Solution Of Helmholtz Equation Research Articles

A Stable Node-Based Smoothed Finite Element Method with Transparent Boundary Conditions for the Elastic Wave Scattering by Obstacles

In this work, a stable node-based smoothed finite element method with TBC (SNS-FEM-TBC) is proposed to solve the scattering of elastic plane waves by a two-dimensional (2D) homogeneous isotropic elastic medium. First, using Helmholtz decomposition, two scalar potential functions are introduced to divide the Navier equation into Helmholtz equations with the coupled boundary conditions for the elastic scattering problem. Second, based on the analytical solutions of Helmholtz equations, TBC operators are deduced. Then, the gradient Taylor expansion is used to construct the stability term to deal with the instability of the original NS-FEM, the gradient of the solution is expressed into a linear formulation through approximating the node-based smoothing domain as a circle. Finally, based on smoothed Galerkin weak formula, the SNS-FEM-TBC formula of linear algebraic system with linear smoothing gradient is derived. Numerical examples show that SNS-FEM can obtain more stable and accurate solutions than standard FEM. Moreover, the convergence rates of [Formula: see text] and [Formula: see text] semi-norm errors of SNS-FEM are faster.

Numerical solutions for Helmholtz equation with stochastic interface based on PML method

In this paper, the stochastic interface for diffraction grating is considered and the model is formulated as the Helmholtz interface problems (HIPs). In order to have more accuracy simulation, PML boundary is used to describe the stochastic interface. Then we develop shape-Taylor expansion for the solution of HIPs, through perturbation method, we obtain the approximate simulations of second and third order. Error estimation and efficient computation of solutions by low-rank approximation are given. Finally, we illustrate these results with numerical simulations.

A modified source-detector configuration for the discrimination between normal and diseased human breast based on the continuous-wave diffuse optical imaging approach: a simulation study.

Breast tumors are among the most common types of tumors that can affect both genders. It may spread to the whole breast without any symptoms. Therefore, the early detection and accurate diagnosis of breast tumors are significantly important. Current approaches for breast cancer screening such as positron emission tomography (PET) and magnetic resonance imaging (MRI) have some limitations of being time and money-consuming. In addition, mammography screening is not recommended for women under forty. Consequently, optical techniques have been introduced as safe and functional alternatives. Diffuse optical imaging is a non-invasive imaging technique that utilizes near-infrared light to examine biological tissues based on measuring the optical transmission and/or reflection at various locations on the tissue surface. In this paper, we propose a modified arrangement between the laser source and the detectors for distinguishing tumors from normal breast tissue. A three-dimensional model of the normal human breast with three types of tumors is developed using a COMSOL simulation software based on the finite element solution of Helmholtz equation to estimate optical fluence distribution. The breast model consists of four layers: skin, fat, glandular, and muscle, where the tumor is included in the glandular layer. Different wavelengths were used to determine the most proper wavelength for the discrimination between the normal tissue and tumor. The obtained results were verified using the receiver operating characteristic (ROC) method. The resultant fluence images show different features between normal breast and breast with tumor especially using 600-nm incident laser as demonstrated by the obtained ROC curves.

Finite difference method based on polynomial interpolation for solving Helmholtz equation

The generalized finite difference formula plays an important role in the meshless method for solving differential equations. The main purpose of this paper is to find the numerical solution of Helmholtz equation, an elliptic partial differential equation describing electromagnetic waves in physics, by using the generalized finite difference formula. Firstly, this paper introduces a simple and practical nodal distribution, which not only guarantees the uniqueness of multivariate polynomial interpolation, but also makes the matrix triangular, so that the constructed basic polynomials can be transformed into Lagrange basis polynomials. Secondly, the generalized finite difference formula is created by polynomial interpolation. Finally, a numerical example of Helmholtz equation under general boundary conditions is given.

Spatial basis interpretation for implementing the acoustic imaging of non-synchronous measurements

Acoustic imaging from non-synchronous measurements is a powerful method that can achieve excellent performance of a densely sampled large array by using an arbitrary prototype array. It has attracted significant interest recently since it is beyond the fundamental limitation of the working frequency determined by the size and microphone spacing of an array. One basic assumption of the non-synchronous measurements theory is the spatial continuity of the acoustic field, which is implemented by introducing the spatial basis function. Nevertheless, a comprehensive investigation of spatial basis function in the context of non-synchronous measurements is still missing. Two kinds of methods are proposed to construct spatial basis functions based on the homogeneous Helmholtz equation solution in this paper. One is the generalized harmonic polynomials (GHPs) and the other is the plane wave expansion (the quasi-uniform plane wave and the non-uniform plane wave). Comparisons of their spectral matrix completion errors show that the proposed construction methods of spatial bases can both yield low errors. Moreover, the construction of the proposed bases is simple to be implemented with only one dimension parameter to be chosen and suitable for the complex sound field. The effectiveness of the proposed bases in the non-synchronous measurements is also validated with simulations and an experiment.

