Solution Of Helmholtz Equation Research Articles

Wave gradiometry and its link with Helmholtz equation solutions applied to USArray in the eastern U.S.

AbstractWave gradiometry is an array processing technique utilizing the shape of seismic wavefields captured by USArray TA stations to determine fundamental wave propagation characteristics. We first explore a compatibility relation that links the displacement spatial gradients to seismogram displacements and velocities through two unknown coefficients, and . These coefficients are solved for through iterative, damped least squares inversion to provide estimates of phase velocity, back azimuth, radiation pattern, and geometrical spreading. We show that the coefficient corresponds to the gradient of logarithmic amplitude, and the coefficient corresponds approximately to the local wave slowness. and vector fields are interpolated to explore a second compatibility relation through solutions to the Helmholtz equation. For most wavefields passing through the eastern U.S., we show that the coefficients are generally orthogonal to the coefficients. Where they are not completely orthogonal, there is a strong positive correlation between and changes in geometrical spreading, which can be further linked with areas of strong energy focusing and defocusing. We finally obtain isotropic Rayleigh wave phase velocity maps for 15 periods between 20 and 150 s, by stacking results from 37 earthquakes, which show a velocity change along the approximate boundary of the early Paleozoic continental margin. We also observe two low‐velocity anomalies, one centered over the central Appalachians where Eocene basaltic volcanism has occurred and the other within the northeastern U.S., possibly associated with the Great Meteor Hotspot track.

Theoretical analysis of interaction between radio pulses and Colorado potato beetles in the potato plant environment

To analyze the distribution of electric intensity of radio-pulse radiation in the plant environment with Colorado potato beetles it was developed the model in the form of the box filled with an isotropic dielectric medium with permittivity and conductivity. The nonstationary problem describing the distribution of electrical intensity of radio-pulse radiation by Maxwell's equations, using Laplace transform reduced to the Helmholtz equation. Solution of Helmholtz equations were obtained by separation of variables. After a series of transformations were obtained by the equation of the electric field components for the plant layer of the Colorado potato beetles. The study of the radio-pulse electromagnetic radiation distribution in the plant environment with Colorado potato beetles and their larvae will determine the necessary biotropic parameters (pulse-modulated frequency, pulse power, pulse period, pulse duration, time of exposure of the plant environment with Colorado potato beetles) radio-pulse energo-informational electromagnetic radiation to suppress reproduction ability of the beetles and destroying their larvae.

The Approximate Solutions of Helmholtz and Coupled Helmholtz Equations on Cantor Sets within Local Fractional Operator

In this paper, we proposed local fractional Laplace decomposition method to solve the
 Helmholtz and coupled Helmholtz equations on Cantor sets within local fractional
 operator. The approximate solutions are obtained by using the local fractional Laplace
 decomposition method, which is the coupling method of local fractional Laplace
 transform and Adomian decomposition method. Illustrative examples are included to
 demonstrate the high accuracy and fast convergence of this new method.

Fast multipole accelerated singular boundary method for the 3D Helmholtz equation in low frequency regime

This paper presents a fast multipole accelerated singular boundary method (SBM) to the solution of the large-scale three-dimensional Helmholtz equation at low frequency. By using a desingularization strategy to directly compute singular kernels in the strong-form collocation discretization, the SBM formulations are derived for the Dirichlet and Neumann problems. A fast multipole method (FMM) is then introduced to expedite the solution process. The CPU time and the memory requirement of the FMM–SBM scheme are both reduced to O(N), where N is the number of boundary nodes. Numerical examples with up to one million unknowns have been tested on a desktop computer. The results clearly illustrate that the proposed strategy appears very efficient and promising in solving large-scale Helmholtz problems.

Monte Carlo simulations of Helmholtz scattering from randomly positioned array of scatterers by utilizing coordinate transformations in finite element method

Electromagnetic scattering from randomly distributed array of scatterers is numerically analyzed by Monte Carlo simulations by utilizing coordinate transformations in the context of finite element method solution of Helmholtz equation. The major goal in proposed approaches is to place transformation media into computational domain by employing the form invariance property of Maxwell’s equations under coordinate transformations, and hence avoiding repeated mesh generation process in multiple realizations of the Monte Carlo method. A simple, single and uniform mesh is used, and only the material parameters of the transformation media are changed with respect to the positions of the objects in each realization. In this manner, computational resources are reduced considerably. The proposed approaches are demonstrated and compared with the standard approach via several numerical simulations. Monte Carlo results are presented in terms of some statistical properties (such as mean, standard deviation, probability density functions approximated by histograms) of radar cross section (RCS) and error values.

Efficient and Accurate Numerical Solutions for Helmholtz Equation in Polar and Spherical Coordinates

AbstractThis paper presents new finite difference schemes for solving the Helmholtz equation in the polar and spherical coordinates. The most important result presented in this study is that the developed difference schemes are pollution free, and their convergence orders are independent of the wave number k. Let h denote the step size, it will be demonstrated that when solving the Helmholtz equation at large wave numbers and considering kh is fixed, the errors of the proposed new schemes decrease as h decreases even when k is increasing and kh > 1.

Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams

A new type of exact solutions of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here.We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system by substituting it to the full 3 dimensional spatial Helmholtz equation.Analyzing the structure of the final equation, we obtain that governing equations for the components of our solution are represented by the proper Riccati equations of complex value, which has no analytical solution in general case.But we find one of the possible exact solutions which is proved to satisfy to such equations for Gaussian beams.

An eight-order accurate numerical method for the solution of 2D Helmholtz equation

This note is concerned with a numerical method for the solution of 2D Helmholtz equation in unit square. The method uses a finite difference approximation in one coordinate space. Similar to the method of line, the method treats the working equation as a system of ordinary differential equation in the remaining independent variable. The method uses a coordinate transformation to decouple the system of ODE. Using this procedure, it is possible to formulate numerical schemes with arbitrary orders of accuracy. Numerical results for an eight-order accurate are presented.

Application of four-point MEGMSOR method for the solution of 2D Helmholtz equations

Application of four-point MEGMSOR method for the solution of 2D Helmholtz equations

Theoretical Introduction and Generation Method of a Novel Nondiffracting Waves: Olver Beams

In this paper, we introduce a new class of scalar nondiffracting Helmholtz-equation solution. We demonstrate that this novel wave-equation solution has some specific orders; among these ordinary Airy beams which are regarded as the zeroth order. Moreover, a general expression of these novel beams, which are named Olver Beams and referred to OBs, is developed. The zeroth and the first high orders of the incident OBs are presented theoretically and numerically in this paper. Yet, based on a computer generated holograms method, the generation’s masks of the Finite OBs in first orders are given in this work. Also, the incident transverse intensity distribution in 1-D and 2-D of the first orders of OBs is performed.

Wave spectral modeling of multidirectional random waves in a harbor through combination of boundary integral of Helmholtz equation with Chebyshev point discretization

A mathematical model is presented to predict the wave spectrum in a complex geometry harbor domain due to the diffraction of multidirectional random waves propagating through entrance of the harbor from the open sea. An accurate description of the harbor geometry, bathymetry and an incident wave spectrum are required. The solution of Helmholtz equation in the bounded and unbounded region is obtained by 2-D boundary integral method with the consideration of partial reflection and corner point approximation. Moreover, the Chebyshev point discretization is applied on boundary of the harbor to improve the accuracy of the numerical scheme. The principle of superposition is adopted to simulate the directional random waves based on the solution of monochromatic incident waves. The form of Mitsuyasu’s frequency spectrum was utilized for the random wave simulation. The wave spectra for multidirectional random waves are analyzed at various recorder stations in the Pohang New Harbor (PNH). Comparison of simulation results with in situ measurement data recorded at various recorder observatories in the PNH demonstrates the feasibility and accuracy of our proposed method. After this study, we analyzed the wave height distribution in the whole PNH domain. Based on the simulation results, some tactic has been introduced to reduce the wave amplitude in the PNH. Our primary results confirm that present numerical model is accurate and flexible to implement on various real harbors to predict the multidirectional random wave diffraction.

