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Presented is a parallel algorithm based on the fast multipole method (FMM) for the Helmholtz equation. This variant of the FMM is useful for computing radar cross sections and antenna radiation patterns. The FMM decomposes the impedance matrix into sparse components, reducing the operation count of the matrix-vector multiplication in iterative solvers to O(N^3/2) (where N is the number of unknowns). The parallel algorithm divides the problem into groups and assigns the computation involved with each group to a processor node. Careful consideration is given to the communications costs. A time complexity analysis of the algorithm is presented and compared with empirical results from a Paragon XP/S running the lightweight Sandia/University of New Mexico operating system (SUNMOS). For a 90,000 unknown problem running on 60 nodes, the sparse representation fits in memory and the algorithm computes the matrix-vector product in 1.26 seconds. It sustains an aggregate rate of 1.4 Gflop/s. The corresponding dense matrix would occupy over 100 Gbytes and, assuming that I/O is free, would require on the order of 50 seconds to form the matrix-vector product.

The exact quantum-mechanical equations for beams of free particles including photons and for beams of Dirac and spin-1 particles in a uniform magnetic field have been derived. These equations present the exact generalizations of the well-known paraxial equations in optics (exact Helmholtz equation for light beams) and particle physics and of the previously obtained paraxial equation for a Dirac particle in a uniform magnetic field. Some basic properties of exact wave eigenfunctions of particle beams have been determined.

By mapping the scalar Helmholtz equation (SHE) to the Schrödinger form we investigate the behavior of $\mathcal{𝒫}\mathcal{𝒯}$ optical structure when the refractive index distribution n admits variation in the longitudinal direction only. Interpreting the Schrödinger equation in terms of a superpotential we determine the supersymmetric partners for n. We also obtain new analytical solutions for the refractive index profiles and provide graphical illustrations for them.

In this letter, we discuss an application of the variational iteration method on obtaining series solution and exact solution of Helmholtz equation with initial conditions. The suggested algorithms are quite efficient and are practically well-suited for use in this class of equation. The proposed iterative schemes find the solution without any need to discretize the variables. The results reveal the advantages of the method for Helmholtz equation.

Propagation of the electromagnetic wave through a system with complex permittivity is investigated. The system consists of two media, one is passive and surrounds the two barrier slabs made out of the second material with complex dielectric constant. Expression for transmission probability is obtained along with equation which allows calculation of the resonant frequency. Analytical expressions for reflection and absorption probabilities are acquired as well. It will be shown, if the frequency is given, that the transmission probability (along with reflection and absorption probabilities) approaches infinity for one particular value of the dielectric constant with negative imaginary part, which is used to describe active media. Furthermore, conditions for appearance of these transmission singularities are obtained.

Symmetric and anti-symmetric trapped modes in a cylindrical tube with a segment of higher density are studied. The problem is reduced to an eigenvalue problem of the spatial Helmholtz equation subject to vanishing Dirichlet boundary condition in the cylindrical coordinate system. Through the domain decomposition method and matching technique, multiple frequency parameters are determined by solving the characteristic equation, and the corresponding n-fold periodic trapped modes can be constructed. It is found that in addition to the fundamental mode, the second- and higher-order trapped modes exist, which depend on the density ratio and length of the inhomogeneity. The local inhomogeneity leads to a decrease of the cutoff frequencies of the homogeneous tube and the corresponding vibration mode is localized near the inhomogeneous segment.

This paper studies an inverse problem of reconstructing a sound-soft obstacle from a homogeneous medium. We deal with it in the framework of statistical inversion and adopt an iterative ensemble Kalman algorithm to reconstruct the boundary. Some numerical examples show that the algorithm is effective and it can recover the shape of the boundary using one or several of the incident waves.

In this paper we study the problem of electromagnetic wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa.³

The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavior of the solutions of the eigenvalue problem. Moreover, the derivation of the transform formulas needed to solve the wave propagation problem involves nontrivial calculations.

The paper provides a complete description on how to construct the solution to the wave propagation problem in a 3-D optical waveguide with cylindrical symmetry. A follow-up article will study the particular cases of a step-index fiber and of a coaxial waveguide. In those cases we will obtain concrete formulas for the field and numerical examples.

