Fourier Pseudospectral Time-domain Method Research Articles

Benchmark cases for urban sound propagation

Urban sound propagation is influenced by multiple reflections on horizontal and vertical surfaces, either specular or diffusive in nature, diffraction around edges and meteorology, causing refraction and scattering of sound waves. This paper initiates multiple benchmark cases for urban sound propagation, with the purpose of comparing the suitability of computational methodologies. The benchmark cases are two-dimensional cross-sections of typical urban geometries, involving all of the effects mentioned above. The sound source is either geometrically screened or is in the line of sight from the receiver's position. When meteorological conditions are included, results obtained from computational fluid dynamics simulations are used. All details of the benchmark cases are concisely described, and results from two numerical methods for outdoor sound propagation are included. Both methods solve the linearized Euler equations (LEE). The first method is the Fourier pseudospectral time-domain method (Fourier-PSTD) implemented in the open source software openPSTD v2.0. The second method is the finite difference time domain method FDTD. A detailed comparison of the results obtained from the two methods is presented.

Application of the Fourier pseudospectral time-domain method in orthogonal curvilinear coordinates for near-rigid moderately curved surfaces.

The Fourier pseudospectral time-domain method is an efficient wave-based method to model sound propagation in inhomogeneous media. One of the limitations of the method for atmospheric sound propagation purposes is its restriction to a Cartesian grid, confining it to staircase-like geometries. A transform from the physical coordinate system to the curvilinear coordinate system has been applied to solve more arbitrary geometries. For applicability of this method near the boundaries, the acoustic velocity variables are solved for their curvilinear components. The performance of the curvilinear Fourier pseudospectral method is investigated in free field and for outdoor sound propagation over an impedance strip for various types of shapes. Accuracy is shown to be related to the maximum grid stretching ratio and deformation of the boundary shape and computational efficiency is reduced relative to the smallest grid cell in the physical domain. The applicability of the curvilinear Fourier pseudospectral time-domain method is demonstrated by investigating the effect of sound propagation over a hill in a nocturnal boundary layer. With the proposed method, accurate and efficient results for sound propagation over smoothly varying ground surfaces with high impedances can be obtained.

Flow and geometrical effects on radiated noise from exhaust pipes computed by the Fourier pseudospectral time-domain method

The current work initializes a systematic numerical study on the combined effects of the geometry surrounding an automotive exhaust pipe, i.e. the ground surface and the automotive body, as well as the influence of the non-uniform velocity and temperature from the exhaust jet, on the acoustic radiation pattern of the pipe. The problem is treated by first solving the RANS equations to obtain the mean velocity and temperature solution of the exhaust jet. The acoustic radiation is computed by solving the linearized Euler (LEE) equations. For an efficient solution of the LEE, a recently developed multi-domain extended Fourier pseudospectral time-domain (PSTD) methodology is used. This method combines the favorable spectral accuracy of the PSTD method with a local grid refinement in the region with high gradients of the mean flow and temperature fields. A filter was found necessary for numerical stability of the PSTD method for this application. For radiation of noise from an exhaust pipe in free field, results from PSTD show a good agreement with reference results, both for cases with and without a jet flow. Results show that the presence of a rigid ground surface and simplified automotive body increases the radiated sound power by 6dB for the lower frequency region, and the effect of the body on the directivity is largest for the higher frequencies. Flow effects slightly increase the shielding effect of the body for all frequencies, but have a main impact behind the exhaust pipe, where low frequencies experience higher levels and a cone of low sound levels characterizes the high frequencies.

Acoustic propagation modeled by the curvilinear Fourier pseudospectral time-domain method

The Fourier pseudospectral time-domain method is an efficient domain-discretization wave-based method to model sound propagation in inhomogeneous bounded media. The method was successfully applied to model atmospheric sound propagation and acoustics in urban environments. One of the limitations of the method is its restriction to a Cartesian grid, confining it to staircase-like geometries. When applying a transform from the Cartesian coordinate system to the curvilinear coordinate system, more arbitrary geometries may be solved by the method. In free field, the frequency dependent accuracy of the curvilinear Fourier pseudospectral time-domain method is investigated as a function of the deformation angle of the grid. Further, the performance of the pseudospectral method with a curvilinear grid as well as a Cartesian grid for scattering of elementary objects as an inclined plate and a cylinder is studied. Finally, sound propagation in a room with non-parallel boundaries and over a building with gabled roof is computed with the pseudospectral method with a curvilinear grid and compared with results obtained from the boundary element method. All computed results are in 2D.

A multi-domain Fourier pseudospectral time-domain method for the linearized Euler equations

The Fourier pseudospectral time-domain (F-PSTD) method is computationally one of the most cost-efficient methods for solving the linearized Euler equations for wave propagation through a medium with smoothly varying spatial inhomogeneities in the presence of rigid boundaries. As the method utilizes an equidistant discretization, local fine scale effects of geometry or medium inhomogeneities require a refinement of the whole grid which significantly reduces the computational efficiency. For this reason, a multi-domain F-PSTD methodology is presented with a coarse grid covering the complete domain and fine grids acting as a subgrid resolution of the coarse grid near local fine scale effects. Data transfer between coarse and fine grids takes place utilizing spectral interpolation with super-Gaussian window functions to impose spatial periodicity. Local time stepping is employed without intermediate interpolation. The errors introduced by the window functions and the multi-domain implementation are quantified and compared to errors related to the initial conditions and from the time iteration scheme. It is concluded that the multi-domain methodology does not introduce significant errors compared to the single-domain method. Examples of scattering from small scale density scatters, sound reflecting from a slitted rigid object and sound propagation through a jet are accurately modelled by the proposed methodology. For problems that can be solved by F-PSTD, the presented methodology can lead to a significant gain in computational efficiency.

