close to each other for both full- and half-space pul-sating sphere model, and decrease very fast as the elementnumber increases. Thus, although piecewise constant ele-ment usually has a bigger discretization error in compari-son with other high-order elements, good results can alsobe obtained for acoustic wave problems by using a finermesh discretization. The accuracy and efficiency of differ-ent BEM approaches are further compared in Table 2 fordifferent wave numbers, where the half model with 5 400piecewise constant triangular elements is used. As for theCBEM approach, the GMRES solver is employed to solve

the problems. Moreover, “Low”, “High” and “Wide” denotethe low-frequency, high-frequency and wideband FMBEMapproaches, respectively. “—” indicates that the solution isnot correct due to the instability of the high-frequency ap-proach for low frequency analysis. It is also demonstratedthat the efficiency of the low-frequency FMBEM decreasesrapidly with increasing wave number k. However, the wide-band version which is obtained by combining the low- andhigh-frequency approaches can solve the problem accuratelyand efficiently for all frequencies under consideration in thisexample.

Table 2 Comparisons of accuracy and efficiency between different BEM approaches

Wave number Relative error/% CPU time/s

k/m−1 CBEM Low High Wide CBEM Low High Wide

0.01 0.059 52 0.059 52 — 0.059 52 98.81 20.39 — 20.44

0.5 0.051 47 0.051 54 0.051 25 0.051 54 98.82 24.39 17.00 24.91

2.5 0.049 28 0.049 45 0.045 37 0.049 69 98.72 32.20 18.14 24.89

5.0 0.069 65 0.069 30 0.065 39 0.065 35 98.94 43.16 26.63 36.94

10.0 0.068 87 0.068 05 0.064 91 0.064 91 99.09 84.61 47.92 47.75

Next, the half-space/semi-infinite acoustic wave prob-lem which is also analyzed in Ref. [26] is used to furtherdemonstrate the efficiency of the present wideband half-space FMBEM approach. The sphere above the infinite rigidplane is discretized by using 10 800 piecewise constant tri-angular elements. The full-space model which is analyzedin the wideband full-space FMBEM approach is obtained bypositioning an identical sphere in the image domain sym-metrical to the infinite plane. The distance d between thecenter of the sphere and the infinite plane increases from 0to 10.0 m. The case of d = 0 denotes that the sphere is cutinto two equal parts by the infinite plane, and only the up-per hemisphere is discretized by using 5 400 piecewise con-stant triangular elements for the present wideband half-spaceFMBEM approach. The GMRES solver here terminates theiterations when the residual is below the tolerance of 10−5,the same as the value used in Ref. [26].

Comparisons of relative errors and CPU times betweenthe half-space and full-space WFMBEM approaches areshown in Table 3, in which “Error/%” denotes the percentagerelative error of the sound pressure on the infinite plane at asample point located at (5.0 m, 0, 0) for k = 0.5 m−1. “Ratio”represents the ratio of total CPU time between the half-spaceand full-space WFMBEM. It is observed that the relativeerrors of the present half-space WFMBEM results are veryclose to those of the full-space WFMBEM results. It is alsoknown that the total CPU times for the present half-spaceWFMBEM are less than the total CPU times used with thefull-space WFMBEM. Moreover, the ratios of CPU times be-tween the half-space and full-space WFMBEM approachesare smaller than those presented in Ref. [26], which is caused

by the fact that the present half-space approach avoids calcu-lating and translating the local expansion coefficients in theimage domain which is required in the approach proposed inRef. [26].

Table 3 Result comparisons for the pulsating sphere examples(k = 0.5 m−1)

Distance Half-space WFMBEM Full-space WFMBEMRatio

d Error/% CPU/s Error/% CPU/s

0 0.051 54 24.91 0.051 74 57.48 0.433

1 1.359 35 68.11 1.361 24 105.50 0.646

2 1.917 32 62.17 1.918 95 93.72 0.663

5 0.830 89 62.23 0.830 16 91.14 0.683

10 0.595 53 62.22 0.597 89 81.97 0.759

5.3 Scattered acoustic field of noise barriers

After verifying the accuracy and efficiency of the presentwideband half-space FMBEM approach through the exam-ples of rectangular box and pulsating sphere, we employ itto test the performances of noise barriers which stand on aninfinite flat ground. Although 2D simulation has often beenapplied for estimating the attenuation effect of noise barrierswith line sources, sometimes it may not be accurate in realapplications as it is based on the assumption that the barri-ers and the sources are straight and infinitely long. Due tothe high computational cost, the 3D CBEM can only be usedto simulate noise barriers with very short lengths. However,with the development of FMBEM, long and curved barrierscan now be modeled.


