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A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems
Computers and Structures xxx (2018) xxx–xxx
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Computers and Structures journal homepage: www.elsevier.com/locate/compstruc
A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems Wenzhen Qu a, Changjun Zheng b, Yaoming Zhang a, Yan Gu c,⇑, Fajie Wang d a
School of science, Shandong University of Technology, Zibo 255049, China Institute of Sound and Vibration Research, Hefei University of Technology, Hefei 230009, China c College of Mathematics, Qingdao University, Qingdao 266071, China d International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 210098, China b
a r t i c l e
i n f o
Article history: Received 26 February 2018 Accepted 1 June 2018 Available online xxxx Keywords: Singular boundary method Fast multipole Acoustic problem Large scale
a b s t r a c t In this paper, we present a new fast meshless method, called as wideband fast multipole singular boundary method (FMSBM), for three-dimensional acoustic problems. The wideband FMSBM applies a partial wave expansion formulation in low frequency regime and a plane wave expansion formulation in high frequency regime. The present method is efficient and accurate for a wider range of frequencies compared with the existing FMSBM approaches. In addition, the method avoids large number of element integrations in the boundary element method for resolving the complicated acoustic model, which can further reduce the computational complexity. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction The boundary element method (BEM) has been regarded as a powerful technique for the numerical solution of problems in computational acoustics governed by the Helmholtz equation [1–6]. The use of the BEM has several advantages over the finite element method (FEM) for problems of interest, especially in requiring only the boundary discretization and the accurate modelling of infinite domains [7–11]. The non-symmetric dense matrices appearing in the solution of the traditional BEM restrict its application to small-scale problems. To break through this bottleneck, the fast multipole method (FMM) [12,13] was introduced to improve the efficiency and reduce the memory requirement of the method [5,14–18]. However, the BEM still encounters a time-consuming issue of a large amount of numerical integrations arising from the discretization of boundary integral equations for large-scale problems [19]. During the past few years, many researchers have paid attention to the meshless methods without requirement of domain and boundary discretization. The method of fundamental solutions (MFS) [20] as a typical meshless boundary collocation approach is a competitive alternative because of its simple mathematical expression and high precision. The MFS accelerated by the FMM and the adaptive cross approximation has been applied to the simulation of ⇑ Corresponding author.
large-scale problems [21,22]. Unfortunately, the traditional MFS encounters the problem of how to place the fictitious boundary outside physical domain, especially for three-dimensional complex boundary. To overcome this drawback, various numerical approaches were developed, such as regularized meshless method (RMM) [23], modified method of fundamental solutions (MMFS) [24], and singular boundary method (SBM) [25]. Among these approaches, the SBM is mathematically simple, easy-to-program, and integration-free, and has been successfully applied to solutions of various physical problems [26–34]. Thanks to these advantages, the SBM approach is less time-consuming and more applicable for complex-shaped three-dimensional domain problems than the BEM. In addition, the SBM eliminates the fictitious boundary in the MFS and becomes numerically more stable than the MFS because of better conditioned interpolation matrix. The SBM approximates physical variables by using a linear combination of the fundamental solution of the governing equation, and its solution also generates a full interpolation matrix as in the BEM. To overcome highly computational cost of the system of equations with the full matrix, Qu et al. [35,36] developed the fast multipole singular boundary method (FMSBM) to reduce the CPU times and memory requirements of the SBM when solving three-dimensional large-scale acoustic problems. The CPU times of the FMSBM in [35] called as traditional approach is increased from OðNÞ (N is the dimensionality of the matrix) to OðN 2 Þ when being used to solve the problems in high-frequency regime. The
E-mail address: [email protected] (Y. Gu). https://doi.org/10.1016/j.compstruc.2018.06.002 0045-7949/Ó 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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diagonal form FMSBM (DF-FMSBM) in [36] is unstable when being applied to low frequency problems. The FMMs based on the partial wave expansion [37] and the plane wave expansion [38] fail in some way outside their preferred frequency regime. In [13], Cheng et al. firstly constructed a wideband FMM by switching the two FMM approaches depending on the level in the tree structure, which can deal with the above mentioned problems. After then, many researches [39–42] used the wideband FMM to accelerate the BEM for the solution of acoustic radiation, acoustic scattering, and acoustic shape sensitivity analysis. In this paper, we combine the wideband FMM and the SBM to construct a wideband FMSBM for three-dimensional acoustic problems. The present approach combines advantages of the traditional FMSBM [35] and the DF-FMSBM [36], which avoids the rapidly increasing CPU times of the former for high-frequency regime and instability of the latter for low-frequency regime. The wideband FMSBM has an OðNÞ efficiency if low-frequency computations dominate and an OðN log NÞ efficiency if high-frequency computations dominate. The numerical results of acoustic pressures for several numerical experiments clearly illustrate that the developed methodology is accurate and efficient. The outline of this paper is organized as follows. Section 2 provides the details of the wideband FMSBM formulations. Section 3 presents three numerical experiments, including a scattering model from a dolphin with no available analytical solution. Section 4 concludes the paper.
For the SBM, the acoustic pressure and its normal gradient can be approximated by using a linear interpolation of the fundamental solution of the Helmholtz equation, which are respectively given as [35] m X /j Gðxi ; yj Þ þ /i U i ;
pðxi Þ ¼
i ¼ 1; 2; . . . ; m;
ð7Þ
j¼1 j–i
m X @Gðxi ; yj Þ /j þ /i Q i ; @nxi j¼1
qðxi Þ ¼
i ¼ 1; 2; . . . ; m;
ð8Þ
j–i
m where fxi gm i¼1 and fyj gj¼1 are respectively collocation and source
points, m the number of boundary points (collocation or source the undetermined coefficients, Gðxi ; yj Þ the fundapoints), f/j gm j¼1 mental solution given by
Gðxi ; yj Þ ¼
1 eikkxi yj k2 ; 4p kxi yj k2
ð9Þ
and U i ; Q i the origin intensity factors expressed as
Ui ¼
Qi ¼
1 ‘i
Z
Si
j 2‘i
þ
Gðxi ; yÞdSy ; 1 ‘i
Z Si
@Gðxi ; yÞ dSy ; @nxi
ð10Þ
ð11Þ
in which ‘i denotes the area of Si , namely, influence domain of the origin intensity factors U i and Q i at source point xi , j is set to
2. Formulations of the wideband FMSBM 2.1. The SBM formulations
j¼
3
1;
for interior problems;
1; for exterior problems:
ð12Þ
In a homogeneous isotropic acoustic medium V 2 R , the propagation of time-harmonic acoustic waves can be described by the Helmholtz equation
With the help of Eqs. (7) and (8), we can form a system of equations as follows
r2 p þ k2 p ¼ 0;
AU ¼ b;
p2V
ð1Þ
where p is the acoustic pressure, and k denotes the wave number expressed as
k ¼ 2pf =c
ð2Þ
in which f is the frequency of acoustic wave, and c is the wave speed. The boundary conditions are imposed as
pðxÞ ¼ p1 ðxÞ; qðxÞ ¼
x 2 S1 ;
@pðxÞ ¼ q1 ðxÞ; @nx
ð3Þ x 2 S2 ;
ð4Þ
in which nx is the outward normal vector at point x, p1 ðxÞ; q1 ðxÞ are known functions, and the whole boundary of the domain V consists of S1 and S2 . The acoustic pressures p for the radiation, scattering and mixed models are respectively equivalent to the following relationships
8 for radiation; > < pR ¼ pT ; p ¼ pS ¼ pT pI ; for scattering; > : pR þ pS ¼ pT pI ; for both;
ð13Þ
in which A is the interpolation matrix of the SBM, U the vector composed by the unknown coefficients, b the boundary condition of interested acoustic problems. 2.2. The wideband FMSBM formulations In this section, we introduce the wideband FMM to accelerate the matrix-vector product in Eq. (13), and the iterative solver of generalized minimal residual method (GMRES) is then employed to solve the system of linear equations of the SBM. The fundamental solution in the wideband FMSBM is respectively expressed as into two parts: (1) a partial wave expansion formulation in low frequency regime; (2) a plane wave expansion formulation in high frequency regime. Details of the implementation of the developed method are described as follows. Fig. 1 plots the source point, collocation point, and points of multipole and local expansions in the
ð5Þ
where the subscripts R; S; I; T respectively denote the radiation, scattering, incident and total waves. In addition, the acoustic pressure p for exterior acoustic wave problems has to satisfy the Sommerfeld radiation condition [43] as follows
@pðrÞ ikpðrÞ ¼ 0; lim r r!1 @r in which i ¼
pffiffiffiffiffiffiffi 1, and r ¼ kxk2 .
