Theoretical and computational aspects of scattering from periodic surfaces: one-dimensional transmission interface

Abstract

We consider the scattering from and transmission through a one-dimensional periodic surface. For this problem, the electromagnetic cases of TE and TM polarization reduce to the scalar acoustic examples. Three different theoretical and computational methods are described, all involving the solution of integral equations and their resulting discrete matrix system of equations for the boundary unknowns. They are characterized by two sample spaces for their discrete solution, coordinate space and spectral space, and labelled by the sampling of the rows and columns of the discretized matrices. They are coordinate-coordinate (CC), the usual coordinate-space method, spectral-coordinate (SC) where the matrix rows are discretized or sampled in spectral space and spectral-spectral (SS) where both rows and columns are sampled in spectral space. The SS method uses a new topological basis expansion for the boundary unknowns.

Equations are derived for infinite surfaces, then specialized and solved for periodic surfaces. Computational results are presented for the transmission problem as a function of roughness, near-grazing incidence as well as many other angles, density and wavenumber ratios. Matrix condition numbers and different sampling methods are considered. An error criterion is used to gauge the validity of the results.

The computational results indicated that the SC method was by far the fastest (by several orders of magnitude), but that it became ill-conditioned for very rough surfaces. The CC method was most reliable, but often required very large matrices and was consequently extremely slow. It is shown that the SS method is computationally efficient and accurate at near-grazing incidence and can be used to fill a gap in the literature. Extensive computational results indicate that both SC and SS are highly robust computational methods. Spectral-based methods thus provide viable computational schemes to study periodic surface scattering.     

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE polarization). The theoretical development is for both infinite and periodic surfaces, the latter equations being derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either the first (CC1) or second kind (CC2). Mixed representation, or SC-type, equations are solved as well as SS equations fully in spectral space.

Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime.

The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest, but it did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence does not occur with SC, then SS should be used, and failing that CC.     

Theoretical and computational aspects of scattering from periodic surfaces: two-dimensional perfectly reflecting surfaces using the spectral-coordinate method

We consider the scattering from a two-dimensional periodic surface. From our previous work on scattering from one-dimensional surfaces (1998 Waves Random Media 8 385) we have learned that the spectral-coordinate (SC) method was the fastest method we have available. Most computational studies of scattering from two-dimensional surfaces require a large memory and a long calculation time unless some approximations are used in the theoretical development. By using the SC method here we are able to solve exact theoretical equations with a minimum of calculation time.

We first derive in detail (part I) the SC equations for scattering from two-dimensional infinite surfaces. Equations for the boundary unknowns (surface field and/or its normal derivative) result as well as an equation to evaluate the scattered field once we have solved for the boundary unknowns. Special cases for the perfectly reflecting Dirichlet and Neumann boundary value problems are presented as is the flux-conservation relation.

The equations are reduced to those for a two-dimensional periodic surface in part II and we discuss the numerical methods for their solution. The two-dimensional coordinate and spectral samples are arranged in one-dimensional strings in order to define the matrix system to be solved.

The SC equations for the two-dimensional periodic surfaces are solved in part III. Computations are performed for both Dirichlet and Neumann problems for various periodic sinusoidal surface examples. The surfaces vary in roughness as well as period and are investigated when the incident field is far from grazing incidence ('no grazing') and when it is near-grazing. Extensive computations are included in terms of the maximum roughness slope which can be computed using the method with a fixed maximum error as a function of the azimuthal angle of incidence, the polar angle of incidence and the wavelength-to-period ratio.

The results show that the SC method is highly robust. This is demonstrated with extensive computations. Furthermore, the SC method is found to be computationally efficient and accurate for near-grazing incidence. Computations are presented for grazing angles as low as 0.01°. In general, we conclude that the SC method is a very fast, reliable and robust computational method to describe scattering from two-dimensional periodic surfaces. Its major limiting factor is high slopes and we quantify this limitation.     

