Acoustic Finite-difference Time-domain Research Articles

A novel 1D-FDTD scheme to solve the nonlinear second-order thermoviscous hydrodynamic model

In this paper, we present a novel and simple Yee Finite-Difference Time-Domain (FDTD) scheme to solve numerically the nonlinear second-order thermoviscous Navier–Stokes and the Continuity equations. In their original form, these equations cannot be discretized by using the Yee’s mesh, at least, easily. As it is known, the use of the Yee’s mesh is recommended because it is optimized in order to obtain higher computational performance and remains at the core of many current acoustic FDTD softwares. In order to use the Yee’s mesh, we propose to rewrite the aforementioned equations in a novel form. To achieve this, we will use the substitution corollary. This procedure is novel in the literature. Although the scheme can be extended to more than one dimension, in this paper, we will focus only on the one-dimensional solution because it can be validated with two analytical solutions to the Burgers equation: the Mendousse mono-frequency solution and the Lardner bi-frequency solution. Numerical solutions are excellently consistent with the analytical solution, which demonstrates the effectiveness of our formulation.

Effect of bat pinna on sensing using acoustic finite difference time domain simulation.

The practicality of the finite-difference time-domain (FDTD) method was confirmed by comparing head-related transfer functions obtained from a three-dimensional (3D) digital model of a bat (Rhinolophus ferrumequinum nippon) head with acoustic experiments using a 3D printed physical model. Furthermore, we simulated the auditory directionality using a 3D digital model that was modified based on the pinna movement of a bat during echolocation and found that the alternating movements of the left and right pinna result in a binaural sound pressure difference for vertical sources. Using the FDTD method, suitable for simulating acoustics in large spaces, we could analyze in detail the binaural echoes that bats receive and the acoustic cues they use for echolocation.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection.

Finite-difference time-domain method has gained increasing interest for room acoustic prediction use. A well-known limitation of the method is a frequency- and direction-dependent dispersion error. In this study, the audibility of dispersion error in the presence of a single surface reflection is measured. The threshold is measured for three different distance conditions with a fixed reflection arrival azimuth angle of 54.7°. The error is placed either in the direct path, or in the reflection path. Additionally a qualitative follow-up experiment to evaluate how the measured thresholds reflect the audibility of error in short room responses is carried out. The results indicate that the threshold varies depending whether the error is in the direct path or in the reflection path. For transient signals the threshold is higher when the error is located in the direct path, whereas for speech signal, the threshold is higher when it is located in the reflection path. Evidence is found that the error is detectable in rendered room responses at the measured threshold levels.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of a single early reflection

Audibility of dispersion error in room acoustic finite-difference time-domain simulation in the presence of absorption of air.

The finite-difference time-domain method has gained increasing interest for room acoustic prediction use. A well-known limitation of the method is a frequency and direction dependent dispersion error. In this study, the audibility of dispersion error in the presence of air absorption is measured. The results indicate that the dispersion error in the worst-case direction of the studied scheme gets masked by the air absorption at a phase velocity error percentage of 0.28% at the frequency of 20 kHz.

Performance analysis of finite-difference time-domain schemes for acoustic simulation implemented on multi-core and many-core processor architectures

There have been many discussions about hardware-accelerations for the large-scale finite-difference time-domain (FDTD) method to deal with various wave dynamics in scientific and engineering computations. Recently, there is growing interest in many-core based parallel computing with graphics processing unit (GPU) or many integrated Core (MIC) accelerators. Therefore, it is worth applying those to large-scale and long-time acoustic simulation for FDTD methods. In this paper, performance analyses are performed for three types of acoustic FDTD schemes, which are FDTD(2,2), FDTD(2,4) and wave equation FDTD (WE-FDTD)(2,2), implemented on GPU, Intel MIC, and multi-core central processing unit (CPU). Here, FDTD(T, S) denotes Tth-order accuracy for the time derivative and Sth-order accuracy for the spatial derivative, respectively. The parallelized FDTD schemes are implemented with Compute Unified Device Architecture (CUDA) and OpenACC, respectively, on GPU, and OpenMP for MIC and multi-core CPU. As a method for performance analysis, we employ the modified roofline model which adds the concept of cache hit ratio to operational intensity. The analysis provides performance comparison results of attainable performance and measured performance for each FDTD scheme and each processor. We performed software optimizations for MIC based on the performance analysis and achieved 78%-91% of attainable performance on MIC.

