Solution Of Helmholtz Equation Research Articles

QSMSOR iterative method for the solution of 2D homogeneous Helmholtz equations

In this paper, we consider the numerical solutions of homogeneous Helmholtz equations of the second order. The Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method is applied to solve linear systems generated form discretization of the second order homogeneous Helmholtz equations using quarter sweep finite difference (FD) scheme. The formulation and implementation of the method are also discussed. In addition, numerical results by solving several test problems are included and compared with the conventional iterative methods.

Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at leastO(hmax⁡5)by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

OPTIMIZATION OF THE METHOD OF AUXILIARY SOURCES FOR 3D SCATTERING PROBLEMS BY USING GENERALIZED IMPEDANCE BOUNDARY CONDITIONS AND LEVEL SET TECHNIQUE

The method of auxiliary sources (MAS) presents a promising alternative to methods based on discretization, currently used for solving scattering problems. The optimal choice of the auxiliary surface and the proper allocation of radiation centers play a crucial role in ensuring accuracy and stability of the MAS. This approach is considered an open issue and can be investigated numerically. In this paper, we propose a systematic and fully automated technique leading to determine the optimal parameters of the MAS for arbitrary shaped obstacles (partially or fully penetrable) for scattering problems. Solving scattering problems with an optimal compromise between accuracy and computational resources has been a requirement of many engineering fields such as inverse problems, microwave imaging, Radar cross section computing and EMC. Numerical techniques based on rigorous formulation like the method of moments (MOM), TLM or FDTD provide accurate results, however in many cases, the computational cost is assessed as prohibitive. The mesh-free methods like the method of auxiliary sources state an auspicious alternative to these techniques. The MAS is a numerical method which was originally developed by a Georgian research group for solving scattering and radiation electromagnetic problems (1-3). The fundamental idea of the MAS is the interchange of boundary conditions and differential equation, which excludes the singularities of the integral equation by shifting the auxiliary sources contour relative to integration one (4). The scattered field is expanded in terms of the fundamental solutions of Helmholtz-equation (5). The boundary value problem is solved by imposing the boundary condition at the scattering surface in the same manner as the standard surface integral methods. Previous researches (6-8) have shown that the appropriate choice of the auxiliary surface and the location of radiation centers is decisive to achieve efficiency of the MAS. The optimal choice of the MAS parameters (auxiliary surface, radiation centers) remains an open issue probed by several scientific manuscripts (9-11, 20, 25). The distribution of the auxiliary sources strongly affects the accuracy and the convergence of the numerical solution, it is shown that to ensure the MAS efficiency, the auxiliary surface should enclose the scattered field singularities as tightly as possible (12, 13, 26). The standard placement of the auxiliary sources is based on empirical conventions and on the caustic hypothesis. As a result, the optimal distribution of the auxiliary sources, for a predefined accuracy is achieved by try-and- error processes or by determining the corresponding caustic surfaces (8, 21-24). These approaches are analytically feasible only when treating problems with canonical geometries (sphere, ellipsoid or infinite plan . . . ). The localization of scattered field singularities for arbitrary-shaped objects is the prevailing breakdown point of the MAS. The salient feature of the proposed technique is the replacement of

Explicit solution of the helmholtz equation in the exterior of nonclosed lipschitz surfaces with an impedance boundary condition

Explicit solution of the helmholtz equation in the exterior of nonclosed lipschitz surfaces with an impedance boundary condition

Propagation of hypergeometric laser beams in a medium with a parabolic refractive index

An expression to describe the complex amplitude of a family of paraxial hypergeometric laser beams propagating in a parabolic-index fiber is proposed. A particular case of a Gaussian optical vortex propagating in a parabolic-index fiber is studied. Under definite parameters, the Gaussian optical vortices become the modes of the medium. This is a new family of paraxial modes derived for the parabolic-index medium. A wide class of solutions of nonparaxial Helmholtz equations that describe modes in a parabolic refractive index medium is derived in the cylindrical coordinate system. As the solutions derived are proportional to Kummer’s functions, only those of them which are coincident with the nonparaxial Laguerre–Gaussian modes possess a finite energy, meaning that they are physically implementable. A definite length of the graded-index fiber is treated as a parabolic lens, and expressions for the numerical aperture and the focal spot size are deduced. An explicit expression for the radii of the rings of a binary lens approximating a parabolic-index lens is derived. Finite-difference time-domain simulation has shown that using a binary parabolic-index microlens with a refractive index of 1.5, a linearly polarized Gaussian beam can be focused into an elliptic focal spot which is almost devoid of side-lobes and has a smaller full width at half maximum diameter of 0.45 of the incident wavelength.

Fast evaluation of system matrices w.r.t. multi-tree collections of tensor product refinable basis functions

An algorithm is presented that for a local bilinear form evaluates in linear complexity the application of the stiffness matrix w.r.t. a collection of tensor product multiscale basis functions, assuming that this collection has a multi-tree structure. It generalizes an algorithm for sparse-grid index sets [R. Balder, Ch. Zenger, The solution of multidimensional real Helmholtz equations on sparse grids, SIAM J. Sci. Comput. 17 (3) (1996) 631–646] and it finds its application in adaptive tensor product approximation methods.

