## 1. Introduction

Numerical methods have been applied to model acoustic phenomena, and the development of computational acoustics allows the simulation of sound generation and propagation in a complex environment. Many classic numerical methods such as the finite difference method (FDM) [

1], the finite element method (FEM) [

2], the boundary element method (BEM) [

3], and other modified methods [

4,

5], have been applied in spectral or temporal acoustic simulations. In particular, meshfree methods are widely applied to solve acoustics problems, because field points used in this method are arbitrarily distributed and the approximation smoothness order is chosen with flexibility. The method of fundamental solutions (MFS) [

6], the multiple-scale reproducing kernel particle method (RKPM) [

7], the element-free Galerkin method (EFGM) [

8], and meshfree methods [

9,

10,

11,

12] are used to address certain acoustic problems, mainly in the frequency domain, and other methods like the equivalent source method (ESM) [

13,

14,

15] solve problems in the time domain. Moreover, there is also a kind of method based on the fundamental solutions to realize the meshfree property, such as the singular boundary method [

16,

17,

18].

The smoothed particle hydrodynamics (SPH) method, as a Lagrangian, meshfree particle method, was independently pioneered by Lucy [

19] and Gingold and Monaghan [

20], for solving astrophysical problems. As a Lagrangian approach, the SPH method has several advantages over the standard grid-based numerical method: (i) the numerical error generated by computing the advection is eliminated, since the advection term is included in the Lagrangian derivative; (ii) complicated domain topologies and moving boundaries are easily represented due to its Lagrangian property, as illustrated in recent reviews by Liu and Liu [

21], Springel [

22], and Monaghan [

23]; (iii) the interface between different mediums can be naturally traced through the particle density, instead of using a special algorithm such as the volume-of-fluid; (iv) it is easy to implement and has a parallel processing ability, for the approximation is implemented in the local support domain instead of the whole computational domain [

24,

25]. Introducing the SPH method to acoustic computation can bring these advantages to this field.

Recently, the SPH method has been gradually used in acoustic computation, and some researchers have attempted to obtain the acoustic field through direct numerical simulation. Wolfe [

26] simulated room reverberation with sound generation and reception, based on a SPH fluid mechanics algorithm, and Hahn [

27] solved the fluid dynamic equations to obtain pressure perturbations during sound propagation. Both of these works can be seen as direct numerical simulations (DNS), based on the SPH method. However, for various acoustic waves in engineering problems, acoustic variables such as the variation in pressure, density, and velocity, are generally small. On the contrary, the values of pressure, density, and velocity exist on a much larger scale than any variation in these variables [

28]. Acoustic wave equations are obtained based on the acoustic variables, to avoid solving fluid dynamic variables. Consequently, solving acoustic wave equations requires a lower computational burden compared to solving the fluid dynamic equations directly, and this approach is widely used in modeling engineering problems.

In our recent work, we proposed the use of the SPH method to solve acoustic wave equations, and tests on the sound propagation and interference simulation were conducted [

29,

30]. Some computational parameters were also discussed [

31,

32]. Based on these tests, the SPH method accurately solved acoustic wave equations, and some parameters were investigated, but only the traditional SPH method was used.

With the advancement of the SPH method, the traditional SPH method is modified or improved to reduce numerical error. Chen et al. [

33,

34] proposed the corrective smoothed particle method (CSPM) on the basis of Taylor series expansion in 1999. The CSPM is effective in reducing numerical error both inside the computational domain and around the boundary, so it has been used in different fields [

35,

36]. Other notable modifications or corrections of the SPH method include the reproducing kernel particle method [

37], the finite particle method (FPM) [

38,

39], moving least square particle hydrodynamics (MLSPH) [

40,

41], and modified smoothed particle hydrodynamics (MSPH) [

42]. The present study focuses on using the CSPM to improve the simulation accuracy of the SPH method for solving acoustic wave equations.