ð6Þ Fig. 1. Expansion points and the boundary nodes in the wideband FMSBM.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
wideband FMSBM, and these points will be used in the following derivation of formulations of the present method. In the wideband FMSBM, the fundamental solution at low frequency is expanded as a partial wave expansion formulation [37]
Gðxi ; yj Þ ¼ h
1 X n X
! m ! ð2n þ 1Þf m n ðk; Oyj Þg n ðk; Oxi Þ;
where points O0 ; x1 ; x2 are provided in Fig. 1, and
Kðn; n0 ; m; m0 Þ ¼ fljl 2 Z; n þ n0 l : even; maxfjm þ m0 j; jn n0 jg 6 l 6 n þ n0 g; ð24Þ
ð14Þ
n¼0 m¼n
n0 nþl
in which h ¼ ik=4p, O is a point close to collocation point yj as
W n;n0 ;m;m0 ;l ¼ ð2l þ 1Þi
n n0 0 0
l 0
l ; m m0
n n0 m m0
ð25Þ
m
shown in Fig. 1, functions f n and g m n are respectively defined as
8 ! m < f m ðk; Oy j Þ ¼ jn ðkr 1 ÞY n ðh1 ; /1 Þ; n ! ð1Þ : g m ðk; Ox Þ ¼ h ðkr ÞY m ðh ; / Þ; i
n
2
n
n
2
ð15Þ
2
ð1Þ
where jn is the nth order spherical Bessel function, hn the nth order Hankel function of the first kind, ðr 1 ; h1 ; /1 Þ and ðr 2 ; h2 ; /2 Þ respectively the spherical coordinates of collocation point y and source ! ! point x with origin point O, namely r 1 ¼ j Oy j and r2 ¼ j Ox j, Y m n the spherical harmonics as
Ym n ðh; /Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn mÞ! m P ðcos hÞeim/ ; ¼ ðn þ mÞ! n
ð16Þ
0
0
n n l n n l both denote the and 0 0 0 m m0 m m0 Wigner 3j symbol [44]. With the help of Eqs. (21)–(23), we recast Eq. (17) as in which
p‘i
)
¼ h
q‘i
1 X n X
! m 2 2 ð2n þ 1ÞLm n ðk; x ÞH n ðk; x xi Þ;
ð26Þ
n¼0 m¼n
where p‘i ¼
P‘
j¼1 pij ,
q‘i ¼
P‘
j¼1 qij ,
and
8 ! 2 > for p‘i ; < f m n ðk; x xi Þ ! m 2 ! Hn ðk; x xi Þ ¼ 2 > : @f mn ðk;x xi Þ for q‘i : @nx
ð27Þ
i
in which P m n denotes the associated Legendre functions. Based on Eq. (14), we can obtain
pij qij
)
¼ h
1 X n X
! m ! ð2n þ 1ÞM m n ðk; Oyj ÞN n ðk; Oxi Þ;
Next, the fundamental solution in the wideband FMSBM at high frequency is expanded into the plane wave expansion [38] as follows
ð17Þ
n¼0 m¼n
where pij ¼ Gðxi ; yj Þ/j , qij ¼ ½@Gðxi ; yj Þ=@nxi /j , ! Nm n ðk; Oxi Þ are respectively expressed as
! m ! Mm n ðk; Oy j Þ ¼ f n ðk; Oyj Þ/j ;
! Mm n ðk; Oyj Þ
and
ð18Þ
ð19Þ
i
! Mm n ðk; Oyj Þ
is called as the multipole moment of the colloin which cation point yj . Then, the moment of a leaf centered at O can be given as ‘ X ! ¼ Mm n ðk; Oyj Þ;
ð20Þ
j¼1
where ‘ represents the number of source points in the leaf. The multipole-to-multipole (M2M), multipole-to-local (M2L), and local-to-local (L2L) translations of the FMM in the low frequency regime are given as 0 Mm n ðk; O Þ ¼
1 X n0 X
X
m0
ð2n0 þ 1Þð1Þ
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ mm0
W n;n0 ;m;m0 ;l f l 1 Lm n ðk; x Þ ¼
1 X n0 X
X
! 0 ðk; O0 OÞMm n0 ðk; OÞ;
ð2n0 þ 1Þð1Þ
¼
ð21Þ
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ
1 X n0 X
X
0
ð2n þ 1Þð1Þ
W n0 ;n;m0 ;m;l f l
1 ! X ! ð1Þ n ^ ; Ox1 Þ ¼ ^v ^ Þ; Tðk; n i ð2n þ 1Þhn ðkjOx1 jÞPn ðn
ð29Þ
ð22Þ
pij qij
)
Z ! ! ! ^ ; Ox1 ÞMðk; n ^ ; x1 xi ÞTðk; n ^ ; Oyj ÞdS; ¼ k Wðk; n
ð30Þ
S0
in which pij ¼ Gðxi ; yj Þ/j , qij ¼ ½@Gðxi ; yj Þ=@nxi /j , and
8 ! 1 > > < eikn^ x xi ! 1 ^ ; x xi Þ ¼ Wðk; n ! 1 > > : @eikn^ x xi @nxi
for pij ;
ð31Þ
for qij ;
! ! b ^ ; Oyj Þ ¼ eikn^ Oyj /j : Mðk; n
ð32Þ
! b ^ ; Oyj Þ is where O is the expansion point as show in Fig. 1, and Mðk; n also called as the multipole moment of the point yj . It should be b is used to distinguish from multipole noted that superscript in M
‘ X
! b ^ ; Oyj Þ; Mðk; n
ð33Þ
j¼1
in which ‘ is the number of source points contained in the leaf. ^ M), ^ multipole-to-local (M2 ^ ^ The multipole-to-multipole (M2 L),
m
! 0 1 ðk; x1 x2 ÞLm n0 ðk; x Þ;
! ! ^ ¼ Ox1 =jOx1 j, and P n denotes the nth degree Legendre where v polynomial. By using Eq. (28), we have
b ^ ; OÞ ¼ Mðk; n
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ mm0
^ in which k ¼ ik=ð16p2 Þ, S0 denotes the boundary of a unit sphere, n is assumed to be the outward unit vector on S0 , and
moment of Eq. (18) for the FMM in low frequency regime. Then, we can construct the moment of a leaf centered at O as
mþm0
! 0 0 0 W n0 ;n;m0 ;m;l g mþm ðk; O0 x1 ÞMm n0 ðk; O Þ; l 2 Lm n ðk; x Þ
ð28Þ
S0
n¼0
8 ! > for pij ; < gm n ðk; Oxi Þ ! m ! Nn ðk; Oxi Þ ¼ > : @gmn ðk;Oxi Þ for qij ; @nx
Mm n ðk; OÞ
Z ! ! ! 1 ^ ; Ox1 Þeikn^ Oyj dS; Gðxi ; yj Þ ¼ k eikn^ x xi Tðk; n
ð23Þ
and local-to-local (^ L2^ L) translations for the FMM at high frequency are respectively expressed as
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
! 0 b b ^ ; OÞ; ^ ; O0 Þ ¼ eikn^ O O Mðk; n Mðk; n
ð34Þ
! ^Lðk; n b ^ ; x1 Þ ¼ Tðk; n ^ ; O 0 x1 Þ Mðk; ^ ; O0 Þ; n
ð35Þ
! 1 2 ^Lðk; n ^ ; x1 Þ; ^ ; x2 Þ ¼ eikn^ x x ^Lðk; n
ð36Þ
^ M, ^ M2 ^ ^ in which points O ; x ; x are plotted in Fig. 1. Based on M2 L, ^ ^ and L2 L translations, Eq. (30) can be rewritten as 0
p‘i q‘i
)
1
2
Z ! ^ ; x2 ÞdS; ^ ; x2 xi Þ^Lðk; n ¼ k Wðk; n
