Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation

The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method (CVBEFM) for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers. To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative, a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures. To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables, a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure. The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers, even for extremely large wavenumbers such as κ = 10 000.


Introduction
The numerical analysis of acoustic problems has various applications in many scientific fields [1] . In recent years, various meshless methods, such as the singular meshless method [2] , the boundary point interpolation method [3] , the singular boundary method [4] , the localized method of fundamental solutions [5] , the localized boundary knot method [6] , the local radial point interpolation method [7] , the Galerkin meshfree method [8] , and the element-free Galerkin method [9] , have been applied to acoustic problems. Boundary integral equations (BIEs) [1,10] are powerful and popular for the numerical solutions of acoustic problems, since they reduce the problem dimensionality and fulfill the radiation condition at infinity in a nature way. Boundary element methods (BEMs) [1] and boundary-type meshless methods [2][3][4]11] possess significant advantages over domain-type methods such as the finite element method for acoustic problems, especially with unbounded domains.
Different from the BIEs for Laplace and elasticity problems, the BIEs for exterior acoustic problems may fail to offer unique solutions at some characteristic wavenumbers of the associated interior problem. Various techniques have been developed to circumvent this annoying shortcoming successfully. The well-known Burton-Miller formulation [12] presents a direct BIE method by using a linear combination of the singular BIE and the associated hypersingular BIE, while the combined field integral equation (CFIE) [13][14] presents an indirect BIE method by using a combination of single and double layer potentials. The applications of the CFIE have been studied mainly to acoustic problems with Dirichlet boundary conditions, rarely to problems with Neumann conditions, and none to problems with mixed conditions [13][14] . In the case of using the CFIE, the computational formulae for Dirichlet and Neumann problems are different, and cannot be unified.
It has been proven theoretically and verified numerically that the Burton-Miller formulation can obtain unique solutions for exterior acoustic problems with arbitrary wavenumbers. The applications of the Burton-Miller formulation have been studied extensively by the BEM [15][16][17] . Although numerous reported studies indicate that BIE-based meshless methods perform excellently in solving boundary value problems such as Laplace and elasticity problems [18] , very few applications of the Burton-Miller formulation to BIE-based meshless methods have been found.
The boundary element-free method (BEFM) [18] is a promising BIE-based meshless method. It not only alleviates the meshing-related shortcomings, but also enhances the precision of BEMs. With the aid of the singular BIE, the BEFM can be applied to acoustic problems free of characteristic wavenumbers [19] . With the aid of the CFIE, the BEFM can be applied to acoustic problems with pure Dirichlet [13] and pure Neumann [14] boundary conditions.
In this paper, a stabilized complex variable moving least-square (CVMLS) approximation [20] is used to approximate the boundary variables in the Burton-Miller formulation, a complex variable BEFM (CVBEFM) for solving the acoustic problems with arbitrary wavenumbers and mixed boundary conditions. The main work of this paper is as follows.
First, a weakly singular version of the Burton-Miller formulation is derived theoretically. The main difficulty in the numerical implementation of the Burton-Miller formulation is to calculate the hypersingular integrals [15][16][17] . Some regularization techniques produce integrals which are restricted to special boundary elements in BEMs or are still hard to calculate numerically. Some regularization techniques are based on the assumptions such as the solid angle is constant and/or the normal derivative of the solid angle on the boundary is zero [15][16] . In this research, by considering the normal derivative of the solid angle and using the singularity subtraction procedures for regularization, the regularization is implemented globally, rather than locally along each element as in some BEMs. Besides, the present weakly singular Burton-Miller formulation successfully gets rid of both strongly singular and hypersingular integrals and astutely avoids the computation of the solid angle and its normal derivative.
Second, a stabilized CVMLS approximation is selected to facilitate the discretization of the weakly singular Burton-Miller formulation. In the BEFM and other BIE-based meshless methods, boundary curvilinear coordinates are used to avoid the singularity in the process of boundary variable approximations, although using curvilinear coordinates is burdensome for some problems. More seriously, using curvilinear coordinates complicates the approximation of the gradients of the boundary variables. The present CVMLS approximation directly uses Cartesian coordinates, and thus facilitates the implementation of the CVBEFM and the ap-proximation of the gradients of the boundary variables in the weakly singular Burton-Miller formulation.
Third, numerical results are provided to demonstrate the effectiveness of the current Burton-Miller CVBEFM for acoustic problems with arbitrary and large wavenumbers. The results show that the CVBEFM in conjunction with the weakly singular Burton-Miller formulation is stable and robust for all wavenumbers including the characteristic wavenumbers. In previously reported numerical results, the applications of meshless methods to acoustic problems focused mainly on k < 10 and seldom on k 100. In this paper, although no extra term is added to eliminate the nasty pollution error existing in many numerical methods [21] , the present CVBEFM can produce satisfactory results for all wavenumbers, even for extremely large wavenumbers such as k = 10 000.

