A modified singular boundary method for three ...

Applied Mathematical Modelling 54 (2018) 189–201

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

A modified singular boundary method for three-dimensional high frequency acoustic wave problems Junpu Li, Wen Chen∗ State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering & Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 210098, China

a r t i c l e

i n f o

Article history: Received 31 May 2017 Revised 1 September 2017 Accepted 14 September 2017 Available online 27 September 2017 Keywords: Modified fundamental solution of Helmholtz equation Singular boundary method Ill-conditioning Helmholtz equation High frequency acoustic wave problems

a b s t r a c t The main purpose of this article is to propose a modified singular boundary method using the modified fundamental solution of Helmholtz equation for simulation of threedimensional high frequency acoustic wave problems. Compared with the standard secondorder discretization methods which usually need to place more than 10–12 grid points in one wavelength per direction, the newly proposed modified singular boundary method only needs 2–3 source points in one wavelength of each direction to produce the accurate solutions with relative error around 1E − 3 level which satisfies most application requirements. It is observed that the present algorithm has similar condition number to the boundary element method and can hereby be solved efficiently by the iterative solver. By adopting the range restricted generalized minimal residual algorithm iterative solver, numerical experiments with 194,400 source points have successfully been achieved on a single laptop for three-dimensional high frequency acoustic wave problems with up to wavenumber 440. © 2017 Elsevier Inc. All rights reserved.

1. Introduction The numerical simulation of the high frequency acoustic wave propagation has a wide variety of applications in diverse science and engineering fields [1–4], such as vehicle noise analysis, underwater sonar imaging detection, high-speed railway vibration and noise, etc. Unlike the Laplace problems, it is computationally very expensive to solve the Helmholtz equation with large wavenumber. This is because of three reasons. Firstly, the Helmholtz problems are usually posed on the infinite domain, which has either to be artificially truncated in the finite element methods (FEM) [5–6] or to encounter fully-polluted interpolation matrix in the boundary element method (BEM) [7–8]. Secondly, one needs to use a much higher sampling frequency than what would be required to obtain the reasonable solution because of the pollution effect [9]. Finally, one has to solve the resulting very large-scale and highly ill-conditioned linear equations [10]. Theoretically, the number of elements in the FEM scales up at least by the cube of kd for three-dimensional (3-D) problems, and in fact even more than that [11], where k represents the wavenumber and d means the maximum diameter of the computational domain. In the BEM [12], the number of elements increases by the square of kd. In order to reduce such computational expenses, several numerical techniques have been proposed in recent years [13–15], among which the common feature is the use of a priori knowledge on the oscillatory nature of the solution in the design of the numerical scheme. ∗

Corresponding author. E-mail address: [email protected] (W. Chen).

https://doi.org/10.1016/j.apm.2017.09.037 0307-904X/© 2017 Elsevier Inc. All rights reserved.

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J. Li, W. Chen / Applied Mathematical Modelling 54 (2018) 189–201

Nomenclature G GSBM GBKM M Ui Rj kd d Jn+ 1 (kr )

basis function of 3D Helmholtz equation fundamental solutions of 3D Helmholtz equation general solutions of 3D Helmholtz equation modified fundamental solution of Helmholtz equation OIF of the 3D Helmholtz equation characteristic radius of range of influence nondimensional wavenumber the maximum diameter of the computational domain the (n + 12 )-order Bessel function of the first kind

ξ ii

the n-order Legendre polynomials. fundamental solutions of 3D Laplace equation OIF of the 3D Laplace equation child node of the source point sj weight value of the child node ξ ii , distance between points sj and xi coordinate of the collocation point xi

2

Pn (cos (θ )) G0 U0i qii rij (rij ,θ ij ,ϕ ij ) (ρii , θ˜ii , ϕ˜ ii )

∇2 u¯ (xm ) βj k N

sj xi

α

p Aj Rerr C f ࢬξ ii − sj − xi = τ ii κii = θi j − θ˜ii

coordinate of the child node ξ ii Laplacian operator computational domain known boundary values unknown coefficients wavenumber boundary nodes number computational boundary source point collocation point adjustment parameter truncation term range of influence average relative errors convergence rate frequency