As the first case, we consider a classical scattering model: an incident plane wave pin ¼ p0 eikz along the positive z axis impinges a rigid sphere with radius R ¼ 1:0 m and center at ð0; 0; 0Þ. The speed of acoustic wave propagation in air is 340 m=s. For this model, the exact solution of the sound pressure is expressed in the following form [40] of polar coordinates ps ðr; hÞ ¼
1 X
n
i ð2n þ 1Þp0
n¼0
ð37Þ
njn1 ðkRÞ ðn þ 1Þjnþ1 ðkRÞ ð1Þ
ð1Þ
nhn1 ðkRÞ ðn þ 1Þhnþ1 ðkRÞ
ð1Þ
hn ðkrÞP n ðcos hÞ;
ð41Þ
S0
P P where p‘i ¼ ‘j¼1 pij , and q‘i ¼ ‘j¼1 qij . Finally, we establish the wideband FMSBM by using the following formulation [40] to transform the moments of the FMM at low frequency to those at high frequency
b ^ ; OÞ ¼ Mðk; n
1 X n X
n
^ m ð2n þ 1Þi Y m n ðnÞM n ðk; OÞ;
ð38Þ
n¼0 m¼n
and using the following equation [40] to convert the local expansion coefficients of the FMM in high frequency regime to those in low frequency regime n
1 Lm n ðk; x Þ ¼
i 4p
Z S0
^ ^ ^ 1 Ym n ðnÞLðk; n; x ÞdS:
ð39Þ
^ and ^ Here, Eqs. (38) and (39) are respectively called as the M2M L2L translations. In the wideband FMSBM, we compute the acoustic pressure and its normal gradient by directly using Eqs. (7) and (8) when the source point yj is close to the collocation point xi . Thus, the origin intensity factors U i and Q i are calculated directly as in the conventional SBM (CSBM). Through the above-mentioned procedures, new SBM formulations accelerated by the wideband FMM have been finally developed. The remaining details of the implementation of the wideband FMSBM, including the upward pass and downward pass, are same as in the wideband FMBEM, which can be found in Ref. [40]. In the upward and downward pass, we require to set the threshold D ¼ 0:2k (k is the acoustic wavelength) of conversion between moments of low-frequency and highfrequency [45]. The switching level s in the tree structure satisfies s
3.1. Scattering from a rigid sphere
sþ1
s
sþ1
d P D and d < D (d and d respectively denote the edge lengths of cells at level s and s þ 1). In addition, a highly illconditioned system of linear equations is formed after the discretization of the present approach when using for acoustic problems at high frequency. Thus, we use the block diagonal preconditioner [46] to improve the convergence rate of the present method.
ð1Þ
in which j and h
are respectively the spherical Bessel and Hankel ð1Þ
functions of the first kind (subscripts of j and h denote their orders), and P n is the nth Legendre polynomial.In this case, p0 is set to 10 Pa and the maximum number of boundary nodes allowed in a childless box is set to 50. The frequency of acoustic wave is assumed to be f ¼ 100 Hz. Fig. 2 plots the error variation of the acoustic pressure at point ð0; 0; 4Þ calculated by the CSBM and the wideband FMSBM with an increasing number of boundary nodes. The numerical results illustrate these two approaches are stable and convergent. Moreover, the wideband FMSBM remains almost the same accuracy as the CSBM. By using the present approach for different frequencies, Table 1 provides the maximum relative errors of acoustic pressures at 400 points distributed on the surface of a sphere with radius r ¼ 4 m and center at the origin. As we can see, the wideband FMSBM can obtain accurate solutions at different frequencies. In Fig. 3, we make a comparison of CPU times consumed by the CSBM and the wideband FMSBM for different frequencies with an increasing number of boundary nodes. The CPU time of the CSBM is independent of the frequency, and thus only one case (100 Hz) is provided. We can observe that the curves of CPU times for the CSBM and the wideband FMSBM for four frequencies have the similar slopes with the trend lines 1 and 2, respectively. Those indicate O(NlogN) efficiency for the wideband FMSBM that is significantly superior to O(N3) for the CSBM. 3.2. Radiation from a car body As the second example, we consider the radiation from a simplified car body [47] as shown in Fig. 4. The dimension of the car is 4:1 m 2:0 m 1:1 m. The speed of acoustic wave propagation in air is 340 m=s. The exact solution [48] of this problem is assumed as
3. Numerical examples In this section, three numerical examples are provided to verify the proposed method. All the codes were written by Fortran 90 and run on a PC with an Intel Core i5-3230M (2.60 GHz) processor with 4 GB memory. The iterative solver (GMRES) will not be terminated until the residue is smaller than the tolerance of 103 [4,40,42]. The following error formula is used to test the accuracy of the proposed method
p pexact Relative error ¼ numerical pexact
ð40Þ
where pnumerical and pexact are respectively the numerical and analytical solutions. In addition, the HyperMesh is used to obtain the coordinates of boundary nodes for numerical examples 2 and 3.
