Spectral Green Research Articles

Quadruple-Dyadic Spectral Green's Functions of Current Sources in Planar Layered Media: Application to Plasmonic Layer-Structures

Quadruple-Dyadic Spectral Green's Functions of Current Sources in Planar Layered Media: Application to Plasmonic Layer-Structures

Spectral Nodal Methodology for Multigroup Slab-Geometry Discrete Ordinates Neutron Transport Problems with Linearly Anisotropic Scattering

In this paper, we propose a numerical methodology for the development of a method of the spectral nodal class that generates numerical solutions free from spatial truncation errors. This method, denominated Spectral Deterministic Method (SDM), is tested as a study of the solutions (spectral analysis) of neutron transport equations in the discrete ordinates (SN) formulation, in slab geometry, multigroup approximation, with linearly anisotropic scattering, considering a heterogeneous domain with fixed-source. The unknowns in the methodology are the cell-edge, and cell average angular fluxes, the numerical values calculated for these quantities coincide with the analytic solution of the equations. These numerical results are shown and compared with the traditional fine-mesh Diamond Difference (DD) method and the coarse-mesh spectral Green's function (SGF) method to illustrate the method's accuracy and stability. The solution algorithms problem is implemented in a computer simulator made in C++ language, the same that was used to generate the results of the reference work.

A NOVEL COARSE-MESH METHOD APPLIED TO NEUTRON SHIELDING PROBLEMS USING THE MULTIGROUP TRANSPORT THEORY IN DISCRETE ORDINATES FORMULATIONS

In this paper, we propose a new deterministic numerical methodology to solve the one-dimensional linearized Boltzmann equation applied to neutron shielding problems (fixed-source), using the transport equation in the discrete ordinates formulation (SN) considering the multigroup theory. This is a hybrid methodology, entitled Modified Spectral Deterministic Method (SDM-M), that derives from the Spectral Deterministic Method (SDM) and Diamond Difference (DD) methods. This modification in the SDM method aims to calculate neutron scalar fluxes with lower computational cost. Two model-problems are solved using the SDM-M, and the results are compared to the coarse-mesh methods SDM, Spectral Green's Function (SGF) and Response Matrix (RM), and the fine-mesh method DD. The numerical results were obtained in the programming language JAVA version 1.8.0_91.

Spectral Green's-function method in driven open quantum dynamics

A novel method based on spectral Green functions is presented for the simulation of driven open quantum dynamics that can be described by the Lindblad master equation in Liouville density operator space. The method extends the Hilbert space formalism and provides simple algebraic connections between the driven and non-driven dynamics in the spectral frequency domain. The formalism shows remarkable analogies to the use of Green functions in quantum field theory such as the elementary excitation energies and the Dyson self-energy equation. To demonstrate its potential, we apply the novel method to a coherently driven dissipative ensemble of 2-level systems comprising a single "active" subsystem interacting with $N$ "passive" subsystems -- a generic model with important applications in quantum optics and dynamic nuclear polarization. The novel method dramatically reduces computational cost compared with simulations based on solving the full master equation, thus making it possible to study and optimize many-body correlated states in the physically realistic limit of an arbitrarily large $N$.

A High‐Order Integral Equation‐Based Solver for the Time‐Dependent Schrödinger Equation

AbstractWe introduce a numerical method for the solution of the time‐dependent Schrödinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light‐matter interaction.The primary computational challenge in using the Volterra formulation is the application of a spacetime history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex‐frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT.Our approach is related to pseudospectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive, high‐order implicit time marching schemes, and naturally includes time‐dependent potentials. We present examples in one and two dimensions showing spectral accuracy in space and eighth‐order accuracy in time for both periodic and free space problems. © 2020 Wiley Periodicals LLC.

