Single layer regularized meshless method for three dimensional exterior acoustic problem

Engineering Analysis with Boundary Elements 77 (2017) 138–144

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Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound

Single layer regularized meshless method for three dimensional exterior acoustic problem

MARK

Lin Liu Shanghai Aircraft Development and Research Institute, Shanghai 201210, PR China

A R T I C L E I N F O

A BS T RAC T

Keywords: Single layer regularized meshless method Helmholtz equation Burton-Miller Dual surface Subtraction and adding-back technique

The Regularized Meshless Method (RMM) is a meshless boundary method. Its source points and physical points are overlapped. The substraction and adding-back technique is utilized to avoid the singularity of the fundamental solution. It is simple and easy to be programmed. But the double layer potential should be adopted in the desingularity technique. Here the single layer potential is employed to circumvent the singularity. The substraction and adding-back technique is succeeded, but the careful selection of particular solution for the null-fields boundary integral equation is chosen to derive the diagonal elements for the Laplace Dirichlet problem. By this particular solution, the diagonal elements can be represented by the single layer potential. Here it is extended to the exterior Helmholtz problem by relationships between Laplace and Helmholtz singularities. The fictitious frequencies are avoided by the Burton-Miller type formula and Dual Surface technique. The accuracy of these methods are shown by three typical examples.

1. Introduction The Method of Fundamental Solutions (MFS) is a typical meshless boundary collocation method. However the choice of source points is arbitrary and without a particular rule. Many Boundary Meshless Methods with source points coincident with physical points have been proposed in the literature. These methods use different techniques to avoid the singularity of fundamental solution. Boundary Node Method (BNM) [1] adopts the interpolation procedure to circumvent the singularity. And it was extended to 2-D interior Helmholtz problem [2]. Boundary Points Method (BPM) [3] uses the ‘moving elements’ to avoid the singularity. Boundary Particle Method (BPM) [4], Boundary Knot Method (BKM) [5] employ an alternative non-singular kernel function to circumvent the singularity. Boundary Distributed Source (BDS) method [6], Improved Boundary Distributed Source (IBDS) [7], Non-Singular Method of fundamental solution [8] remove the singularities by distributed source over areas (for 2D) or volumes (for 3D) covering the source points. Regularized Meshless Method (RMM) which uses the desingularization of subtracting and adding back technique was proposed by Young et al. [9] for 2-D Laplace problem, and then applied to different problems [10–13]. This method was later extended to 2-D [14] and 3D [15] exterior acoustic problem. In RMM the double layer potential was adopted as the fundamental solution for the convenience of using null-fields boundary integral equation to desingularize the fundamental solution for Laplace equation. Then Helmholtz equation fundamental solution is represented by its direct relation with Laplace equation. In this paper the substraction and adding-back technique is succeeded,

but the careful selection of particular solution for the null-fields boundary integral equation is chosen to derive the diagonal elements for the Laplace Dirichlet problem. By this particular solution, the diagonal elements can be represented by the single layer potential [16]. Here it is extended to Helmholtz problem. This paper is also similar to the idea of Singular Boundary Method [17,18] to get the magnitude of singular source, which also uses the single layer potential as the fundamental solution. This method also extended to 2-D interior [19] and exterior [20] acoustic problem. In the SBM [17] or ISBM [18], the inverse interpolation technique (IIT) was adopted to get the singular source magnitude for the Dirichlet problem. They got the Neumann problem singular source magnitude by the subtraction and adding-back technique in null-fields boundary integral equation firstly, then integrated the solution to achieve the Dirichlet problem singular source magnitude, the constant for the integration was derived by the inverse interpolation from the domain points. In this paper the Dirichlet problem singular source magnitude is directly derived from the null-field integral equation by the subtraction and adding-back technique using the single layer potential, without the needing of inverse interpolation. Recently the explicit empirical formula for the diagonal elements has been proposed [21]. It will be compared in numerical example 5.1. In the following sections, the theory of the Single Layer Regularized Meshless Method (SRMM) for exterior acoustic problem is introduced. The Burton-Miller technique and Dual Surface technique are adopted to avoid the non-uniqueness. Then three typical examples show the validation of these methods.

http://dx.doi.org/10.1016/j.enganabound.2017.02.001 Received 8 August 2016; Received in revised form 2 February 2017; Accepted 2 February 2017 0955-7997/ © 2017 Elsevier Ltd. All rights reserved.

