Domain Green Function Research Articles

Directional decomposition of the acoustic wave equation for fluids and metafluids in spherical geometries, with application to transformational acoustics

A new directional decomposition of the acoustic 3D wave equation is derived for spherically symmetric geometries, where the wave fields do not need to possess such a symmetry. This provides an alternative basis for various applications of techniques like invariant embedding and time domain Green functions in spherically symmetric geometries. Contrary to previous results on spherical wave splittings, the new decomposition is given in a very explicit form. The wave equation considered incorporates effects from radially varying compressibility and density, but also from anisotropic density, a property of certain so called metafluids. By applying the new spherical wave splitting, we show that all spherically symmetric acoustic metafluid cloaks are diffeomorphic images of a homogeneous and isotropic spherical ball of perfect fluid.

DISPERSION ANALYSIS OF DOUBLE-SIDED OPEN PERIODIC MEDIA USING INHOMOGENEOUS PLANE WAVE EXCITATION

Double-sided open periodic structures are analyzed using inhomogeneous plane wave scattering. The leaky and surface wave modes of several unit cells of different structures are computed using the poles of generalized reflection and transmission coefficients of inhomogeneous plane waves in the spectral domain. It is shown that the reflection and transmission coefficients of the zeroth order Floquet mode contain the poles of the Green's function of the complex stratified periodic structure. The properties of evanescent mode amplification as well as super resolution near field imaging in a wire medium are addressed. A balanced leaky wave antenna unit cell with double-sided radiation feature is introduced and it is shown that, in contrast to grounded structures, total absorption in lossless non- chiral double-sided open unit cells is not feasible as long as the behavior of the unit cell is well described by its fundamental mode. In homogeneous open dielectric structures, the spectral representation of the Green's function leads to a series of propagating discrete surface wave modes, together with a continuous integral representation indicating the branch cut in the Riemann sheet (1). By introducing possibly inhomogeneous periodicities into these structures, the analysis of the whole configuration can be reduced to the analysis of one period known as unit cell. This unit cell has open boundaries at the unbounded sides, and periodic boundary conditions are assumed at its side walls, see Fig. 1. In homogeneous layered dielectric media, the spectral domain Green's functions have closed form (2). However, inhomogeneous complex objects do not have any closed form Green's function. Therefore, numerical methods are employed to compute the eigenvalues of these structures. In the numerical computation of open boundary structures, often perfectly matched layers (PML) are chosen as absorbing boundary condition (3). As a result, the spectral domain Green's function can be expressed in terms of the summation of three different groups of modes namely Berenger modes, leaky modes and surface wave modes (4). Among these modes, Berenger modes are complex modes which are more concentrated in the PML layer, while leaky modes are complex modes more concentrated in the guiding structure (5). For the analysis of complex objects immersed in or placed on top of a layered medium, the computation of leaky complex modes is possible with several methods. The most common approach is based on eigenvalue decomposition of the system matrix resulting from a numerical model (6-8). Another robust method is trying to find the stationary points of stored energy derived from physical observations. Since eigenvalue decomposition is a numerically expensive procedure, a physics based method for the computation of complex and surface wave modes of planar periodic configurations was addressed in (9, 10). This method was based on the scattering of inhomogeneous plane waves from grounded unit cells and the eigenmodes were explored by exciting the poles of the reflection coefficient.

A quasianalytical time domain solution for scattering from a homogeneous sphere.

A transient spherical multipole expansion-like solution for acoustic scattering from a spherical object is derived within a mesh-free and singularity-free time domain integral equation (TDIE) framework for the sound-soft, sound-rigid and penetrable cases. The method is based on an expansion of the time domain Green's function that allows independent evaluation of spatial and temporal convolutions. The TDIE system is solved by descretizing the integral equations in space and time, forming a matrix system via the method of moments, and solving the system with the marching on in time algorithm. Spatial discretization using tesseral harmonics leads to closed form expressions for spatial integrals, and use of a strictly band limited temporal interpolant permits efficient, accurate computation of temporal convolutions via numerical quadrature. The accuracy of these integrations ensures late time stability and accuracy of the deconvolution data. Results presented demonstrate the accuracy and convergence of the approach for broadband simulations compared with Fourier transformed analytical data.