Modelling of sound propagation to three-dimensional urban courtyards using the extended fourier PSTD method

Noise from road traffic propagates to acoustically shielded areas as roadside courtyard by multiple reflection and diffraction paths in a complex three-dimensional (3D) environment. The computation of noise levels and assessment of candidate noise mitigation measures for these areas has up to now been based upon two-dimensional (2D) geometrical assumptions. Here, a recently developed efficient wave-based method, the extended Fourier pseudospectral time-domain (PSTD) method, is used to investigate the necessity of a 3D model. For frequencies up to 500 Hz and low traffic velocities of 30 km/h and 50 km/h, a road traffic noise configuration of an urban street canyon with or without cross streets and a closed roadside courtyard is compared to the 2D configuration as studied previously. It can be concluded that the contribution of distant sources is overpredicted by the 2D configuration. As noise mitigation measures, additional façade absorption, façade screens and roof screens have been studied. Results show that the 2D configuration underpredicts the effect of façade mitigation measures, by maximum 1.5 dB(A) for the absorption case and 4.4 dB(A) for the screens case. The effect of roof screens is overpredicted up to 1.7 dB(A). Given these deviations and the found deviations between the 3D configurations of street canyon with and without cross streets, the need for a 3D model can be concluded to be strongly configuration dependent. The 3D model is finally used to investigate the effect of a façade opening to the courtyard, which could lead to up to 10 dB(A) higher noise levels as compared to the noise propagating over the roof level and may prohibit the use of these courtyards as quiet areas. Absorption in the façade opening can significantly limit this negative effect.

THE EXTENDED FOURIER PSEUDOSPECTRAL TIME-DOMAIN (PSTD) METHOD FOR FLUID MEDIA WITH DISCONTINUOUS PROPERTIES

The Fourier pseudospectral time-domain (PSTD) method is an attractive method to efficiently model wave propagation through a weakly inhomogeneous fluid medium. The method fails, however, for fluid media with discontinuous properties. This failure is due to the well known Gibbs phenomenon which arises when Fourier transforming a discontinuous function. The extended Fourier PSTD method developed here can accurately and efficiently model wave propagation through weakly inhomogeneous fluid media with discontinuities in the media properties. Rather than using Fourier transforms to calculate the spatial derivatives in the wave equation, the method developed here uses a generalized eigenfunction expansion for which no Gibbs phenomenon arises. Two approaches to solving the resulting time-domain problem are explored and the accuracy of the method is demonstrated. Finally, results from an example calculation of transmission through the water-air interface are shown.

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.

A Fourier Pseudospectral Method for Some Computational Aeroacoustics Problems

A Fourier pseudospectral time-domain method is applied to wave propagation problems pertinent to computational aeroacoustics. The original algorithm of the Fourier pseudospectral time-domain method works for periodical problems without the interaction with physical boundaries. In this paper we develop a slip wall boundary condition, combined with buffer zone technique to solve some non-periodical problems. For a linear sound propagation problem whose governing equations could be transferred to ordinary differential equations in pseudospectral space, a new algorithm only requiring time stepping is developed and tested. For other wave propagation problems, the original algorithm has to be employed, and the developed slip wall boundary condition still works. The accuracy of the presented numerical algorithm is validated by benchmark problems, and the efficiency is assessed by comparing with high-order finite difference methods. It is indicated that the Fourier pseudospectral time-domain method, time stepping method, slip wall and absorbing boundary conditions combine together to form a fully-fledged computational algorithm.

Electromagnetic simulations of metallic materials using the mapped fourier PSTD algorithm

AbstractPreviously, the regular Fourier pseudospectral time‐domain (PSTD) algorithm was recognized as mainly applicable to simulations of low‐contrast dielectric materials. In this paper, we discuss the difficulties in simulating high‐contrast and particularly metallic materials using the Fourier PSTD method. To overcome these difficulties, we use a nonuniform grid obtained from a specially designed mapping curve that can reduce the Gibbs phenomenon in the mapped space. The convergence property of this technique is then examined. We show dramatic improvements compared with regular Fourier PSTD and finite‐difference time‐domain (FDTD) methods on a uniform grid. As an example application, we apply our method to the modeling of the 2D scattering of a chrome‐projection lithography mask. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 569–575, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20699

Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media

A multidomain pseudospectral time-domain (PSTD) method with a newly developed well-posed PML is introduced as an accurate and flexible tool for the modeling of electromagnetic scattering by 2-D objects buried in an inhomogeneous lossy medium. Compared with the previous single-domain Fourier PSTD method, this approach allows for an accurate treatment of curved geometries with subdomains, curvilinear mapping, and high-order Chebyshev polynomials. The effectiveness of the algorithm is confirmed by an excellent agreement between the numerical results and analytical solutions for perfectly conducting as well as permeable dielectric cylinders. The algorithm has been applied to model various ground-penetrating radar (GPR) applications involving curved objects in a lossy half space with an undulating surface. This multidomain PSTD algorithm is potentially a very useful tool for simulating antennas near complex objects and inhomogeneous media.