Engineering Analysis with Boundary Elements 77 (2017) 138–144

L. Liu

2. Formulation of single layer regularized meshless method

The formulas of

(1)

subjected to the following boundary conditions

p (x ) = p (x ),

q (x ) =

x ∈ ΓD (Dirichlet boundary condition )

∂p (x ) = q (x ), ∂n

x ∈ ΓN (Neumann boundary condition )

(2) (3)

where Ω is a bounded domain with boundary Γ = ΓD + ΓN , n presents the outward normal, k = ω / c is the wavenumber, ω is the angular frequency, c is the wave speed in the medium Ω, p is the complex valued amplitude of radiated and/or scattered wave.

⎧ pR = pT , if radiation ⎪ if scattering , p = ⎨ pS = pT − pI , ⎪p ⎩ R + S = pT − pI , if both

⎧∑N α G (x , s ), xi ∈ Ω i j ⎪ j =1 j p (x i ) = ⎨ N ⎪∑ j =1, j ≠ i αj G (xi , sj ) + αi G (xi , si ), xi ∈ Γ ⎩

⎧ N ∂G (xi , sj ) , xi ∈ Ω ⎪∑ j =1 αj ∂n xi ∂p (xi ) ⎪ q (x i ) = =⎨ ∂G (xi , sj ) ∂n xi ∂G (xi , si ) ⎪ N + αi , xi ∈ Γ ⎪∑ j =1, j ≠ i αj ∂n ∂n xi xi ⎩

∂G (xi , sj ) ∂n xi

eikr r =

r3

Directly from the Burton-Miller concept, we can construct BurtonMiller type regularized meshless method.

⎧ ⎛ ⎞ ⎪∑N αj ⎜G (xi , sj ) + γ BM ∂G (xi , sj ) ⎟ , xi ∈ Ω ⎪ j =1 ⎜⎝ ∂nsj ⎟⎠ ⎪ ⎞ ⎛ ⎪ p (xi ) = ⎨∑Nj =1, j ≠ i αj ⎜G (xi , sj ) + γ BM ∂G (xi , si ) ⎟ ⎪ n ∂ ⎠ ⎝ si ⎪ ⎞ ⎛ ∂ G ( x , s ) ⎪ i i BM ⎟, xi ∈ Γ ⎪+ αi ⎜G (xi , si ) + γ ∂nsi ⎠ ⎝ ⎩

(5)

(6)

〈(xi − sj ), n xi 〉(ikr − 1)

(14)

(8) Where γ BM = i / k according to [25].

is the fundamental solution and the physical normal derivative of three-dimensional Helmholtz equation, r = ∥ xi − sj ∥, 〈, 〉 denotes the inner product. If the collocation points and source points coincide, the singularities are encountered. However, unlike the potential problem, the source intensity factors can't be calculated directly from the Helmholtz fundamental solutions by analytical-numerical technique. Fortunately, the Helmholtz fundamental solutions have a similar order of singularities as the related Laplace fundamental solutions. Hence the corresponding relationships can be represented by the following asymptotic expressions [22]

∂G (xi , si ) ∂GL (xi , si ) = , ∂n xi ∂n xi

xi → si

∂G (xi , si ) ∂GL (xi , si ) = , ∂nsi ∂nsi

xi → si

4. Dual-surface single-layer regularized meshless method The basic idea is to generate a virtual second surface inside the structure (Fig. 1) by shifting the original points sj along the element normal to a “virtual” surface sjDS using a distance δDS which depends on the wavelength.

(9)

xi → si

(10)

(11)

∂ 2G (xi , si ) ∂ 2GL (xi , si ) k2 = + G (xi , si ), ∂n xi ∂nsi ∂n xi ∂nsi 2

xi → si

(13)

⎧ 2 ⎛ ⎞ ⎪∑N αj ⎜ ∂G (xi , sj ) + γ BM ∂ G (xi , sj ) ⎟ , xi ∈ Ω ⎜ =1 j ⎪ ∂n xi ∂nsj ⎟⎠ ⎝ ∂n xi ⎪ ⎛ ∂G (x , s ) ∂p (xi ) ⎪ N ∂ 2G (xi , sj ) ⎞ i j q (x i ) = = ⎨∑ ⎟ αj ⎜⎜ + γ BM =1, ≠ j j i ∂n xi ⎪ ∂n xi ∂nsj ⎟⎠ ⎝ ∂n xi ⎪ ⎞ ⎛ ∂G (xi, si ) 2 ⎪ BM ∂ G (xi, si ) xi ∈ Γ ⎪ + αi ⎜ ∂nx + γ ∂nx ∂ns ⎟ , i i i ⎠ ⎝ ⎩

(7)

G (xi , si ) = GL (xi , si ) + ik ,

have been derived in [23].

