Two-dimensional Helmholtz Research Articles

A Chebyshev collocation method for directly solving two-dimensional ocean acoustic propagation in linearly varying seabed.

It is one of the most concerning problems in hydroacoustics to find a method that can calculate the acoustic propagation accurately and adapt to the variation of the seabed. Currently, the one-dimensional spectral method has been employed to address the simplified ocean acoustic propagation model successfully. However, due to the model's application limitations and approximation error, it poses challenges when attempting to solve real-world ocean acoustic fields. Hence, there is a crucial need to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation, without relying on a simplified model. In previous work, we achieved successful solutions for the two-dimensional Helmholtz equation within a rectangular domain, utilizing a collocation-type spectral method. Taking into account the fluctuations in the actual seabed, we introduce a Chebyshev collocation spectral method to directly tackle the two-dimensional ocean acoustic propagation problem, which could solve the case of a seabed with linear variation, sound velocity variation and inhomogeneous medium situation. After comparative verification, the calculation result of the two-dimensional spectral method is more accurate than traditional mature models such as Kraken and COUPLE. By eliminating model constraints and enlarging the solution range, this spectral method holds immense potential in real marine environments.

Numerical investigation of two-dimensional fractional Helmholtz equation using Aboodh transform scheme

Purpose This paper aims to present a numerical investigation for two-dimensional fractional Helmholtz equation using the Aboodh integral homotopy perturbation transform scheme (AIHPTS). Design/methodology/approach The proposed scheme combines the Aboodh integral transform and the homotopy perturbation scheme (HPS). This strategy is based on an updated form of Taylor’s series that yields a convergent series solution. This study analyzes the fractional derivatives in the context of Caputo. Findings This study illustrates two numerical examples and calculates their approximate results using AIHPTS. The derived findings are also presented in tabular form and graphical representations. Research limitations/implications In addition, He’s polynomials are calculated using HPS, so the minimal computational outcome is a defining feature of this method and gives a competitive advantage over other series solution techniques. Originality/value Numerical data and graphical illustrations for different fractional order levels confirm the proposed method’s successful performance. The results show that the proposed approach is speedy and straightforward to execute on fractional-ordered models.

An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation

Abstract In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.

A fast algorithm for the two-dimensional Helmholtz transmission problem with large multiple scattering configurationsa).

We develop an efficient three-stage algorithm for simulating multiple acoustic scattering by two-dimensional configurations comprising large numbers of penetrable scatterers. Our approach is based on a boundary integral equation reformulation of the Helmholtz transmission partial differential equation, and a reduction of the boundary integral system for computationally efficient evaluation of wave interactions between scatterers. A key ingredient of our algorithm is to represent the interactions between scatterers using expansions of cylindrical wavefunctions. For large numbers of scatterers, this approach facilitates the application of the fast multipole method, leading to linear complexity of the algorithm with respect to the number of scatterers. Numerical results demonstrate the efficiency of our algorithm for configurations containing a few hundred to hundreds of thousands of individual scatterers.

Weighted means and an analytic characterization of discs

Weighted means are obtained for solutions of the two-dimensional Helmholtz and modified Helmholtz equations and also for harmonic functions. The presence of a logarithmic weight reduces the coefficient in the last two mean value identities. A new theorem characterizing disks in the Euclidean plane R 2 \mathbb {R}^2 analytically is proved; it is based on the weighted mean value property of solutions to the modified Helmholtz equation.

PDE-Based Two-Dimensional Radiomagnetotelluric forward Modelling Using Vertex-Centered Finite-Volume Scheme

An efficient finite-volume algorithm, based on the vertex-centered technique, is proposed for solving two-dimensional radiomagnetotelluric forward modeling. Firstly, we derive the discrete expressions of the radiomagnetotelluric Helmholtz-type equation and the corresponding mixed boundary conditions using the vertex-centered finite-volume technique. Then, the corresponding approximate solutions of the radiomagnetotelluric forward problem can be calculated by applying the finite-volume scheme to treat the boundary conditions. Secondly, we apply the finite-volume algorithm to solve two-dimensional Helmholtz equations and the resistivity half-space model. Numerical experiments demonstrate the high accuracy of the proposed approach. Finally, we summarize the radiomagnetotelluric responses through a numerical simulation of a two-dimensional model, which enables qualitative interpretation of field data. Furthermore, our numerical method can be extended and implemented for three-dimensional radiomagnetotelluric forward modeling to achieve more accurate computation.

A matrix-free parallel two-level deflation preconditioner for two-dimensional heterogeneous Helmholtz problems

We propose a matrix-free parallel two-level deflation method combined with the Complex Shifted Laplacian Preconditioner (CSLP) for two-dimensional heterogeneous Helmholtz problems encountered in seismic exploration, antennas, and medical imaging. These problems pose challenges in terms of accuracy and convergence due to scalability issues with numerical solvers. Motivated by the limitations imposed by excessive computational time and memory constraints when employing a sequential solver with constructed matrices, we parallelize the two-level deflation method without constructing any matrices. Our approach utilizes preconditioned Krylov subspace methods and approximates the CSLP preconditioner with a parallel geometric multigrid V-cycle. For the two-level deflation, standard inter-grid deflation vectors and further high-order deflation vectors are considered. As another main contribution, the matrix-free Galerkin coarsening approach and a novel re-discretization scheme as well as high-order finite-difference schemes on the coarse grid are studied to obtain wavenumber-independent convergence. The optimal settings for an efficient coarse-grid problem solver are investigated. Numerical experiments of model problems show that the wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

A Legendre Spectral-Element Method to Incorporate Topography for 2.5D Direct-Current-Resistivity Forward Modeling

An effective and accurate solver for the direct-current-resistivity forward-modeling problem has become a cutting-edge research topic. However, computational limitations arise due to the substantial amount of data involved, hindering the widespread use of three-dimensional forward modeling, which is otherwise considered the most effective approach for identifying geo-electrical anomalies. An efficient compromise, or potentially an alternative, is found in two-and-a-half-dimensional (2.5D) modeling, which employs a three-dimensional current source within a two-dimensional subsurface medium. Consequently, a Legendre spectral-element algorithm is developed specifically for 2.5D direct-current-resistivity forward modeling, taking into account the presence of topography. This numerical algorithm can combine the complex geometric flexibility of the finite-element method with the high precision of the spectral method. To solve the wavenumber-domain electrical potential variational problem, which is converted into the two-dimensional Helmholtz equation with mixed boundary conditions, the Gauss–Lobatto–Legendre (GLL) quadrature is employed in all discrete quadrilateral spectral elements, ensuring identical Legendre polynomial interpolation and quadrature points. The Legendre spectral-element method is applied to solve a two-dimensional Helmholtz equation and a resistivity half-space model. Numerical experiments demonstrate that the proposed approach yields highly accurate numerical results, even with a coarse mesh. Additionally, the Legendre spectral-element algorithm is employed to simulate the apparent resistivity distortions caused by surface topographical variations in the direct-current resistivity Wenner-alpha array. These numerical results affirm the substantial impact of topographical variations on the apparent resistivity data obtained in the field. Consequently, when interpreting field data, it is crucial to consider topographic effects to the extent they can be simulated. Moreover, our numerical method can be extended and implemented for a more accurate computation of three-dimensional direct-current-resistivity forward modeling.

A network model for can-annular combustors using a multidimensional transfer matrix based on the mode matching method

Heavy-duty land-based gas turbines with can-annular combustors typically exhibit a set of thermoacoustic modes with similar frequencies, due to their rotational symmetry and the weak coupling between neighbouring cans. The Bloch boundary condition is often used to simplify these combustors. Because the widely used (quasi-)one-dimensional model cannot naturally consider high-order modal components in the three-dimensional acoustic field, it is valid only when the frequency is sufficiently low and the length of the can-to-can interaction gap is sufficiently short. In this paper, we present a theoretical model using the mode-matching method to consider high-order modal components. A reflection coefficient matrix is used to accurately describe the acoustic characteristics of the gap area. This reflection coefficient matrix is found to depend on the nondimensional cross-sectional area ratio, the Helmholtz number based on the length of the gap area, and the Bloch number. To facilitate a full low-order network modelling of the high-frequency thermoacoustic modes in can-annular combustors, a flame transfer matrix is also used to describe the modal coupling effect across the flame. Based on this model, the thermoacoustic problem of a four-flame can-annular combustor is studied as an example. It is shown that the frequency and growth rate of the thermoacoustic modes can be predicted accurately using the reflection coefficient matrix and flame transfer matrix. The effects of high-order modal components across the flame and in the gap areas are studied separately. The results show that the model containing both of the two coupling effects has the widest range of validity. These couplings can be ignored if only low-frequency plane acoustic waves are considered but need to be carefully considered for higher frequency modes. Moreover, when multiple flames are considered, asymmetric flame distributions are found to more easily lead to modal coupling compared to symmetric flame distributions. All results are validated by comparing with the numerical solutions of the full two-dimensional Helmholtz equation using the finite element method.