Dispersion analysis of leaky guided waves in fluid-loaded waveguides of generic shape

A fully coupled 2.5D formulation is proposed to compute the dispersive parameters of waveguides with arbitrary cross-section immersed in infinite inviscid fluids.The discretization of the waveguide is performed by means of a Semi-Analytical Finite Element (SAFE) approach, whereas a 2.5D BEM formulation is used to model the impedance of the surrounding infinite fluid. The kernels of the boundary integrals contain the fundamental solutions of the space Fourier-transformed Helmholtz equation, which governs the wave propagation process in the fluid domain. Numerical difficulties related to the evaluation of singular integrals are avoided by using a regularization procedure. To improve the numerical stability of the discretized boundary integral equations for the external Helmholtz problem, the so called CHIEF method is used.The discrete wave equation results in a nonlinear eigenvalue problem in the complex axial wavenumbers that is solved at the frequencies of interest by means of a contour integral algorithm. In order to separate physical from non-physical solutions and to fulfill the requirement of holomorphicity of the dynamic stiffness matrix inside the complex wavenumber contour, the phase of the radial bulk wavenumber is uniquely defined by enforcing the Snell–Descartes law at the fluid–waveguide interface.Three numerical applications are presented. The computed dispersion curves for a circular bar immersed in oil are in agreement with those extracted using the Global Matrix Method. Novel results are presented for viscoelastic steel bars of square and L-shaped cross-section immersed in water.

A Cell-Integrated Semi-Lagrangian Semi-Implicit Shallow-Water Model (CSLAM-SW) with Conservative and Consistent Transport

Abstract A Cartesian semi-implicit solver using the Conservative Semi-Lagrangian Multitracer (CSLAM) transport scheme is constructed and tested for shallow-water (SW) flows. The SW equations solver (CSLAM-SW) uses a discrete semi-implicit continuity equation specifically designed to ensure a conservative and consistent transport of constituents by avoiding the use of a constant mean reference state. The algorithm is constructed to be similar to typical conservative semi-Lagrangian semi-implicit schemes, requiring at each time step a single linear Helmholtz equation solution and a single application of CSLAM. The accuracy and stability of the solver is tested using four test cases for a radially propagating gravity wave and two barotropically unstable jets. In a consistency test using the new solver, the specific concentration constancy is preserved up to machine roundoff, whereas a typical formulation can have errors many orders of magnitude larger. In addition to mass conservation and consistency, CSLAM-SW also ensures shape preservation by combining the new scheme with existing shape-preserving filters. With promising SW test results, CSLAM-SW shows potential for extension to a nonhydrostatic, fully compressible system solver for numerical weather prediction and climate models.

Fast acoustic scattering predictions with non-uniform potential flow effects

Acoustic scattering simulations with non-uniform potential flow effects are performed by two formulations described in the literature, valid under specific flow conditions. Both formulations require the solution of the Helmholtz equation for the acoustic scattering computation and the solution of the Laplace equation for the steady potential base flow. These equations are solved using the boundary element method (BEM), and the fast multipole method (FMM) is used to accelerate the matrix-vector products arising from the Helmholtz and Laplace BEM equations. An assessment of the solutions obtained by the two formulations is presented for a verification test case where analytical and full unsteady Euler solutions are available. Results of acoustic scattering for large-scale simulations of realistic airframe configurations are presented. Good agreement is found between Taylor’s low Mach number scattering formulation and the solution of the full unsteady Euler equations for high-frequency simulations of plane waves scattering from a rigid cylinder. The computational cost per iteration of the FMM–BEM for the acoustic scattering around a full airplane configuration discretized by 550,000 boundary elements in a non-uniform potential flow is 19 s.

SEMI-ANALYTICAL SOLUTIONS OF THE 3-D HOMOGENEOUS HELMHOLTZ EQUATION BY THE METHOD OF CONNECTED LOCAL FIELDS

We advance the theory of the two-dimensional method of connected local flelds (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cube's surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges and eight on the vertices (corners). The local fleld within the unit cell is expanded in a truncated spherical Fourier- Bessel series. From this representation we develop a closed-form, 3D local fleld expansion (LFE) coe-cients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Green's function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Green's function after propagating a distance of ten wavelengths are well under ten percent.

Harmonic wave excitation in a semi infinite medium

In the present paper, we obtained the most general solution of the one-dimensional partial differential equation for harmonic wave excitation in a semi infinite medium, by computing the symmetry groups using the general prolongation formula for their infinitesimal generators of a groups of transformations based on the technique given by Olver([4], [5]) in explicit form. In the recent year the authors Bao, Wei and Zhao [1], Kurus [3], Ramahi and Seydou [6], Ranosava [7] Sneddon and Read [8] worked for the solution of Helmholtz equation.