3. Burton-miller type regularized meshless method

where xi is the i-th physical point, sj is the j-th source point located on the physical boundary, αj the j-th unknown intensity of the distributed source at sj, N the numbers of source points and

eikr

∂ 2GL (xi, si ) ∂n xi ∂nsi

(4)

where the subscripts T, R and I denote the total, radiation and incidence wave respectively. By the single layer fundamental solutions, the approximate solutions p(x) and q(x) of exterior acoustic problem can be expressed as follows:

G (xi , sj ) =

and

As well already known that Eqs. (5), (6) encounters the nonuniqueness problems when the wave number k is the eigen-frequency of the corresponding interior problem. Many techniques to avoid the non-uniqueness exist in the literature. These techniques can be classified into two categories. One is to add additional restriction to get the unique solution, such as the CHIEF method [24] to add more points in the domain, which also satisfy the Helmholtz equation; The other one is to add the damping in the original equation, and shift the fictitious eigen-frequencies to the complex plane [25]. Many methods can be considered as this category. Burton-Miller Method [26] adds the imaginary double layer integral equation to the original one. Dualsurface method was original utilized in the electromagnetic scattering problem [27], and then extended to the acoustic scattering problem [28], which adapts the imaginary surface to shift the fictitious frequencies. Here we adapt these two damping methods with the regularized meshless method to overcome the non-uniqueness.

From the boundary value problem Helmholtz equation in 3D domain Ω exterior to a closely boundary Γ

▽2p (x ) + k 2p (x ) = 0, x ∈ Ω

∂GL (xi, si ) ∂nsi

(12)

where GL (xi , sj ) is the fundamental solutions of Laplace equation, GL (xi , sj ) = 1/ r in 3D problems. The derivation of GL (xi , si ) and ∂GL (xi, si ) are shown in Appendix A.

Fig. 1. Dual-surface model scheme.

∂n xi

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⎧∑N α (G (x , s ) + γ DSG (x , s DS )), xi ∈ Ω i j i j j =1 j ⎪ ⎪ N DS p (x i ) = ⎨ ∑ αj (G (xi , sj ) + γ DSG (xi , sj )) ⎪ j =1, j ≠ i ⎪ + α (G (x , s ) + γ DSG (x , s DS )), xi ∈ Γ ⎩ i i i i i

∂p ∂pinc + scattered = 0, ∂r ∂r

Al = −po (2l + 1) i l

•

(22)

(23)

(24)

The observer point is at (1.5, 0, 0). In order to calibrate SRMM, RRMSE with boundary node number by the explicit empirical formula for Dirichlet problem diagonal elements [21] are compared in Fig. 3. The comparisons of single layer regularized meshless method (SRMM), Burton-Miller method, Dual Surface method and analytical results are shown in Fig. 4 for the rigid sphere and Fig. 5 for the soft sphere. It can be observed that the SRMM encountered fictitious frequencies at k = nπ , n = 1, 2, 3, … for the rigid sphere. Both Burton-Miller method and Dual Surface method overcome the non-uniqueness. The fictitious frequencies are not appeared in the range of the calculation for the soft sphere. 5.2. Bean shaped surface radiation

(17)

In this case, the bean shaped surface [30] in Fig. 6 is defined by the following equation:

This plane wave can be represented as a series of spherical harmonics as follows,

( y + 0.3 cos(πz /R ))2 x2 + + z 2 = R2 0.64(1 − 0.1 cos(πz / R )) 0.64(1 − 0.4 cos(πz / R ))

N

(18)

(25)