Numerical solutions for point-source high frequency Helmholtz equation through efficient time propagators for Schrödinger equation

For high frequency Helmholtz equations with point-source conditions, geometrical optics provides an efficient way to compute the solutions without ‘pollution effect’ that is usually unavoidable in finite difference and finite element methods. However, geometrical optics generally can only provide locally valid approximations for the wavefield near the primary source. In order to obtain globally valid wavefield, we will propagate the wavefield with efficient time propagators for an appropriate time-dependent Schrödinger equation. The geometrical optics is first applied to compute locally valid approximations for the wavefield, then the wavefield is propagated through the time-dependent Schrödinger equation until its change is small enough in the domain of interest, which results in the globally valid approximation for the solution to the Helmholtz equation in the domain of interest. The adapted time propagator for the time-dependent Schrödinger equation is a Strang operator splitting based pseudospectral approach, which assures the overall efficiency of the method since the pseudospectral approach is unconditionally stable such that large time step sizes are allowed to propagate the wavefield efficiently, and the number of points per wavelength is independent of the frequency. Both two-dimensional and three-dimensional numerical experiments are presented to demonstrate the method.

Exploring fractional order 2‐D Helmholtz equation using finite difference scheme through the bat optimization algorithm

This endeavor proposes an effective implementation of a hybrid technique for computing the approximate solution of fractional order Helmholtz equation, with Dirichlet boundary conditions. The novel scheme is an amalgamation of the traditional finite difference method with the bat optimization algorithm (BOA). This nature‐inspired optimization technique simulates the echolocation behavior of foraging bats, which ascertain the surroundings through the echo of their emitted sound pulse. Systematically, the deliberated fractional order system is altered into an integer order partial differential equation by virtue of linearized expansion of Laplace transformation. Subsequently, the attained system is processed through the proposed innovative finite difference optimization technique (FDOT) the numerical discussions. Furthermore, some experiments are carried out in order to expound the effective execution and application of the technique. In addition, the convergence and accuracy of the scheme are also delineated numerically, via statistical inference. The significant outcomes of the analysis reveal the advantageousness of the proposed numerical scheme, which can efficiently handle the complexities of the fractional order 2‐D Helmholtz equation.

Radon-to-Helmholtz mappings and nonlinear diffraction tomography.

This paper addresses a number of approximate, analytically invertible solutions of the scalar Helmholtz equation. Primary attention is devoted to the Glauber approximation (GA) derived for the far-field pattern. It is shown that the GA has the form of a nonlinear Radon-to-Helmholtz (RtH) mapping, which transforms a sinogram of the scattering potential into an approximate solution of the Helmholtz equation. A proposal of how to construct a position space counterpart of the GA is formulated. Also, it is established that a paraxial version of the Glauber model coincides, up to an inessential constant factor, with a momentum-space representation of the Mazar-Felsen propagator, which describes forward-scattered waves. For weakly scattering objects, these solutions are reduced to the conventional Born/Rytov approximations, which may, however, preserve the parametrization and sampling formats of the original nonlinear models. Since all RtH mappings are analytically invertible, they can be applied to the (nonlinear) diffraction tomography of penetrable objects. In particular, the Glauber model, which has been largely ignored for years, is shown to provide efficient inversion of synthetic data. The resulting tomograms clearly outperform the Born inversions, even for moderately scattering potentials.

An efficient approach for solution of fractional-order Helmholtz equations

In this article, a hybrid technique called the homotopy perturbation Elzaki transform method has been implemented to solve fractional-order Helmholtz equations. In the hybrid technique, the Elzaki transform method and the homotopy perturbation method are amalgamated. Three problems are solved to validate and demonstrate the efficacy of the present technique. It is also demonstrated that the results obtained from the suggested technique are in excellent agreement with the results by other techniques. It is shown that the proposed method is efficient, reliable and easy to implement for various related problems of science and engineering.

Exponential node clustering at singularities for rational approximation, quadrature, and PDEs.