Efficient Analysis of Large Aperiodic Antenna Arrays Using the Domain Green's Function Method

An efficient method-of-moments (MoM) based domain decomposition technique, viz., the domain Green's function method (DGFM), is presented for analyzing large antenna arrays. The DGFM is a perturbation technique where mutual coupling between array elements is accounted for during the formulation of an active impedance matrix for each domain/array element. The active current distribution on the entire array geometry is obtained by solving the smaller matrix equations related to the elements, and not that of the problem as a whole. This leads to a significant saving in both runtime and memory usage. The method also takes into account the edge effects attributed to the finite size of the array, complex excitations with nonlinear phase shift and is not limited to periodic array configurations. The DGFM is an approximation and assumes a slowly varying current distribution between domains. A novel way to mitigate the aforementioned, by including secondary coupling effects, is also discussed. Furthermore, an efficient active impedance matrix fill strategy is presented where the active impedance matrix summation is truncated to include only a certain number of terms. Parallelization using both distributed and shared memory programming models have also been applied to the DGFM, to further optimize runtime and memory usage.

Motion Performance Analysis of Deep Water Pipe-Laying Vessel in Regular Wave

Based on 3-D potential flow theory, sources and sinks are brought in, distributed upon the wet surface of ship hull. Three-dimensional non-speed frequency domain Green function is also introduced to determine the flow field velocity potential. As a result, added mass, damping coefficient and ship motion response of this deep water pipe-laying vessel are calculated under the regular wave condition. After compared these calculation results to model tank test data, we can see that the theoretical calculation fit well with experimental data. These calculations could provide prediction and guidance for vessel design and marine engineering operation.

Comment on the Breakdown of Van Vleck's Propagator and Its Implications for the Energy Domain Green's Function

Van Vleck's propagator is perhaps one of the most fundamental objects of semiclassical theory. Its Laplace transform gives the energy Green's function G(x, x′, E), which is the starting point for the derivation of Gutzwiller trace formula. The validity of Van Vleck's formula for chaotic and mixed systems is therefore of crucial importance, especially since during the derivation of the trace formula it is assumed to be valid for infinite time. In the last two years there has been numerical evidence for the time-scale validity of the semiclassical propagator when the underlying dynamics is chaotic. The time-scales were longer that previously predicted by Tabor et al. An explanation for this long time validity based on the multidimensional stationary phase approximation was recently proposed by Sepúlveda-Tomsovic-Heller. In this contribution we would like to review these recent results and comment on their implications in the energy domain and particularly in the validity of the trace formula.

Evaluation of time domain electromagnetic fields radiated by constant velocity moving particles traveling along an arbitrarily shaped cross‐section waveguide using frequency domain Green's functions

A technique for the accurate computation of the time domain electromagnetic fields radiated by a charged distribution traveling along an arbitrarily shaped waveguide region is presented. Based on the transformation (by means of the standard Fourier analysis) of the time‐varying current density of the analyzed problem to the frequency domain, the resulting equivalent current is further convolved with the dyadic electric and magnetic Green's functions. Moreover, we show that only the evaluation of the transverse magnetic modes of the structure is required for the calculation of fields radiated by particles traveling in the axial direction. Finally, frequency domain electric and magnetic fields are transformed back to the time domain, just obtaining the total fields radiated by the charged distribution. Furthermore, we present a method for the computation of the wakefields of arbitrary cross‐section uniform waveguides from the resulting field expressions. Several examples of charged particles moving in the axial direction of such waveguides are included.

Complex Time Representation of the Sommerfeld-Type Integrals and Its Application to 1-D Periodic Structures

The concept of complex time is employed to derive the time domain representation of the Green's function of a periodic structure for the first time. The spatial domain Green's function of the periodic structure in the frequency domain is separated into singular and nonsingular terms. The nonsingular terms are approximated by using exponential functions, resulting in the well known complex time representation of the time domain Green's function. In order to find a closed form expression for the singular terms, a novel time domain representation of the Sommerfeld-type integrals is derived. The proposed procedure is applied to the computation of the time domain Green's function of a one dimensional periodic structure. Numerical experiments demonstrate the high accuracy of this method.

Analytical and numerical approximations for the lossy on-axis impulse response of a circular piston

In biological tissues, the frequency dependence of attenuation and speed of sound for ultrasound propagation is described by the power law wave equation, which is a partial differential equation with fractional time derivatives. As demonstrated previously, the time domain Green's function for the power law wave equation combined with the Rayleigh-Sommerfeld integral is an effective reference for calculations of the lossy on-axis impulse response of a circular piston. Using the result obtained from this reference, two different approximations to the lossy on-axis impulse response are evaluated. The first approximation is an analytical expression that is proportional to the difference between two cumulative distribution functions for maximally skewed stable probability densities. The second approximation numerically convolves the lossless impulse response with a maximally skewed stable probability density function. The results show that both approximations achieve relatively small errors. Furthermore, the analytical approximation provides an excellent estimate for the arrival time of the lossy impulse response, whereas the departure time of the lossy impulse response is more difficult to characterize due to the heavy tail of the maximally skewed stable probability density function. Both approximations are rapidly calculated with the STABLE toolbox. [supported in part by NIH Grant R01 EB012079.]

Spatial Domain Green's Functions of Layered Media Using a New Method for Sommerfeld Integrals

A simplified approach for accurate and efficient computation of infinite domain Sommerfeld integrals (SI) associated with spatial domain Green's functions of layered media is described in this article. Integrand in SI excluding Bessel function is expressed as sum of complex exponentials using the matrix pencil method (MPM) which requires fewer terms than when we include oscillating Bessel functions. By using a novel three term representation for small arguments and classical large argument formulas of Bessel functions, analytical expressions for computing integrals along infinite domain SI tails are derived. The newly derived analytical formulas use the same MPM expansions for any given set of radial distance parameter ρ, enabling us to efficiently solve closed form Green's functions in layered media.

An Efficient Analysis Method for Cylindrical Conformal Microstrip Antenna Fed by Microstripline

Firstly, a domain-division solution is provided in this paper, which can solve Hankel function&#x2019;s singular problem of Sommerfeld integral in calculating the spatial domain Green function for cylindrically stratified media and realize the computation for all elements of <svg style="vertical-align:-0.0pt;width:12.375px;" id="M1" height="11.6" version="1.1" viewBox="0 0 12.375 11.6" width="12.375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.537)"><path id="x1D44D" d="M698 636l-541 -596q60 -5 176 -5q91 0 138.5 5t69.5 24q44 36 85 124l29 -15q-38 -125 -64 -173h-559l-9 16l545 598h-182q-81 0 -109 -8t-48 -31q-26 -29 -55 -103l-29 3q23 86 42 200h22q11 -16 21 -20.5t34 -4.5h428z" /></g> </svg>-matrix in the method of moment. Then, the arbitrary shape cylindrical conformal microstrip antenna (CCMA) fed by microstripline is theoretically analysed using the RWG basis function. Numerical results are presented in the form of mutual coupling between two current modes and electromagnetic characteristics of a CCMA fed by microstripline. The results show that, comparing with the planar counterpart microstrip antenna, the reflection coefficient of the CCMA increases, the current on the microstripline varies sharply, and the backward radiation field appears. However, the frontward radiation field of the CCMA is similar to the planar case.

A free surface frequency domain green function with viscous dissipation and partial reflections from side walls

The free-surface Green function method is widely used in solving the radiation or diffraction problems caused by a ship or ocean structure oscillating on the waves. In the context of inviscid potential flow, hydrodynamic problems such as multi-body interaction and tank side wall effect cannot be properly dealt with based on the traditional free-surface frequency domain Green function method, in which the water viscosity is omitted and the energy dissipation effect is absent. In this paper, an open-sea Green function with viscous dissipation was presented within the theory of visco-potential flow. Then the tank Green function with a partial reflection from the side walls in wave tanks was formulated as a formal sum of open-sea Green functions representing the infinite images between two parallel side walls of the source in the tank. The new far-field characteristics of the tank Green function is vitally important for improving the validity of side-wall effects evaluation, which can be used in supervising the tank model tests.

Closed-Form Green's Function Representations for Mutual Coupling Calculations Between Apertures on a Perfect Electric Conductor Circular Cylinder Covered With Dielectric Layers

Closed-form Green's function (CFGF) representations are developed for tangential magnetic current sources to calculate the mutual coupling between apertures on perfectly conducting circular cylinders covered with dielectric layers. The new representations are obtained by first rewriting the corresponding spectral domain Green's function representations in a different form (so that accurate results for electrically large cylinders, and along the axial line of a cylinder can be obtained). Then, the summation over the cylindrical eigenmodes is calculated efficiently. Finally, the resulting expressions are transformed to the spatial domain using a modified two-level generalized pencil of function method. Numerical results are presented showing good agreement when compared to CST Microwave Studio results.

Verification and validation of ShipMo3D ship motion predictions in the time and frequency domains

This paper compares frequency domain and time domain predictions from the ShipMo3D ship motion library with observed motions from model tests and sea trials. ShipMo3D evaluates hull radiation and diffraction forces using the frequency domain Green function for zero forward speed, which is a suitable approach for ships travelling at moderate speed (e.g., Froude numbers up to 0.4). Numerical predictions give generally good agreement with experiments. Frequency domain and linear time domain predictions are almost identical. Evaluation of nonlinear buoyancy and incident wave forces using the instantaneous wetted hull surface gives no improvement in numerical predictions. Consistent prediction of roll motions remains a challenge for seakeeping codes due to the associated viscous effects.

