Discrete Plane Wave Research Articles

A Discrete Version of Plane Wave Solutions of the Dirac Equation in the Joyce Form

We construct a discrete version of the plane wave solution to a discrete Dirac-Kähler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under which a discrete plane wave solution satisfies a discrete Joyce equation are discussed.

Total field scattering field formulation in a non-uniform grid by means of FDTD-DPW

ABSTRACTFinite-difference time-domain discrete plane wave technique (FDTD-DPW) is a numerical method which by using total the field/scattered field scheme allows one to propagate a plane wave quasi-perfectly isolated. This means a propagation in the total field domain without reflexions to the scattered field domain on the order of machine precision (−300 dB). This technique is valid for any angle of propagation, and for any gridcell aspect ratio. In this work, we extend this numerical illumination procedure for uniform three-dimensional grids to domains that are meshed by a non-uniform grid. To do so, we introduce some modifications to the original FDTD-DPW technique.

Analytical modelling for predicting the sound field of planar acoustic metasurface

An analytical model is built to predict the acoustic fields of acoustic metasurfaces. The acoustic fields are investigated for a Gaussian sound beam incident on the acoustic metasurfaces. The Gaussian sound beam is decomposed into a set of discrete elementary plane waves. The diffraction caused by the acoustic metasurfaces can be obtained using this analytical model, which is validated with the numerical simulations for the different incident angles of the Gaussian sound beam. This model overcomes the limitation of the method based on the generalised Snell's law which can only predict the direction of a specific diffracted order. Actually, this analytical model can be also used to predict the sound fields of acoustic metasurfaces under any incident sound if its Fourier transforms exist. This conclusion is demonstrated by studying the sound field for a point sound source incident on the acoustic metasurface. The acoustic admittances of acoustic metasurfaces are required in the calculation of the analytical model. Therefore, a numerical method for obtaining the effective acoustic admittances is proposed for the structurally complex metasurfaces without the analytical expressions of material properties, such as equivalent density and sound speed.

Analytical Expression for the Time-Domain Green’s Function of a Discrete Plane Wave Propagating in a 3-D FDTD Grid

In this paper, a closed-form expression for the time-domain dyadic Green’s function of a discrete plane wave (DPW) propagating in a 3-D finite-difference time-domain (FDTD) grid is derived. In order to verify our findings, the time-domain implementation of the DPW-injection technique is developed with the use of the derived expression for 3-D total-field/scattered-field (TFSF) FDTD simulations. This implementation requires computations of the time-domain Green’s function of DPW with the use of multiple-precision arithmetic. Then, excitations at the TFSF interface can be computed as a time-domain convolution of a source function with Green’s function of DPW. The developed time-domain implementation of the DPW-injection technique demonstrates the leakage error across the TFSF interface around the numerical noise level that verifies the correctness of the derivation.

COMPREHENSIVE RCS SIMULATION OF DISPERSIVE MEDIA USING SO-FDTD-DPW METHOD

Perfectly Matched Layer (PML) is modeled by Split-Field FDTD (SF-FDTD) in order to simulate Radar Cross Section (RCS) of a plasma slab. PML is used as an absorbing boundary, and discrete plane wave (DPW) is employed to generate plane wave. DPW method has a power isolation of −300 dB between scattered-field and total-field regions. The dispersive media is modelled by shift- operator FDTD. In this article, the SO-FDTD and DPW are combined, and it is proved that this combination shows a good stability. Finally, two different plasma profiles (exponential and polynomial) are used to prove reflection coefficient of a conductive layer can be reduced by choosing true profile of covering layer. By using Near-to-Far-Field Transformation, all fields are transferred to far-field region to calculate RCS.