Fig. 2. Relative errors of acoustic pressure at point ð0; 0; 4Þ.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx Table 1 The maximum relative errors of acoustic pressures with different frequencies. Frequency (Hz) Number of boundary nodes Maximum relative error
200 868 2.84546E02
1000 3208 2.65426E02
pðx; y; zÞ ¼
2000 12,892 3.28533E02
5000 51,022 4.65420E02
zeikr i ; 1 þ kr r2
ð42Þ
Fig. 3. CPU times consumed by the CSBM and the wideband FMSBM.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y2 þ z2 . The Neumann boundary condition is imposed on the surface of the car, which can be easily derived by using Eqs. (4) and (42). In this model, we distribute 101,414 boundary nodes on the surface of the car body. The maximum number of boundary nodes allowed in a childless box is set to 120. Fig. 5 shows the numerical error surface of acoustic pressures at frequencies f ¼ 300 Hz and 2000 Hz in a square domain f2 6 x 6 2; 2 6 y 6 2; z ¼ 2:5g, which is located in front of the car body. It can be found that the maximum errors of acoustic pressures at two frequencies 300 Hz and 2000 Hz are less than 0.2% and 2% respectively. In addition, we investigate relative errors of numerical results influenced by the present method with different maximum number (Max-Num) of boundary nodes allowed in a childless box. Table 2 lists the comparison of maximum relative errors and CPU times of acoustic pressures at the frequency
Fig. 4. The sketch of a simplified car body.
Fig. 6. The sketch of a dolphin.
(a) f = 300 Hz
(b) f = 2000 Hz
Fig. 5. The numerical error surface of acoustic pressures at frequencies (a) 300 Hz and (b) 2000 Hz in a square domain f2 6 x; y 6 2; z ¼ 2:5g.
Table 2 The maximum relative errors and CPU times of acoustic pressures with different Max-Num. Max-Num Maximum relative error CPU time (s)
40 1.89186E03 4.37482E+03
80 1.89233E03 4.10790E+03
120 1.89270E03 4.01226E+03
160 1.89325E03 4.19948E+03
200 1.88774E03 4.96435E+03
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
f ¼ 300 Hz in the square domain (see Fig. 5) with an increasing Max-Num. In the implementation of the present method, the tree structure with different levels are produced due to the different Max-Num, which leads to a slight effect on the accuracy of the method as seen in Table 2. Moreover, we can see that CPU time first decreases and then increases when increasing the Max-Num. Thus, it is necessary to test the wideband FMSBM program for different Max-Num on a specific computer for a typical problem for
determining the optimal Max-Num, and then one can employ for similar applications. Such testing has to be performed on different computer platforms for better efficiency. 3.3. Radiation and scattering from a dolphin As illustrated in Fig. 6, we study the radiation and scattering from a dolphin [49] of which dimension is 2 m in length, 0.73 m
Fig. 7. The distribution of 123,550 boundary nodes on the surface of a dolphin.
(a) f = 1000 Hz
(b) f = 2000 Hz
(c) f = 3000 Hz
(d) f = 4000 Hz
Fig. 8. The relative errors of acoustic pressures on the surface of a sphere when considering four different frequencies f ¼ 1000; 2000; 3000; 4000 Hz.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
in width, and 0.55 m in height. The speed of acoustic wave propagation in water is 1473 m=s. For this example, two cases are considered: (1) acoustic radiation model with known analytical solution; (2) acoustic scattering model with no available analytical solution. We have same following conditions for both cases: (a) 123,550 boundary nodes are distributed on the surface of the dolphin as shown in Fig. 7; (b) the maximum number of boundary nodes allowed in a childless box is set to 200 for the present method. Case 1: ; zÞ ¼ eikr =r is imposed on The Dirichlet boundary condition pð x; y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; y ; zÞ 2 þ z2 and ðx the surface of the dolphin, in which r ¼ x2 þ y denotes a boundary point. Thus, the exact solution is given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðx; y; zÞ ¼ eikr =r [48], and r ¼ x2 þ y2 þ z2 . This is an exterior radiation problem. In this simulation, four different frequencies f ¼ 1000; 2000; 3000; 4000 Hz are respectively considered. Fig. 8 plots the relative errors of acoustic pressures on the surface of a sphere with radius r ¼ 3 m and center at the origin. We can observe that the numerical errors calculated by the present method increase when the frequency increases. However, the satisfied results are still obtained even as f ¼ 4000 Hz. All these results indicate that the wideband FMSBM is a competitive method for solving the acoustic radiation problem with a complex boundary.
Fig. 9. Real part of acoustic pressure along the x-axis calculated by using the wideband FMBEM and the wideband FMSBM.
7
Case 2: The scattering from the dolphin is considered as the second case. We assume that the incident wave is a plane wave pin ¼ 1000eikx in the þx direction, and the boundary of the dolphin is rigid. The analytical solution is not available in this case. Thus, a comparison of numerical results obtained by the wideband FMSBM and FMBEM [40] is provided. The surface of the dolphin is discretized into 123,550 constant triangular elements for the wideband FMBEM. Figs. 9 and 10 respectively show the real and imaginary parts of acoustic pressures calculated by the wideband FMSBM and FMBEM. We can observe from these two figures that the numerical results of acoustic pressures obtained by the wideband FMSBM agree pretty well with those of the wideband FMBEM, indicating that the proposed method works well for the problem with complex geometric boundary and large number of boundary nodes. 4. Conclusions A wideband fast multipole singular boundary method (FMSBM) is developed to reduce the computational complexity and the memory requirement of the SBM by combining the wideband FMM and the SBM. In the proposed method, the fundamental solution is expanded as a partial wave expansion formulation in low frequency regime and a plane wave expansion formulation in high frequency regime. The wideband FMSBM combines advantages of the traditional FMSBM and the DF-FMSBM, which avoids the rapidly increasing CPU times of the former for high-frequency regime and instability of the latter for low-frequency regime. Moreover, compared with the wideband FMBEM, the proposed method distributes the boundary nodes on the surface and has no requirement of boundary elements, and meanwhile it is integration-free for the off-diagonal elements of the interpolation matrix. Numerical results clearly illustrate the potential of the wideband FMSBM for the solution of acoustic problems with complicated boundary, including the scattering model of a dolphin with no available analytical solution. However, the formulation of the present approach requires better strategies to control the number of iterations when solving the acoustic problems in high-frequency regime. In the further work, we will introduce a dual-level technique [50] for the present method to solve the highly ill-conditioning resulting from fullypopulated interpolation matrix at higher frequency. In addition, the proposed method based on the direct differentiation technique can be applied to the solution of acoustic shape sensitivity analysis. These are under intense investigation. Acknowledgements The work described in this paper was supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2017BA003) and the Doctoral Research Foundation of Shandong University of Technology (Grant No. 4041/416031). References
Fig. 10. Imaginary part of acoustic pressure along the x-axis calculated by using the wideband FMBEM and the wideband FMSBM.
