Mathematical Problems in Engineering

Volume 2015, Article ID 348314, 7 pages

http://dx.doi.org/10.1155/2015/348314

## SPH Simulation of Acoustic Waves: Effects of Frequency, Sound Pressure, and Particle Spacing

^{1}School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China^{2}Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093, USA^{3}Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science and Technology, Wuhan 430074, China^{4}School of Engineering, University of Liverpool, The Quadrangle, L69 3GH Liverpool, UK

Received 30 August 2014; Revised 15 November 2014; Accepted 15 November 2014

Academic Editor: Kim M. Liew

Copyright © 2015 Y. O. Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Acoustic problems consisting of multiphase systems or with deformable boundaries are difficult to describe using mesh-based methods, while the meshfree, Lagrangian smoothed particle hydrodynamics (SPH) method can handle such complicated problems. In this paper, after solving linearized acoustic equations with the standard SPH theory, the feasibility of the SPH method in simulating sound propagation in the time domain is validated. The effects of sound frequency, maximum sound pressure amplitude, and particle spacing on numerical error and time cost are then subsequently discussed based on the sound propagation simulation. The discussion based on a limited range of frequency and sound pressure demonstrates that the rising of sound frequency increases simulation error, and the increase is nonlinear, whereas the rising sound pressure has limited effects on the error. In addition, decreasing the particle spacing reduces the numerical error, while simultaneously increasing the CPU time. The trend of both changes is close to linear on a logarithmic scale.

#### 1. Introduction

Some classic numerical methods such as the finite element method (FEM) [1, 2], the boundary element method (BEM) [3], and other modified or coupled methods [4–6] are widely used for acoustic simulations. However, these mesh-based methods are not ideal for solving acoustic problems consisting of a variety of media or with deformable boundaries.

Meshfree methods can handle such complicated problems. The method of fundamental solutions (MFS) [7], the multiple-scale reproducing kernel particle method (RKPM) [8], the element-free Galerkin method (EFGM) [9], and other meshfree methods [10, 11] have been applied to these acoustic problems.

The smoothed particle hydrodynamics (SPH) method, as a meshfree, Lagrangian method, was first independently pioneered by Lucy [12] and Gingold and Monaghan [13] to solve astrophysical problems in 1977. In addition, the SPH method has been used in many different fields [14–16]. It not only has most advantages of a meshfree method, but also is suitable for solving problems with material separation or large ranges of density as illustrated in recent reviews by Li and Liu [17], Springel [18], M. B. Liu and G. R. Liu [19], and Monaghan [20] due to its Lagrangian property. Introducing the SPH method to acoustic computation also brings its advantages to some fields like bubble acoustic, combustion noise, sound propagation in multiphase flows, and so on.

With the advance of the SPH method in acoustic simulation, some research literatures [21, 22] discussed solving fluid dynamic equations to simulate sound waves. In addition, we published a conference paper [23] that used the SPH method to solve linearized acoustic equations for modeling sound propagation and interference. Numerical results showed that the SPH method was capable of accurately modeling sound propagation, but the effects of frequency and sound pressure on the SPH simulation need further discussion. Therefore, the present paper focuses on discussing the effect of frequency and sound pressure of the acoustic waves on the numerical error caused by the SPH simulation.

The present paper is organized as follows. In Section 2, the standard SPH theory is used to solve the linearized acoustic equations. In Section 3, a one-dimensional sound propagation model is built. In Section 4, a numerical experiment is given based on standard SPH algorithms, and the effect of frequency, sound pressure, and particle spacing on the simulation is analyzed with considering the changes of particle spacing and Courant number. Section 5 summarizes the results of this work.

#### 2. SPH Formulations of Sound Waves

##### 2.1. Basic Formulations of SPH

As a meshfree, Lagrangian particle method, the SPH method is an important method widely used in recent years. Formulations in the SPH theory are represented in a particle approximation form. The properties of each particle are computed using an interpolation process over its neighboring particles [24]. In this way, the integral of a field function can be represented aswhere is the kernel approximation operator, is a function of the vector is the volume of the integral, is the smoothing kernel, and is the smoothing length.

The particle approximation for the function at particle within the support domain can be written aswhere and are the position of particles and , is the number of particles in the computational domain, is the mass of particle , , and is the distance between particle and particle .

Similarly, the gradient of function at particle is obtained aswhere .

##### 2.2. SPH Formulations of Sound Waves

In acoustic simulation, the governing equations for constructing SPH formulations are the laws of continuity, momentum, and state. The simplest and most common acoustical problem occurs under some assumptions. On one hand, the medium is lossless and at rest, so an energy equation is unnecessary; on the other hand, a small departure from quiet conditions occurs as follows:where is the fluid density, is the quiescent density which does not vary in time and space, is the change of density, is the instantaneous pressure at time of the fluid, is the quiescent pressure, is the sound pressure, is the speed of sound, and is the flow velocity. , , and are taken to be small quantities of first order.

By discarding second-order terms in the acoustic equations, the linearized continuity, momentum, and state equations (for ideal air) governing sound waves are obtained as

Applying the SPH particle approximation (see (3)) to (7), the particle approximation equation of the continuity of acoustic waves is written aswhere is the quiescent density which does not vary in time and space, is the quiescent pressure, and .

The momentum equation in SPH method is obtained as

Particle approximation of the equation of state is

The cubic spline function given by Monaghan and Lattanzio [25] is used as the smoothing kernel in this paper, which is written aswhere is in one-dimensional space, , is the distance between two particles, and is the smoothing length which defines the influence area of smoothing function .

The second-order leap-frog integration [26] is used in the present paper. All-pair search approach [24], as a direct and simple algorithm, is used to realize the neighbor particles searching in the acoustic wave simulation.

#### 3. Sound Propagation Model

In order to evaluate the effect of sound pressure and frequency, a one-dimensional sound wave which propagates in a pipe with uniform cross section is used. The acoustic model is shown in Figure 1. The sound pressure at is plotted with a solid line, while the sound pressure at s is plotted with a dashed line.