Rational approximations of functions with singularities can converge at a root-exponential rate if the poles are exponentially clustered. We begin by reviewing this effect in minimax, least-squares, and AAA approximations on intervals and complex domains, conformal mapping, and the numerical solution of Laplace, Helmholtz, and biharmonic equations by the “lightning” method. Extensive and wide-ranging numerical experiments are involved. We then present further experiments giving evidence that in all of these applications, it is advantageous to use exponential clustering whose density on a logarithmic scale is not uniform but tapers off linearly to zero near the singularity. We propose a theoretical model of the tapering effect based on the Hermite contour integral and potential theory, which suggests that tapering doubles the rate of convergence. Finally we show that related mathematics applies to the relationship between exponential (not tapered) and doubly exponential (tapered) quadrature formulas. Here it is the Gauss–Takahasi–Mori contour integral that comes into play.

Approximate Solution of Time-Fractional Helmholtz and Coupled Helmholtz Equations

The approximate analytical solution of the time-fractional Helmholtz and coupled Helmholtz equations have been acquired successfully via Residual Power Series Method (RPSM). The approximate solutions obtained by RPSM are compared with the exact solutions through different graphics and tables. The fractional derivatives are described in the Caputo sense. The numerical results demonstrate that the new method is quite accurate and readily implemented.

Universal dispersion curves of a planar waveguide with an exponential graded-index guiding layer and a nonlinear cladding

A planar waveguide consisting of three layers is considered. The guiding layer is assumed of exponentially graded index of refraction. The cover layer is a nonlinear material of Kerr-type. The refractive index distribution of the film layer changes as an exponential function from the guiding layer to the substrate. The solutions of Helmholtz equation are found. They are written in terms of three parameters a, b and V. The solutions in the guiding layer and substrate are found as Bessel functions of order Vb. The characteristic equation is derived and the dispersion curves are plotted and analyzed. A set of attracting features are found such as there is no cut-off thickness corresponding to a symmetric waveguide structure. The b-values do not exceed unity. This means the dispersion curves refer to guided modes.

The Fourier extension operator of distributions in Sobolev spaces of the sphere and the Helmholtz equation

The purpose of this paper is to characterize the entire solutions of the homogeneous Helmholtz equation (solutions in ℝd) arising from the Fourier extension operator of distributions in Sobolev spaces of the sphere $H^\alpha (\mathbb {S}^{d-1}),$ with α ∈ ℝ. We present two characterizations. The first one is written in terms of certain L2-weighted norms involving real powers of the spherical Laplacian. The second one is in the spirit of the classical description of the Herglotz wave functions given by P. Hartman and C. Wilcox. For α > 0 this characterization involves a multivariable square function evaluated in a vector of entire solutions of the Helmholtz equation, while for α < 0 it is written in terms of an spherical integral operator acting as a fractional integration operator. Finally, we also characterize all the solutions that are the Fourier extension operator of distributions in the sphere.

Helmholtz solving and performance optimization in global/regional assimilation and prediction system

Despite efficient parallelism in the solution of physical parameterization in the Global/Regional Assimilation and Prediction System (GRAPES), the Helmholtz equation in the dynamic core, with the increase of resolution, can hardly achieve sufficient parallelism in the solving process due to a large amount of communication and irregular access. In this paper, optimizing the Helmholtz equation solution for better performance and higher efficiency has been an urgent task. An optimization scheme for the parallel solution of the Helmholtz equation is proposed in this paper. Specifically, the geometrical multigrid optimization strategy is designed by taking advantage of the data anisotropy of grid points near the pole and the isotropy of those near memory equator in the Helmholtz equation, and the Incomplete LU (ILU) decomposition preconditioner is adopted to speed up the convergence of the improved Generalized Conjugate Residual (GCR), which effectively reduces the number of iterations and the computation time. The overall solving performance of the Helmholtz equation is improved by thread-level parallelism, vectorization, and reuse of data in the cache. The experimental results show that the proposed optimization scheme can effectively eliminate the bottleneck of the Helmholtz equation as regards the solving speed. Considering the test results on a 10-node two-way server, the solution of the Helmholtz equation, compared with the original serial version, is accelerated by 100×, with one-third of iterations reduced.

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

We consider the Helmholtz equation $$-\Delta u+V \, u - \lambda \, u = f $$ on $${\mathbb {R}}^n$$ where the potential $$V:{\mathbb {R}}^n\rightarrow {\mathbb {R}}$$ is constant on each of the half-spaces $${\mathbb {R}}^{n-1}\times (-\infty ,0)$$ and $${\mathbb {R}}^{n-1}\times (0,\infty )$$ . We prove an $$L^p-L^q$$ -Limiting Absorption Principle for frequencies $$\lambda >\max \, V$$ with the aid of Fourier Restriction Theory and derive the existence of nontrivial solutions of linear and nonlinear Helmholtz equations. As a main analytical tool we develop new $$L^p-L^q$$ estimates for a singular Fourier multiplier supported in an annulus.