The largest dimension of this bean is (in the z-axis direction) 2R. We test the radiation by a pulsating bean with R=2. The Dirichlet boundary condition on the surface is produced by a point source of sphere dilatation wave with unit intensity located at the coordinate origin,

where jl is the spherical Bessel function, Pl is the Legendre function of order l, and i = −1 . The scattered wave can be represented as follow, N

l =0

jl (ka ) hl (ka )

ptotal = pinc + pscattered

Two different boundary conditions on the sphere will be conducted. One is that sphere boundary is rigid to test Neumann formula; The other one is that the sphere boundary is soft to test the Dirichlet formula. We model a sphere of radius a = 1m . The fluid medium surrounding the sphere is air with sound speed c = 343m/s and mean density ρo = 1.21 kg/m3. The wave number at a frequency ω is given as k = ω / c . A plane wave of amplitude po=1 propagating in the direction of +z direction is scattered by sphere centered at the origin (0, 0, 0). The sphere in the presence of the plane wave is shown in Fig. 2. An incident plane wave of amplitude po traveling in +z direction is given by

∑ Al hl (kr ) Pl (cos(θ ))

at r = a

The total sound pressure at any field point is the sum of the incident and scattered pressures,

5.1. Sphere plane wave scattering

pscattered =

- For the soft boundary condition, the total pressure on the sphere surface is zero. That is,

Al = −po (2l + 1) i l

5. Numerical examples and discussions

l =0

(21)

Therefore,

Here we choose γ DS = i and δ DS = λ /8 according to [29]. Compared with Burton-Miller type method, one of the advantages is obvious: no additional singularities are arisen.

∑ (2 l + 1) i ljl (kr ) Pl (cos(θ ))

ljl −1 (ka ) − (l + 1) jl +1 (ka ) lhl −1 (ka ) − (l + 1) hl +1 (ka )

pinc + pscattered = 0,

(16)

pinc = po

(20)

Therefore, (15)

⎧ DS ⎞ ⎛ ⎪∑N αj ⎜ ∂G (xi , sj ) + γ DS ∂G (xi , sj ) ⎟ , xi ∈ Ω ⎟ ⎪ j =1 ⎜⎝ ∂n xi ∂n xi ⎠ ⎪ ⎛ ∂G (xi , sj ) ∂G (xi , sjDS ) ⎞ ∂p (xi ) ⎪ N ⎟⎟ q (x i ) = = ⎨∑ j =1, j ≠ i αj ⎜⎜ + γ DS ∂n xi ∂n xi ⎪ ⎝ ∂n xi ⎠ ⎪ DS ⎞ ⎛ ∂G (x , s ) ⎪ i i DS ∂G (xi , si ) ⎟ , xi ∈ Γ ⎪ + αi ⎜ ∂nxi + γ ∂n xi ⎠ ⎝ ⎩

pinc = po eikz

at r = a

(19)

where hl is the spherical Hankel function of the first kind of order l, The coefficient Al is determined from the boundary condition.

•

-For the rigid boundary condition, the total normal velocity on the sphere surface is zero. That is,

Fig. 3. RRMSE of explicit empirical formula method [21] and SRMM for Soft sphere plane wave scattering total pressure comparisons at (1.5,0,0).

Fig. 2. Plane wave scattered by a sphere.

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Analytical SRMM Burton-Miller Dual Surface

1

Analytical SRMM Burton-MIller Dual Surface

0.6 image

real

0.4

0.2

P

P

0.5

0

-0.2

0

-0.4

2

4

6

k

8

10

12

14 -0.6

Fig. 4. Rigid sphere plane wave scattering total pressure comparisons at (1.5,0,0).

2

4

k

6

8

Fig. 7. Bean shaped surface radiation comparisons at (R,0,0).

1.4 1.2 1 Analytical SRMM Burton-Miller Dual Surface

0.8

P

0.6 0.4 0.2 0 -0.2

Fig. 8. Two sphere radiation model.

-0.4 2

4

6

k

8

10

12

pressure contours at the z=0 plane for the non-dimensional wavenumbers ka = 1, 2 . The numbers on the contours indicate the related values of the non-dimensional pressure amplitude p /(z o vo ). Both of the Burton-Miller method and Dual Surface method results are in good agreement with the BEM results [31] and SBM results [32]. Also we can observe that the Dual Surface method performs better than BurtonMiller method nearly to the boundary.

14

Fig. 5. Soft sphere plane wave scattering total pressure comparisons at (1.5,0,0).