Theoretical and computational aspects of scattering from rough surfaces: one-dimensional perfectly reflecting surfaces

Abstract

We discuss the scattering of acoustic or electromagnetic waves from one-dimensional rough surfaces. We restrict the discussion in this report to perfectly reflecting Dirichlet surfaces (TE polarization). The theoretical development is for both infinite and periodic surfaces, the latter equations being derived from the former. We include both derivations for completeness of notation. Several theoretical developments are presented. They are characterized by integral equation solutions for the surface current or normal derivative of the total field. All the equations are discretized to a matrix system and further characterized by the sampling of the rows and columns of the matrix which is accomplished in either coordinate space (C) or spectral space (S). The standard equations are referred to here as CC equations of either the first (CC1) or second kind (CC2). Mixed representation, or SC-type, equations are solved as well as SS equations fully in spectral space.

Computational results are presented for scattering from various periodic surfaces. The results include examples with grazing incidence, a very rough surface and a highly oscillatory surface. The examples vary over a parameter set which includes the geometrical optics regime, physical optics or resonance regime, and a renormalization regime.

The objective of this study was to determine the best computational method for these problems. Briefly, the SC method was the fastest, but it did not converge for large slopes or very rough surfaces for reasons we explain. The SS method was slower and had the same convergence difficulties as SC. The CC methods were extremely slow but always converged. The simplest approach is to try the SC method first. Convergence, when the method works, is very fast. If convergence does not occur with SC, then SS should be used, and failing that CC.     

Decompose Boolean Matrices with Correlation Clustering

One of the tasks of data science is the decomposition of large matrices in order to understand their structures. A special case of this is when we decompose relations, i.e., logical matrices. In this paper, we present a method based on the similarity of rows and columns, which uses correlation clustering to cluster the rows and columns of the matrix, facilitating the visualization of the relation by rearranging the rows and columns. In this article, we compare our method with Gunther Schmidt’s problems and solutions. Our method produces the original solutions by selecting its parameters from a small set. However, with other parameters, it provides solutions with even lower entropy.     

Lattice Boltzmann computational fluid dynamics in three dimensions

The recent development of the lattice gas method and its extension to the lattice Boltzmann method have provided new computational schemes for fluid dynamics. Both methods are fully paralleled and can easily model many different physical problems, including flows with complicated boundary conditions. In this paper, basic principles of a lattice Boltzmann computational method are described and applied to several three-dimensional benchmark problems. In most previous lattice gas and lattice Boltzmann methods, a face-centered-hyper-cubic lattice in four-dimensional space was used to obtain an isotropic stress tensor. To conserve computer memory, we develop a model which requires 14 moving directions instead of the usual 24 directions. Lattice Boltzmann models, describing two-phase fluid flows and magnetohydrodynamics, can be developed based on this simpler 14-directional lattice. Comparisons between three-dimensional spectral code results and results using our method are given for simple periodic geometries. An important property of the lattice Boltzmann method is that simulations for flow in simple and complex geometries have the same speed and efficiency, while all other methods, including the spectral method, are unable to model complicated geometries efficiently.     

Rigorous and approximate methods for modeling wave scattering from a locally perturbed perfectly conducting surface

Rigorous and approximate methods are considered for solving the problem of harmonic plane wave scattering from a plane surface arbitrarily perturbed along one dimension on a finite interval. This problem is treated using the Fredholm integral equations of the second kind and the Kirchhoff and Rayleigh approximations. The estimates of the computational efficiency of the integral equation method and the Rayleigh approximation are compared by calculating fields scattered from random rough surfaces in the resonance region (i.e., when the roughness height is comparable to or smaller than the incident wavelength) for an arbitrary incidence of a plane wave. Scattering patterns calculated using the integral equations and the Kirchhoff approximation are discussed in the case of large-scale random rough surface scattering. Particular attention is paid to scattering at near-grazing incidence.     

Domain compression via anisotropic metamaterials designed by coordinate transformations

We introduce a spatial coordinate transformation technique to compress the excessive white space (i.e. free-space) in the computational domain of finite methods. This approach is based on the form-invariance property of Maxwell’s equations under coordinate transformations. Clearly, Maxwell’s equations are still satisfied inside the transformed space, but the medium turns into an anisotropic medium whose constitutive parameters are determined by the coordinate transformation. The proposed technique can be employed to reduce the number of unknowns especially in high-frequency applications wherein a finite method requires an electrically-large computational domain. After developing the analytical background of this technique, we report some numerical results for finite element simulations of electromagnetic scattering problems.     