Explaining neural network predictions of acoustic fields in ideal single- and multi-path environments

Neural networks are an increasingly popular tool for solving a wide variety of problems. Given some up-front computational effort, the complex patterns which exist in large datasets can be embedded in an explicit, though difficult to interpret, neural-network ‘formula’ for future predictions. As reliance on neural networks increases, so does the demand for their explainability, since practitioners must create safe and unbiased methods. In this presentation, an effort is made to better explain and understand neural networks which are trained to provide solutions to the point-source Helmholtz-equation in axisymmetric single-path, two-path, and multi-path (ideal waveguide) environments having constant sound speed. This analysis emphasizes source frequencies in the 100s of Hz, depths up to 500 m, and ranges up to 1 km for sound speeds near 1500 m/s. In all cases, the neural networks' intermediate computational values are compared to those of the corresponding analytical Helmholtz-equation solution used to generate the neural networks' training dataset. The effects of the choice of neural network inputs, noise-like inputs, and correlated input features are also considered. The insights developed from this investigation provide insights for the extension of neural network predictions of acoustic fields to more complex environments. [Work supported by the NDSEG fellowship program.]

Evidence for Symmetry Breaking in the Ground State of the Si(111)‐7 × 7: The Origin of Its Metal–Insulator Transition

Scanning tunneling microscopy (STM) of the Si(111)‐7 × 7 reconstructed surface at 5 K reveals that the highly symmetric STM image found at room temperature does not reflect the lowest energy, ground state of this system. Instead, the ground state has certain adatom charge densities distorted by ≈0.2–0.4 Å from the symmetric STM charge density positions observed at 300 K. This agrees with adatom ion core displacements in both a Patterson map and two additional phase‐reconstructed atom maps. Such ion core distortions are consistent with the Jahn–Teller effect associated with the threefold mirror symmetry of the Si(111)‐7 × 7 and its antiphase, faulted structure. The nodal properties of the electronic states of an underlying 7 × 7 honeycomb that arise from threefold symmetry are determined from solutions of the 2D inhomogeneous Helmholtz equation, and provide insight to the observed temperature‐dependent charge density changes. It is shown that an excited state of the honeycomb cells of a Si(111)‐7 × 7 support the higher symmetry charge densities observed at room temperature and the presence of strong electron–phonon coupling. The role of symmetry and anharmonicity in defining the structure and electronic interactions in the Si(111)‐7 × 7 are discussed.

On convergence analysis and numerical solutions of local fractional Helmholtz equation

Local fractional q-homotopy analysis transform method (q -HATM) is employed to solve the local fractional Helmholtz equation. Uniqueness and convergence analysis of the method is investigated by Banach’s fixed point theory. Solutions are expressed in the form of rapidly series with fast computable basics by Mathematica software. Reliability analysis is provided. Computational results display that LFq-HATM is an efficient and powerful method to obtain solutions to the present equation and has the potential to be applicable to other related fractional-order systems.

Autoproducts in and near acoustic shadow zones created by barriers.

The autoproducts are nonlinear mathematical constructs developed from acoustic fields with non-zero bandwidth. When averaged through the field's bandwidth, the autoproducts may mimic a genuine acoustic field at frequencies that are lower or higher than the original field's bandwidth. The resulting opportunity to extend signal processing to user-selectable below- or above-band frequencies is intriguing for many signal processing algorithms. Based on prior work, the limitations of the autoproducts' mimicry of out-of-band fields are understood when the in-band acoustic field is well-represented by ray acoustics. Thus, the focus in this study is on autoproducts in acoustic shadow zones behind barriers containing only diffracted acoustic fields where a sum of ray-path contributions is not an adequate field description. Diffraction is expected to be a detriment to autoproduct techniques due to its sensitivity to frequency. Two ideal shadow-zone environments with exact analytic Helmholtz-equation solutions are considered: Sommerfeld's half-plane problem, also known as knife-edge diffraction, and Mie scattering from a sphere with ka = 40, where k is the wavenumber and a is the sphere's radius. With the exception of the shadow regions, autoproducts experience only mild degradation in field-mimicry performance when compared to what the ray-based theory would predict.

Parallel controllability methods for the Helmholtz equation

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy, convergence and scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures.