Parabolic Equation Method Research Articles

An inter-model comparison of parabolic equation methods for sound propagation from wind turbines.

The modeling of sound propagation for land-based wind turbines is a complex task that takes various parameters into account. Not only do the wind speed and wind direction affect the noise received at a certain position by changing the refraction of the sound, but also the terrain complexity, ground impedance, and receiver position relative to the source and ground all affect propagation. These effects are seen by the reflections of the sound at the ground surface causing interference of sound waves, or by the receiver being positioned in and out of noise shadow zones in the upwind far field position, or in steep terrain irregularities. Several sound propagation models with different levels of fidelity have been developed through time to account for these effects. This paper will focus on two different parabolic equation models, the Beilis-Tappert Parabolic Equation and the Generalized Terrain Parabolic Equation, through theoretical studies of varying terrain complexity, ground impedance, and sound speed profiles (upwind, downwind, and no wind). In addition, the propagation models are validated through spectral comparisons to noise measurements from two different campaigns considering loudspeaker noise and wind turbine noise, respectively.

A feature-compressed multi-task learning U-Net for shallow-water source localization in the presence of internal waves

The spatiotemporal variation of sound speed profile caused by internal waves usually causes the mismatch in the Matched-Field Processing (MFP) source localization. A feature-compressed multi-task learning U-Net with Convolutional Block Attention Module (MTL-UNET-CBAM) is proposed to estimate the range and depth of underwater sources in the South China Sea environment with the presence of internal waves. To handle the mismatch caused by internal waves, the temperature sensor chain data are used to reconstruct the two-dimensional sound speed profiles (2D-SSPs) based on the 2D advection model. Then, 2D-SSPs are used to generate the training set with the parabolic equation method. Sensitivity analysis is investigated to examine the effects of sound speed profile mismatch on the source localization performance. The simulation result shows the higher robustness of MTL-UNET-CBAM to the sound speed profile mismatch compared with the conventional Matched-Field Processing (CMFP) method. Experiment data in the South China Sea also used to validate the source localization performance of MTL-UNET-CBAM.

Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

Embedding antennas with tilted beam patterns into parabolic wave equation‐based models for tunnel propagation

AbstractAccurate radio wave prediction models are necessary for the innovation of mobile communication technology and the deployment of wireless sensor networks in railways. The parabolic wave equation (PWE) method is one commonly used method to predict the electromagnetic characteristic of long railway tunnels. However, the advent of novel antennas and technologies (e.g., beam steering, beamforming) has posed new challenges to PWE‐based models such as effective means of taking tilted beam patterns into account. This study addresses such a challenge by using tilted Gaussian beams to embed antennas with oriented patterns or tilted beam directions into PWE‐based models. The proposed approach does not reduce the simulation efficiency of PWE itself. The validity and practicability of the technique are demonstrated by comparing against ray‐tracing as well as experimental measurements in various tunnel environments.

Block-Centered Finite-Difference Methods for Time-Fractional Fourth-Order Parabolic Equations

The block-centered finite-difference method has many advantages, and the time-fractional fourth-order equation is widely used in physics and engineering science. In this paper, we consider variable-coefficient fourth-order parabolic equations of fractional-order time derivatives with Neumann boundary conditions. The fractional-order time derivatives are approximated by L1 interpolation. We propose the block-centered finite-difference scheme for fourth-order parabolic equations with fractional-order time derivatives. We prove the stability of the block-centered finite-difference scheme and the second-order convergence of the discrete L2 norms of the approximate solution and its derivatives of every order. Numerical examples are provided to verify the effectiveness of the block-centered finite-difference scheme.

Convergence analysis of space-time domain decomposition method for parabolic equations

Space-time domain decomposition methods have been successfully applied to solve linear systems that arise from the discretization of time-dependent problems. In this paper, we present a space-time pure multiplicative Schwarz method for solving linear parabolic equations. With this method, the systems of equations are first coupled together and then solved by preconditioned GMRES, so the solutions at arbitrary time steps can be obtained simultaneously at a time. Under some mild assumptions, we develop an optimal convergence theory and show that the convergence rate is bounded independently of the mesh sizes, the subdomain partition and the window size. Some numerical results are reported to illustrate the optimality and scalability of the proposed method. Moreover, the numerical comparison also shows that our method has a faster convergence rate.