Giladi [16] proposed an asymptotically derived BEM where the kernel function is the product of a smooth amplitude and an oscillatory phase factor. This strategy has high-order accuracy, whose required element number in per wavelength direction is far less than the BEM. However, the singular integration is still inevitable which brings heavy computing loads in high-dimensional high-frequency cases. Kim [17] suggested an asymptotic decomposition (AD) approach. Numerical experiments show that this algorithm scheme is accurate enough when placing 4 and 5 grid points per wavelength. However, like the asymptotically derived BEM mentioned above, the AD approach is only applicable to the problems whose solution property is already partly known. Chen [18–19] proposed the boundary knot method (BKM) which uses the non-singular general solutions of the Helmholtz equation as the kernel function. Numerical investigations illustrate that two boundary nodes per wavelength in one direction is usually enough to yield the highly accurate solution for 3D high-frequency acoustic wave problems. However, severely illconditioned interpolation matrix constrains its application for large-scale problems. The singular boundary method (SBM) [20–22] is a strong-form boundary collocation algorithm and has clear boundaries with the boundary element method (BEM) and the method of fundamental solution (MFS) [23–25]. The key technique in the SBM is to replace the singularity at origin of the fundamental solution by the origin intensity factor (OIF). It is noted that the SBM is easily applicable to complex-shaped domain problems. Although extensive studies have been reported in the literatures for the SBM simulation of potential theory [26], low frequency Helmholtz equation [27], time-dependent diffusion problems [28], etc. However, few has been done to analyze the Helmholtz equation with high wavenumber. As the boundary collocation methods, the BKM and the SBM share some common properties but have their own merits and demerits as well. The merits of the BKM are the low sampling frequency and high numerical accuracy. However, severe ill-condition of its interpolation matrix obstructs its use with the iterative solver and hereby constrains its application to large-scale problems. On the other hand, the advantage of the SBM is the high numerical stability but requires placing at


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least 6 source points in one wavelength per direction to obtain the reasonable solution in simulation of 3-D acoustic wave problems, which causes heavy computation load for large-scale computation. In this article, a modified fundamental solution of the Helmholtz equation to replace the standard fundamental solution in the SBM is proposed. We call the SBM based on the modified fundamental solution as the modified singular boundary method (MSBM) in the study. The modified fundamental solution can be considered a tradeoff between the standard fundamental solution and the general solution. By the help of the modified fundamental solution, the merits of the BKM and the SBM are combined to achieve reasonable solution accuracy in a low sampling frequency while keeping the condition number (L2-norm) sufficiently low. The article is organized as follows: Section 2 introduces the modified fundamental solution of Helmholtz equation and its application to the SBM formulation. Section 3 investigates the MSBM through two benchmark examples. Finally, some conclusions and outlook are provided in Section 4 and Section 5, respectively. 2. Formulation of the modified singular boundary method The governing equation of the propagation of the steady acoustic wave in the isotropic medium can be expressed as the Helmholtz equation with constant wavenumber. The 3D Helmholtz equation on Dirichlet boundary conditions can be written as

∇ 2 u ( xm ) + k2 u ( xm ) = 0, u(xm ) = u¯ (xm ),

( x m ) ∈ ,

(1)

( xm ) ∈ ,

(2)

∇2

where denotes the Laplacian operator, k is the wavenumber. u¯ (xm ) means the known boundary values on the boundary , u(xm ) represents the physical value in domain at point xm , xm is a computational point in domain . In the SBM and the BKM, a linear combination of basis functions corresponding to different boundary source points sj is adopted to approximate the physical variable u(xm ) as below,

u ( xm ) =

N

β j G(xm , s j ),

(3)

j=1

where β j is the unknown coefficient and G the basis function. In the BKM, G = GBKM , where GBKM = sin(rkr ) is the general solutions of 3D Helmholtz equation, k the wavenumber, r the distance between source point and collocation point. In the SBM, we take G = GSBM as the basis function, where GSBM =

eikr r

is the fundamental solutions of 3-D Helmholtz equation ikr

adopted in the BEM and the SBM. Considering that the fundamental solution GSBM = e r has singularity at origin, the SBM uses the OIF [29–30] to replace the singular terms in the interpolation matrix. The SBM formulation with the standard fundamental solution can be expressed as

u¯ (xi ) =

N

β j GSBM (xi , s j ) + βiU i ,

(4)

j=1, j=i

where u¯ (xi ) denotes the known physical variable on boundary , Ui represents the OIF of the 3D Helmholtz equation. On the other hand, because the non-singular general solution is adopted in the BKM, the BKM formulation can be written as