Audibility of dispersion error in room acoustic finite-difference time-domain simulation as a function of simulation distance.

Finite-difference time-domain (FDTD) simulation has been a popular area of research in room acoustics due to its capability to simulate wave phenomena in a wide bandwidth directly in the time-domain. A downside of the method is that it introduces a direction and frequency dependent error to the simulated sound field due to the non-linear dispersion relation of the discrete system. In this study, the perceptual threshold of the dispersion error is measured in three-dimensional FDTD schemes as a function of simulation distance. Dispersion error is evaluated for three different explicit, non-staggered FDTD schemes using the numerical wavenumber in the direction of the worst-case error of each scheme. It is found that the thresholds for the different schemes do not vary significantly when the phase velocity error level is fixed. The thresholds are found to vary significantly between the different sound samples. The measured threshold for the audibility of dispersion error at the probability level of 82% correct discrimination for three-alternative forced choice is found to be 9.1 m of propagation in a free field, that leads to a maximum group delay error of 1.8 ms at 20 kHz with the chosen phase velocity error level of 2%.

Performance analysis of SSE and AVX instructions in multi-core CPUs and GPU computing on FDTD scheme for solid and fluid vibration problems

In this work a unified treatment of solid and fluid vibration problems is developed by means of the Finite-Difference Time-Domain (FDTD). The scheme here proposed takes advantage from a scaling factor in the velocity fields that improves the performance of the method and the vibration analysis in heterogenous media. Moreover, the scheme has been extended in order to simulate both the propagation in porous media and the lossy solid materials. In order to accurately reproduce the interaction of fluids and solids in FDTD both time and spatial resolutions must be reduced compared with the set up used in acoustic FDTD problems. This aspect implies the use of bigger grids and hence more time and memory resources. For reducing the time simulation costs, FDTD code has been adapted in order to exploit the resources available in modern parallel architectures. For CPUs the implicit usage of the advanced vectorial extensions (AVX) in multi-core CPUs has been considered. In addition, the computation has been distributed along the different cores available by means of OpenMP directives. Graphic Processing Units have been also considered and the degree of improvement achieved by means of this parallel architecture has been compared with the highly-tuned CPU scheme by means of the relative speed up. The speed up obtained by the parallel versions implemented were up to 3 (AVX and OpenMP) and 40 (CUDA) times faster than the best sequential version for CPU that also uses OpenMP with auto-vectorization techniques, but non includes implicitely vectorial instructions. Results obtained with both parallel approaches demonstrate that massive parallel programming techniques are mandatory in solid-vibration problems with FDTD.

Modeling underwater impact pile driving noise in shallow, inhomogeneous channels

The broadband synthesis of a parabolic equation (PE) propagation model for shallow water acoustic propagation in inhomogeneous channels is presented to account for the noise produced by impact pile driving. The PE model utilizes sediment information obtained from boring measurements and detailed bathymetry to model range dependent propagation in the Columbia River between Portland, OR and Vancouver, WA. The impact pile driving source is modeled in two ways: first, as a reverberating impulse source that emits Mach-wave radiation [Reinhall, Dahl, J. Acoust. Soc. Am. 130, 1209 (2011)]; and second, with a structural acoustic finite-difference time-domain (FDTD) model [Shahab, Woolfe, and Hastings, J. Acoust. Soc. Am. 130, 2558 (2011)]. Model results using both source models are shown to be in good agreement with acoustic measurements of test pile operations in the Columbia River at multiple locations from 10 to 800 meters from the pile driving source. Implications for noise levels in river systems with varying bottom sediment characteristics are presented and discussed. [This research is supported with funding from the Oregon Department of Transportation.]