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Contents lists available at ScienceDirect
Computers and Structures journal homepage: www.elsevier.com/locate/compstruc
A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems Wenzhen Qu a, Changjun Zheng b, Yaoming Zhang a, Yan Gu c,⇑, Fajie Wang d a
School of science, Shandong University of Technology, Zibo 255049, China Institute of Sound and Vibration Research, Hefei University of Technology, Hefei 230009, China c College of Mathematics, Qingdao University, Qingdao 266071, China d International Center for Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing 210098, China b
a r t i c l e
i n f o
Article history: Received 26 February 2018 Accepted 1 June 2018 Available online xxxx Keywords: Singular boundary method Fast multipole Acoustic problem Large scale
a b s t r a c t In this paper, we present a new fast meshless method, called as wideband fast multipole singular boundary method (FMSBM), for three-dimensional acoustic problems. The wideband FMSBM applies a partial wave expansion formulation in low frequency regime and a plane wave expansion formulation in high frequency regime. The present method is efficient and accurate for a wider range of frequencies compared with the existing FMSBM approaches. In addition, the method avoids large number of element integrations in the boundary element method for resolving the complicated acoustic model, which can further reduce the computational complexity. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction The boundary element method (BEM) has been regarded as a powerful technique for the numerical solution of problems in computational acoustics governed by the Helmholtz equation [1–6]. The use of the BEM has several advantages over the finite element method (FEM) for problems of interest, especially in requiring only the boundary discretization and the accurate modelling of infinite domains [7–11]. The non-symmetric dense matrices appearing in the solution of the traditional BEM restrict its application to small-scale problems. To break through this bottleneck, the fast multipole method (FMM) [12,13] was introduced to improve the efficiency and reduce the memory requirement of the method [5,14–18]. However, the BEM still encounters a time-consuming issue of a large amount of numerical integrations arising from the discretization of boundary integral equations for large-scale problems [19]. During the past few years, many researchers have paid attention to the meshless methods without requirement of domain and boundary discretization. The method of fundamental solutions (MFS) [20] as a typical meshless boundary collocation approach is a competitive alternative because of its simple mathematical expression and high precision. The MFS accelerated by the FMM and the adaptive cross approximation has been applied to the simulation of ⇑ Corresponding author.
large-scale problems [21,22]. Unfortunately, the traditional MFS encounters the problem of how to place the fictitious boundary outside physical domain, especially for three-dimensional complex boundary. To overcome this drawback, various numerical approaches were developed, such as regularized meshless method (RMM) [23], modified method of fundamental solutions (MMFS) [24], and singular boundary method (SBM) [25]. Among these approaches, the SBM is mathematically simple, easy-to-program, and integration-free, and has been successfully applied to solutions of various physical problems [26–34]. Thanks to these advantages, the SBM approach is less time-consuming and more applicable for complex-shaped three-dimensional domain problems than the BEM. In addition, the SBM eliminates the fictitious boundary in the MFS and becomes numerically more stable than the MFS because of better conditioned interpolation matrix. The SBM approximates physical variables by using a linear combination of the fundamental solution of the governing equation, and its solution also generates a full interpolation matrix as in the BEM. To overcome highly computational cost of the system of equations with the full matrix, Qu et al. [35,36] developed the fast multipole singular boundary method (FMSBM) to reduce the CPU times and memory requirements of the SBM when solving three-dimensional large-scale acoustic problems. The CPU times of the FMSBM in [35] called as traditional approach is increased from OðNÞ (N is the dimensionality of the matrix) to OðN 2 Þ when being used to solve the problems in high-frequency regime. The
E-mail address: [email protected] (Y. Gu). https://doi.org/10.1016/j.compstruc.2018.06.002 0045-7949/Ó 2018 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
diagonal form FMSBM (DF-FMSBM) in [36] is unstable when being applied to low frequency problems. The FMMs based on the partial wave expansion [37] and the plane wave expansion [38] fail in some way outside their preferred frequency regime. In [13], Cheng et al. firstly constructed a wideband FMM by switching the two FMM approaches depending on the level in the tree structure, which can deal with the above mentioned problems. After then, many researches [39–42] used the wideband FMM to accelerate the BEM for the solution of acoustic radiation, acoustic scattering, and acoustic shape sensitivity analysis. In this paper, we combine the wideband FMM and the SBM to construct a wideband FMSBM for three-dimensional acoustic problems. The present approach combines advantages of the traditional FMSBM [35] and the DF-FMSBM [36], which avoids the rapidly increasing CPU times of the former for high-frequency regime and instability of the latter for low-frequency regime. The wideband FMSBM has an OðNÞ efficiency if low-frequency computations dominate and an OðN log NÞ efficiency if high-frequency computations dominate. The numerical results of acoustic pressures for several numerical experiments clearly illustrate that the developed methodology is accurate and efficient. The outline of this paper is organized as follows. Section 2 provides the details of the wideband FMSBM formulations. Section 3 presents three numerical experiments, including a scattering model from a dolphin with no available analytical solution. Section 4 concludes the paper.
For the SBM, the acoustic pressure and its normal gradient can be approximated by using a linear interpolation of the fundamental solution of the Helmholtz equation, which are respectively given as [35] m X /j Gðxi ; yj Þ þ /i U i ;
pðxi Þ ¼
i ¼ 1; 2; . . . ; m;
ð7Þ
j¼1 j–i
m X @Gðxi ; yj Þ /j þ /i Q i ; @nxi j¼1
qðxi Þ ¼
i ¼ 1; 2; . . . ; m;
ð8Þ
j–i
m where fxi gm i¼1 and fyj gj¼1 are respectively collocation and source
points, m the number of boundary points (collocation or source the undetermined coefficients, Gðxi ; yj Þ the fundapoints), f/j gm j¼1 mental solution given by
Gðxi ; yj Þ ¼
1 eikkxi yj k2 ; 4p kxi yj k2
ð9Þ
and U i ; Q i the origin intensity factors expressed as
Ui ¼
Qi ¼
1 ‘i
Z
Si
j 2‘i
þ
Gðxi ; yÞdSy ; 1 ‘i
Z Si
@Gðxi ; yÞ dSy ; @nxi
ð10Þ
ð11Þ
in which ‘i denotes the area of Si , namely, influence domain of the origin intensity factors U i and Q i at source point xi , j is set to
2. Formulations of the wideband FMSBM 2.1. The SBM formulations
j¼
3
1;
for interior problems;
1; for exterior problems:
ð12Þ
In a homogeneous isotropic acoustic medium V 2 R , the propagation of time-harmonic acoustic waves can be described by the Helmholtz equation
With the help of Eqs. (7) and (8), we can form a system of equations as follows
r2 p þ k2 p ¼ 0;
AU ¼ b;
p2V
ð1Þ
where p is the acoustic pressure, and k denotes the wave number expressed as
k ¼ 2pf =c
ð2Þ
in which f is the frequency of acoustic wave, and c is the wave speed. The boundary conditions are imposed as
pðxÞ ¼ p1 ðxÞ; qðxÞ ¼
x 2 S1 ;
@pðxÞ ¼ q1 ðxÞ; @nx
ð3Þ x 2 S2 ;
ð4Þ
in which nx is the outward normal vector at point x, p1 ðxÞ; q1 ðxÞ are known functions, and the whole boundary of the domain V consists of S1 and S2 . The acoustic pressures p for the radiation, scattering and mixed models are respectively equivalent to the following relationships
8 for radiation; > < pR ¼ pT ; p ¼ pS ¼ pT pI ; for scattering; > : pR þ pS ¼ pT pI ; for both;
ð13Þ
in which A is the interpolation matrix of the SBM, U the vector composed by the unknown coefficients, b the boundary condition of interested acoustic problems. 2.2. The wideband FMSBM formulations In this section, we introduce the wideband FMM to accelerate the matrix-vector product in Eq. (13), and the iterative solver of generalized minimal residual method (GMRES) is then employed to solve the system of linear equations of the SBM. The fundamental solution in the wideband FMSBM is respectively expressed as into two parts: (1) a partial wave expansion formulation in low frequency regime; (2) a plane wave expansion formulation in high frequency regime. Details of the implementation of the developed method are described as follows. Fig. 1 plots the source point, collocation point, and points of multipole and local expansions in the
ð5Þ
where the subscripts R; S; I; T respectively denote the radiation, scattering, incident and total waves. In addition, the acoustic pressure p for exterior acoustic wave problems has to satisfy the Sommerfeld radiation condition [43] as follows
@pðrÞ ikpðrÞ ¼ 0; lim r r!1 @r in which i ¼
pffiffiffiffiffiffiffi 1, and r ¼ kxk2 .