Restricting angles of incidence to improve super resolution in time reversal focusing that uses metamaterial properties of a resonator array.

Focusing waves with a spatial extent smaller than a half wavelength (i.e., super resolution or sub diffraction limit) is possible using resonators placed in the near field of time reversal (TR) focusing. While a two-dimensional (2D) Helmholtz resonator array in a three-dimensional reverberant environment has limited ability to produce a high-resolution spatial focus in the TR focusing of audible sound, it is shown that acoustic waves propagating out-of-plane with the resonator array are not as strongly affected by the smaller effective wavelength induced by the resonator array, partially negating the effect of the resonators. A physical 2D waveguide is shown to limit the out-of-plane propagation, leading to improved resolution. It is also shown that post processing using an orthogonal particle velocity decomposition of a spatial scan of the focusing can filter out-of-plane particle motion in the near field of the array, which bypasses the effect of the unwanted third spatial dimension of propagation. The spatial resolution in a reverberant environment is shown to improve in the presence of a 2D Helmholtz resonator array and then further improve by adding a 2D waveguide. The resolution among the resonator array is better still without using a waveguide and instead using the partial-pressure reconstruction.

BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT HELMHOLTZ BVPs IN 2D

In this paper, we construct boundary-domain integral equations (BDIEs) of the Dirichlet and mixed boundary value problems for a two-dimensional variable-coefficient Helmholtz equation. Using an appropriate parametrix, these problems are reduced to several BDIE systems. It is shown that the BVPs and the formulated BDIE systems are equivalent. Fredholm properties and unique solvability and invertibility of BDIE systems are investigated in appropriate Sobolev spaces.

A robust approach for computing solutions of fractional-order two-dimensional Helmholtz equation

The Helmholtz equation plays a crucial role in the study of wave propagation, underwater acoustics, and the behavior of waves in the ocean environment. The Helmholtz equation is also used to describe propagation through ocean waves, such as sound waves or electromagnetic waves. This paper presents the Elzaki transform residual power series method (E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T-RPSM) for the analytical treatment of fractional-order Helmholtz equation. To develop this scheme, we combine Elzaki transform (E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T) with residual power series method (RPSM). The fractional derivatives are described in Caputo sense. The E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}$$\\end{document}T is capable of handling the fractional order and turning the problem into a recurrence form, which is the novelty of our paper. We implement RPSM in such a way that this recurrence relation generates the results in the form of an iterative series. Two numerical applications are considered to demonstrate the efficiency and authenticity of this scheme. The obtained series are determined very quickly and converge to the exact solution only after a few iterations. Graphical plots and absolute error are shown to observe the authenticity of this suggested approach.

Learning domain-independent Green’s function for elliptic partial differential equations

Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain-independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients.

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side

A Modified Residual Power Series Method for the Approximate Solution of Two-Dimensional Fractional Helmholtz Equations

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method (ST-RPSM). This scheme uses a combination of the Shehu transform (ST) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare ST-RPSM results with exact solutions.

Comparisons between direct interpolation and reciprocity techniques of the boundary element method for solving two-dimensional Helmholtz problems