A Semi-Analytical Method for the Solution of Helmholtz Equation

This note is concerned with a semi-analytical method for the solution of 2-D Helmholtz equation in unit square. The method uses orthogonal functions to project the problem down to finite dimensional space. After the projection, the problem simplifies to that of obtaining solutions for second order constant coefficient differential equations which can be done analytically. Numerical results indicate that the method is particularly useful for very high wave numbers.

Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

AbstractThis paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number. Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense. Simulated examples are given to illustrate the conclusion.

Parallel solution of high frequency Helmholtz equations using high order finite difference schemes

High frequency Helmholtz equations pose a challenging computational task, especially at high frequencies. This work examines the parallel solution of such problems, using 2nd, 4th and 6th order finite difference schemes. The examples include problems with known analytic solutions, enabling error evaluation of the different schemes on various grids, including problems on L-shaped domains. The resulting linear systems are solved with the block-parallel CARP-CG algorithm which succeeded in lowering the relative residual, indicating that it is a robust and reliable parallel solver of the resulting linear systems. However, lowering the error of the solution to reasonable levels with practical mesh sizes is obtained only with the higher order schemes. These results corroborate the known limitations of the 2nd order scheme at modeling the Helmholtz equation at high frequencies, and they indicate that CARP-CG can also be used effectively with high order finite difference schemes. Furthermore, the parallel scalability of CARP-CG improves when the wave number is increased (on a fixed grid), or when, for a fixed wave number, the grid size is decreased.

Application of He's homotopy perturbation method for multi‐dimensional fractional Helmholtz equation

PurposeThis purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.Design/methodology/approachBy using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).FindingsThis result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.Originality/valueThe most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.

Conformal mapping for the Helmholtz equation: Acoustic wave scattering by a two dimensional inclusion with irregular shape in an ideal fluid

The complex variables method with mapping function was extended to solve the linear acoustic wave scattering by an inclusion with sharp/smooth corners in an infinite ideal fluid domain. The improved solutions of Helmholtz equation, shown as Bessel function with mapping function as the argument and fractional order Bessel function, were analytically obtained. Based on the mapping function, the initial geometry as well as the original physical vector can be transformed into the corresponding expressions inside the mapping plane. As all the physical vectors are calculated in the mapping plane (η,η), this method can lead to potential vast savings of computational resources and memory. In this work, the results are validated against several published works in the literature. The different geometries of the inclusion with sharp corners based on the proposed mapping functions for irregular polygons are studied and discussed. The findings show that the variation of angles and frequencies of the incident waves have significant influence on the bistatic scattering pattern and the far-field form factor for the pressure in the fluid.

The radial basis integral equation method for solving the Helmholtz equation

A meshless method for the solution of Helmholtz equation has been developed by using the radial basis integral equation method (RBIEM). The derivation of the integral equation used in the RBIEM is based on the fundamental solution of the Helmholtz equation, therefore domain integrals are not encountered in the method. The method exploits the advantage of placing the source points always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green’s identities and the remaining equations are the derivatives of the first equation with respect to space coordinates. Radial basis function (RBF) interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). The accuracy and robustness of the method has been tested on some analytical solutions of the problem. Two different RBFs have been used, namely augmented thin plate spline (ATPS) in 2D and f(R)=R4ln(R) augmented by a second order polynomial. The latter has been found to produce more accurate results.

Approximation of entire function solutions of the Helmholtz equation having slow growth

In this paper, we study the Chebyshev polynomial approximation of entire solutions of Helmholtz equations in in Banach spaces ( space, Hardy space and Bergman space). Some bounds on generalized order of entire solutions of Helmholtz equations of slow growth have been obtained in terms of the coefficients and approximation errors using function theoretic methods.

Application of NHPM for solving Helmholtz equation

In this paper, an analytic solution of Helmholtz equation is obtained by using a new modification of homotopy perturbation method NHPM. Theoretical considerations are discussed. Comparison of the results of applying NHPM with those of homotopy perturbation method HPM leads to significant consequences. The new HPM usually result in the exact solution, even for the problems which HPM leads to an approximate solution. Two examples are presented to demonstrate advantages of the NHPM in comparison with HPM.

Three-dimensional acoustic scattering by layered media: a novel surface formulation with operator expansions implementation

The scattering of acoustic waves by irregular structures plays an important role in a wide range of problems of scientific and technological interest. This contribution focuses on the rapid and highly accurate numerical approximation of solutions of Helmholtz equations coupled across irregular periodic interfaces meant to model acoustic waves incident upon a multi-layered medium. We describe not only a novel surface formulation for the problem in terms of boundary integral operators (Dirichlet–Neumann operators), but also a Boundary Perturbation methodology (the Method of Operator Expansions) for its numerical simulation. 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 avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of Boundary Integral/Element Methods. The approach is a generalization to multiple layers of Malcolm & Nicholls' Operator Expansions algorithm for dielectric structures with two layers. As with this precursor, this approach is efficient and spectrally accurate.