Dispersion analysis of the meshless local boundary integral equation and radial basis integral equation methods for the Helmholtz equation

Numerical solutions of the Helmholtz equation suffer from pollution effect especially for higher wavenumbers. The major cause for this is the dispersion error which is defined as the relative phase difference between the numerical solution of the wave and the exact wave. The dispersion error for the meshless methods can be a priori determined at an interior source node assuming that the potential field obeys a harmonic evolution with the numerical wavenumber.In this paper, the dispersion errors, in the solution of 2D Helmholtz equation, for two different meshless methods are investigated, the local boundary integral equation method and the radial basis integral equation method. Radial basis functions, with second order polynomials and frequency-dependent polynomial basis vectors are used for the interpolation of the potential field in both methods. The results have been found to be of comparable accuracy with other meshless approaches reported in the literature.

Solutions of helmholtz equation in complex domain

A harmonic equation and the Helmholtz equation are elliptic type equations and describe important physical processes (the first – stationary, the second - stationary and dynamic). Effective solutions of boundary value problems for harmonic equation (in different regions in the plane) are constructed by the methods of the theory of analytic functions of a complex variable. These methods can not be applied directly to solving problems for the Helmholtz equation. In the scientific literature, solutions of boundary value problems for this equation are known only in certain areas that are represented by cumbersome formulas. In the paper, using the solution of the Helmholtz equation in a circle through the functions (not analytical) of complex variable and conformal mapping of a given area on the circle, a general approach to building a solution of the corresponding boundary value problem is formulated. An important prerequisite for presenting this solution as functional series is finding the solution of harmonic equation in a given region that satisfies the given boundary conditions and an analytic function in this region respectively. The solutions of the Helmholtz equation in the plane with an elliptic hole and half-plane are constructed. For effective formulation of boundary value prob­lems and finding analytic functions in these areas, systems of basic functions in the corresponding spaces of analytic functions are found.

Exponential splitting for [formula omitted]-dimensional paraxial Helmholtz equation with high wavenumbers

This paper explores applications of the exponential splitting method for approximating highly oscillatory solutions of the n-dimensional paraxial Helmholtz equation. An eikonal transformation is introduced for oscillation-free platforms and matrix operator decompositions. It is found that the sequential, parallel and combined exponential splitting formulas possess not only anticipated algorithmic simplicity and efficiency, but also the accuracy and asymptotic stability required for highly oscillatory wave computations.

Fokas integral equations for three dimensional layered-media scattering

The scattering of acoustic waves by periodic structures is of central importance in a wide range of problems of scientific and technological interest. This paper describes a rapid, high-order numerical algorithm for simulating solutions of Helmholtz equations coupled across irregular (non-trivial) interfaces meant to model acoustic waves incident upon a multiply layered medium. Building upon an interfacial formulation from previous work, we describe an Integral Equation strategy inspired by recent developments of Fokas and collaborators for its numerical approximation. The method requires only the discretization of the layer interfaces (so that the number of unknowns is an order of magnitude smaller than volumetric approaches), while it requires neither specialized quadrature rules nor periodized fundamental solutions characteristic of many popular Boundary Integral/Element Methods. As with previous contributions by the authors on this formulation, this approach is efficient and spectrally accurate for smooth interfaces.