PurposeThis work compares the performance of the three boundary element techniques for solving Helmholtz problems: dual reciprocity, multiple reciprocity and direct interpolation. All techniques transform domain integrals into boundary integrals, despite using different principles to reach this purpose.Design/methodology/approachComparisons here performed include the solution of eigenvalue and response by frequency scanning, analyzing many features that are not comprehensively discussed in the literature, as follows: the type of boundary conditions, suitable number of degrees of freedom, modal content, number of primitives in the multiple reciprocity method (MRM) and the requirement of internal interpolation points in techniques that use radial basis functions as dual reciprocity and direct interpolation.FindingsAmong the other aspects, this work can conclude that the solution of the eigenvalue and response problems confirmed the reasonable accuracy of the dual reciprocity boundary element method (DRBEM) only for the calculation of the first natural frequencies. Concerning the direct interpolation boundary element method (DIBEM), its interpolation characteristic allows more accessibility for solving more elaborate problems. Despite requiring a greater number of interpolating internal points, the DIBEM has presented higher-quality results for the eigenvalue and response problems. The MRM results were satisfactory in terms of accuracy just for the low range of frequencies; however, the neglected higher-order primitives impact the accuracy of the dynamic response as a whole.Originality/valueThere are safe alternatives for solving engineering stationary dynamic problems using the boundary element method (BEM), but there are no suitable comparisons between these different techniques. This paper presents the particularities and detailed comparisons approaching the accuracy of the three important BEM techniques, aiming at response and frequency evaluation, which are not found in the specialized literature.

A sonic black hole of a rectangular cross-section

In this study, a rectangular cross-section sonic black hole (SBH) is explored and a one-dimensional model equation for the acoustic field inside it is presented. The model equation's applicability is verified through a comparison with the corresponding two-dimensional Helmholtz equation. A general analytical solution, which takes into account the SBH's quadratically varying internal cross-section, is derived using linearly independent confluent Heun functions. To investigate the acoustic wave behavior in the SBH, the wave splitting method is utilized to obtain forward and backward acoustic pressure waves, and the reflection coefficient is calculated. To confirm the validity of the analytical solution, a comparison is made with numerical results obtained from the Riccati equation. Within the context of this study, a detailed discussion is provided regarding the applicability of the analytical solution.

A Study of the Effects of the Shape Parameter and Type of Data Points Locations on the Accuracy of the Hermite-Based

Theoretical approximation ideas served as the driving force behind the research. one can see that the shape parameter's behavior is driven by the kind of problem and the analytical standards that are applied. the primary issue here is not just how it impacts the interpolant's accuracy but also how quickly it converges, or how quickly the error reduces as the number of data nodes rises. Hence, this article considers two globally supported and positive radial kernels and three different patterns of data point locations on the same computational domain. The research specifically studied the effects of the shape parameter and the type of data points locations on the accurate performance of an Hermite-based symmetric approach. The two-dimensional Helmholtz equation and the two-dimensional Poisson equation were used as test functions. The problems were first solved on the three different types of data point locations using linear Laguerre-Gaussians and then, the linear Matern. In each case, the graph of the error against the shape parameter was drawn to enable easy identification of the optimal value of the shape parameter. One important result indicated that, an improved accuracy cannot be achieved without the appropriate value of the shape parameter irrespective of the type of data site used.

Algorithmic Solutions in the Problem of Evaluating the Quality of Communication with an Unmanned Aerial Vehicle

Algorithmic solutions have been developed to assess the quality of communication with an unmanned aerial vehicle. The quality of communication is determined by the probability of a symbol error at the input of an error-correcting encoder, the a priori calculation of which is performed in the approximation of Rice’s communication channel model. The main attention in the solution is given to the determination of attenuation due to the combined influence of the terrain and tropospheric refraction on the energy parameters of the line ground communication point–unmanned aerial vehicle. The attenuation was calculated by jointly solving a pair of parabolic equations formed from the two-dimensional Helmholtz equation in narrow- and wide-angle approximations by the pseudospectral method of separating the steps of a pair of parabolic equations. In the features of the main stages of the algorithmic implementation of the developed solution are highlighted, and its performance is tested on specific test examples.

Derivation of an in-plane amplitude equation and its paraxial approximation for elastic guided waves in plates

In addressing the construction of beam solutions for elastic guided waves in plates, we propose expressions of the amplitude function for the Lamb wave and the shear horizontal plate wave. The amplitude function is governed by an in-plane amplitude equation describing plate waves that has a same form as the two-dimensional Helmholtz equation. Therefore, beam solutions can be constructed within the same framework as this Helmholtz equation. The paraxial approximation is then applied to this in-plane amplitude equation. On the basis of this approximation, Gaussian beam solutions for both the Lamb and shear horizontal plate waves are derived. The effects of beam width, wavenumber, and guided-wave mode upon the wavefields are examined through several numerical examples.