ð6Þ Fig. 1. Expansion points and the boundary nodes in the wideband FMSBM.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
wideband FMSBM, and these points will be used in the following derivation of formulations of the present method. In the wideband FMSBM, the fundamental solution at low frequency is expanded as a partial wave expansion formulation [37]
Gðxi ; yj Þ ¼ h
1 X n X
! m ! ð2n þ 1Þf m n ðk; Oyj Þg n ðk; Oxi Þ;
where points O0 ; x1 ; x2 are provided in Fig. 1, and
Kðn; n0 ; m; m0 Þ ¼ fljl 2 Z; n þ n0 l : even; maxfjm þ m0 j; jn n0 jg 6 l 6 n þ n0 g; ð24Þ
ð14Þ
n¼0 m¼n
n0 nþl
in which h ¼ ik=4p, O is a point close to collocation point yj as
W n;n0 ;m;m0 ;l ¼ ð2l þ 1Þi
n n0 0 0
l 0
l ; m m0
n n0 m m0
ð25Þ
m
shown in Fig. 1, functions f n and g m n are respectively defined as
8 ! m < f m ðk; Oy j Þ ¼ jn ðkr 1 ÞY n ðh1 ; /1 Þ; n ! ð1Þ : g m ðk; Ox Þ ¼ h ðkr ÞY m ðh ; / Þ; i
n
2
n
n
2
ð15Þ
2
ð1Þ
where jn is the nth order spherical Bessel function, hn the nth order Hankel function of the first kind, ðr 1 ; h1 ; /1 Þ and ðr 2 ; h2 ; /2 Þ respectively the spherical coordinates of collocation point y and source ! ! point x with origin point O, namely r 1 ¼ j Oy j and r2 ¼ j Ox j, Y m n the spherical harmonics as
Ym n ðh; /Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn mÞ! m P ðcos hÞeim/ ; ¼ ðn þ mÞ! n
ð16Þ
0
0
n n l n n l both denote the and 0 0 0 m m0 m m0 Wigner 3j symbol [44]. With the help of Eqs. (21)–(23), we recast Eq. (17) as in which
p‘i
)
¼ h
q‘i
1 X n X
! m 2 2 ð2n þ 1ÞLm n ðk; x ÞH n ðk; x xi Þ;
ð26Þ
n¼0 m¼n
where p‘i ¼
P‘
j¼1 pij ,
q‘i ¼
P‘
j¼1 qij ,
and
8 ! 2 > for p‘i ; < f m n ðk; x xi Þ ! m 2 ! Hn ðk; x xi Þ ¼ 2 > : @f mn ðk;x xi Þ for q‘i : @nx
ð27Þ
i
in which P m n denotes the associated Legendre functions. Based on Eq. (14), we can obtain
pij qij
)
¼ h
1 X n X
! m ! ð2n þ 1ÞM m n ðk; Oyj ÞN n ðk; Oxi Þ;
Next, the fundamental solution in the wideband FMSBM at high frequency is expanded into the plane wave expansion [38] as follows
ð17Þ
n¼0 m¼n
where pij ¼ Gðxi ; yj Þ/j , qij ¼ ½@Gðxi ; yj Þ=@nxi /j , ! Nm n ðk; Oxi Þ are respectively expressed as
! m ! Mm n ðk; Oy j Þ ¼ f n ðk; Oyj Þ/j ;
! Mm n ðk; Oyj Þ
and
ð18Þ
ð19Þ
i
! Mm n ðk; Oyj Þ
is called as the multipole moment of the colloin which cation point yj . Then, the moment of a leaf centered at O can be given as ‘ X ! ¼ Mm n ðk; Oyj Þ;
ð20Þ
j¼1
where ‘ represents the number of source points in the leaf. The multipole-to-multipole (M2M), multipole-to-local (M2L), and local-to-local (L2L) translations of the FMM in the low frequency regime are given as 0 Mm n ðk; O Þ ¼
1 X n0 X
X
m0
ð2n0 þ 1Þð1Þ
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ mm0
W n;n0 ;m;m0 ;l f l 1 Lm n ðk; x Þ ¼
1 X n0 X
X
! 0 ðk; O0 OÞMm n0 ðk; OÞ;
ð2n0 þ 1Þð1Þ
¼
ð21Þ
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ
1 X n0 X
X
0
ð2n þ 1Þð1Þ
W n0 ;n;m0 ;m;l f l
1 ! X ! ð1Þ n ^ ; Ox1 Þ ¼ ^v ^ Þ; Tðk; n i ð2n þ 1Þhn ðkjOx1 jÞPn ðn
ð29Þ
ð22Þ
pij qij
)
Z ! ! ! ^ ; Ox1 ÞMðk; n ^ ; x1 xi ÞTðk; n ^ ; Oyj ÞdS; ¼ k Wðk; n
ð30Þ
S0
in which pij ¼ Gðxi ; yj Þ/j , qij ¼ ½@Gðxi ; yj Þ=@nxi /j , and
8 ! 1 > > < eikn^ x xi ! 1 ^ ; x xi Þ ¼ Wðk; n ! 1 > > : @eikn^ x xi @nxi
for pij ;
ð31Þ
for qij ;
! ! b ^ ; Oyj Þ ¼ eikn^ Oyj /j : Mðk; n
ð32Þ
! b ^ ; Oyj Þ is where O is the expansion point as show in Fig. 1, and Mðk; n also called as the multipole moment of the point yj . It should be b is used to distinguish from multipole noted that superscript in M
‘ X
! b ^ ; Oyj Þ; Mðk; n
ð33Þ
j¼1
in which ‘ is the number of source points contained in the leaf. ^ M), ^ multipole-to-local (M2 ^ ^ The multipole-to-multipole (M2 L),
m
! 0 1 ðk; x1 x2 ÞLm n0 ðk; x Þ;
! ! ^ ¼ Ox1 =jOx1 j, and P n denotes the nth degree Legendre where v polynomial. By using Eq. (28), we have
b ^ ; OÞ ¼ Mðk; n
n0 ¼0m0 ¼n0 l2Kðn;n0 ;m;m0 Þ mm0
^ in which k ¼ ik=ð16p2 Þ, S0 denotes the boundary of a unit sphere, n is assumed to be the outward unit vector on S0 , and
moment of Eq. (18) for the FMM in low frequency regime. Then, we can construct the moment of a leaf centered at O as
mþm0
! 0 0 0 W n0 ;n;m0 ;m;l g mþm ðk; O0 x1 ÞMm n0 ðk; O Þ; l 2 Lm n ðk; x Þ
ð28Þ
S0
n¼0
8 ! > for pij ; < gm n ðk; Oxi Þ ! m ! Nn ðk; Oxi Þ ¼ > : @gmn ðk;Oxi Þ for qij ; @nx
Mm n ðk; OÞ
Z ! ! ! 1 ^ ; Ox1 Þeikn^ Oyj dS; Gðxi ; yj Þ ¼ k eikn^ x xi Tðk; n
ð23Þ
and local-to-local (^ L2^ L) translations for the FMM at high frequency are respectively expressed as
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
! 0 b b ^ ; OÞ; ^ ; O0 Þ ¼ eikn^ O O Mðk; n Mðk; n
ð34Þ
! ^Lðk; n b ^ ; x1 Þ ¼ Tðk; n ^ ; O 0 x1 Þ Mðk; ^ ; O0 Þ; n
ð35Þ
! 1 2 ^Lðk; n ^ ; x1 Þ; ^ ; x2 Þ ¼ eikn^ x x ^Lðk; n
ð36Þ
^ M, ^ M2 ^ ^ in which points O ; x ; x are plotted in Fig. 1. Based on M2 L, ^ ^ and L2 L translations, Eq. (30) can be rewritten as 0
p‘i q‘i
)
1
2
Z ! ^ ; x2 ÞdS; ^ ; x2 xi Þ^Lðk; n ¼ k Wðk; n