Problem description and hypersingular Burton-Miller formulation
The acoustic problem in a bounded homogeneous medium Ω ⊂ R 2 or an unbounded homogeneous medium Ω = R 2 /(Ω ∪ Γ) is governed by the following acoustic wave equation: where u(x) is the sound pressure, k is the wavenumber, and Γ = ∂Ω is the boundary of Ω. For exterior problems, u should fulfill the radiation condition at infinity as lim r→∞ √ r( ∂u ∂r − iku) = 0, where r = |x|, and i = √ −1. If the normal vector n x on the boundary Γ points from Ω into Ω for interior problems and points from Ω into Ω for exterior problems, the solution of both interior and exterior acoustic problems (1) can be unified as follows: while the associated singular BIE is where q(x) = ∂u(x) ∂nx is the acoustical flux on x, u * (x, y) is the fundamental solution to Eq. (1), u in (y) is the incident wave, and c(y) is a coefficient determined by the boundary geometry at y.
Physically, c(y) denotes the solid angle in the acoustic domain [1,[15][16] . Of course, we have c(y) = 1/2 when Γ is smooth around y. The singular BIE (3) is valid for both interior and exterior problems. However, if the normal vector n x is defined pointing from Ω into Ω for both interior and exterior problems, it is not trivial to obtain a unified singular BIE [1,19] .
Derivating the singular BIE (3) with respect to n y yields the hypersingular BIE as follows: where q in (y) = ∂uin(y) ∂ny . It should be stressed that the term ∂c(y) ∂ny u(y) should appear on the right-hand side of Eq. (4), since the normal derivative ∂c(y) ∂ny is usually not equal to zero when y ∈ Γ [15][16] .
Both the singular BIE (3) and the hypersingular BIE (4) hold for the exterior and interior acoustic problems, but their solutions may fail for the exterior problems at some characteristic wavenumbers of the associated interior problem. Since the characteristic wavenumbers of the two BIEs are different, one can use the following hypersingular Burton-Miller formulation: where y ∈ Γ, and α and β are coupling constants which can be taken as α = 1 and β = i/max(k, 1) to ensure the valid solutions for arbitrary wavenumbers [1,22] . Clearly, both singular and hypersingular BIEs are special cases of the hypersingular Burton-Miller formulation (5).