Wideband Circularly Polarized Antenna With In-Lens Polarizer for High-Speed Communications

In this contribution, a broadband G-band leaky-wave (LW) fed lens antenna with an integrated dielectric grid polarizer is presented. The proposed wideband polarizer unit cell geometry enables its fabrication at frequencies higher than 100 GHz, presenting high transmission properties and low axial ratio (AR). A quasi-analytic technique based on multilayer spectral Green's function combined with a numerical Floquet modes' solver is used to optimize the lens aperture efficiency and AR. The proposed technique is validated via full-wave (FW) simulations. A design is proposed in low dielectric permittivity material, achieving FW simulated aperture efficiency higher than 75% over 44% relative bandwidth, and an AR lower than 3 dB over 35% relative bandwidth. The antenna is able to achieve multiple directive circularly polarized (CP) beams when fed by a focal plane array, preserving the AR bandwidth. A prototype has been fabricated and measured, exhibiting an excellent agreement with quasi-analytic and FW simulations.

A response matrix spectral nodal method for energy multigroup X,Y-geometry discrete ordinates problems in non-multiplying media

A new spectral nodal method is proposed for coarse-mesh numerical solution of energy multigroup X,Y-geometry discrete ordinates (SN) transport problems in non-multiplying media. This method uses the Response Matrix (RM) technique to solve analytically the multigroup one-dimensional transverse-integrated SN nodal equations with approximations only for the node-edge angular fluxes in the transverse leakage terms. We approximate these leakage terms by constants, hence we refer to the present spectral nodal method as the RM-Constant Nodal (RM-CN) method. We show in numerical calculations that the offered RM-CN method generates as accurate solutions as the companion spectral Green's function-constant nodal (SGF-CN) method, whose mathematical preliminaries are essentially the same, except that the multigroup RM-CN algorithm is both algebraically and computationally simpler than the SGF-CN method.

The Adjoint Discrete Ordinates Solution Applied to Direct and Inverse Neutral Particle X, Y-Geometry Source-Detector Problems

Solving the adjoint transport equation allows obtaining maps of the neutral-particle importance to an objective function, e.g., a detector response. In source-detector problems, the adjoint interior source is defined as the macroscopic absorption cross section of the material the detector is made of. Determining the intensity of a source of particles in each energy group (causes) is possible by solving an inverse source-detector problem, given the source location and the detector response (effects). In this work, we present the application of an adjoint technique to solve energy multigroup X,Y-geometry direct and inverse transport problems. In order to obtain the importance maps, we have extended the adjoint spectral Green's function constant-nodal (SGF*-CN) method to numerically solve energy multigroup adjoint transport problems in the discrete ordinates formulation. Described here are the methodology to obtain the discretized SGF*-CNequations and the adjoint partial one-node block inversion scheme used to iteratively solve these equations. We present numerical results to two test problems to illustrate the accuracy of the present methodology.

A Fast Back-Projection Approach to Diffraction Tomography for Near-Field Microwave Imaging

A novel fast back-projection (FBP) approach is presented as a more robust alternative to conventional diffraction tomography (DT) for near-field three-dimensional (3-D) microwave imaging. Like DT, the FBP algorithm utilizes the spectral Green's function and stationary phase solution in the forward model, enabling the implementation of the fast Fourier transform to achieve real-time image generation with the same efficiency as DT. However, if the Green's function or the antenna gain pattern has any nulls within the imaging volume, the DT can have singularities that give rise to unbalanced images. The FBP not only naturally avoids the singularity problem and obtains balanced images through the point spread function, but is also much less sensitive to noise due to the matched-filter property of back-projection. 3-D simulations and measurement of a monostatic near-field free-space imaging scenario are presented to demonstrate the efficiency of the proposed FBP method in comparison with DT, and the improved uniformity and noise resistance of the image.