6. Conclusion The Regularized Meshless Method (RMM) is a meshless boundary method. Its source points and physical points are overlapped. The substraction and adding-back technique is utilized to avoid the singularity of the fundamental solution. It is simple and easy to be programmed. But the double layer potential should be adopted in the desingularity technique. Here the single layer potential is employed to circumvent the singularity. The substraction and adding-back technique is succeeded, but the careful selection of particular solution for the null-fields boundary integral equation is chosen to derive the diagonal elements for the Laplace Dirichlet problem. By this particular solution, the diagonal elements can be represented by the single layer potential. In this paper it is extended to the exterior Helmholtz problem by relationships between Laplace and Helmholtz singularities. The fictitious frequencies are avoided by the Burton-Miller type formula and dual surface technique. The accuracy of these methods are shown by three typical examples for different boundary conditions.

Fig. 6. Bean model.

namely, the analytical solution is p = eikr / r . Fig. 7 plots the real part and imaginary part of the computed pressure at (R, 0, 0). It can be observed that the both Burton-Miller method and Dual Surface method perform well with the analytical solution. 5.3. Two sphere radiation The radiation problem from two unit spheres and with centers at a distance of 4 and radius a=1, vibrating with uniform radial velocity vo is considered here, as shown in Fig. 8. Figs. 9 and 10 show the equal

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L. Liu

Fig. 9. Two sphere radiation at ka=1, (a) is the Burton-Miller method result, (b) is the Dual Surface method result.

Fig. 10. Two sphere radiation at ka=2, (a) is the Burton-Miller method result, (b) is the Dual Surface method result.

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Appendix A. The derivation of GL (xi , si ) in Eq. (9) and ∂GL (xi, si ) in Eq. (10) ∂n xi Followed by Young et al. [9], the null-fields of the boundary integral equation on the direct method is used to derive the GL (xi , si ).

⎛

∫Γ ⎜⎝u (s) ∂GL∂(nxs, s)

0=

− GL (x, s )

∂u (s ) ⎞ ⎟ dΓ (s ), ∂ns ⎠

x ∈ Ωe

(A.1)

where the superscript e denotes the exterior domain. If the particular solution u (s ) = 1 and ∂u (s )/∂ns = 0 are chosen, the Eq. (A.1) can be rewritten as following:

∫Γ ∂GL∂(nxs, s) dΓ (s) = 0,

x ∈ Ωe

(A.2)

When the collocation point x approaches the boundary, we can discretize Eq. (A.2) as follows: N

∂GL (xi , sj )

∑

∂nsj

j =1

Sj = 0,

x∈Γ (A.3)

where Sj is the element area around sj,

lim sj → xi

∂GL (xi , sj )

∂GL (xi , sj )

+

∂n xi

∂nsj

∂GL (xi, sj ) ∂nsj

=−

〈(xi − sj ), nsj 〉 r3

. When the source point sj moves close to the collocation point xi, we have

=0 (A.4)

So we get N

∂GL (xi , si ) 1 = ∂n xi Si

∑

Sj

∂GL (xi , sj )

j =1, j ≠ i

∂nsj

(A.5)

If the nodes on the boundary are uniformly distributed, then

∂GL (xi , si ) = ∂n xi

N

∂GL (xi , sj )

∑ j =1, j ≠ i

∂nsj

(A.6)

Now we choose other particular solution u(s) for the Laplace equation to derive GL (xi , si ).

w≔x + y + z

(A.7)

u (s ) = 〈nx , (ws − wx )〉

(A.8)

∂u (s ) = 〈▿u (s ), ns 〉 = 〈nx , ns 〉 ∂ns

(A.9)

Then the null-fields Eq. (A.1)

⎛

∫Γ ⎜⎝〈nx , (ws − wx )〉 ∂GL∂(nxs, s)

⎞ − GL (x, s )〈nx , ns 〉⎟ dΓ (s ) = 0, ⎠

x ∈ Ωe

(A.10)

When the collocation point x approaches the boundary, we can discretize Eq. (A.10) as follows:

GL (xi , si ) =

1 Si

⎧ ∂G (x , s ) ⎫ L i j Sj ⎨ 〈n xi , (wsj − wxi )〉 − GL (xi , sj )〈n xi , nsj 〉⎬ ⎭ ⎩ ∂nsj j =1, j ≠ i N

∑

⎪

⎪

⎪

⎪

(A.11)

If the nodes on the boundary are uniformly distributed, then N

∑

GL (xi , si ) =

j =1, j ≠ i

⎧ ∂G (x , s ) ⎫ L i j ⎨ 〈n xi , (wsj − wxi )〉 − GL (xi , sj )〈n xi , nsj 〉⎬ n ∂ sj ⎭ ⎩ ⎪

⎪

⎪

⎪

(A.12)

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