Regularization
In addition, Proof The singular BIE for the Laplace equation Δv(x) = 0 is [1] Setting v(x) = 1 yields Eq. (6). Besides, since the normal derivative of a double layer potential is continuous across Γ [1] , i.e., lim y→y,ỹ∈Ω which shows that Eq. (7) is true. Taking the derivative of Eq. (9) with n y and sending x → Γ yield i.e., Hence, in view of Eqs. (6) and (7), we have which shows that Eq. (8) is true. (5) can be regularized as the following weakly singular BIE :
where H are the first kind Hankel functions of orders zero and one, respectively.
Thus, as x → y, i.e., r → 0, the singularities in all integrals on the left-hand side of Eq. (10) can be canceled completely. As a result, all integrals on the left-hand side of Eq. (10) are always regular.
Remark 2 Let c = 0.577 215 664 9 · · · be the Euler constant. As r = |x − y| → 0, u * (x, y) can be expanded as [19] from which it can be verified that Thus, the two integrals on the right-hand side of Eq. (10) have weak logarithmic singularities as x → y. Remark 3 Equation (10)  ∂ny , and thus astutely avoids the computation of the two terms. In previous studies, it was usually assumed that c(y) ≡ 1/2 and/or ∂c(y) ∂ny ≡ 0. Such assumptions are not required in the present formulation (10).

Boundary variable approximations
To solve the weakly singular Burton-Miller formulation (10) To obtain the meshless approximation of the boundary function u(z), we define where z * can either be z or a node z I in the influence domain of z, a j (z) is the coefficient, and p j (z) is the conjugate of the basis function p j (z). As proven in Refs. [20] and [23], the shifted and scaled function p j (z) = (z − z e ) j /h j can enhance the stability and performance of the CVMLS approximation, where z e is fixed in the influence domain of z. Minimizing where u I is the approximation to the nodal value u(z I ), the set ∧(z) w I (z)p k (z I )p j (z I ) for k, j = 0, 1, · · · , m, and B j (z) = w I (z)p j (z I ) for j = 0, 1, · · · , m and = 1, 2, · · · τ . Inserting Eq. (19) into Eq. (18) yields the CVMLS approximation of u(z) as follows: where the CVMLS shape function is Similarly, the CVMLS approximation of the boundary function q(z) is where q I is the approximation to the nodal value q(z I ).

Discretization for mixed boundary conditions
Using Eqs. (20) and (22), the weakly singular Burton-Miller formulation (10) can be discretized as the following linear system of algebraic equations: All integrals on the left-hand side of Eq. (23) are always regular and can be calculated by using the standard Gaussian quadrature. The integrals on the right-hand side of Eq. (23) contain weak logarithmic singularities and can be calculated by the logarithmical Gaussian quadrature or the improved integration procedure [19] .
Since the CVMLS shape function Φ j (x) lacks the delta function property [18,20] , i.e., Φ j (x I ) = δ jI , we have u I = u(x I ) and q I = q(x I ). Thus, as in other meshless methods [24][25][26][27][28] , the boundary conditions in the present CVBEFM must be dealt with carefully. For problems in potential theory and linear elasticity, the boundary conditions in the boundary node method can be satisfied by coupling discretized BIEs in terms of approximate nodal variables, together with equations relating these approximations to their exact values through the MLS approximation [18] . For acoustic problems, an analogous technique is adopted in this study.
The mixed boundary conditions for the acoustic wave equation (1) are where u and q are known. In this case, the unknown boundary data are q on Γ u and u on Γ q . To simplify the representation, we assume that the first N u boundary nodes {x j } Equations (23), (26), and (27)  . Finally, the pressure u in the acoustic medium can be approximated by using Eq. (2) as follows:

Discretization for pure Dirichlet or Neumann boundary conditions
The procedure in Subsection 4.2 is valid for mixed boundary conditions including pure Dirichlet or Neumann types. This procedure makes that the number of algebraic equations is 2N and the unknowns comprise both u I and q I for all boundary nodes {x I } N I=1 ⊂ Γ. In the cases of pure Dirichlet boundary condition (i.e., Γ q = ∅) or pure Neumann boundary condition (i.e., Γ u = ∅), the number of algebraic equations can be reduced into N . In this section, the detailed formulation is presented for pure Dirichlet boundary condition. The corresponding formulation for pure Neumann boundary condition is similar.
For Eq. (1), if the given boundary condition is pure Dirichlet type, i.e., the unknown function is just the acoustical flux q on Γ. In this case, since u is known on Γ, the approximation for u in Eq. (20) is no longer required. Thus, with Eq. (22) only, the weakly singular Burton-Miller formulation (10) can be discretized as follows: which can be solved directly to obtain the N unknowns { q I } N I=1 . Finally, the pressure u in the acoustic medium can be approximated as follows: ∂u * (x, y) ∂n x dΓ x + u in (y), y ∈ Ω or Ω .