ADI spectral collocation methods for parabolic problems

We discuss the Crank–Nicolson and Laplace modified alternating direction implicit Legendre and Chebyshev spectral collocation methods for a linear, variable coefficient, parabolic initial-boundary value problem on a rectangular domain with the solution subject to non-zero Dirichlet boundary conditions. The discretization of the problems by the above methods yields matrices which possess banded structures. This along with the use of fast Fourier transforms makes the cost of one step of each of the Chebyshev spectral collocation methods proportional, except for a logarithmic term, to the number of the unknowns. We present the convergence analysis for the Legendre spectral collocation methods in the special case of the heat equation. Using numerical tests, we demonstrate the second order accuracy in time of the Chebyshev spectral collocation methods for general linear variable coefficient parabolic problems.     

Basin of attraction of cycles of discretizations of dynamical systems with SRB invariant measures

Computer simulations of dynamical systems arediscretizations, where the finite space of machine arithmetic replaces continuum state spaces. So any trajectory of a discretized dynamical system is eventually periodic. Consequently, the dynamics of such computations are essentially determined by the cycles of the discretized map. This paper examines the statistical properties of the event that two trajectories generate the same cycle. Under the assumption that the original system has a Sinai-Ruelle-Bowen invariant measure, the statistics of the computed mapping are shown to be very close to those generated by a class of random graphs. Theoretical properties of this model successfully predict the outcome of computational experiments with the implemented dynamical systems.     

一种二阶混合有限体元格式的GAMG预条件子

针对一种含跳系数椭圆问题的二阶混合有限体元格式,讨论求解相应离散系统PGMRES法的预条件子构造问题.通过严格的理论分析,建立分层基下该二阶混合有限体元刚度矩阵和二次有限元刚度矩阵的谱等价关系,并利用关于二次有限元刚度矩阵的一种基于分层思想的GAMG预条件子,为二阶混合有限体元刚度矩阵设计一种高效GAMG预条件子.数值结果验证理论分析的正确性和新预条件子的高效性与稳定性.     

On Jacobian matrices for flows

We present a general method for constructing numerical Jacobian matrices for flows discretized on a Poincaré surface of section. Special attention is given to Hamiltonian flows where the additional constraint of energy conservation is explicitly taken into account. We demonstrate the approach for a conservative dynamical flow and apply the technique for the general detection of periodic orbits.     

Some theoretical aspects of elastic wave modeling with a recently developed spectral element method

A spectral element method has been recently developed for solving elastodynamic problems. The numerical solutions are obtained by using the weak formulation of the elastodynamic equation for heterogeneous media, based on the Galerkin approach applied to a partition, in small subdomains, of the original physical domain. In this work, some mathematical aspects of the method and the associated algorithm implementation are systematically investigated. Two kinds of orthogonal basis functions, constructed with Legendre and Chebyshev polynomials, and their related Gauss-Lobatto collocation points are introduced. The related integration formulas are obtained. The standard error estimations and expansion convergence are discussed. An element-by-element pre-conditioned conjugate gradient linear solver in the space domain and a staggered predictor/multi-corrector algorithm in the time integration are used for strong heterogeneous elastic media. As a consequence, neither the global matrices nor the effective force vector is assembled. When analytical formulas are used for the element quadrature, there is even no need for forming element matrix in order to further save memory without losing much in computational efficiency. The element-by-element algorithm uses an optimal tensor product scheme which makes this method much more efficient than finite-element methods from the point of view of both memory storage and computational time requirements. This work is divided into two parts. The first part mainly focuses on theoretical studies with a simple numerical result for the Che-byshev spectral element, and the second part, mainly with the Legendre spectral element, will give the algorithm implementation, numerical accuracy and efficiency analyses, and then the detailed modeling example comparisons of the proposed spectral element method with a pseudo-spectral method, which will be seen in another work by Lin, Wang and Zhang.     

Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems

This paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws. We consider problems with parametric uncertainty in initial conditions and model coefficients, whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables. The stochastic spectral method relies on multi-resolution schemes where the stochastic domain is discretized using tensor-product stochastic elements supporting local polynomial bases. A Galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution. Hyperbolicity of the resulting Galerkin system is analyzed. A finite volume scheme with a Roe-type solver is used for discretization of the spatial and time variables. An original technique is introduced for the fast evaluation of approximate upwind matrices, which is particularly well adapted to local polynomial bases. Efficiency and robustness of the overall method are assessed on the Burgers and Euler equations with shocks.     