u¯ (xi ) =

N

β j GBKM (xi , s j ),

(5)

j=1

with either Eq. (4) or (5), we can obtain the unknown coefficient β j . Consequently, the physical variable u(xm ) at any point in domain can be easily calculated by Eq. (3). Because the non-singular general solution is adopted in the BKM, we only need to place 2 source points in each wavelength per direction to obtain the highly accurate solution. However, severely ill-conditioned interpolation matrix obstructs its use of the iterative solver and hereby constrains its application to large-scale problems. On the other hand, by introducing the concept of the OIF, the SBM avoids the singular integration and has the similar condition number to the BEM, but the SBM with the standard fundamental solution still requires to place at least 6 source points in each wavelength per direction to achieve the reasonable solution accuracy, which brings rather heavy computing loads for 3D high-frequency acoustic wave problems. It is reasonable to think the combination of merits of the BKM and the SBM. With this idea in mind, we propose a modified fundamental solution of the Helmholtz equation to replace the standard fundamental solution in the SBM. By the application of the modified fundamental solution, we can obtain a reasonable solution accuracy in a low sampling frequency while keeping the conditioning number sufficiently low. The modified fundamental solution M can be expressed as

M = (1 − α )

sin(kr ) cos(kr ) +α , α ∈ [0, 1] r r

(6)

192


where M means the modified fundamental solution of the Helmholtz equation and α ∈ [0, 1] represents the adjustment parameter. It can be noted from Eq. (6) that when α = 1, M equals to the real part of the GSBM , i.e., M = real(GSBM ), and when α = 0, M = GBKM . Actually, we take a tradeoff between the real(GSBM ) and u¯ (k ) as the basis function of the modified singular boundary method, which is controlled by the adjustment parameter u(k). Thus, the MSBM may inherit merits from both the BKM and the SBM at the same time. The MSBM formulation can hereby be expressed as

sin(kri j ) cos(kri j ) β j (1 − α ) · +α· + βi ((1 − α ) · k + α · U i ),

N

u¯ (xi ) =

ri j

j=1, j=i N

2 U = + Ri i

p

j=1, j=i n=1

ri j

(7)

ψ jn ,

π R2i

(8)

where

ψ j1 = ψ j3 = ψ j5 = ψ j7 = 0, ψ j2 = ψ j4 = ψ j6 =

π R4j 3sin2 (ϕi j ) 4ri3j

2

−1 ,

π R6j 35sin4 (ϕi j ) 8ri5j

8

5π R8j 231 64ri7j

16

(9)

− 5sin2 (ϕi j ) + 1 ,

sin (ϕi j ) − 6

(10)

(11)

189 4 21 2 sin (ϕi j ) + sin (ϕi j ) − 1 , 8 2

(12)

−→ where rij means the distance between source point sj and collocation point xi , φ ij is the angle between the vector xi s j and the normal nj at the boundary point sj , p means the truncation term, Aj represents the corresponding range of influence,

Rj =

Aj

π denotes the characteristic radius of range of influence.

To balance the numerical efficiency and accuracy of the OIF, the truncation term p is taken 5 in this study. For more details about the OIF, see the Refs [29–30]. The derivation of Eqs. (8)–(12) is given in Appendix. A, and the code of the OIF formulation can be found in the Singularity toolbox V1.0 at following website: http://dx.doi.org/10.13140/RG.2.2.13985.51040. Finally, we evaluate the unknown coefficient β j by Eqs. (7) and (8). The physical variable u(xm ) at any point inside the domain can hereby be evaluated by Eq. (13)

u ( xm ) =

N

sin(krm j ) cos(krm j ) β j (1 − α ) · +α· , rm j

j=1

rm j

(13)

3. Numerical results and discussions Without specific instructions, the range restricted generalized minimal residual algorithm (RRGMRES) [31–34] solver which has the regularization function is adopted to solve the resulting discretization equations. The results reported below are calculated on a laptop with Intel Core i7-4710MQ 2.50 GHz Processor and 16GB RAM. kd (k the wavenumber and d the maximum diameter of the computational domain) denotes the nondimensional wavenumber. The numerical accuracy is depicted by the average relative errors (Rerr).