Numerical examples
Three examples are solved in this section to show the efficiency of the present Burton-Miller CVBEFM. In this study, the quadratic basis function is used, and the radius of the influence domain for meshfree approximation is 2.5h. To evaluate the performance of the method, we define an average relative error as follows: where u(x j ) and u h (x j ) are analytical and numerical results of u(x) on the N t points x j , respectively.

Example 1
The first example solves an exterior acoustic problem in an infinitely long cylinder with the radius a = 2.0. In the polar coordinate system (r, θ), the Neumann boundary condition is imposed as ∂u ∂n = e iθ , and the analytical solution is [29] u(r, θ) = ae iθ H , r a, 0 θ < 2π.
To study the performance of the CVBEFM for all wavenumbers, the weakly singular versions of the singular BIE (i.e., α = 1 and β = 0 in Eq. (10)), the hypersingular BIE (i.e., α = 0 and β = 1 in Eq. (10)), and the Burton-Miller formulation (10) are independently used in the CVBEFM. Figure 1 depicts the results of |u(r, θ)| at (4.0, π/4) with respect to the wavenumber k obtained by using 20 boundary nodes, Fig. 2 depicts the error e(u), while Fig. 3 depicts the condition number of the system matrices. From Figs. 1 and 2, we can find that the singular BIE performs worse as k approximates to 1.915 852, 3.507 789, 5.086 727, 6.661 836, 8.235 318, and 9.808 106, the hypersingular BIE performs worse as k approximates to 2.666 592, 4.269 718, 5.855 423, 7.435 340, and 9.012 799, while the Burton-Miller formulation produces satisfactory results in all cases. Additionally, from Fig. 3, we can find that singular and hypersingular BIEs may produce large condition numbers when k approximates to these characteristic wavenumbers, but the Burton-Miller formulation produces well-conditioned system matrices in all cases. Consequently, the CVBEFM in conjunction with the Burton-Miller formulation is robust to avoid the influence of characteristic wavenumbers.
In Fig. 4, the convergence of the Burton-Miller CVBEFM is displayed for k = 1, 1.915 852, 2.666 592, 3.507 789, 4.269 718, 5.086 727, 5.855 423, 6.661 836, 7.435 34, 8.235 318, 9.012 799, 9.808 106, and 10. As shown in the figure, the average related error e(u) decreases monotonously and quickly when the nodal spacing h decreases. Thus, the Burton-Miller CVBEFM is able to produce stable convergence results with high experimental convergence rate, about 3 even for characteristic wavenumbers. However, the accuracy does not change monotonously when the wavenumber increases. The reason is unclear and requires further research.
In Fig. 5, the absolute error |u − u h | at (r, θ) = (4.0, π/4) of the Burton-Miller CVBEFM obtained by using 128 boundary nodes is displayed for some wavenumbers. The results of the BEM [29] obtained by using 128 boundary elements and the BPIM [3] obtained by using 128 boundary nodes are also displayed for comparison. We can find that the precision of the CVBEFM is higher than the precision of either the BEM or the BPIM. Especially, the precision of the CVBEFM is several orders of magnitude higher than that of the BEM. It is well-known that the solutions of acoustic problems are highly oscillatory for large wavenumbers. Then, because of the nasty pollution error, in many numerical techniques, the node number must increase much faster than the wavenumber so as to obtain satisfactory numerical results [21] . To study the performance of the Burton-Miller CVBEFM for extremely large wavenumbers, Fig. 6   In Table 1, the relative error |u − u h |/|u| at four points and the average relative error e(u) of the Burton-Miller CVBEFM are displayed for various wavenumbers k. Although the number of boundary nodes is set to be N = k in this analysis, from Table 1, we can find that the present CVBEFM can produce satisfactory results, even for extremely large wavenumbers such as k = 10 000.

Example 2
The second example solves an exterior acoustic problem describing an incident plane wave u in (r, θ) = e ikr cos θ scattered by an infinitely long cylinder with the radius a = 2.0. The Dirichlet boundary condition is imposed for the scattered wave field u s as u s = 0, and the analytical solution is [30] u s (r, θ) = − J 0 (ka) where r a, 0 θ < 2π, J t is the Bessel function of order t, and H (1) t is the Hankel function of the first kind of order t. The total wave field for this example is u = u s + u in . Figure 8 shows the distributions of the scattered wave field u s and the associated absolute error |u s − u sh | obtained by the Burton-Miller CVBEFM with 200 boundary nodes for k = 5π. We can find that the error is less than 7 × 10 −3 , which indicates that the present CVBEFM obtains very efficient results. In Fig. 9, the convergence of the Burton-Miller CVBEFM is displayed for k = 1, π, 2π, 4π, and 8π. Again, the average related error e(u) decreases monotonously and quickly with the decrease in the nodal spacing h, and thus the Burton-Miller CVBEFM is able to produce stable convergence results.
In Table 2, the relative error |u − u h |/|u| at four points and the average relative error e(u) of the Burton-Miller CVBEFM are displayed for various wavenumbers k. These numerical results are obtained by using N = 4k and a = 1.0. We can find from Table 2 that the CVBEFM can produce satisfactory results even for large wavenumbers. π π π π Fig. 9 Results of e(u) at different k for the second example (color online)

Example 3
The third example considers an interior acoustic problem with mixed boundary conditions in a unit circle centered at the origin. Dirichlet and Neumann boundary conditions are imposed on the upper and lower halves of the boundary, respectively. The analytical solution is u(x 1 , x 2 ) = sin(πx 1 )e (π 2 −k 2 ) x 2 .
In Fig. 10, the results of u are presented along the curve given by x 1 = 0.5 cos θ and x 2 = 0.5 sin θ with θ ∈ [0, 2π]. The numerical results are obtained by the Burton-Miller CVBEFM using 100 boundary nodes for k = 10, 20, and 30. Clearly, the numerical results agree well with analytical results. The convergence is presented in Fig. 11, from which we can see that the method is fast convergent.

Conclusions
This paper discusses the meshless numerical analysis of acoustic problems by introducing the Burton-Miller formulation to develop a Burton-Miller CVBEFM for improving the quality of numerical solutions, particularly in the neighborhood of characteristic wavenumbers.
By the utilization of the normal derivative of the solid angle and singularity subtraction procedures, a weakly singular Burton-Miller formulation is derived theoretically to transform both strongly singular and hypersingular integrals into weakly singular integrals. The weakly singular Burton-Miller formulation is valid for both interior and exterior acoustic problems with mixed boundary conditions, admits a unique solution for all wavenumbers, and avoids the computation of the solid angle and its normal derivative. Due to these merits, the CVBEFM is efficient for all wavenumbers, and all integrals in the CVBEFM are at most weakly singular.
In the meshless discretization process of the weakly singular Burton-Miller formulation, the CVMLS approximation is used to approximate unknown boundary variables, which further facilitates the meshless implementation, especially the approximation of boundary variable gradients in the weakly singular Burton-Miller formulation. Numerical examples testify that the present Burton-Miller CVBEFM only requires boundary nodes and is stable and effective for acoustic problems with arbitrary wavenumbers. Although the performance of meshless methods in previously reported numerical results has been shown mostly for k < 10 and seldom for k 100, this research shows that the present CVBEFM can produce satisfactory results for all wavenumbers, even for extremely large wavenumbers such as k = 10 000.

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