The implementation of boundary conditions in the SPH method is not as straightforward as in the grid-based numerical models. Historically, this characteristic has been regarded as a weak point of the particle method [

43,

44]. Several approaches have been proposed to treat boundary conditions for computational fluid dynamics. Among them, it is feasible to use virtual particles [

45] to implement the boundary conditions. These virtual particles are allocated on and outside the boundary, as shown in several related works [

46,

47]. So far, a limited implementation of acoustic boundaries has been observed, and the rigid acoustic boundary developed from the solid boundary in fluid dynamics is shown in [

26,

27,

29,

30,

31,

32]. The boundary treatment technique for different acoustic boundaries is important for acoustic numerical analysis by the Lagrangian meshfree method.

The present paper is organized as follows: In

Section 2, the CSPM formulations are provided to solve the acoustic wave equations. In

Section 3, a hybrid meshfree-FDTD (finite difference time domain) method is proposed for acoustic boundary treatment. In

Section 4, a sound propagation model is used to validate the CSPM algorithm, and the effects of different computational parameters are discussed. In

Section 5, soft, rigid, and absorbing boundaries are used to simulate sound reflection and transmission, and numerical results are compared with theoretical solutions.

## 3. Hybrid Meshfree-FDTD Method for Boundary Treatment

The meshfree method suffers from the problem that not enough particles in the support domain can be used for computation at the boundary. In the present paper, the FDTD method is introduced and is combined with the virtual particle technique, and thus, a technique based on the meshfree-FDTD hybrid method for acoustic boundary treatment is accordingly constructed. The feasibility and validity of the meshfree-FDTD hybrid method is verified by simulating sound propagation in pipes with boundaries.

Since the FDTD method was proposed by Yee [

52] in 1966, it has received wide recognition and has been used to solve problems in many different research fields. The FDTD method can solve fundamental equations in the time domain. In this paper, for building the hybrid method, the FDTD method proposed by Wang [

53], which was used to simulate an underwater acoustic boundary and the virtual particle technique, are combined.

In the hybrid meshfree-FDTD boundary treatment, three types of particles need to be built before computation, namely the fluid particle, the boundary particle, and the virtual particle. During the computation, the numerical methods chosen for these three kinds of particles are shown below.

The hybrid meshfree-FDTD boundary treatment technique uses the meshfree method to obtain the parameter value of fluid and boundary particles, and the FDTD method to solve the parameter value of virtual particles.

Figure 1 is the sketch of the treatment of particles on a wall boundary when using the meshfree-FDTD method. The number of particles is represented by

i. Virtual particles can be obtained through extending boundary particles to the outside of the computation region, and the distribution of virtual particles is parallel to the boundary particles, with equal particle spacing. The number of layers can be chosen according to the scale of the support domain. There should be enough virtual particles in the support domain for boundary particles, and virtual particles outside the support domain are unnecessary. In the present work, we use three layers of virtual particles to build the boundary.

For the soft boundary, the boundary conditions are:

The formulation for the sound pressure of the virtual particles is written as:

Assuming that the velocity perpendicular to the surface is

u, the velocity of the virtual particles (e.g., particle

i + 1) according to the central difference scheme should be calculated from the momentum equation as:

which can be written as:

where the superscript

n represents the temporal index, ∆

t is the time step, and ∆

x is the particle spacing.

For the rigid boundary, the normal component of the pressure gradient on the surface equals zero when the wave is vertically incident with the boundary. Therefore, for the rigid case, the sound pressure

δp satisfies the following:

where

**n** represents the normal direction of the surface. According to the finite difference scheme, we have:

which can be written as:

For the absorbing boundary condition, the popular first-order absorbing boundary condition (ABC) proposed by Mur is used in the present work. Assuming that the ABC is located at

x =

x_{i}, sound propagates from the left side to the right side. The ABC can be written as:

where the field parameter

f can be

δp,

u_{x}, or

u_{y} in this equation. This leads to a different expression for virtual particles than that used in the meshfree-FDTD hybrid method, as follows:

The field parameter f in this equation can be δp or **v**.