The Malyuzhinets—Popov diffraction problem revisited

In this paper, the high-frequency diffraction of a plane wave incident along a planar boundary turning into a smooth convex contour, so that the curvature undergoes a jump, is asymptotically analyzed. An approach modifying the Fock parabolic-equation method is developed. Asymptotic formulas for the wavefield in the illuminated area, shadow, and the penumbra are derived. The penumbral field is characterized by novel and previously unseen special functions that resemble Fock’s integrals.

Regional infrasonic observations from surface explosions – influence of atmospheric variations and realistic terrain

SUMMARY A pair of 1-ton, conventional surface explosions were conducted at the Nevada National Security Site in the fall of 2020 producing seismoacoustic signatures observable hundreds of kilometres from the source location. Regional infrasonic observations include tropospheric ducting at large distances to the south, a wide stratospheric waveguide with signals observed more than 700 km to the east, and anomalous arrivals in the stratospheric shadow zone. Notable differences in propagation between the events are identified despite the explosions being conducted just two days apart due to a sharp temporal shift in the tropospheric winds as well as structural changes in the stratospheric winds. Propagation simulations of the two events have been completed using a combination of ray tracing and parabolic equation (PE) methods. Simulations have been conducted to quantify the impact of the temporal variations in the atmosphere as well as the influence of terrain on propagation. Temporal variations in reflection locations are found to produce notable changes in downrange propagation due to spatially varying terrain features. Finite frequency effects modelled by the PE are found to predict ensonification not included in corresponding 2D ray tracing simulations. Notable variations in predicted signal amplitude are found due to focusing by along-path and cross-path terrain gradients; though, the later of these is only modelled using fully 3D ray tracing analysis.

A Sliced Parabolic Equation Method to Characterize Maritime Radio Propagation

For maritime broadband communications, atmospheric ducts can enable beyond line-of-sight communications or cause severe interference. Due to the strong spatial–temporal variability of atmospheric conditions in near-shore areas, atmospheric ducts have inherent spatial heterogeneity and suddenness. This paper aims to evaluate the effect of horizontally inhomogeneous ducts on maritime radio propagation through theoretical analysis and measurement validation. To make better use of meteorological reanalysis data, we design a range-dependent atmospheric duct model. Then, a sliced parabolic equation algorithm is proposed to improve the prediction accuracy of path loss. We derive the corresponding numerical solution and analyze the feasibility of the proposed algorithm under the range-dependent duct conditions. A 3.5 GHz long-distance radio propagation measurement is utilized to verify the algorithm. The spatial distribution characteristics of atmospheric ducts in the measurements are analyzed. Based on actual duct conditions, the simulation results are consistent with the measured path loss. The proposed algorithm outperforms the existing method during the multiple duct periods. We further investigate the influence of different duct horizontal characteristics on the received signal strength.

ПРИБЛИЖЕННЫЕ АСИМПТОТИЧЕСКИЕ ВЫРАЖЕНИЯ ЭЛЕКТРОМАГНИТНОГО ПОЛЯ ИСТОЧНИКОВ ВБЛИЗИ ГЛАДКОЙ ВЫПУКЛОЙ ПРОВОДЯЩЕЙ ПОВЕРХНОСТИ ВРАЩЕНИЯ

Approximate asymptotic solution is developed for the electromagnetic field due of a surface magnetic and electric currents located near an arbitrary smooth convex conducting surface of rotation of large size. The expressions of field are valid in a near-surface layer with a thickness of the order of the wavelength and have the form of a sum of a series of azimuthal harmonics and an integral over a continuous spectrum of eigenfunctions of the field. The coefficients of the series are expressed by integrals from the product of a given distribution of sources with eigenfunctions. For eigenfunctions, both general integral representations and closed expressions in terms of Airy functions are obtained, which are uniformly valid along surface of a rotation for the case of one pole by the parabolic equation method. The field expressions take into account the inseparability of the field by the sum in term of E- and H-type fields, are uniformly valid in the near-surface layer, excluding the vicinity of the poles of the rotation surface, and have no discontinuities on the caustics of surface rays. The expressions obtained were used to calculate the mutual admittances between two annular slots on a semi-infinite and smoothly truncated conical surface. Numerical results obtained by the proposed and rigorous methods for a semi-infinite cone were compared. It is shown that in the case of in-phase annular slots in a semi-infinite cone, the obtained asymptotic expression for mutual admittance coincides with the main term of the asymptotic of a strict solution based on the method of eigenfunctions.