Dispersion analysis of plane wave discontinuous Galerkin methods

SUMMARYThe plane wave DG (PWDG) method for the Helmholtz equation was introduced and analyzed in [GITTELSON, C., HIPTMAIR, R., AND PERUGIA, I. Plane wave discontinuous Galerkin methods: analysis of the h‐version. Math. Model. Numer. Anal. 43 (2009), 297–331] as a generalization of the so‐called ultra‐weak variational formulation, see [O. CESSENAT AND B. DESPRéS, Application of an ultra weak variational formulation of elliptic PDEs to the two‐dimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299]. The method relies on Trefftz‐type local trial spaces spanned by plane waves of different directions, and links cells of the mesh through numerical fluxes in the spirit of DG methods.We conduct a partly empirical dispersion analysis of the method in a discrete translation‐invariant setting by studying the mismatch of wave numbers of discrete and continuous plane waves traveling in the same direction. We find agreement of the wave numbers for directions represented in the local trial spaces. For other directions, the PWDG methods turn out to incur both phase and amplitude errors. This manifests itself as a pollution effect haunting the h‐version of the method. Our dispersion analysis allows a quantitative prediction of the strength of this effect and its dependence on the wave number and number of plane waves. Copyright © 2014 John Wiley & Sons, Ltd.

Closed-Form Evaluation of the MIMO Channel Spatial Covariance

The full spatial covariance matrix of the multiple-input multiple-output channel is an important quantity that is used in channel modeling, communication system signal processing, and performance analysis. This paper formulates a closed-form expression for this covariance matrix based on the average multipath propagation characteristics and the radiation patterns of the antennas in the transmit and receive arrays. Simplifications to the formulation are established for cases where the propagation can be represented using discrete plane waves or where the power angular spectrum can be modeled using clusters that satisfy uniform, truncated Gaussian, or truncated Laplacian distributions. Numerical studies demonstrate application of the formulation as well as its accuracy under different scenarios.

Numerical Simulation of Reverberation Chamber Parameters Affecting the Received Power Statistics

This paper presents a reverberation chamber (RC) numerical modeling based on a discrete plane wave representation. In particular, the attention is focused on those parameters that may influence the field statistics. It is shown that a random choice of the angle of incidence, the initial phase, and the polarization in a particular range and a proper choice of the number of the plane waves allow us to reproduce the same behavior of experimental data. Both experimental and numerical field data are checked applying the Anderson–Darling (AD) test. More precisely, we count the frequencies where the received power statistics is rejected by the adopted goodness-of-fit test. This paper shows a correspondence between numerical and experimental parameters affecting the occurrence of the rejected frequencies, also giving a probability density function for the observed occurrence. The analysis is completed showing that the AD test rejection does not compromise the chamber use for typical radiated emission or radiated susceptibility tests.

A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail. Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Green's function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.

Sparse Pseudo Inverse of the Discrete Plane Wave Transform

An algorithm is presented for the sparse pseudo inversion of discrete forms of the plane wave transform. The starting point for the sparse pseudo inverse is a new sparse representation of the discrete plane wave transform (DPWT). While other sparse representations of the DPWT exist, the sparse representation developed here is shown to be amenable to sparse factorization. This feature is used to develop an efficient, error-controlled pseudo-inverse of the DPWT. Representative numerical examples are provided to illustrate the general properties of the proposed algorithms. Areas for additional work are identified, and potential applications of the results are discussed.

Discrete scattering theory: Green’s function for a square lattice

It is well known that, under certain circumstances, discrete plane waves can propagate through lattices. Waves can also be generated by oscillating one point in the lattice: the corresponding solution of the governing partial difference equations is the discrete Green’s function, g mn . The far-field behaviour of g mn is obtained using three methods: textbook derivations are corrected and a formula for g nn as a Legendre function is derived. The low-frequency behaviour of g mn is also obtained using Mellin transforms. These results are useful in the development of a discrete scattering theory.

The partition of unity finite element method for short wave acoustic propagation on non-uniform potential flows

A novel numerical method is proposed for modelling time-harmonic acoustic propagation of short wavelength disturbances on non-uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non-uniform. Many wavelengths can be included within a single element leading to ultra-sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid-based schemes. Numerical results for lined uniform two-dimensional ducts and for non-uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.