In the following numerical analysis of noise barriers,the coefficient c in Eq. (46) is also set to 5 for single preci-sion, and the tolerance for convergence of the GMRES solveris set to 10−3. The barrier and ground surfaces are modeledas rigid surfaces, i.e., normal velocities are specified as zeroon them, and a unit point source is placed 10.0 m away fromthe barrier (i.e., a = 10.0 m) and 1.0 m above the ground. Inthe first noise barrier example as depicted in Fig. 11, the sizeof the barrier is set to 10.0 m × 1.0 m × 5.0 m (l × w × h),and 30 720 piecewise constant triangular elements are usedto discretize the barrier surface. Figures 12 and 13 illustratethe sound pressure level (SPL) distributions on the plane ofx3 = 0 when the excitation frequencies of the point sourceare 20.0 Hz and 100.0 Hz, respectively.

Fig. 11 A straight noise barrier model

Fig. 12 Noise distribution around the straight barrier (excitationfrequency: 20 Hz)

Fig. 13 Noise distribution around the straight barrier (excitationfrequency: 100 Hz)

A quarter-circular noise barrier as shown in Fig. 14 withw = 1.0 m, h = 5.0 m and α = π/4 is chosen as the secondnoise barrier example. 38 747 piecewise constant triangularelements are applied to discretize the barrier surface. TheSPL distributions on the plane of x3 = 0 are illustrated inFigs. 15 and 16 for the frequencies of 20.0 Hz and 100.0 Hz,

respectively. From Figs. 12, 13, 15 and 16, one can findthat the performance of noise barrier greatly depends on itsshapes. The results and performances of the present wide-band half-space FMBEM in simulations of the noise barriersare summarized in Table 4. It is observed that the presentwideband half-space FMBEM approach is efficient and ableto handle such large-scale practical problems.

Fig. 14 3D and 2D views for a quarter-circular noise barrier model

Fig. 15 Noise distribution around the curved barrier (excitationfrequency: 20 Hz)

Fig. 16 Noise distribution around the curved barrier (excitationfrequency: 100 Hz)


Table 4 Result comparisons for the noise barrier example

Number of SPL atCPU/s

elements receiver/dB

Straight barrier (20 Hz) 30 720 54.7 195.8

Straight barrier (100 Hz) 30 720 43.8 326.2

Curved barrier (20 Hz) 38 747 52.5 402.8

Curved barrier (100 Hz) 38 747 43.2 682.1

6 Conclusions

A novel wideband fast multipole boundary element approachto 3D half-space/plane-symmetric acoustic wave problemshas been presented in this paper. Numerical examples clearlydemonstrate the accuracy, efficiency and potential of thepresent approach for solving large-scale half-space an alsoplane-symmetric acoustic wave problems.

The half-space fundamental solution is used in theboundary integral equations, which ensures that the treestructure required in the fast multipole algorithm can be builtfor the boundary elements of the structure in the real domainonly. Furthermore, a set of symmetric relations betweenthe multipole expansion coefficients of the real and imagedomain are derived, and the half-space fundamental solu-tion is modified for the purpose of applying such symmet-ric relations to avoid calculating, translating and saving themultipole/local expansion coefficients of the image domain.Also, the wideband fast multipole algorithm is implementedso that the present method is accurate and efficient for bothhigh- and low-frequency half-space acoustic wave problems.The Burton–Miller method is employed to conquer the non-uniqueness difficulty which is observed in solving exterioracoustic wave problems based on the conventional boundaryintegral formulation.

Further studies can be implemented for more com-plicated and practical half-space/plane-symmetric acousticwave problems by the developed code. Although, this paperis concentrated on the acoustic state analysis, the approachcan also be applied easily to the design and sensitivity anal-ysis of half-space/plane-symmetric acoustic wave problems.