A new base of wind turbine noise measurement data and its application for a systematic validation of sound propagation models

Abstract. Extensive measurements in the area of wind turbines were performed in order to validate a sound propagation model which is based on the Crank–Nicolson parabolic equation method. The measurements were carried out over a flat grass-covered landscape and under various environmental conditions. During the measurements, meteorological and wind turbine performance data were acquired and acoustical data sets were recorded at distances of 178, 535 and 845 m from the wind turbine. By processing and analysing the measurement data, validation cases and input parameters for the sound propagation model were derived. The validation includes five groups that are characterised by different sound propagation directions, i.e. downwind, crosswind and upwind conditions in varying strength. In strong upwind situations, the sound pressure levels at larger distances are overestimated because turbulence is not considered in the modelling. In the other directions, the model reproduces the measured sound propagation losses well in the overall sound pressure level and in the third octave band spectra. As in the recorded measurements, frequency-dependent maxima and minima are identified, and losses generally increase with increasing distance and frequency. The agreement between measured and modelled sound propagation losses decreases with distance. The data sets used in the validation are freely accessible for further research.

Canonical Euler splitting method for parabolic partial functional differential algebraic equations

A novel canonical Euler splitting method is presented for semilinear composite stiff parabolic partial functional differential algebraic equations with initial and Dirichlet boundary conditions. The original partial differential problems are transformed into the semi-discrete problems by spatial discretization, and then the canonical Euler splitting method is employed to solve the resulting semi-discrete problems. Under appropriate assumptions, the stability and convergence theories of this method are established. A series of numerical experiments are given to illustrate the effectiveness of this method and the correctness of theoretical results.Numerical results also demonstrate that the constructed method can significantly improve the calculation speed.

HDG method for linear parabolic integro-Differential equations

This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time apriori error bounds. To handle the integral term, an extended Ritz-Volterra projection is introduced, which helps in achieving optimal order convergence of O(hk+1) for the semi-discrete problem when polynomials of degree k≥0 are used to approximate both the solution and the flux variables. Further, element-by-element post-processing is proposed, and it is established that it achieves convergence of the order O(hk+2) for k≥1. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.

Analysis of a full-discrete two-grid discontinuous Galerkin method for nonlinear parabolic equations

Analysis of a full-discrete two-grid discontinuous Galerkin method for nonlinear parabolic equations

Electromagnetic Effective-Degree-of-Freedom Prediction With Parabolic Equation Method

The performance of a multiple-input-multiple-output (MIMO) wireless communication system can usually be indicated by its effective degree of freedom (EDOF). The parabolic equation (PE) method is widely used to predict electromagnetic (EM) propagation loss for its simplicity and efficiency. A novel PE-based EM channel model is first established for acquiring the EDOF in an electrically large and complex environment. At first, the accuracy of the proposed PE Model is verified by comparing it with the dyadic Green’s function. It is shown that the PE Model produces the same results as the analytical one. Then, the connection between the limit of the EM EDOF and the optimal source/receiver numbers is analyzed in the inhomogeneous environment with obstacles. Finally, a complex urban environment is analyzed and the sensitivity analysis is made for the EM EDOF. Numerical results show that the EM EDOF will be stable when the number of transmitting/receiving array sources reaches a certain point. The proposed PE model can be used as an efficient tool for evaluating communication channels in complex environments.