Consistent modeling of boundaries in acoustic finite-difference time-domain simulations

The finite-difference time-domain method is one of the most popular for wave propagation in the time domain. One of its advantages is the use of a structured staggered grid, which makes it simple and efficient on modern computer architectures. A drawback, however, is the difficulty in approximating oblique boundaries, having to resort to staircase approximations. In many scattering problems this means that the grid resolution required to obtain an accurate solution is much higher than what is dictated by propagation in a homogeneous material. In this paper zero boundary data are considered, first for the velocity and then the pressure. These two forms of boundary conditions model perfectly rigid and pressure-release boundaries, respectively. A simple and efficient method to consistently model curved rigid boundaries in two dimensions was developed in Tornberg and Engquist [J. Comput. Phys. 227, 6922-6943 (2008)]. Here this treatment is generalized to three dimensions. Based on the approach of this method, a technique to model pressure-release surfaces with second order accuracy and without additional restriction on the timestep is also introduced. The structure of the standard method is preserved, making it easy to use in existing solvers. The effectiveness is demonstrated in several numerical tests.

An efficient realization of frequency dependent boundary conditions in an acoustic finite-difference time-domain model

The finite-difference time-domain (FDTD) method provides a simple and accurate way of solving initial boundary value problems. However, most acoustic problems involve frequency dependent boundary conditions, and it is not easy to include such boundary conditions in an FDTD model. Although solutions to this problem exist, most of them have high computational costs, and stability cannot always be ensured. In this work, a solution is proposed based on “mixing modelling strategies”; this involves separating the FDTD mesh and the boundary conditions (a digital filter representation of the impedance) and combining them into a global solution. This solution is based on an interaction model that involves wave digital filters. The proposed method is validated with several test cases.

Acoustic finite-difference time-domain simulation considering the fluctuation of sound pressure caused by heat energy : Wave propagation in a narrow tube with temperature gradient

Acoustic finite-difference time-domain simulation considering the fluctuation of sound pressure caused by heat energy : Wave propagation in a narrow tube with temperature gradient

Development of Sound Localization System with Tube Earphone Using Human Head Model with Ear Canal

In this study, we propose a new acoustic model including the human ear canal and a thin tube earphone. The use of a tube earphone enables simultaneous listening of both virtual and real surrounding sound. First, we perform acoustic FDTD (finite difference time domain) simulations using an MRI head model with ear canals. The calculated external impedance viewed from the eardrum numerically shows that the influence of the inserted tube is small. A listening experiment with six subjects also confirms the effectiveness of a tube earphone. Second, we calculate HRTFs (head-related transfer functions) for eight directions in the horizontal plane to realize sound localization with a tube earphone. We also design inverse filters based on the propagation calculations including the characteristics of tube earphones. Finally we evaluate the localization system by another listening experiment with six subjects. The results reveal that the applicability of a system with tube earphones and inverse filters, particularly for the front directions.

Acoustic wave propagation in atmospheric boundary layer above a rough terrain

In this paper we will present the latest development in the numerical simulation of acoustic wave propagation in the lowest atmospheric boundary layer above a rough terrain with a finite difference time domain (FDTD) algorithm. It is of great interest to investigate the effect of dynamic turbulent boundary layer meteorology coupled with surface terrain on acoustic wave propagation in frequency bands relevant to battlefield sensor systems. Turbulent mixing in the vicinity of the earth–air boundary layer is the key factor contributing to acoustic signal fluctuations such as upward refraction, scattering, high attenuation, and coherence degradations. Using first principles acoustic wave equation expressions for a moving fluid will allow us to couple acoustic models with time varying, 3-D large eddy simulation (LES) results. The specific meteorological parameters are vector wind velocity, temperature, pressure, and density as a function of space and time. Our acoustic simulation uses a coupled system of first-order temporal and spatial differential equations relating wind velocity, pressure, and density variations. This system of equations was evaluated with an explicitly time-stepping finite difference numerical approach. The coupling of LES and acoustic FDTD simulation will allow us to directly identify spectral portions of turbulent boundary layer meteorology contributing to short- and long-term acoustic signal variations.