An Extended Linear Discontinuous Method for One-group Fixed Source Discrete Ordinates Problems with Isotropic Scattering in Slab Geometry

Nowadays, the obtainment of an accurate numerical solution of fixed source discrete ordinates problems is relevant in many areas of engineering and science. In this work, we extend the hybrid Finite Element Spectral Green's Function method (FEM-SGF), originally developed to solve eigenvalue diffusion problems, for fixed source problems using as a mathematical model, the discrete ordinates formulation in one energy group with isotropic scattering in slab geometry. This new method, Extended Linear Discontinuous Discrete Ordinates (ELD-SN), is based on the use of neutron balance equations and the construction of a hybrid auxiliary equation. This auxiliary equation combines a linear discontinuous approximation and spectral parameters to approximate the neutron angular flux inside the cell. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of our methodology. ELD-SN method is free from spatial truncation errors in S2 quadrature, and generate good results in the other quadrature sets. This method is more accurate than the conventional Diamond Difference (DD) and Linear Discontinuous (LD) methods, but surpassed by the Spectral Green's Function (SGF) method, for quadrature order greater than two.

A composite spatial grid spectral Green's function method for one speed discrete ordinates eigenvalue problems in two-dimensional Cartesian geometry

Spectral Green's function nodal methods (SGF) are well established as a class of coarse mesh methods. For this reason, they are widely used in the solution of neutron transport problems in discrete ordinates formulation (SN). When compared with fine mesh methods, SGF are considered efficient, as solutions are as accurate as, using a smaller number of spatial nodes, reducing floating point operations. However, the development of spectral-nodal methods for XY Cartesian geometries, has been limited due to (a) difficulties in implementing efficient computational algorithms and, (b) high algebraic and computational costs. This is because these methods need to use NBI-type (One-Node Block Inversion) sweep schemes. The composite spatial grid methods were developed to overcome these challenges. In this work, we describe a composite spatial grid spectral-nodal method to solve one-speed discrete ordinate eigenvalue problems in XY Cartesian geometry with isotropic scattering. The discretization is developed into two stages and two 1D problems coupled by transverse leakage terms in each domain region are obtained. In order to converge toward the numerical solution, we used an alternating-direction iterative technique and a modified source iteration sweep scheme. Also, we used the conventional power method to estimate the problem's dominant eigenvalue. Numerical results for benchmark problems are presented to illustrate the accuracy and performance of the developed method. This approach offers more accurate and efficient results for integral quantities if compared with others SGF methods.

Application of an adjoint technique to one–speed [formula omitted]–geometry source–detector transport problems in the discrete ordinates formulation using a spectral nodal method

Described here is an application of an adjoint technique for solving source–detector discrete ordinates (SN) transport problems by using a spectral nodal method. For one–speed X,Y–geometry adjoint SN transport problems, considering linearly anisotropic scattering in non–multiplying media, the coarse–mesh adjoint spectral Green's function–constant nodal (SGF†–CN) method is presented. The discretized SGF†–CN equations are solved by using the adjoint partial one–node block inversion (NBI) iterative scheme. The partial adjoint NBI scheme performs SN transport sweeps from the four corners of the rectangular domain. One uses the most recent estimates for the adjoint node–edge average fluxes in the outgoing directions, to solve the resulting adjoint SN problem in that node for all the incoming adjoint node–edge average fluxes, which constitute the outgoing adjoint node–edge average fluxes for the adjacent nodes in the direction of the transport sweep. Numerical results are given to illustrate the method's accuracy in coarse–mesh adjoint SN calculations and some advantages of using the adjoint technique in source–detector problems.

Pattern synthesis for multi‐layered perforated dielectric superstrate loaded patch antenna with equal E / H beamwidth

The pattern synthesis based on the spectral Green's functions for multi-layered perforated dielectric loaded patch antenna is presented. It is conducted by optimising the dielectric superstrate parameters, i.e. relative permittivity (εr) and thickness. To solve the problem of the optimised εr unavailable from market, perforated dielectric superstrate (PDS) is utilised. The equivalence between PDS and homogeneous dielectric superstrate (HDS) in spectral domain is investigated and the effective relative permittivity (εreff) of PDS is studied. Moreover, by introducing εreff, the general optimisation procedure for HDS loaded antenna is also applicable for PDS loaded antenna, which reduces the computational cost significantly. To validate their method, a two-layer PDS loaded patch antenna with E/H beamwidth equalisation was designed, fabricated and measured. Simulations and measurements show that the synthesised antenna has satisfactory performances over the ITU-RS.1068-1994 band from 13.75 to 14 GHz. The half 10 dB E/H beamwidths of 19°/20°, −27 dB cross polarisation and −15.5 dB side lobe level have been achieved.