Efficient physics‐based recurrent neural network model for radio wave propagation in tunnels at 2.4 GHz

AbstractThe vector parabolic equation (VPE) method has been widely applied to the modelling of radio wave propagation in tunnel environments. However, dense spatial discretizations are generally required for VPE simulations to achieve acceptable accuracy. This results in high computational costs for large‐size tunnels and high‐frequency wireless systems. This letter presents an efficient physics‐based recurrent neural network (RNN) model for modeling radio wave propagation in tunnels. The proposed RNN‐based model can leverage coarse‐mesh VPE to generate high‐fidelity received signal strength and achieve significant computational savings. The validity of the proposed approach is demonstrated against fine‐mesh VPE simulations in various tunnel cases.

CNN‐based received signal strength prediction: A frequency conversion scheme

AbstractTo support the design and deployment of emerging high‐frequency wireless systems in various applications such as intelligent transportation, advanced received signal strength (RSS) simulation tools for radio coverage prediction are needed. In railway transportation, the vector parabolic equation (VPE) method is a commonly used approach for radio wave propagation modelling in large guiding structures such as tunnels. However, running VPE simulations at high frequencies in such an environment is computationally expensive, as the discretization parameters are closely related to the wavelength. In this letter, the authors propose a convolutional neural network‐based model that can efficiently provide RSS prediction at high frequencies by leveraging data obtained from VPE simulations at a lower frequency as an anchor. This proposed model can significantly improve the efficiency of VPE for high‐frequency RSS predictions and has been validated against conventional VPE in various tunnel cases.

An Iterative High-Accuracy ADI Method for the 3D Parabolic Equation

The alternating direction implicit parabolic equation (ADI-PE) method and the Crank–Nicolson parabolic equation (CN-PE) method have been widely used for solving the 3D parabolic equation (3D-PE) in radio wave propagation. The ADI-PE method is more computationally efficient than the CN-PE method. The accuracy of the ADI-PE method is improved by the higher-order Mitchell–Fairweather (MF)-ADI method. This paper presents an iterative high-accuracy (IHA)-ADI method for the 3D parabolic equation. A derivation of the proposed method is presented. The convergence and stability of the proposed method are estimated. Several numerical examples are considered to illustrate the advantages of the proposed method. The results of error analysis and a comparative study show that the proposed method is unconditionally stable and computationally efficient. The proposed method is more numerically accurate than the MF-ADI method.

High-performance computation toward large-scale underwater acoustics modelling

The complex nature of the ocean has long presented a challenge for researchers in the field of oceanography, including in the area of underwater acoustics. As a result, significant efforts have been made to develop accurate numerical models to better understand and study the ocean. Established models such as Acoustic Toolbox and Range-dependent Acoustic Models have proven to be effective for modelling sound propagation. However, these models were designed to run on single-core computers, and there is potential to optimise their performance on modern systems. The use of General-purpose graphics processing units (GPGPUs) and Single Instruction/Multiple Data (SIMD) computing units is one way to achieve this optimization. In addition to improving the computational speed of established models, these optimized models can also contribute to implementing more sophisticated modelling approaches, such as broad-band modelling. In this study, we will discuss the optimized performance of established codes, particularly those based on parabolic equation methods (PE), and also discuss the results of advanced modelling techniques.

A Hybrid 3-D-ADI-TDPE/DGTD Method for Multiscale Target Echo Simulation in Large-Scale Complex Environments

In this article, we propose a novel hybrid method for multiscale target echo simulation in large-scale complex environments. The three-dimensional-alternating direction implicit-time domain parabolic equation (3-D-ADI-TDPE) method with refractive index terms and the discontinuous Galerkin time-domain (DGTD) method are combined to solve the propagation of transient electromagnetic waves in large-scale regions and backscattering in multiscale target regions, respectively. At the same time, to realize the connection of two regions and the prediction of target echo, the transfer functions of wave propagation and target backscattering are obtained by the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> transform (ZT). In addition, the target echo process is transformed into a cascade of transfer functions to realize simulation modularization and parallel processing. We also discuss the performance of the proposed method in terms of accuracy, memory consumption, and CPU processing time. Results indicate that the processing time required by the 3-D-ADI-TDPE/DGTD hybrid method is more than 11 times less than that of DGTD alone. Moreover, the ZT improves the flexibility of the hybrid method and provides a new concept for the simulation technology of radio wave propagation.