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

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


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][5][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.
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][15][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 2 Mathematical Problems in Engineering 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.

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 (r) can be represented as where ⟨⋅⟩ is the kernel approximation operator,  is a function of the vector r, Ω is the volume of the integral,  is the smoothing kernel, and ℎ is the smoothing length.The particle approximation for the function (r) at particle  within the support domain can be written as where r  and r  are the position of particles  and ,  is the number of particles in the computational domain,   is the mass of particle ,   = (  , ℎ), and   is the distance between particle  and particle .
Similarly, the gradient of function (r) at particle  is obtained as where

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,  0 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,  0 is the quiescent pressure,  is the sound pressure,  0 is the speed of sound, and u is the flow velocity., , and u 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 as where  0 is the quiescent density which does not vary in time and space,  0 is the quiescent pressure, and u  = u  − u  .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 as where   is 1/ℎ in one-dimensional space,  = /ℎ,  is the distance between two particles, and ℎ 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.

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  = 0 is plotted with  Sound pressure of the source of sound wave in Figure 1 is where  represents time (propagation starts when  = 0),  is the geometric position,  Amp is the maximum sound pressure of the acoustic wave,  is the circular frequency of the acoustic wave,  = / 0 is the wave number, and the sound speed  0 is 340 m/s.The sound propagates from  < 0 to  ≥ 0, with the SPH computational region going from −10 m to 70 m.In the computation, the region of the sound source is from −10 m to 0 m, which is determined by the theoretical solutions, while the region of sound propagation is from 0 m to 70 m, which is obtained by the SPH method.At the end of the sound propagation domain, the particles are set free in the computation.Since the SPH computational region is far longer than the sound propagation distance, any effects caused by ignoring this boundary will not propagate to the sound propagation domain and therefore can be neglected.The simulation results at the time  = 0.1 s are used to compare with the theoretical solutions.A similar numerical model is used in our recent conference paper [23].
In the evaluation of numerical results, the nondimensional simulation error of the sound pressure is defined by where  *  is the simulation pressure of the particle  and (  ) is the theoretical pressure at the position of particle .

Numerical Experiment.
A sound propagation model is built to confirm agreement between SPH simulation results and the theoretical solution.The parameters used in the numerical experiment of the sound waves are listed in Table 1.The physical time required for the modeling was 141.6 seconds.
A comparison between the SPH results and theoretical solutions of sound pressure at  = 0.1 s is shown in Figure 2. The theoretical solution is plotted with a solid line while the SPH results are plotted with red points.In order to clearly identify the numerical results, the points are plotted at intervals of 12.
As shown in Figure 2, the sound pressure of the sound wave changes along with the position, and a peak and a valley appear in the final wave at  = 0.1 s.Comparing with the analytical data, it can be seen that the SPH method can accurately simulate the sound pressure of an acoustic wave as the values of sound pressure can be obtained precisely, and the simulation error is 1.6 × 10 −4 .The trend of both the SPH results and the theoretical solutions also has a similar tendency.

Effect of Frequency.
Different sound propagation models with the circular frequency of sound changing from 10 rad/s to 200 rad/s are simulated, and the effect of circular frequency is discussed.In the discussion, different computational parameters, namely, the particle spacing and the Courant number, are also considered. Figure 3 gives the numerical error for different computational cases with different  and particle spacing.Figure 4 shows the sound pressure error for different cases with different values of circular frequency and Courant number.Each point in both figures stands for a computational case.The Courant number used in the computational cases in Figure 3 is 0.05, and the particle spacing used in all cases in Figure 4 is 0.06 m.As can be seen from Figure 3,  pre increases along with the increase of , and the increase of  pre is nonlinear.Along with the increase of , the increase of  pre is larger.On the other hand, we can see from the line graph that  pre is much larger than other positions when  is 200 rad/s.The numerical error grows faster when the particle spacing is larger.
As shown in Figure 4,  pre increases along with the increase of .What is more, three lines are represent three different Courant numbers, namely, three different choices of time step when the particle spacing is the same. pre obtained from three different Courant numbers has almost the same value when  is under 100 rad/s.After that, some difference is shown among three lines when  becomes larger, but the Courant number affects the effect of  less in general.

Effect of Sound
Pressure.In the present section, different sound propagation models with the maximum sound pressure amplitude changing from 10 Pa to 200 Pa are simulated, and the effects of sound pressure are discussed.At the same time, different computational parameters, namely, the particle spacing and the Courant number, are also used in the simulation.The  of the sound in all computational cases in this section is 50 rad/s.
Figure 5 illustrates the changes of sound pressure error when the numerical model uses different  Amp and particle spacing.Figure 6 describes the changes of error with different  Amp and different Courant numbers.Each point in both figures stands for a computational case.The Courant number used in all computational cases in Figure 5 is 0.05, and the particle spacing used in all cases in Figure 6 is 0.06 m.
It can be seen from the line graph that  pre has limited change, which is a little increase, along with the increase of  Amp .The trend of the line is approximately linear when  As can be seen from Figure 7, there is an approximately linear relation between  pre and Δ with double logarithmic coordinates.As Δ increases,  pre also increases.This result indicates a convergence of the simulation, as a smaller Δ leads to a more accurate result.Note that the errors generated with the three different Courant numbers all have similar values.A linear fitted curve obtained by using all  pre results produces a slope of approximately 1.4 for the fitted line.
The required CPU cost in terms of time to perform the SPH simulation corresponding to different Δ values is given in Figure 8. Three different point types were also used to represent different Courant numbers in the computation.The  also shows the linear fit lines for each Courant number, as well as the calculated slopes.
The graph shows that the CPU time logarithmically decreases from hundreds of seconds to about one second as Δ increases from 0.03 m to 0.3 m.This decreasing trend is roughly linear, with an absolute slope value of approximately 2.5.Although the CPU time is different for each Courant number, the trends are similar.Thus, an appropriate particle spacing value can be decided within a certain range by comparing the results in Figures 7 and 8 for a given application.

Conclusions
In this work, the meshfree, Lagrangian SPH method is used to solve linearized acoustic equations.The feasibility of the SPH method in simulating acoustic waves is evaluated using a numerical experiment of one-dimensional sound propagation model.The effect of frequency, sound pressure, and particle spacing is then subsequently discussed based on the numerical error with considering the Courant number.
(1) The results of computational cases, with the acoustic wave frequency changes from 10 rad/s to 200 rad/s, show that the rising of sound frequency increases simulation error, and the increase is nonlinear.
(2) As the maximum sound pressure amplitude changes from 10 Pa to 200 Pa, the rising sound pressure increases the simulation error.However, the resulting change in error is limited and marginal.
(3) Decreasing the particle spacing in the computation reduces the numerical error, while simultaneously increasing the CPU time.The trend of both changes is close to linear on a logarithmic scale.The Courant number has little effect on the error generated with different sound frequency or sound pressure, whereas the particle spacing has a more significant influence on both properties.

Figure 5 :
Figure 5: Sound pressure error versus maximum sound pressure amplitude for different particle spacings.

Figure 6 :
Figure 6: Sound pressure error versus maximum sound pressure amplitude for different Courant numbers.

Figure 8 :
Figure 8: CPU time versus particle spacing for different Courant numbers.

Table 1 :
Model parameters and computational parameters for the numerical experiment.