## 4. Sound Propagation Simulation with CSPM

#### 4.1. Sound Propagation Model

Sound propagation along ducts with different boundaries are discussed, as shown in

Figure 2. In this model, sound propagates from

x < 0 m to

x ≥ 0 m, and the positive direction of the

x-axis denotes the direction of sound propagation. The CSPM computational region is from −50 m to 150 m, and the propagation time is 0.25 s.

The sound pressure of the acoustic wave [

54] in the ducts is written as:

where

t denotes time,

x is the geometric position in the propagation direction,

ω is the angular frequency of the sound wave, and

k =

ω/

c_{0} is the wave number. In addition, the sound speed

c_{0} = 340 m/s, and

ω = 340 rad/s.

#### 4.2. Verification of the Meshfree Algorithm

Table 1 lists the computational parameters that are used in the CSPM algorithm for sound propagation. The Courant-Friedriches-Lewy number is written as

C_{CFL} for short, and

C_{CFL} =

u∆

t/∆

x.

In order to verify the algorithm, the CSPM algorithm is built to solve acoustic wave equations for sound propagation modeling. The costing central processing unit (CPU) time for the CSPM is 55.6 s, with the performance of computation measured on the Intel Core i3-3240 with RAM 4.00 GB (Gigabyte Technology Co., Ltd., New Taipei City, Taiwan). Then, the simulation results are compared to theoretical solutions, as shown in

Figure 3. In this figure, solid lines demonstrate theoretical solutions, and points represent the CSPM simulation results. For clearly identifying different points, they are plotted at intervals of 14 grid points.

From the figure, it can be seen that several peaks and valleys appear in the graph between −50 m and 150 m. The algorithm is able to model the sound propagation process, and the CSPM simulation results are in good agreement with the theoretical solutions.

#### 4.3. Discussion on Computational Parameters

In this section, the effects of the initial particle spacing and the time step on the accuracy of the present CSPM method is discussed. The method is compared with theoretical solutions. The numerical accuracy is evaluated with the relative root mean square errors (

L_{error}) and the maximum error (

M_{error}), which are given as follows:

where

$\delta p(i)$ and

$\delta \overline{p}(i)$ are simulation results and theoretical solutions at particle

i, and

N is the total number of particles in the computation domain.

The convergence rate (

R_{error}) for

L_{error} and

M_{error} are evaluated as:

where

Error represents

L_{error} or

M_{error}, and

NP_{max} and

NP_{min} represent the maximum and minimum number of particles in the computational domain, respectively.

Sound propagation with particle spacing changing from 0.02 to 0.10 m is computed. Then, the CSPM numerical error of the sound pressure according to Equations (40) and (41), is shown in

Figure 4. In the computation, the ratio of the particle spacing to the smoothing length remains the same.

From the figure, it can be seen that, when the particle spacing increases, L_{error} and M_{error} increase gradually. L_{error} and M_{error} are the smallest at particle spacing ∆x = 0.02 m, which are 1.5 × 10^{−3} and 0.047 Pa, respectively. When at particle spacing ∆x = 0.10 m, L_{error} and M_{error} reach 0.126 and 1.17 Pa, respectively. The convergence rate for L_{error} and M_{error} is about 1.997 and 1.998, respectively, and 1.998 on average. In conclusion, the CSPM algorithm shows a good convergence in the simulation of sound propagation.

Similarly, the numerical error of sound propagation by using the CSPM algorithms a with different

C_{CFL} is also discussed. When the

C_{CFL} changes from 0.05 to 0.32,

L_{error} and

M_{error} are computed, as shown in

Figure 5.

As can be seen from the figure, in the region of 0.05 ≤ C_{CFL} ≤ 0.28, with the increasing of C_{CFL}, L_{error} and M_{error} increase slowly. When C_{CFL} is equal to 0.05, L_{error} and M_{error} are 0.02 and 0.19 Pa, respectively. When C_{CFL} is 0.28, L_{error} and M_{error} are 0.03 and 0.20 Pa, respectively. Moreover, when C_{CFL} is greater than 0.28, L_{error} and M_{error} increase sharply with increased C_{CFL}. In general, according to the present case, for maintaining computational accuracy and efficiency in the numerical simulation, C_{CFL} is preferably set as under 0.28.