Acknowledgements Many thanks must be given to Prof. ToshiroMatsumoto and Dr. Toru Takahashi of Nagoya University for theirgreat academic helps during the graduate study of the first author.

Appendix: Symmetric relations of multipole moments

In order to obtain the symmetric relations of low-frequency mo-ments, we employ (r, θ, φ) to represent the spherical coordinates of

the vector−→Oy as shown in Fig. 17. Hence, Im

n (k,−→Oy) can be given as

Imn (k,−→Oy) = cm

n jn(kr)Pmn (cos θ)e−imφ. (A1)

Fig. 17 The spherical coordinates of the vector−→Oy

Moreover, its normal derivative can be expressed as

∂Imn (k,−→Oy)

∂n(y)=

3∑i=1

∂Imn (k,−→Oy)

∂yini(y), (A2)

where ni(y) is the Cartesian component of the normal vector n(y).By using the chain rule of differentiation, ∂Im

n /∂yi can be obtainedas

∂Imn

∂y1= cm

n

[− cos φ

rjn(kr)Pm+1

n−1 (cos θ) +meiφ

r sin θjn(kr)Pm

n (cos θ)

−k sin θ cos φ jn+1(kr)Pmn (cos θ)

]e−imφ, (A3)

∂Imn

∂y2= cm

n

[− sinφ

rjn(kr)Pm+1

n−1 (cos θ) − imeiφ

r sin θjn(kr)Pm

n (cos θ)

−k sin θ sinφ jn+1(kr)Pmn (cos θ)

]e−imφ, (A4)

∂Imn

∂y3= cm

n

[n − mr

cos θ jn(kr)Pmn (cos θ) − k cos θ jn+1(kr)Pm

n (cos θ)

+sin θ

rjn(kr)Pm+1

n (cos θ)]e−imφ. (A5)

Therefore, without any loss of generality for half-space/plane-symmetric problems, if we consider the infinite/symmetry plane as

x3 = const, the spherical coordinates of the mirror vector−→Oy can be

written as (r,π − θ, φ), then it is easy to obtain

Imn (k,−→Oy) = (−1)n+mIm

n (k,−→Oy), (A6)

∂Imn (k,−→Oy)

∂y1= (−1)n+m ∂Im

n (k,−→Oy)

∂y1, (A7)

∂Imn (k,−→Oy)

∂y2= (−1)n+m ∂Im

n (k,−→Oy)

∂y2, (A8)

∂Imn (k,−→Oy)

∂y3= (−1)n+m+1 ∂Im

n (k,−→Oy)

∂y3. (A9)

Considering the following mirror vector

n(y) = (n1(y), n2(y),−n3(y)), (A10)

we can obtain


∂Imn (k,−→Oy)

∂n(y)= (−1)n+m ∂Im

n (k,−→Oy)

∂n(y). (A11)

Thus, we have the following symmetric relations

Mmn (k,O) = (−1)n+mMm

n (k,O). (A12)Next, we will derive the symmetric relations for high-

frequency moments. As depicted in Fig. 18, sphere O is a unitsphere for integration in real domain and sphere O is its mirrorimage. Moreover, P1 and P2 are the integration points on sphereO, P3 and P4 are the symmetric points of P1 and P2 on sphere O.Therefore, it is easy to obtain

kkkP3 ·−→Oy = kkkP1 ·

−→Oy, (A13)

kkkP4 ·−→Oy = kkkP2 ·

−→Oy, (A14)

kkkP3 · n(y) = kkkP1 · n(y), (A15)

kkkP4 · n(y) = kkkP2 · n(y), (A16)

where kkkP1 , kkkP2 , kkkP3 and kkkP4 are the outward unit vectors at P1, P2,P3 and P4.

Fig. 18 The unit sphere and its mirror image

Considering the definition of high-frequency moments(Eq. (29)), we can easily obtain the following symmetric relationsfor high-frequency moments

F(k, kkkP3 ,O) = F(k, kkkP1 ,O), (A17)

F(k, kkkP4 ,O) = F(k, kkkP2 ,O). (A18)

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a wideband fast multipole boundary element method for half-space/plane-symmetric acoustic wave...

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