A Meshless Method for Helmholtz Equation Based on RBF and TSVD

In this paper, an improved meshless method for numerical solution of Helmholtz equation is presented. To overcome the unacceptable huge matrix condition number, we combine some Tikhonov regularization tools with radial basis function to solve the equations. Compared to the finite element method, a better result in numerical experiments will be approached using the proposed method. Furthermore, higher accuracy can be obtained by adopting suitable radial basis functions which is impossible for traditional RBF meshless method.

A Compressible Nonhydrostatic Cell-Integrated Semi-Lagrangian Semi-Implicit Solver (CSLAM-NH) with Consistent and Conservative Transport

Abstract A cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic solver for the fully compressible moist Euler equations in two-dimensional Cartesian (x–z) geometry is presented. The semi-implicit CISL solver uses the inherently conservative semi-Lagrangian multitracer transport scheme (CSLAM) and a new flux-form semi-implicit formulation of the continuity equation that ensures numerically consistent transport. The flux-form semi-implicit formulation is based on a recent successful approach in a shallow-water equations (SWE) solver (CSLAM-SW). With the new approach, the CISL semi-implicit nonhydrostatic solver (CSLAM-NH) is able to ensure conservative and consistent transport by avoiding the need for a time-independent mean reference state. Like its SWE counterpart, the nonhydrostatic solver presented here is designed to be similar to typical semi-Lagrangian semi-implicit schemes, such that only a single linear Helmholtz equation solution and a single call to CSLAM are required per time step. To demonstrate its stability and accuracy, the solver is applied to a set of three idealized test cases: a density current (dry), a gravity wave (dry), and a squall line (moist). A fourth test case shows that shape preservation of passive tracers is ensured by coupling the semi-implicit CISL formulation with existing shape-preserving filters. Results show that CSLAM-NH solutions compare well with other existing solvers for the three test cases, and that it is shape preserving.

A coordinate transformation approach for efficient repeated solution of Helmholtz equation pertaining to obstacle scattering by shape deformations

A computational model is developed for efficient solutions of electromagnetic scattering from obstacles having random surface deformations or irregularities (such as roughness or randomly-positioned bump on the surface), by combining the Monte Carlo method with the principles of transformation electromagnetics in the context of finite element method. In conventional implementation of the Monte Carlo technique in such problems, a set of random rough surfaces is defined from a given probability distribution; a mesh is generated anew for each surface realization; and the problem is solved for each surface. Hence, this repeated mesh generation process places a heavy burden on CPU time. In the proposed approach, a single mesh is created assuming smooth surface, and a transformation medium is designed on the smooth surface of the object. Constitutive parameters of the medium are obtained by the coordinate transformation technique combined with the form-invariance property of Maxwell’s equations. At each surface realization, only the material parameters are modified according to the geometry of the deformed surface, thereby avoiding repeated mesh generation process. In this way, a simple, single and uniform mesh is employed; and CPU time is reduced to a great extent. The technique is demonstrated via various finite element simulations for the solution of two-dimensional, Helmholtz-type and transverse magnetic scattering problems.

Multigrid Method for Solution of 3D Helmholtz Equation Based on HOC Schemes

A higher order compact difference (HOC) scheme with uniform mesh sizes in different coordinate directions is employed to discretize a two- and three-dimensional Helmholtz equation. In case of two dimension, the stencil is of 9 points while in three-dimensional case, the scheme has 27 points and has fourth- to fifth-order accuracy. Multigrid method using Gauss-Seidel relaxation is designed to solve the resulting sparse linear systems. Numerical experiments were conducted to test the accuracy of the sixth-order compact difference scheme with Multigrid method and to compare it with the standard second-order finite-difference scheme and fourth-order compact difference scheme. Performance of the scheme is tested through numerical examples. Accuracy and efficiency of the new scheme are established by using the errors normsl2.

Gaussian Beam Methods for the Helmholtz Equation

In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of nontrapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number k, both for single beams and superposition of beams. The main result is that the relative local L 2 error in the beam approximations decay as k −N/2 independent of dimension and presence of caustics for N th order beams.