Rer r (u ) =

NT m=1

|u(m ) − u¯ (m )|2

NT

|u¯ (m )|2 ,

(14)

m=1

where u¯ (m ) and u(m) mean the analytical and numerical solutions at xm , respectively, NT represents the test points number. To characterize the convergence of the numerical results, Eq. (15) is given

C = −2

ln(Er ror (N1 )) − ln(Er ror (N2 )) , ln(N1 ) − ln(N2 )

(15)

where Error(N1 ) and Error(N2 ) denote numerical errors against the source points number N1 and N2 , respectively. Example 1. Consider a 3D acoustic wave problem in the unit cube domain (with f = 2ckπ , c = 340 m/s). The test points are placed on the sphere surface with radius 0.4 m, and the analytical solution is given by

∇ 2 u(x, y, z ) + k2 u(x, y, z ) = 0, (x, y, z ) ∈ . u¯ (x, y, z ) = cos(kx ) + cos(ky ) + cos(kz ), (x, y, z ) ∈


193

Table 1 MSBM simulation results against wavenumber k. Nodes N

9600

48,600

88,640

101,400

135,0 0 0

194,400

Wavenumber k Dimensionless wavenumber kd Frequency f (Hz) Nodes/Wavelength (N/λ) Iteration number Rerr

100 173 5411 2.51 200 4.3E − 3

220 381 11,905 2.57 200 3.1E − 3

290 502 15,693 2.59 250 1.08E − 2

320 554 17,316 2.55 350 1.04E − 2

370 641 20,022 2.55 350 9.68E − E − 3

440 762 23,810 2.57 350 1.44E − 2

Table 2 MSBM simulation results against the source points number. Nodes N

α

600 2400 5400 Condition number (L2-norm)

9600

α =1E − 3 α =1 α =0

3.9E + 3 4.3E + 1 3.3E + 15

2.1E + 4 1.4E + 2 5.2E + 20

1.0E + 4 5.4E + 1 1.5E + 19

1.6E + 4 1.0E + 2 5.0E + 19

Table 3 The average relative error and condition number (L2-norm) of the MSBM against α .

α

1E + 0

1E − 1

1E − 2

1E − 3

1E − 8

0

Condition number (L2-norm) Rerr

4.8E + 2 3.5E − 1

4.0E + 2 7.0E − 2

2.2E + 3 1.2E − 2

1.8E + 4 6.6E − 3

4.6E + 8 1.9E − 4

3.9E + 18 1.3E − 5

Table 4a Numerical results of the MSBM (α = 1E − 3). Nodes N

384

864

1536

2400

Wavenumber k Nodes/Wavelength (N/λ) Rerr (α = 1E − 3) CPU time (s) Storage requirements (Mb)

20 2.51 3.60E − 2 0.038 0.29

30 2.51 1.33E − 2 0.092 2.65

40 2.51 1.11E − 2 0.269 9.09

50 2.51 3.54E − 2 0.583 25.3

Table 4b Numerical results of the SBM with the standard fundamental solutions. Nodes N

2400

5400

9600

15,0 0 0

Wavenumber k Nodes/Wavelength (N/λ) Rerr CPU time (s) Storage requirements (Mb)

20 6.28 6.42E − 3 1.01 50.4

30 6.28 3.91E − 2 8.06 283

40 6.28 2.04E − 2 33.82 950

50 6.28 1.06E − 2 198.98 1935

Case 1. The numerical characteristics of the MSBM are investigated in this case.√The results of the MSBM with α = 1E − 3 against the wavenumber k(m − 1 ) and nondimensional wavenumber kd (d = 3m) are listed in Table 1. It can be observed from Table 1 that the MSBM only needs to place 2–3 source points in each wavelength per direction to obtain the reasonable solution accuracy. The proposed modified fundamental solution enables the MSBM to overcome the issue of high sampling frequency in simulation of Helmholtz equation with high wavenumber. Case 2. The condition number (L2-norm) and the influence of the adjustment parameter α are investigated in this case. In order to avoid the possible influence of the iterative solver, it should be stressed that the Gaussian solver is adopted in this case. Table 2 is listed to show the condition number (L2-norm) of the MSBM with α = 1E − 3, α = 1 and α = 0, respectively, against the source points number under k = 20m − 1 . The average relative error and condition number (L2-norm) against the adjustment parameter α are listed in Table 3 under source points number 2400 and k = 40m − 1 (N/λ = 3.14). It can be observed that the condition number (L2-norm) and accuracy of the MSBM both increase with the decreasing of the adjustment parameter α . Actually, the MSBM tends to the BKM when α is taken 0, and degenerates into the SBM with the standard fundamental solution when α is taken 1. Secondly, Table 4a and b are listed to show the different numerical efficiency and storage requirements of the MSBM and the SBM to obtain the reasonable numerical solution accuracy.

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J. Li, W. Chen / Applied Mathematical Modelling 54 (2018) 189–201 Table 5a The numerical accuracy of the different methods against the wavenumber k. Wavenumber k

15

30

45

60

75

N/λ MSBM MSBM MSBM MSBM BKM BEM MFS SBM

12.57 5.27E − 6 8.72E − 6 1.56E − 4 1.74E − 2 3.16E − 7 5.38E − 3 3.23E − 13 2.26E − 3

6.28 2.08E − 4 6.72E − 3 2.75E − 2 7.97E − 2 1.76E − 6 5.93E − 2 1.07E − 11 3.91E − 2

4.19 1.39E − 4 4.31E − 4 3.47E − 3 2.21E − 1 6.14E − 6 4.01E − 1 1.34E − 10 2.51E − 2

3.14 3.30E − 4 1.05E − 2 8.58E − 2 6.31E − 1 5.56E − 5 1.49E − 1 6.97E − 9 4.92E − 1

2.51 5.42E − 3 7.71E − 3 2.51E − 2 4.40E − 1 7.96E − 4 4.33E − 1 8.12E − 6 1.23E − 1

(α =1E − 5) (α =1E − 3) (α =1E − 2) (α =1)

Table 5b The condition number (L2-norm) of the different methods against the wavenumber k. Wavenumber k

15

30

45

60

75


12.57 2.02E + 6 2.02E + 4 2.01E + 3 5.79E + 2 9.72E + 19 2.60E + 1 1.85E + 19 2.18E + 1

6.28 1.10E + 6 1.41E + 4 5.03E + 3 6.98E + 3 3.46E + 19 7.02E + 1 7.34E + 18 8.90E + 1

4.19 7.34E + 5 2.42E + 4 4.05E + 3 1.99E + 3 5.53E + 19 1.74E + 2 2.27E + 18 1.76E + 3

3.14 5.30E + 5 1.96E + 4 3.00E + 3 7.64E + 2 1.34E + 19 5.61E + 1 1.91E + 18 1.09E + 2

2.51 6.62E + 5 1.59E + 4 1.61E + 3 9.44E + 2 2.06E + 18 2.26E + 1 3.24E + 17 1.13E + 2

(α =1E − 5) (α =1E − 3) (α =1E − 2) (α =1)

Table 5c The CPU time costs (seconds) of the different methods against the wavenumber k. Wavenumber k

15

30

45

60

75


12.57 3.58 2.94 3.27 3.86 4.31 57.75 7.02 8.26

6.28 3.40 2.95 3.17 3.85 4.31 57.36 7.45 7.35

4.19 3.42 3.14 3.21 3.89 4.20 56.80 7.42 7.35

3.14 3.31 3.03 3.20 4.03 4.27 58.07 7.30 7.46

2.51 3.35 3.05 3.30 4.06 4.13 57.42 7.27 7.53

(α =1E − 5) (α =1E − 3) (α =1E − 2) (α =1)

It can be observed form Table 4a and b that the modified fundamental solution has obvious advantage over the standard fundamental solution in terms of numerical efficiency. When the wavenumber is taken k = 50, the required boundary nodes number, CPU time and storage requirements of the MSBM is 16%, 2.9‰ and 1.3%, respectively, of those of the SBM with the standard fundamental solution. And this advantage will be more obvious with the increasing of wavenumber. In theory, the operation efficiency (with Gaussian solver) and storage requirements of the boundary collocation method is O(N3 ) and O(N2 ), respectively. The storage requirements and computation load increase very fast with the boundary nodes increasing. That is the reason why it is so important to find a new strategy with both low sampling frequency and low condition number in simulation Helmholtz equation with large wavenumber. Thirdly, Table 5a–c are listed to show the different numerical accuracy, condition number (L2-norm), and CPU time (seconds) of the MSBM, the BKM, the SBM, the BEM and the MFS (with the cube surface having length 2 as the fictitious boundary), respectively, against the wavenumber k (with 5400 source points or boundary elements). It is noted that the numerical accuracy and condition number of the MSBM are different with varying adjustment parameters α , while the CPU time is substantially stable. In theory, the selecting of the adjustment parameter is dependent on nodes/wavelength (N/λ). When the sampling frequency is taken N/λ ∈ [2, 3], it can be observed that α = 1E − 3 can meets most of the computational requirements. On one hand, it can be found from Table 5a–c that although the MFS and the BKM can obtain highly accurate numerical solution in very low sampling frequency, the condition numbers (L2-norm) of them are very large. On the other hand, the BEM and the SBM have low condition number. However, it is observed that both the two methods cannot obtain the reasonable numerical solution accuracy when we place 2.5 source points in each wavelength per direction. Finally, the convergence curves of the BKM, the BEM, the SBM and the MSBM with varied adjustment parameters α under k = 25m − 1 are plotted in Fig. 1.


195

Fig. 1. The convergence curve of the MSBM, the SBM, the BKM and the BEM against boundary nodes number.

On one hand, it can be found from Fig. 1 that the MSBM with α = 1E − 10, α = 1E − 3 and α = 1 converge stably, while the BKM is in fact equal to the MSBM with α = 0 appears unstably due to highly ill-conditioned interpolation matrix. For large-scale problems, such ill-conditioned discretization matrix could derail the numerical solution. On other hand, in comparison with the SBM using the standard fundamental solution, the MSBM with α = 1E − 3 and α = 1E − 10 perform better on both convergence rate and numerical accuracy. Therefore, α plays a role somehow like the regularization parameter to adjust the balance between the conditioning of the interpolation matrix and the resulting solution accuracy. Actually, it can be observed from Fig. 1 that the effect of the regularization is quite obvious even a very small α is taken. Case 3. The convergence of the MSBM adopting the RRGMRES iterative solver against the iteration number is investigated in this case. The convergence curves of the MSBM solution under 9600 source points and k = 100m − 1 (with N/λ = 2.51) against the iteration number are plotted in Fig. 2. Fig. 2 shows that the numerical error decreases rapidly in the initial 10 iterations, and all converge to the reasonable error within 100 iterations. It can be found that the smaller the adjustment parameter α is, the slower the convergence speed of the error curve is observed. This phenomenon is because the adjustment parameter decides the condition number of the resulting interpolation matrix, which affects directly the numerical solving efficiency of the RRGMRES solver. In conclusions, the adjustment parameter α can be considered a regularization factor for the MSBM. When the adjustment parameter α becomes smaller, the MSBM is more tend to the BKM and away from the SBM with the standard fundamental solution. And both the condition number (L2-norm) of the interpolation matrix and the resulting solution accuracy increase. Thus, we can achieve possibly a reasonable solution accuracy with a low sampling frequency while keeping the condition number sufficiently low by selecting an appropriate adjustment parameter α in the modified fundamental solution. Example 2. The main purpose of this example is to examine the MSBM to the complex-shaped domain problems. Consider interior acoustic wave pressure problems with c = 340 m/s, k = 2πc f m−1 . The source points are placed on the surface of the following submarine-shaped domain with body size 120m × 12m × 12m as shown in Fig. 3. Case 1. The governing equation of this case is given by.

2 ∇ u(x, y, z ) + k2 u(x, y, z ) = √0, (x, √y, z ) ∈ √ u¯ (x, y, z ) = √1 J 1 (kr ) + cos( 33 kx + 33 ky + 33 kz ), (x, y, z ) ∈ , r = x2 + y2 + z2 kr 2

where J 1 (kr ) represents the 2

1 2 -order

Bessel function of the first kind.

The test points are placed on a line (x, y, z) = ( − 3: 0.03: 3, 0, 0). Fig. 4 depicts the analytical solution and the MSBM results with α = 1e − 3. 115,846 source points are used for this kd = 1200 case (d = 120 m), i.e., f = 541 Hz, and the maximum iteration number of the RRGMRES solver is set to be 200.

196


Fig. 2. The convergence curve of the MSBM with varied α against iteration number.

Fig. 3. The submarine-shaped domain.

Fig. 4. The MSBM and analytical solutions in Example 2 case 1.


197

Fig. 5. The relative error distribution of the MSBM solution in Example 2 case 2.

The numerical experiments show that the MSBM solution consumes 5.15E + 4 s CPU time, and the average relative error is 2.46%. It can be found from Fig. 4 that the MSBM solution is in good agreement with the analytical solution with total dimensionless wavenumber up to kd = 1200 (d = 120 m). And it should be stressed that the average number of source points in each wavelength per direction is about 3 in this case with the wavenumber k = 10 m − 1 . This is almost down to the bottom line of the minimal sampling frequency allowed by the Shannon’s sampling theorem, i.e., in order for an analog signal to be fully reconstructed, the sampling frequency should not be less than twice of the highest frequency in the analog signal spectrum. Case 2. In this case, we consider the interior acoustic pressure problem having complex series solution, the governing equation is given by

2 ∇ u ( xm ) + k2 u ( xm ) = 0, ( xm ) ∈ u¯ (xm ) =

50

n=0

√1 J 1 kr n+ 2

(kr )Pn (cos(θ )), (xm ) ∈

where, Jn+ 1 (kr ) represents the (n + 12 )-order Bessel function of the first kind, Pn (cos (θ )) means the n-order Legendre 2

polynomials. We take source points number N = 115,846, α = 1e − 3, kd = 480 (d = 120 m), i.e., f = 216 Hz, and the maximum iteration number of the RRGMRES solver is set to be 300. We plot Fig. 5 to show the relative error distribution of the MSBM solution on plane z = 0m. The numerical reports show that the MSBM consumes 6.82E + 4 s CPU time to obtain the reasonable numerical results, the average relative error of the MSBM solution in Fig. 5 is 7.25E − 4, and the maximum relative error is 0.15%. Through this example, it can be found that the proposed modified fundamental solution of Helmholtz equation is still valid for complex-shaped domain problems, where the numerical efficiency and the numerical stability are almost not affected. 4. Conclusions In this article, a modified singular boundary using the modified fundamental solution of Helmholtz equation for 3-D high frequency acoustic wave problems is proposed. By the help of the newly proposed modified fundamental solution, numerical investigations show that the condition number (L2-norm) of the MSBM significantly drops compared with the BKM, while the solution accuracy remains little affected. In comparison with the SBM using standard fundamental solution, the modified fundamental solution enables the MSBM to place 2–3 source points in each wavelength per direction to obtain the accurate solution with relative error around 1E − 3 level, which satisfies most application requirements. And it should be stressed

198


that based on the Shannon’s sampling theorem, placing 2 source points in each wavelength is the minimum requirement to produce the correct solution. In addition, it is noted that the MSBM has similar condition number to the BEM under the same number of boundary nodes. This makes the resulting discretization equations of the MSBM be efficiently solved by the iterative solver [35]. Numerical experiments show that the MSBM performs well for the tested cases and appears promising to solve the largescale high frequency acoustic wave problems. It can be noted that the MSBM yields accurate solution for high frequency acoustic wave problems up to k = 440 in unit cube domain when the boundary nodes number is taken 194,400. In stark contrast, the wavenumber upper limit is about k = 185 in the SBM with the standard fundamental solutions. In addition, the strategy is also successfully tested to the interior acoustic pressure simulation of a submarine model with dimensionless total wavenumber kd = 1200 (d = 120 m). In conclusions, the most important significance of the proposed modified fundamental solutions is that by introducing the concept of the adjustment parameter α , it is possible to artificially adjusts the numerical characteristics of the MSBM. We can achieve possibly reasonable solution accuracy in a low sampling frequency while keeping the condition number sufficiently low by selecting an appropriate adjustment parameter α in the modified fundamental solution. 5. Outlook As a novel proposed numerical methodology, the MSBM still encounters the following problems to be further addressed: (1) Determine the optimal adjustment parameter α in the modified fundamental solution of Helmholtz equation in terms of nodes/wavelength (N/λ); (2) Further accelerate the solution process by using an appropriate fast algorithm [36–39]; (3) Select an appropriate preconditioning technique to reduce the iteration number of the RRGMRES iterative solver. Acknowledgments The work was supported by the National Science Funds of China (Grant Nos. 11302069, 11372097, 11572111), the 111 Project (Grant No. B12032), the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX17_0488) and the Postgraduate Scholarship Program from the China Scholarship Council (Grant No. 201706710107). Appendix A. The derivation of Eqs. (8)–(12) In previous studies [29], it is demonstrated that the SBM with Eq. (A1) has the explicit error bounds, and the discrete form of Eq. (A1) was given in the Ref [29].