As the first case, we consider a classical scattering model: an incident plane wave pin ¼ p0 eikz along the positive z axis impinges a rigid sphere with radius R ¼ 1:0 m and center at ð0; 0; 0Þ. The speed of acoustic wave propagation in air is 340 m=s. For this model, the exact solution of the sound pressure is expressed in the following form [40] of polar coordinates ps ðr; hÞ ¼
1 X
n
i ð2n þ 1Þp0
n¼0
ð37Þ
njn1 ðkRÞ ðn þ 1Þjnþ1 ðkRÞ ð1Þ
ð1Þ
nhn1 ðkRÞ ðn þ 1Þhnþ1 ðkRÞ
ð1Þ
hn ðkrÞP n ðcos hÞ;
ð41Þ
S0
P P where p‘i ¼ ‘j¼1 pij , and q‘i ¼ ‘j¼1 qij . Finally, we establish the wideband FMSBM by using the following formulation [40] to transform the moments of the FMM at low frequency to those at high frequency
b ^ ; OÞ ¼ Mðk; n
1 X n X
n
^ m ð2n þ 1Þi Y m n ðnÞM n ðk; OÞ;
ð38Þ
n¼0 m¼n
and using the following equation [40] to convert the local expansion coefficients of the FMM in high frequency regime to those in low frequency regime n
1 Lm n ðk; x Þ ¼
i 4p
Z S0
^ ^ ^ 1 Ym n ðnÞLðk; n; x ÞdS:
ð39Þ
^ and ^ Here, Eqs. (38) and (39) are respectively called as the M2M L2L translations. In the wideband FMSBM, we compute the acoustic pressure and its normal gradient by directly using Eqs. (7) and (8) when the source point yj is close to the collocation point xi . Thus, the origin intensity factors U i and Q i are calculated directly as in the conventional SBM (CSBM). Through the above-mentioned procedures, new SBM formulations accelerated by the wideband FMM have been finally developed. The remaining details of the implementation of the wideband FMSBM, including the upward pass and downward pass, are same as in the wideband FMBEM, which can be found in Ref. [40]. In the upward and downward pass, we require to set the threshold D ¼ 0:2k (k is the acoustic wavelength) of conversion between moments of low-frequency and highfrequency [45]. The switching level s in the tree structure satisfies s
3.1. Scattering from a rigid sphere
sþ1
s
sþ1
d P D and d < D (d and d respectively denote the edge lengths of cells at level s and s þ 1). In addition, a highly illconditioned system of linear equations is formed after the discretization of the present approach when using for acoustic problems at high frequency. Thus, we use the block diagonal preconditioner [46] to improve the convergence rate of the present method.
ð1Þ
in which j and h
are respectively the spherical Bessel and Hankel ð1Þ
functions of the first kind (subscripts of j and h denote their orders), and P n is the nth Legendre polynomial.In this case, p0 is set to 10 Pa and the maximum number of boundary nodes allowed in a childless box is set to 50. The frequency of acoustic wave is assumed to be f ¼ 100 Hz. Fig. 2 plots the error variation of the acoustic pressure at point ð0; 0; 4Þ calculated by the CSBM and the wideband FMSBM with an increasing number of boundary nodes. The numerical results illustrate these two approaches are stable and convergent. Moreover, the wideband FMSBM remains almost the same accuracy as the CSBM. By using the present approach for different frequencies, Table 1 provides the maximum relative errors of acoustic pressures at 400 points distributed on the surface of a sphere with radius r ¼ 4 m and center at the origin. As we can see, the wideband FMSBM can obtain accurate solutions at different frequencies. In Fig. 3, we make a comparison of CPU times consumed by the CSBM and the wideband FMSBM for different frequencies with an increasing number of boundary nodes. The CPU time of the CSBM is independent of the frequency, and thus only one case (100 Hz) is provided. We can observe that the curves of CPU times for the CSBM and the wideband FMSBM for four frequencies have the similar slopes with the trend lines 1 and 2, respectively. Those indicate O(NlogN) efficiency for the wideband FMSBM that is significantly superior to O(N3) for the CSBM. 3.2. Radiation from a car body As the second example, we consider the radiation from a simplified car body [47] as shown in Fig. 4. The dimension of the car is 4:1 m 2:0 m 1:1 m. The speed of acoustic wave propagation in air is 340 m=s. The exact solution [48] of this problem is assumed as
3. Numerical examples In this section, three numerical examples are provided to verify the proposed method. All the codes were written by Fortran 90 and run on a PC with an Intel Core i5-3230M (2.60 GHz) processor with 4 GB memory. The iterative solver (GMRES) will not be terminated until the residue is smaller than the tolerance of 103 [4,40,42]. The following error formula is used to test the accuracy of the proposed method
p pexact Relative error ¼ numerical pexact
ð40Þ
where pnumerical and pexact are respectively the numerical and analytical solutions. In addition, the HyperMesh is used to obtain the coordinates of boundary nodes for numerical examples 2 and 3.
