The sound propagation in a wedge-shaped waveguide with perfectly reflecting boundaries is one of the few range-dependent problems with an analytical solution. This provides a benchmark for the theoretical and computational studies on the simulation of ocean acoustic applications. We present a direct finite volume method (FVM) simulation for the ideal wedge problem, and both time and frequency domain results are analyzed. We also study the broadband problem with large-scale parallel simulations. The results presented in this paper validate the accuracy of the numerical techniques and show that the direct FVM simulation could be applied to large-scale complex acoustic applications with a high performance computing platform.
The research on the sound propagation problem has a multidisciplinary and practical significance over oceanology, biology, shipbuilding, military affairs, and many other science and engineering subjects [
The construction of a computational solution for this problem has been extensively investigated over the last decades. Underwater sound propagation can be mathematically described by the wave equation; thus, essentially different computational models use different approximations or discretization methods. In overall, there are seven types of computational methods to solve the sound propagation problem: the spectral or fast field program (FFP) [
Running benchmark problems and comparing the numerical results are important and necessary steps for validating a computational solution. The ideal wedge problem describes the sound propagation in a wedge-shape waveguide, and it is one of the benchmarks proposed by Acoustical Society America (ASA) [
Although these fast computational methods were successfully used in explaining many observed ocean acoustic phenomena, there are still lots of scientific issues which cannot be accurately addressed by the approximated approach. For realistic problems, we usually have to solve a two-way wave equation with complex geometry. For this purpose, the numerical approach which directly discretizes the governing equations should be adopted. Direct FDM and FEM simulations [
Over the past few decades, high performance computing (HPC) techniques have made great achievements and the simulation time is possible to be significantly reduced through large-scale parallel computing, which means that the traditional direct simulation approach may become applicable for realistic ocean acoustic problems on a modern HPC platform.
In this paper, we design an acoustic numerical solver based on the finite volume method (FVM) and present the full numerical results of the direct FVM simulations for sound propagation in an ideal wedge. Both time domain and frequency domain results are analyzed; furthermore, we also model the broadband problem. The broadband probem is relatively difficult to simulate through the approximated approach; therefore, it is seldom studied over the last decades. The method proposed in this paper has its advantages for complex applications such as irregular boundary problems or broadband problems.
The remainder of the paper is organized as follows: the numerical techniques used in this paper are introduced in Section
In this section, we give a synopsis of the ASA wedge benchmark and an introduction to the numerical techniques used in our simulations.
In the ideal wedge problem investigated by this paper, a pressure-release sea surface and a rigid or pressure-release bottom constitute the main boundaries of the 2D wedge. Both the surface and the bottom are perfectly reflecting boundaries and in the opposite the left boundary has to be set to an nonreflecting (absorbing) boundary. Although, in the real ocean environment, the wedge angle is very small, in this paper, we set it to
The sketch of the ideal wedge environment used in the numerical experiments. The line source is located at 400 m in range and 200 m in depth. The source frequency is 25 Hz.
The 2D wedge problem is a vertical section of the 3D problem. The 3D wedge problem may have two types of wave sources. One is a spatial point locating in the field; the other takes a form of horizontal infinite straight line across the section parallel to the apex [
According to Luo et al. [
Luo et al. [
Under the 2D cylindrical coordinates, the ideal wedge problem could be expressed as
The analytical solution to the Helmholtz equation with rigid and/or pressure-release bottom boundary conditions becomes
The normal-mode methods and other frequency domain methods focus on the solution of the Helmholtz equation. These approaches solve the sound propagation process in the frequency domain. Nevertheless, through the direct FVM simulations, we can directly obtain the computational solution of the original wave equation.
To numerically solve the wave equation, we use an open source CFD toolbox released by the OpenCFD Ltd., named OpenFOAM. The equation is discretized through the finite volume method which locally satisfied the physical conservation laws through the integral over a control volume. For the spatial discretization terms, various predefined schemes are selectively applied and the temporal terms are discretized using a simple Euler scheme. Finally, the wave equation is reduced to a linear system; thus, using the iterative solvers in OpenFOAM, we can get the solutions for the equations at every time step. Solvers in the toolbox include the conjugate (PCG) and biconjugate gradient (PBiCG) methods. More details are presented in the OpenFOAM Manual. The boundary conditions for sea surface, bottom, and the left vertical numerical boundary are described numerically [ The rigid bottom: The pressure-release bottom: The nonreflecting boundary:
The numerical schemes used to descretize the wave equation and the boundary conditions are listed in Table
Numerical schemes used for discretizing the wave equation and the boundary conditions.
Differential operators | Numerical schemes |
---|---|
|
Euler |
|
Euler |
|
Gauss linear |
|
Gauss linear |
Interpolation schemes | Linear |
Solver and preconditioner used for solving
|
Settings |
---|---|
Solver | PCG |
Preconditioner | DIC |
In the following section, we first give a description to the platform and the parameters of the numerical experiments. Then, we give the numerical results for the periodic single frequency source problem with different bottom boundaries. The problem with a broadband pulse source is also considered and tested. Finally, the large-scale parallel simulation is applied and analyzed.