We consider the three-dimensional scalar problem of acoustic propagation in a muffler. We develop and analyze a Fredholm-type formulation for a stationary fluid in the time-harmonic setting. We prove a homogenization result for a muffler containing periodically perforated ducts. Essentially, the perforated boundaries become completely transparent when the period of perforations, which is assumed to be of the same order as the size of perforations, tends to zero. We also derive a homogenized impedance condition when the perforated duct is coated by an absorbing material. We present numerical examples in two dimensions, obtained from coupling finite elements in the muffler with modal decompositions in the inlet and outlet ducts, which demonstrate the limiting validity of the theoretical results.

For the scalar wave equation in periodic structures, we establish rigorously the manifestation of a spectral bandgap as a frequency interval of exponentially small transmission of energy through a slab of the crystal when the thickness of the slab tends to infinity. We extend the result to slabs that contain a planar defect for frequencies that do not coincide with guided mode frequencies for the defect. For one-dimensional crystals, we prove that the transmission approaches a nonzero value at guided mode frequencies and give an explicit formula for the transmission. The main analytical tool in the multidimensional case is the Calderón boundary integral projectors for the Helmholtz equation.

We consider the Helmholtz transmission problem with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. In particular, the (weighted) $H^1$ norm of the solution is bounded by the $L^2$ norm of the source term, independently of the wavenumber. These bounds then imply the existence of a resonance-free strip beneath the real axis. The main novelty is that the only comparable results currently in the literature are for smooth, convex obstacles with strictly positive curvature, while here we assume only Lipschitz regularity and star-shapedness with respect to a point. Furthermore, our bounds are obtained using identities first introduced by Morawetz (essentially integration by parts), whereas the existing bounds use the much-more sophisticated technology of microlocal analysis and propagation of singularities. We also adapt existing results to show that if the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible.

Based on the implicit calculus equation modeling approach, this paper proposes a speculative concept of the potential and wave operators on negative dimensionality. Unlike the standard partial differential equation (PDE) modeling, the implicit calculus modeling approach does not require the explicit expression of the PDE governing equation. Instead the fundamental solution of physical problem is used to implicitly define the differential operator and to implement simulation in conjunction with the appropriate boundary conditions. In this study, we conjecture an extension of the fundamental solution of the standard Laplace and Helmholtz equations to negative dimensionality. And then by using the singular boundary method, a recent boundary discretization technique, we investigate the potential and wave problems using the fundamental solution on negative dimensionality. Numerical experiments reveal that the physics behaviors on negative dimensionality may differ on positive dimensionality. This speculative study might open an unexplored territory in research.

The efficient numerical solution of the exterior Helmholtz equation modeling the acoustic scattering by an obstacle is considered. The exterior problem is approximated by truncating the domain with a rectangular boundary and using either a perfectly matched layer or a second-order absorbing boundary condition to reduce reflections from the artificial boundary. The scattering problem is solved iteratively by using a domain embedding method with an efficient separable preconditioner. With this approach, it is possible to realize the GMRES iterations in a low-dimensional subspace and use the partial solution method to solve the linear systems with the preconditioner. The accuracy and the efficiency of the proposed solution technique are studied with numerical experiments.

We analyze the dispersion properties of elements obtained by a discontinuous Galerkin method with Lagrange multipliers. The dispersion analysis of these elements presents a challenge in that the Lagrange multiplier degrees of freedom are directional, and hence an unbounded mesh is made up of more than one repeating pattern. Two approaches to overcome this difficulty are presented. The similarity in the two sets of results offers mutual validation of the two approaches.

The upper frequency limit of vibroacoustic calculations with finite element methods is usually concluded from resolution rules for the minimal wavelength encountered in the problem. Here, we derive general resolution rules that account for the pollution effect in finite element solutions of time-harmonic equations. These rules are given for the Helmholtz equation and the Bernoulli beam equation. The latter are based on an analysis of numerical dispersion for finite difference solutions. The theoretical results are given in the broader context of industrial vibroacoustic computations in the medium-frequency range. The governing equations of deterministic vibroacoustic computations and statistical energy analysis are reviewed with the goal to indicate, respectively, upper and lower frequency bounds for the applicability of either model. From the discussion of priorities in industrial application, open questions for theoretical investigations are deduced.

This article deals with the use of the Ultra Weak Variational Formulation to solve Helmholtz equation and time harmonic Maxwell equations.

The method, issued from domain decomposition techniques, lies in partitioning the domain into subdomains with the use of adapted interface conditions. Going further than in domain decomposition, we make so that the problem degenerates into an interface problem only.