(xi ) − U0i =

N j=1, j=i

G0 xi , s j A j ,

Ai

1 where G0 (xi , s j ) = |x −s , (xi ) = i j|

(A1)

G0 (xi , s )d(s ), Aj is the range of influence, U0i the OIF of the Laplace equation on Dirichlet

boundary conditions. Based on multipole expansion [40–41], the following multipole expansion formula is given ∞

ρn 1 = Pn (cos(τ )), r r n+1

(A2)

n=0

where r is the distance between points P and Q, τ the angle between the vectors P and Q, as given in Fig. A1. Pn (u) represents the Legendre polynomial of degree n. We reformulate Eq. (A1) as

U0i =

G0 (xi , s )d(s ) +

Ai

N j=1, j=i A j

G0 (xi , s )d(s ) − Ai

N j=1, j=i

G0 xi , s j A j .

(A3)

By the help of Eq. (A2), the middle part of Eq. (A3) can be rewritten as

G0 (xi , s )d(s ) =

Aj

=

p ∞ ii=1 n=1

qii ρiin P ri j n+1 n

where p = ∞, A j =

∞ ii=1

p ∞ ii=1 n=0

qii ρiin P ri j n+1 n

(cos(τii ))

, (cos(τii )) + G0 xi , s j A j

(A4)

qii , ξ ii denotes the child node of the source point sj within the range of influence Aj , qii is the weight

value of the child nodes ξ ii , ρ ii the distance between source point sj and its child node ξ ii , rij the distance between the source point sj and collocation point xi , ࢬξ ii − sj − xi = τ ii , as given in Fig. A2.


199

Fig. A1. The local polar coordinate system for nodes Q and P.

Fig. A2. The local polar coordinate system of the source point sj .

For each range of influence Aj , we establish a local polar coordinate system with origin at point sj . The coordinate of the collocation point xi in this local polar coordinate system is (rij , θ ij , ϕ ij ), and the child node ξ ii of the source point sj is coordinated as (ρii , θ˜ii , ϕ˜ ii ), as given in Fig. A2. Based on Eq. (A4) and Eq. (A3) can be reformulated as follows:

U0i =

Ai

G0 (xi , s )d(s ) +

N

p ∞

j=1, j=i n=1 ii=1

Ai

qii ρiin Pn (cos(τii )) r n+1 ij

.

(A5)

To evaluate efficiently Eq. (A5), the following simplified conditions are introduced:

(1) The range of influence Aj is simplified as a round plane with characteristic radius R j =

Aj

π with center sj ;

(2) The equivalent round plane is perpendicular to the outward normal nj at the source point sj as given in Fig. A3.

200


Fig. A3. The equivalent range of influence Aj .

Based on above simplified conditions, the first part of Eq. (A.5) can be simplified as

G0 (xi , s )d(s ) =

Ri 0

Ai

2π r dr = 2π √ x2 + r 2

x2 + Ri 2 − x ,

(A6)

as shown in Fig. A3. When x = 0, we have

G0 (xi , s )d(s )

Ai

=

Ai

2 . Ri

(A7)

The second part of Eq. (A5) can hereby be simplified as p ∞ p N N qii ρiin Pn (cos(τii )) = ψ jn , rinj +1

j=1, j=i n=1 ii=1

(A8)

j=1, j=i n=1

when n = 1,

ψ j1 = 0,

(A9)

when n = 2,

ψ j2 = 1 ri3j

=

∞ ii=1

qii ρii2 r3

r j 2π 0 0

ij

r

3 2

3 3

cos2 (τii ) −

1 2

=

∞ ii=1

qii ρii2 r3 ij

3 2

sin (ϕi j )cos2 (κii ) −

sin (ϕi j cos2 (κ ) − 12 )drdκ = 2 2

2

π R4j 4r 3

ij

3sin2 (ϕi j ) 2

−1

1 2

,

(A10)

where κii = θi j − θ˜ii , the equivalent range of influence Aj is placed on x − sj − y plane, as given in Fig. A2. φ ij means the angle −→ between vector xi s j and the normal nj at the source point sj (i.e., z axis of Fig. A2). It is noted that

lim r→0

1 cos(kr ) = lim . r→0 r r

(A11)

The OIF of the Laplace equation U0i can serve as the OIF of the Helmholtz equation Ui , i.e.,

U0i = U i .

(A12)

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