Fig. 2. Relative errors of acoustic pressure at point ð0; 0; 4Þ.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx Table 1 The maximum relative errors of acoustic pressures with different frequencies. Frequency (Hz) Number of boundary nodes Maximum relative error
200 868 2.84546E02
1000 3208 2.65426E02
pðx; y; zÞ ¼
2000 12,892 3.28533E02
5000 51,022 4.65420E02
zeikr i ; 1 þ kr r2
ð42Þ
Fig. 3. CPU times consumed by the CSBM and the wideband FMSBM.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where r ¼ x2 þ y2 þ z2 . The Neumann boundary condition is imposed on the surface of the car, which can be easily derived by using Eqs. (4) and (42). In this model, we distribute 101,414 boundary nodes on the surface of the car body. The maximum number of boundary nodes allowed in a childless box is set to 120. Fig. 5 shows the numerical error surface of acoustic pressures at frequencies f ¼ 300 Hz and 2000 Hz in a square domain f2 6 x 6 2; 2 6 y 6 2; z ¼ 2:5g, which is located in front of the car body. It can be found that the maximum errors of acoustic pressures at two frequencies 300 Hz and 2000 Hz are less than 0.2% and 2% respectively. In addition, we investigate relative errors of numerical results influenced by the present method with different maximum number (Max-Num) of boundary nodes allowed in a childless box. Table 2 lists the comparison of maximum relative errors and CPU times of acoustic pressures at the frequency
Fig. 4. The sketch of a simplified car body.
Fig. 6. The sketch of a dolphin.
(a) f = 300 Hz
(b) f = 2000 Hz
Fig. 5. The numerical error surface of acoustic pressures at frequencies (a) 300 Hz and (b) 2000 Hz in a square domain f2 6 x; y 6 2; z ¼ 2:5g.
Table 2 The maximum relative errors and CPU times of acoustic pressures with different Max-Num. Max-Num Maximum relative error CPU time (s)
40 1.89186E03 4.37482E+03
80 1.89233E03 4.10790E+03
120 1.89270E03 4.01226E+03
160 1.89325E03 4.19948E+03
200 1.88774E03 4.96435E+03
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
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W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
f ¼ 300 Hz in the square domain (see Fig. 5) with an increasing Max-Num. In the implementation of the present method, the tree structure with different levels are produced due to the different Max-Num, which leads to a slight effect on the accuracy of the method as seen in Table 2. Moreover, we can see that CPU time first decreases and then increases when increasing the Max-Num. Thus, it is necessary to test the wideband FMSBM program for different Max-Num on a specific computer for a typical problem for
determining the optimal Max-Num, and then one can employ for similar applications. Such testing has to be performed on different computer platforms for better efficiency. 3.3. Radiation and scattering from a dolphin As illustrated in Fig. 6, we study the radiation and scattering from a dolphin [49] of which dimension is 2 m in length, 0.73 m
Fig. 7. The distribution of 123,550 boundary nodes on the surface of a dolphin.
(a) f = 1000 Hz
(b) f = 2000 Hz
(c) f = 3000 Hz
(d) f = 4000 Hz
Fig. 8. The relative errors of acoustic pressures on the surface of a sphere when considering four different frequencies f ¼ 1000; 2000; 3000; 4000 Hz.
Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002
W. Qu et al. / Computers and Structures xxx (2018) xxx–xxx
in width, and 0.55 m in height. The speed of acoustic wave propagation in water is 1473 m=s. For this example, two cases are considered: (1) acoustic radiation model with known analytical solution; (2) acoustic scattering model with no available analytical solution. We have same following conditions for both cases: (a) 123,550 boundary nodes are distributed on the surface of the dolphin as shown in Fig. 7; (b) the maximum number of boundary nodes allowed in a childless box is set to 200 for the present method. Case 1: ; zÞ ¼ eikr =r is imposed on The Dirichlet boundary condition pð x; y pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; y ; zÞ 2 þ z2 and ðx the surface of the dolphin, in which r ¼ x2 þ y denotes a boundary point. Thus, the exact solution is given as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pðx; y; zÞ ¼ eikr =r [48], and r ¼ x2 þ y2 þ z2 . This is an exterior radiation problem. In this simulation, four different frequencies f ¼ 1000; 2000; 3000; 4000 Hz are respectively considered. Fig. 8 plots the relative errors of acoustic pressures on the surface of a sphere with radius r ¼ 3 m and center at the origin. We can observe that the numerical errors calculated by the present method increase when the frequency increases. However, the satisfied results are still obtained even as f ¼ 4000 Hz. All these results indicate that the wideband FMSBM is a competitive method for solving the acoustic radiation problem with a complex boundary.
Fig. 9. Real part of acoustic pressure along the x-axis calculated by using the wideband FMBEM and the wideband FMSBM.
7
Case 2: The scattering from the dolphin is considered as the second case. We assume that the incident wave is a plane wave pin ¼ 1000eikx in the þx direction, and the boundary of the dolphin is rigid. The analytical solution is not available in this case. Thus, a comparison of numerical results obtained by the wideband FMSBM and FMBEM [40] is provided. The surface of the dolphin is discretized into 123,550 constant triangular elements for the wideband FMBEM. Figs. 9 and 10 respectively show the real and imaginary parts of acoustic pressures calculated by the wideband FMSBM and FMBEM. We can observe from these two figures that the numerical results of acoustic pressures obtained by the wideband FMSBM agree pretty well with those of the wideband FMBEM, indicating that the proposed method works well for the problem with complex geometric boundary and large number of boundary nodes. 4. Conclusions A wideband fast multipole singular boundary method (FMSBM) is developed to reduce the computational complexity and the memory requirement of the SBM by combining the wideband FMM and the SBM. In the proposed method, the fundamental solution is expanded as a partial wave expansion formulation in low frequency regime and a plane wave expansion formulation in high frequency regime. The wideband FMSBM combines advantages of the traditional FMSBM and the DF-FMSBM, which avoids the rapidly increasing CPU times of the former for high-frequency regime and instability of the latter for low-frequency regime. Moreover, compared with the wideband FMBEM, the proposed method distributes the boundary nodes on the surface and has no requirement of boundary elements, and meanwhile it is integration-free for the off-diagonal elements of the interpolation matrix. Numerical results clearly illustrate the potential of the wideband FMSBM for the solution of acoustic problems with complicated boundary, including the scattering model of a dolphin with no available analytical solution. However, the formulation of the present approach requires better strategies to control the number of iterations when solving the acoustic problems in high-frequency regime. In the further work, we will introduce a dual-level technique [50] for the present method to solve the highly ill-conditioning resulting from fullypopulated interpolation matrix at higher frequency. In addition, the proposed method based on the direct differentiation technique can be applied to the solution of acoustic shape sensitivity analysis. These are under intense investigation. Acknowledgements The work described in this paper was supported by the Natural Science Foundation of Shandong Province of China (Grant No. ZR2017BA003) and the Doctoral Research Foundation of Shandong University of Technology (Grant No. 4041/416031). References
Fig. 10. Imaginary part of acoustic pressure along the x-axis calculated by using the wideband FMBEM and the wideband FMSBM.
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Please cite this article in press as: Qu W et al. A wideband fast multipole accelerated singular boundary method for three-dimensional acoustic problems. Comput Struct (2018), https://doi.org/10.1016/j.compstruc.2018.06.002