Study of the motions of fishing vessels by a time domain panel method

In this paper, a wide variety of computed motion results is presented for three existing fishing vessels. In order to do that, time domain computations of 3D ship motions are performed with a time domain Green's function. The computational method adopted is based on a previously developed one, whose numerical scheme here is subjected to modifications that increase its robustness and overall efficiency, so that it can be applied to calculate the motions of fishing vessels. The results are then compared with simulations using WAMIT for the zero speed case, and a strip theory method is used to determine the effect of forward speed. Results are presented for head seas, quartering head waves and following waves with three distinct Froude numbers.

Verification and validation of ShipMo3D ship motion predictions in the time and frequency domains

This paper compares frequency domain and time domain predictions from the ShipMo3D ship motion library with observed motions from model tests and sea trials. ShipMo3D evaluates hull radiation and diffraction forces using the frequency domain Green function for zero forward speed, which is a suitable approach for ships travelling at moderate speed (e.g., Froude numbers up to 0.4). Numerical predictions give generally good agreement with experiments. Frequency domain and linear time domain predictions are almost identical. Evaluation of nonlinear buoyancy and incident wave forces using the instantaneous wetted hull surface gives no improvement in numerical predictions. Consistent prediction of roll motions remains a challenge for seakeeping codes due to the associated viscous effects.

INTEGRAL-EQUATION ANALYSIS OF FREQUENCY SELECTIVE SURFACES USING EWALD TRANSFORMATION AND LATTICE SYMMETRY

In this paper, we present the space-domain integral- equation method for the analysis of frequency selective surfaces (FSS), consisting of an array of periodic metallic patches or a metal screens perforated periodically with arbitrarily shaped apertures. The computation of the spatial domain Green's function is accelerated by the Ewald transformation. The geometric model is simplifled by the lattice symmetry, so that the unknowns are greatly reduced. Time of fllling MOM matrix and solving linear system is dramatically reduced. Our technique shows much higher e-ciency when compared with the available commercial software and the existing methods published.

Efficient Determination of the Poles and Residues of Spectral Domain Multilayered Green's Functions That are Relevant in Far-Field Calculations

In this work the total least squares algorithm (TLSA) is applied to the determination of the proper and improper poles of spectral domain multilayered Green's functions that are closer to the branch point, and to the determination of the residues at these poles. The introduction of an adequate transformation in the spectral domain permits that the TLSA provides accurate values of the poles and residues, regardless of the proximity of the poles to the branch point. It is shown that the poles and residues supplied by the TLSA can be used to write the far field of the spatial domain Green's functions in terms of closed-form expressions that are reliable in a wide variety of scenarios.

Fast Integral Equation Solution Techniques for Planar-3D Structures in Multilayered Media

In this contribution, a hybrid space/spectral domain integral equation approach is presented for the characterization of quasi-3D structures in multilayered media. The verti- cal conductors for the quasi-3D components like vias, through holes, airbridges, flnite dielectric regions etc. can cross an arbitrary number of dielectric layers with arbitrary vertical discretiza- tion. The electrodynamic behavior of the vertical currents is incorporated by extended spectral domain Green's functions derived by analytical integrations over the vertical direction and by a summation over the Cartesian wavenumbers which must be computed only once and stored in a database. For short lateral distances, the couplings are computed with a subtraction tech- nique based on asymptotic representations of the extended Green's functions, whereas for larger distances and for group coupling computations in context with fast matrix vector product evalu- ations, adaptive integration path deformations with extended Laguerre quadrature methods are applied. For the solution of the linear systems of equations, a GMRES solver was implemented together with a diakoptic preconditioner based on the group decomposition of larger structures, combined with a reversed Cuthill McKee reordering applied to the submatrix pattern of the system matrix. With all these techniques, the analysis of complex quasi-3D structures can be carried out with the same performance than pure planar microstrip and/or slotline structures.

Analysis of Microstrip Dipole Antennas on a Layered Metamaterial Substrate

A numerical method is developed to analyze microstrip dipole antennas on a planar layered metamaterial substrate based on periodic "method of moments" (MoM). In the developed numerical method the spectral domain Green function in conjunction with the "array scanning method" (ASM) is applied. The substrate is made of planar periodic metallic patches inserted in a two layer grounded dielectric slab. Effects of metamaterial loading on input impedance of dipole antennas is investigated using the developed numerical method. It is shown that with metamaterial loading antenna miniaturization is achieved.