The accuracy of the numerical techniques used in this paper is measured with the ASA ideal wedge problem. The specific features of this problem are shown in Figure
Generally, the accuracy of the numerical simulation is dependent on its mesh density. A convergence analysis with different mesh desities is illustrated in Figure
Comparison of the Fourier transform of the FVM result
The section at 150-m depth
127200 cells
508800 cells
2035200 cells
In this paper, the analytical solution is shown in terms of transmission loss (TL) in units of dB re 1 m, which is defined as
The analytical solution to the Helmholtz equation is a frequency domain function. The direct FVM simulation outputs the time series of the pressure field
The locations of ridges and troughs of the FVM results are same with the analytical solution and remain invariant while the number of cells is increasing. This analysis shows that a mesh with 508800 cells is sufficient for the convergence.
The troughs of the curves in 2 indicate the destructive interference of the wave coming from the source and each reflected waves. Looking into the details, we may find that some numerical troughs (i.e., the one marked with an “X” in Figure
Comparing those figures with Figure
The overview of the pressure field
Here, we consider the ideal wedge problem drawn in Figure
The asymptotic stability of the FVM result is evident in Figure
In Figure
Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the rigid bottom problem: (a) TL result at depth of 30 m; (b) TL result at depth of 150 m.
30-m depth result
150-m depth result
Over both of the two receiver lines, the locations of ridges and troughs in range are the same between the FVM result and the analytical solution. Also, the error of the FVM simulation at most ridges has been controlled within 1 dB.
In this experiment, we consider the ideal wedge problem with a pressure-release bottom. Two snapshots of the spatial result of the direct FVM simulation to the problem are shown in Figure
The overview of the pressure field
The results of the pressure-release bottom problem also show a behavior of temporal convergence. The wave propagation has reached the stable status after the time of 1.0 s. Observing the curves’ right ends, we find a different reflection mode with that in the simulation for the rigid bottom problem.
In Figure
Comparisons of the Fourier transform of the FVM simulation and the analytical solution to the pressure-release bottom problem: (a) TL result at depth of 30 m; (b) TL result at depth of 150 m.
30-m depth result
150-m depth result
From the results shown in Figure
In this section, the method of the direct FVM simulation is applied to the broadband problem which is seldom studied in the research on the approximated approach during the past few dacades. The ideal wedge waveguide is also used, and the results are analyzed after the experiments.
The broadband problem solved in this paper uses the initial and boundary conditions from the ASA benchmark. It has a single square pulse source lasting for 0.1 s in the same ideal wedge waveguide as shown in Figure
The simulation for the single pulse source problem in the ideal wedge geometry. The bottom boundary is set rigid. The t value of each snapshot is listed in the subtitles of the figures. The small blue rings around the source in (g) and (h) are the secondary reflections on the edge of the source.
The tone scale of the figures indicates the sound pressure
The computation of the direct FVM simulation for the ideal wedge problem has been parallelized. The experiments use the broadband pulse sound source and the rigid bottom condition. The time step is set as
The clock time versus the number of processors.
508800 cells
8140800 cells
From Figures
In this paper, we propose the platform and techniques of a direct FVM simulation for the sound propagation problem. The techniques of this simulation are applied to an ideal wedge problem characterized by a homogeneous water column, a pressure-release sea surface, a rigid or pressure-release bottom, and a periodic single frequency or a broadband pulse line source. The accuracy of the simulation is analyzed by comparing the numerical results with an analytical solution. A series of parallel computing experiments on solving the broadband problem is implemented.
We have studied the time and frequency domain results of the FVM simulation for the ideal wedge problem and have compared the results with the analytical solution proposed by Luo et al. [
Both the periodic single frequency source and the broadband pulse source are simulated and analyzed. We have investigated the time and frequency domain results. The platform and techniques of the simulation show a huge adaptability to these two different sources. By observing the physical phenomena, the asymptotic stability of the solutions to the periodic problem is considered. This fact helps us ensure the integration interval of the Fourier transform in the accuracy analysis. The numerical pulse generated by and propagated in the simulation for the broadband problem behaves well.
The large-scale parallel tests are implemented. The results on up to 384 processors show a good scalability. The techniques of the direct FVM simulation could be easily applied to a parallel environment with hundreds of processors, and this application could significantly reduce the CPU time in the simulation.
In summary, a direct FVM simulation for the ideal wedge problem with a homogeneous wedge column, a pressure-release surface, a rigid or pressure-release bottom, and a periodic or single pulse line source is proposed. With the experiments and analysis, this simulation method shows its application prospects in the complicated simulations on ocean acoustics with higher performance computing platforms.
The authors declare that there are no competing interests regarding the publication of this paper.
This project is supported by National Natural Science Foundation of China (Grant nos. 61303071 and 61120106005) and the Open Fund from the State Key Laboratory of High Performance Computing (nos. 501503-01 and 201503-02).