# Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique

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## Abstract

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## 1. Introduction

## 2. CSPM Formulations for Sound Waves

#### 2.1. Basic Concepts of SPH

**r**) at particle i used in the SPH method can be written as a summation of neighboring particles, as:

**r**, W

_{ij}= W(

**r**

_{i}−

**r**

_{j}, h), W is the smoothing kernel function, h is the smoothing length defining the influence area of the smoothing function, N indicates the number of particles in the support domain, m

_{j}is the mass of particle j, and ρ

_{j}is the density. In the SPH convention, the kernel approximation operator is marked by the angle bracket < >.

_{D}= 1/h, 15/(7πh

^{2}), and 3/(2πh

^{3}), respectively.

#### 2.2. Acoustic Wave Equations in Lagrangian Form

**v**is the flow velocity, P is the pressure, t is time, and c is the speed of sound.

**v**are supposedly small, which can be expressed as:

_{0}and p

_{0}remain the same during the computation, they can also be expressed as:

#### 2.3. CSPM Formulations for Acoustic Waves

#### 2.3.1. Particle Approximation of the Continuity Equation

**v**

_{ij}=

**v**

_{i}−

**v**

_{j}. Considering Equation (7), the continuity equation can also be written as:

#### 2.3.2. Particle Approximation of the Momentum Equation

#### 2.3.3. Particle Approximation of the Equation of State

#### 2.3.4. Corrective Smoothed Particle Method

**r**) is assumed to be sufficiently smooth in a domain that contains

**r**, the Taylor series expansion for f(

**r**) in the vicinity of ξ can be written as:

**ξ**, and integrating over the support domain, the following formulation can be obtained:

**r**) at

**ξ**can be written as:

**r**−

**ξ**, h) with $\nabla $W(

**r**−

**ξ**, h), a corrective kernel approximation for the first derivative of f(

**r**) at

**ξ**can be written as:

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

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

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

_{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:

**v**.

## 4. Sound Propagation Simulation with CSPM

#### 4.1. Sound Propagation Model

_{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

_{CFL}for short, and C

_{CFL}= u∆t/∆x.

#### 4.3. Discussion on Computational Parameters

_{error}) and the maximum error (M

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

_{error}) for L

_{error}and M

_{error}are evaluated as:

_{error}or M

_{error}, and NP

_{max}and NP

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

_{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.

_{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.

_{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.

## 5. Application of Different Acoustic Boundaries

#### 5.1. Soft Boundary

#### 5.2. Rigid Boundary

#### 5.3. Absorbing Boundary

## 6. Conclusions

- The CSPM method is proposed to simulate sound propagation in the time domain by solving acoustic wave equations. Numerical results agree well with theoretical solutions in the modeling of sound propagation in pipes.
- The CSPM method exhibits good convergence, while maintaining a constant ratio of the particle spacing to the smoothing length. According to the present work, the convergence rate is about 1.998 and the CCFL is suggested to be under 0.28.
- A hybrid meshfree-FDTD method is developed and used as an acoustic boundary treatment technique for the meshfree method, and different boundaries are built for virtual particles by using this technique.
- The sound propagation and reflection computed with soft, rigid, and absorbing boundaries, agree well with theoretical solutions for modeling sound propagation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**The sketch of simulating the acoustic boundary using the hybrid meshfree-finite difference time domain (FDTD) method.

**Figure 3.**Sound pressure computation between corrective smoothed particle method results and theoretical solutions.

**Figure 5.**Sound pressure error versus C

_{CFL}in the CSPM computation; (

**a**) L

_{error}and (

**b**) M

_{error}.

**Figure 6.**Comparison between CSPM results and exact solutions with sound reflecting from the soft boundary: (

**a**) t = 0 s; (

**b**) t = 0.20 s; (

**c**) t = 0.40 s; (

**d**) t = 0.50 s; (

**e**) t = 0.60 s and (

**f**) t = 0.80 s.

**Figure 7.**Comparison between CSPM results and exact solutions with sound reflecting from the rigid boundary: (

**a**) t = 0.40 s; (

**b**) t = 0.80 s.

**Figure 8.**Comparison between CSPM results and exact solutions with sound propagation through absorbing boundary: (

**a**) t = 0.40 s; (

**b**) t = 0.45 s; (

**c**) t = 0.50 s and (

**d**) t = 0.60 s.

**Table 1.**Parameters for the corrective smoothed particle method (CSPM) algorithm for sound propagation modeling.

Computational Parameters | Values |
---|---|

∆x | 0.04 m |

h | 0.058 m |

Kernel Type | Cubic Spline |

C_{CFL} | 0.10 |

c_{0} | 340 m/s |

ρ_{0} | 1.0 kg/m^{3} |

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**MDPI and ACS Style**

Zhang, Y.O.; Li, X.; Zhang, T. Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique. *Math. Comput. Appl.* **2017**, *22*, 26.
https://doi.org/10.3390/mca22010026

**AMA Style**

Zhang YO, Li X, Zhang T. Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique. *Mathematical and Computational Applications*. 2017; 22(1):26.
https://doi.org/10.3390/mca22010026

**Chicago/Turabian Style**

Zhang, Yong Ou, Xu Li, and Tao Zhang. 2017. "Modeling Sound Propagation Using the Corrective Smoothed Particle Method with an Acoustic Boundary Treatment Technique" *Mathematical and Computational Applications* 22, no. 1: 26.
https://doi.org/10.3390/mca22010026