Exact global boundary condition with reduced computational complexity for grid truncation in acoustic finite difference time domain simulations

The numerical analysis of transient acoustic radiation by and scattering from inhomogeneous bodies is often carried out using the finite difference time domain (FDTD) method. However, FDTD grids must be truncated with absorbing boundary conditions when the structure of interest resides in an unbounded medium. Local boundary conditions have to be imposed on a convex outer boundary, thereby often inflating the problem size. This drawback is alleviated if the grid is truncated using exact boundary conditions that rely on a retarded-time boundary integral (RTBI) [L. Ting and M. J. Miksis, J. Acoust. Soc. Am. 80, 1825–1827 (1986)]. Unfortunately, for a 3-D computational domain comprising of O(N3) FDTD cells, where N denotes the number of cells per unit length, the evaluation of the RTBI using classical methods requires O(N4) operations per time step, which renders the RTBI approach impractical. In this presentation, it will be shown that this computational bottleneck can be alleviated by employing the plane wave time domain algorithm [A. A. Ergin et al., J. Comput. Phys. 146, 157–180 (1998)] using which the RTBI can be evaluated in O(N2<th>log2<th>N) operations. Numerical results that demonstrate the efficacy and reduced complexity of the method will be presented.

Simple conformal methods for finite-difference time-domain modeling of pressure-release surfaces

The finite-difference time-domain (FDTD) method provides a simple and accurate means of simulating a wide range of acoustic wave propagation problems. Unfortunately, the method has a voracious appetite for computational resources. For example, to accurately model scattering from a continuously varying pressure-release boundary, an FDTD grid is typically required that has a much finer discretization than is necessary to model propagation in a homogeneous space. Such a fine discretization can become prohibitive when considering large-scale problems. Two simple conformal techniques are presented for acoustic FDTD simulations of problems involving pressure-release boundaries. These techniques, which rely upon splitting velocity cells adjacent to the pressure-release boundary, significantly improve the accuracy of the results over those of the standard “staircase” representation of the boundary. These methods permit the use of a coarser FDTD grid than would otherwise be practical and yet add negligible computational cost. The improved accuracy of these split-cell techniques is shown for both a spherical scatterer and a spherical resonator.

The complementary operators method applied to acoustic finite-difference time-domain simulations

The complementary operators method (COM) has recently been introduced as a mesh-truncation technique for open-domain radiation problems in electromagnetics. The COM entails the construction of two solutions that employ absorbing boundary conditions (ABCs) with complementary behavior, i.e., the reflection coefficients associated with the two ABCs are exactly opposite each other. The average of these solutions then yields a new solution in which the errors caused by artificial reflections from the termination of grid are nearly eliminated. In this work, COM is introduced for the finite-difference time-domain (FDTD) solution of acoustics problems. The development of COM is presented in terms of Higdon’s absorbing boundary operators, but generalization to non-Higdon operators is straightforward. The effectiveness of COM in comparison to other absorbing boundary conditions is demonstrated with numerical experiments in two and three dimensions.

Implementation of transparent sources embedded in acoustic finite-difference time-domain grids

The finite-difference time-domain (FDTD) method is a simple but powerful numerical method which has been used to perform a wide variety of complex simulations. One of the considerations in using this method is modeling the source of the incident field. When the physical source of acoustic energy is located within the FDTD grid it has typically been modeled as a “hard” source for which a source function determines the field at a source node. One drawback of a hard source is that it acts itself as a scatterer. Moreover, its scattering cross section is dictated by the spatial step size within the grid; thus, scattering is independent of the underlying physical size of the source. In this paper a scheme is presented that permits the creation of completely transparent sources. These sources, although more expensive to implement than traditional hard sources, radiate the same field as a hard source, but do not scatter energy.

A Classification of Split Difference Methods for Hyperbolic Equations in Several Space Dimensions

A Classification of Split Difference Methods for Hyperbolic Equations in Several Space Dimensions