Technique for analysing cascaded frequency selective surface screens with dissimilar lattice geometries

The paper presents an efficient technique for analysing cascaded frequency selective surfaces (FSS) with non-commensurate periodicities by using an iterative approach based on the discrete plane wave spectrum (PWS) expansion. In this method, the tangential electric fields at the interfaces of the screens are first expanded in a set of transverse electric and transverse magnetic waves, and the responses of the individual FSS screens are derived for each of these waves. The transmission and reflection characteristics of the composite structure are then computed in an iterative manner, circumventing the need for matrix inversions. Such inversions can be extremely expensive in terms of CPU time and memory for non-commensurate FSS, because the analysis of the structures can no longer be reduced to that of a single cell by invoking Floquet's theorem. Representative numerical examples are presented to illustrate the computational advantage gained by using the proposed approach as apposed to a conventional scattering matrix cascading technique, which is strictly valid only for composites with commensurate periodicities.

Three dimensional modelling and numerical analysis of super-radiant harmonic emission in an FEL (optical klystron)

A full 3-D analysis of super-radiant (bunched electron) free electron harmonic radiation is presented. A generalized form of the FEL pendulum equation was derived and numerically solved. Both spectral and phasor formulation were developed to treat the radiation in the time domain. In space the radiation field is expanded in terms of either a set of free space discrete modes or plane waves. The numerical solutions reveal some new distinctly 3-D effects to which we provide a physical explanation.

Third-Order Interactions in Kelvin Ship-Wave Systems

This paper considers the possible existence of third-order resonant interactions in Kelvin ship-wave systems. Such interactions will generally occur when two discrete plane wave systems are present, but Phillips's criterion for third-order interactions is applied to the Kelvin pattern to show that there is no resonant interaction, except possibly along the cusp line. A more accurate analysis is then performed in the vicinity of the cusp line, using triple-integral solutions of the third-order free-surface condition, and it is shown that the resulting solutions are divergent on the cusp line, for large distances behind the disturbance. It is concluded that significant nonlinear interactions can occur on the cusp line, involving energy transfer among adjacent wave numbers.

Holographic Interferometry of Transparent Objects with Illumination Derived from Phase Gratings

Laser speckle in holographic interferograms of three-dimensional transparent objects is eliminated at the expense of limiting the available viewing directions to a finite number. This is accomplished by using a phase grating to divide the object wave into a number of discrete plane waves propagating in various directions. Experimental results of the application of this technique to the study of natural convection above a heated wire are presented. Fringe resolution obtained by this technique is compared with that obtained using plane wave or diffuse illumination. A brief analysis of fringe localization in holographic interferometry is presented in the discussion.

On the Rayleigh assumption in scattering by a periodic surface

AbstractIn treating plane wave scattering by a periodic surface, Lord Rayleigh (10) assumed that the discrete, outgoing and evanescent plane wave representation for the scattered field was valid on the surface itself. Recently, this Rayleigh assumption has been questioned and criticized. For the surface y = b cos kx on which the total field vanishes, Petit and Cadilhac(8) have demonstrated its invalidity when Kb > 0·448. The present paper discusses scattering of a wave, incident from y > 0, by an analytic periodic surface y = f(x) ( – ∞ < x < ∞), and shows that the Rayleigh assumption is valid if and only if the solution can be continued analytically across the boundary at least to the line y = minf(x). Conformal mapping and results relating to the analytic continuation of solutions to elliptic partial differential equations reduce the problem to one involving the location of singularities and critical points of a potential Green's function. Provided that the perturbation of the surface from a plane is sufficiently gentle, the validity of the Rayleigh assumption is established. For the surface y = b cos kx it is shown that the assumption is valid if Kb < γ, where γ is a positive number no greater than 0·448, the precise value of which is unknown. Possible extensions of the analysis to different or more general situations are suggested.