The spectral nodal method applied to multigroup [formula omitted] neutron transport problems in One-Dimensional geometry with Fixed–Source

The nodal methods, as deterministic models, form a class of numerical methods developed to generate accurate numerical solutions of the Boltzmann equation for neutron transport. These methods are algebraically and computationally more laborious than the traditional deterministic fine-mesh methods like the Diamond Difference method (DD). However, their numerical solution for traditional coarse-mesh problems is more accurate. For this reason, the nodal methods and their algorithms for direct and iterative solution schemes have been the subject of extensive research. In this paper we propose a simpler methodology for the development of a method of spectral nodal class which is tested as an initial study of the solutions (spectral analysis) of the neutron transport equation in the formulation of discrete ordinates (SN), in one-dimensional geometry, multigroup energy approximation, isotropic scattering and considering homogeneous and heterogeneous domains. These results are compared with the traditional fine-mesh DD method and the spectral nodal methods, spectral Green's function (SGF) and Response Matrix (RM) to test their numerical accuracy, stability and consistency.

An analytical nodal method for energy multi-group discrete ordinates transport calculations in two-dimensional rectangular geometry

A spectral nodal method for energy multi-group X, Y-geometry, discrete ordinates (SN) problems in non-multiplying medium is developed. This analytical coarse-mesh method is referred to as the multi-group spectral Green's function - constant nodal (SGF-CN) method. The SGF-CN method uses the multi-group SGF method for numerically solving the one-dimensional transverse-integrated SN nodal equations with constant approximations for the transverse leakage terms. As the energy-group transfer scattering source terms are treated analytically in the offered method, the only approximations occur in the group transverse leakage terms. Numerical results are given to illustrate the method's accuracy for coarse-mesh calculations.

Useful Physical Images and Algorithms for Vector Dyadic Green's Functions [Wireless Corner

This article gathers useful, simple algorithms and their physical interpretations for field solutions from incremental sources in three-dimensional (3-D) spatial, two-dimensional (2-D) spectral, and one-dimensional (1-D) spectral domains. The interpretations of the 1-D spectral Green's functions are visualized in space as fields from current sheets, tubes, and shells for the planar, circular cylindrical, and spherical cases, respectively. A joint algorithm is presented for solving the multilayer case for all three cases. Similarly, field problems involving cylindrical objects or bodies of revolution (BOR) are structured into spectrums of 2-D spatial solutions from line sources and ring sources, respectively. The formulations and physical images are pedagogical and open up for new creative ways of teaching electromagnetic (EM) field theory as well as structuring numerical algorithms for field solutions that take known symmetries into account. It is also shown that the 3-D spatial Green's functions can be approximated to improve physical interpretation by omitting higher-order 1/r terms when r > 2λ.

Efficient and Accurate Computation of Electric Field Dyadic Green’s Function in Layered Media

Concise and explicit formulas for dyadic Green's functions, representing the electric and magnetic fields due to a dipole source placed in layered media, are derived in this paper. First, the electric and magnetic fields in the spectral domain for the half space are expressed using Fresnel reflection and transmission coefficients. Each component of electric field in the spectral domain constitutes the spectral Green's function in layered media. The Green's function in the spatial domain is then recovered involving Sommerfeld integrals for each component in the spectral domain. By using Bessel identities, the number of Sommerfeld integrals are reduced, resulting in much simpler and more efficient formulas for numerical implementation compared with previous results. This approach is extended to the three-layer Green's function. In addition, the singular part of the Green's function is naturally separated out so that integral equation methods developed for free space Green's functions can be used with minimal modification. Numerical results are included to show efficiency and accuracy of the derived formulas.