The new formulation is equivalent to the weak formulation. The discretization process is a Galerkin one. A possible advantage of the UWVF applied to wave equations is that we use the physical approach that consists in approximating the solution with plane waves. The formulation allows to use a very large mesh as compared to the frequency, on the contrary to the Finite Element Method when applied to time harmonic equations. Furthermore, the convergence analysis shows the method is a high order one: the order evolves as the square root of the number of degrees of freedom.

Discrete methods of numerical analysis have been used successfully for decades for the solution of problems involving wave diffraction, etc. However, these methods, including the finite element and boundary element methods, can require a prohibitively large number of elements as the wavelength becomes progressively shorter. In this work, a new type of interpolation for the acoustic field is described in which the usual conventional shape functions are modified by the inclusion of a set of plane waves propagating in multiple directions. Including such a plane wave basis in a boundary element formulation has been found in the current work to be highly successful. Results are shown for a variety of classical scattering problems, and also for scattering from nonconvex obstacles. Notable results include a conclusion that, using this new formulation, only approximately 2.5 degrees of freedom per wavelength are required. Compared with the 8 to 10 degrees of freedom normally required for conventional boundary (and finite) elements, this shows the marked improvement in storage requirement. Moreover, the new formulation is shown to be extremely accurate. It is estimated that for 2D Helmholtz problems, and for a given computational resource, the frequency range allowed by this method is extended by a factor of three over conventional direct collocation Boundary Element Method. Recent successful developments of the current method for plane elastodynamics problems are also briefly outlined.

Continuous interpolation of the sound pressure is favored in most applications of boundary element methods for acoustics. Few papers are known to the authors in which discontinuous elements are applied. Mostly they were used because they guarantee C¹ continuity of the geometry at element edges. The effect of superconvergence is known for boundary element collocation on discontinuous elements. This effect is observed if the collocation points are located at the zeroes of orthogonal functions, e.g. at the zeroes of the Legendre polynomials.

In this paper, we start with a review of related work. Then, formulation of discontinuous elements and position of nodes on the element are presented and discussed. One parameter controls the location of nodes on the element. The major part of the paper consists of the investigation of the computational example of a long duct. For that, the numeric solution is compared with the analytic solution of the corresponding one-dimensional problem. Error dependence in terms of frequency, element size and location of nodes on discontinuous elements is reported. It will be shown that the zeroes of the Legendre polynomials account for an optimal position of nodes. Similar results are observed for triangular elements. It can be seen that the error in the Euclidean norm changes by one or two orders of magnitude if the location of nodes is shifted over the element. It can be seen that the optimal location varies with the wave-number although remaining in the vicinity of the zeroes of orthogonal functions. The irregular mesh of a sedan cabin compartment accounts for the second example. Optimal choice of node position is confirmed for this example. One of the key results of this paper is that discontinuous boundary elements perform more efficiently than continuous ones, in particular for linear elements. This, however, further implies that nodes are located at the zeroes of orthogonal functions on the element.

Numerical solutions of the Helmholtz equation and the one-way Helmholtz equation are compared in the weak range dependence limit, where the overall range distance is increased while the range dependence is weakened. It is observed that the difference between the solutions of these two equations persists in this limit. The one-way Helmholtz equation involves a square root operator and it can be further approximated by various one-way models used in underwater acoustics. An operator marching method based on the Dirichlet-to-Neumann map and a local orthogonal transform is used to solve the Helmholtz equation.

To derive the integral representation for the axial wave describing the interference of near-axial waves in an arbitrary deep-water waveguide in a range-independent medium in long-range acoustic propagation in the ocean, it is necessary to construct the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a narrow strip having a width of order ω^-1/2, where ω is a cyclic frequency. In a range-independent (separable) case the desired solutions coincide with the principal term of the uniform asymptotic expansion as ω→∞ of normal modes when a mode number is a value of order of unity. In this paper, the solutions of the homogeneous Helmholtz equation concentrated near the waveguide axis in a range-independent ocean and which decrease exponentially outside a strip containing the axis are constructed in the form admitting generalization to the case of a range-dependent medium. The solutions are represented as the product of exponentials and parabolic cylinder functions whose arguments are infinite series in powers of ω^-1/2. Coefficients of these series are found from a recurrent system of partial differential equations up to terms allowing to get a residual (difference between the left-hand side of the Helmholtz equation and zero) of order ω^-1/2. Numerical results are obtained for medium parameters corresponding to the Munk canonical sound-speed profile.