八重准周期排列的短沟槽结构减阻机理分析

设计构建了一排和三排阵列的八重准周期短条纹沟槽减阻结构,以及作为对比研究的无序和周期结构,并采用雷诺Navier-Stokes方程和RANGκ-ε湍流模型,系统计算了这些结构表面的湍流边界层状态和总应力,模拟结果显示:八重准周期沟槽结构相对于周期和无序结构具有更优的减阻效应,且为三排阵列时的减阻效果明显优于单排阵列,这一结果得到了减阻实验的验证,通过分析比较不同结构的流体边界层特性发现,八重准周期结构可有效抑制附面层的涡强度,减小湍流耗散速率,保持流体条纹相的稳定性,结合二维光栅的夫琅禾费衍射波模型分析表明,八重准周期结构可减弱衍射谱在大角度方向上的谱强度,揭示出该结构抑制流体相干扰动波扩展的物理机制,并与流场分析结果相符合。     

Electromagnetic properties of random material

This work addresses the relationship between grain properties and the permeability and permittivity spectra of non-crystalline materials or aerosols.

The scattered multipolar fields about a single sphere are related to the polarizability of a random collection of such spheres. Using the Clausius-Mossotti relation the effective permeability and permittivity spectra of an amorphous material is determined for arbitrary permittivity and permeability of the individual spheres, packing density, and sphere size. Although the author considers the spectra over a range where the product of the external wavevector and sphere radius is kept small, typically less than one-tenth, the product of the internal wavevector and sphere radius is unconstrained and seen to have a large effect on predicted spectra.

The result is a variety of possible spectral types which include resonances, relaxations and certain complex. conglomerate spectra that have been measured and far which no direct explanation is otherwise available.     

Cryptographic Algorithm Using Newton-Raphson Method and General Bischi-Naimzadah Duopoly System

Image encryption is an excellent method for the protection of image content. Most authors used the permutation-substitution model to encrypt/decrypt the image. Chaos-based image encryption methods are used in this model to shuffle the rows/columns and change the pixel values. In parallel, authors proposed permutation using non-chaotic methods and have displayed good results in comparison to chaos-based methods. In the current article, a new image encryption algorithm is designed using combination of Newton-Raphson’s method (non-chaotic) and general Bischi-Naimzadah duopoly system as a hyperchaotic two-dimensional map. The plain image is first shuffled by using Newton-Raphson’s method. Next, a secret matrix with the same size of the plain image is created using general Bischi-Naimzadah duopoly system. Finally, the XOR between the secret matrix and the shuffled image is calculated and then the cipher image is obtained. Several security experiments are executed to measure the efficiency of the proposed algorithm, such as key space analysis, correlation coefficients analysis, histogram analysis, entropy analysis, differential attacks analysis, key sensitivity analysis, robustness analysis, chosen plaintext attack analysis, computational analysis, and NIST statistical Tests. Compared to many recent algorithms, the proposed algorithm has good security efficiency.     

An acoustic shadowgraph trial to detect convection in the Arctic

The EU Framework 5 project CONVECTION aims to understand convection processes in the Greenland Sea. By studying water motion close to the surface we hope to determine how convection is linked to atmospheric conditions and local surface features.

The usual methods of studying such processes in the ocean are by taking multiple soundings of conductivity, temperature and pressure or towing a large chain measuring temperature and salinity through a cross-section of ocean. These have the disadvantage of yielding information only while the research vessel is in the area. We have employed an alternative acoustic method that can provide data for long periods using semi-permanent moorings.

The acoustic shadowgraph method relies on the fact that when an acoustic signal propagates through a region containing convective irregularities the temperature variations along the path cause the signal amplitude to fluctuate. Unlike tomography, the shadowgraph does not require travel times to be measured and so the equipment can be much cheaper.

This paper describes the experimental apparatus, its testing and deployment on Vesteris Bank in the Greenland Sea in October 2001 and its recovery in April 2002. It also gives an overview of some of the acoustic intensity results and shows how they can be interpreted to yield estimates of sub-surface convection velocities.