A hybrid spectral nodal method for multigroup X,Y-geometry discrete ordinates eigenvalue problems

A hybrid spectral nodal method is developed for the solution of multigroup discrete ordinates (SN) eigenvalue problems with isotropic scattering in X,Y geometry. In this method, the spectral diamond - spectral Green's function (SD-SGF) hybrid scheme, originally developed for numerically solving multigroup SN eigenvalue problem in slab geometry with no spatial truncation error, is generalized to solve the energy multigroup transverse-integrated SN nodal equations with constant approximation for the group transverse leakage terms. The resulting SD-SGF-constant nodal (SD-SGF-CN) method is more accurate than conventional nodal methods for criticality calculations because it treats both the scattering and fission source terms exactly; the only approximation involves the transverse leakage terms. This is in contrast to the classic multigroup SN nodal methods, wherein the group transverse leakage terms, the scattering source and the fission source are all approximated, hence decoupling angular directions and energy groups in the transverse integrated SN nodal equations. We solve the SD-SGF-CN equations using the partial one-node block inversion (NBI) iterative scheme (inner iterations) for each estimate of the dominant eigenvalue in the conventional power method (outer iterations). We show numerical results to illustrate the accuracy of the present multigroup SD-SGF-CN method for coarse-mesh calculations.

Finite Element Method with Spectral Green's Function in Slab Geometry for Neutron Diffusion in Multiplying Media and One Energy Group

The physical phenomenon of neutrons transport associated with eigenvalue problems appears in the criticality calculations of nuclear reactors and can be treated as a diffusion process. This paper presents a new method to solve eigenvalue problems of neutron diffusion in slab geometry and one energy group. This formulation combines the Finite Element Method, considered an intermediate mesh method, with the Spectral Green's Function Method, which is free of truncation errors, and it is considered a coarse mesh method. The novelty of this formulation is to approach the spatial moments of the neutron flux distribution by the first-order polynomials obtained from the spectral analysis of diffusion equation. The approximations provided by the new formulation allow obtaining accurate results in coarse mesh calculations. To validate the method, we compare the results obtained with the methods described in the literature, specifically the Diamond Difference method. The accuracy and the computational performance of the proposed formulation were characterized by solving benchmarks problems with a high degree of heterogeneity.

Resonant-state-expansion Born approximation with a correct eigen-mode normalisation

The Born approximation (Born 1926 Z. Phys. 38 802) is a fundamental result in physics, it allows the calculation of weak scattering via the Fourier transform of the scattering potential. As was done by previous authors (Ge et al 2014 New J. Phys. 16 113048) the Born approximation is extended by including in the formula the resonant-states (RSs) of the scatterer. However in this study unlike previous studies the included eigen-modes are correctly normalised with dramatic positive consequences for the accuracy of the method. The normalisation of RSs used in the previous RS expansion Born approximation or resonant-state expansion (RSE) Born approximation made in Ge et al (2014 New J. Phys. 16 113048) has been shown to be numerically unstable in Muljarov et al (2014 arXiv:1409.6877) and by analytics here. The RSs of the system can be calculated using my recently discovered RSE perturbation theory for dispersive electrodynamic scatterers (Muljarov et al 2010 Europhys. Lett. 92 50010; Doost et al 2012 Phys. Rev. A 85 023835; Doost et al 2013 Phys. Rev. A 87 043827; Armitage et al 2014 Phys. Rev. A 89; Doost et al 2014 Phys. Rev. A 90 013834) and normalised correctly to appear in spectral Green's functions and hence the RSE Born approximation via the flux-volume normalisation which I recently rigorously derived in Armitage et al (2014 Phys. Rev. A 89), Doost et al (2014 Phys. Rev. A 90 013834), Doost (2016 Phys. Rev. A 93 023835). In the case of effectively one-dimensional systems I find a RSE Born approximation alternative to the scattering matrix method.