Dispersion free wave splittings for structural elements

Wave splittings are derived for three types of structural elements: membranes, Timoshenko beams, and Mindlin plates. The Timoshenko beam equation and the Mindlin plate equation are inherently dispersive, as is each Fourier component of the membrane equation in an angular decomposition of the field. The distinctive feature of the wave splittings derived in the present paper is that, in homogeneous regions, they transform the dispersive wave equations into simple one-way wave equations without dispersion. Such splittings have uses both for radial scattering problems in the 2D cases and for scattering problems in dispersive media. As an example of how the splittings may be applied, a direct scattering problem is solved for a membrane with radially varying density. The imbedding method is utilized, and agreement is obtained with an FE simulation.     

Structured matrix algorithms for inverse scattering on the line

Abstract In this article the Marchenko integral equations leading to the solution of the inverse scattering problem for the 1-D Schr?dinger equation on the line are solved numerically. The linear system obtained by discretization has a structured matrix which allows one to apply FFT based techniques to solve the inverse scattering problem with minimal computational complexity. The numerical results agree with exact solutions when available. A proof of the convergence of the discretization scheme is given. Keywords Structured matrix systems, 1-D inverse scattering, Marchenko integral equation     

Composed Scattering Model for Direct Volume Rendering

Based on the equation of transfer in transport theory of optical physics,a new volume rendering model,called composed scattering model(CSM),is presented.In calculating the scattering term of the equation,it is decomposed into volume scattering intensity and surface scattering intensity,and they are composed with the boundary detection operator as the weight function.This proposed model differs from the most current volume rendering models in the aspect that in CSM segmentation and illumination intensity calculation are taken as two coherent parts while in existing models they are regarded as two separate ones.This model has been applied to the direct volume rendering of 3D data sets obtained by CT and MRI.The resultant images show not only rich details but also clear boundary surfaces.CSM is demonstrated to be an accurate volume rendering model suitable for CT and MRI data sets.     

基于改进简化天空光照模型的天空绘制算法

通过使用Hoffman等的算法进行天空效果的绘制,会产生两个问题：在绘制的天空效果图的中央产生一条肉眼可视的黑带;散射系数不能随着观察者视点位置和太阳的位置变化而变化。通过改变Rayleigh散射方程中与散射方向有关因子的系数,解决使用Hoffman等的算法在绘制天空效果时产生黑带的问题;通过建立Mie散射方程方向系数与观察者视点位置和太阳位置之间的线性方程,解决散射系数不能随视点位置和太阳位置变化的问题。结果表明,改进后绘制的天空效果将更适用于飞行模拟器实时仿真。     

Quantum computing via the Bethe ansatz

We recognize quantum circuit model of computation as factorisable scattering model and propose that a quantum computer is associated with a quantum many-body system solved by the Bethe ansatz. As an typical example to support our perspectives on quantum computation, we study quantum computing in one-dimensional nonrelativistic system with delta-function interaction, where the two-body scattering matrix satisfies the factorisation equation (the quantum Yang–Baxter equation) and acts as a parametric two-body quantum gate. We conclude by comparing quantum computing via the factorisable scattering with topological quantum computing.     

Comparison of homogeneous and heterogeneous modeling of transient scattering from dispersive media directly in the time domain

Accurate modeling of pulse propagation and scattering in dispersive medium is a problem in many disciplines (i.e. electromagnetics and acoustics). The inclusion of an additional term in the wave equation (the derivative of the convolution between the causal time-domain propagation factor and the acoustic pressure) that takes into account the dispersive nature of the medium is utilized to make these problems tractable. The resulting modified wave equation (either homogeneous or heterogeneous) is applicable to either linear or non-linear propagation. For the case of an acoustic wave propagating in a two-dimensional heterogeneous dispersive medium, a finite-difference time-domain (FDTD) representation of the modified linear wave equation can been used to solve for the acoustic pressure. The method is applied to the case of scattering from and propagating through a 2D infinitely long cylinder with real world material properties. It is found that ignoring the heterogeneity in the medium can lead to significant error in the propagated/scattered field.     

Computing the Bidirectional Scattering of a Microstructure Using Scalar Diffraction Theory and Path Tracing

Most models for bidirectional surface scattering by arbitrary explicitly defined microgeometry are either based on geometric optics and include multiple scattering but no diffraction effects or based on wave optics and include diffraction but no multiple scattering effects. The few exceptions to this tendency are based on rigorous solution of Maxwell's equations and are computationally intractable for surface microgeometries that are tens or hundreds of microns wide. We set up a measurement equation for combining results from single scattering scalar diffraction theory with multiple scattering geometric optics using Monte Carlo integration. Since we consider an arbitrary surface microgeometry, our method enables us to compute expected bidirectional scattering of the metasurfaces with increasingly smaller details seen more and more often in production. In addition, we can take a measured microstructure as input and, for example, compute the difference in bidirectional scattering between a desired surface and a produced surface. In effect, our model can account for both diffraction colors due to wavelength-sized features in the microgeometry and brightening due to multiple scattering. We include scalar diffraction for refraction, and we verify that our model is reasonable by comparing with the rigorous solution for a microsurface with half ellipsoids.     

Hypersingular integral equations not needed in the impedance problem in scattering theory

We propose a new approach to the analysis of the impedance problem for the Helmholtz equation in the exterior of a body (obstacle) in two and three dimensions. This approach can be called ‘method of interior boundaries’, because an additional boundary is introduced inside the scattering body. An appropriate boundary condition is specified on the additional boundary. The solution of the problem is obtained in the form of a single-layer potential on the whole boundary. The density in the potential satisfies the uniquely solvable Fredholm equation of the second kind and can be computed by standard codes. In fact, our method holds for any positive wave numbers. The Neumann problem is a particular case of our model. The method of hypersingular integral equations widely used in the scattering theory is more complicated and has several shortcomings in comparison with our approach.     

随机粗糙面的一种新双站散射模型

描述了一种对改进的IEM模型的扩展双站散射模型。我们发现在非相干功率的表达式中,交叉分量和补充分量都由两项组成:一项与误差函数无关,而另一项与误差函数有关。后一项可看成是由误差函数引起的修正项。模型预测值和测量数据的良好吻合显示了这种扩展双站散射模型的有效性。     

Even- and odd-parity kinetic equations of particle transport. 1: Algebraic and centered forms of the scattering integral

We consider an equivalent formulation of the linear kinetic transport equation for neutral particles (neutrons, photons) as a system of two equations for even and odd parts of the distribution function. The particle scattering integral of even- and odd-parity transport equations is converted into a non-linear algebraic form and into a centered form. In the algebraic form of the integral we clearly identify the net result of two opposite processes, i.e., particle scattering from a beam and into the beam. In the centered form of the integral the principal terms of scattering processes are canceled out. An iterative method is proposed for the solution of the system of even- and odd-parity equations with these forms of the scattering integral. Convergence of iterations is studied for a one-dimensional plane problem.     

A method to account for shoot scale clumping in coniferous canopy reflectance models

The three-dimensional structure of a coniferous shoot gives rise to multiple scattering of light between the needles of the shoot, causing the shoot spectral reflectance to differ from that of a flat leaf. Forest reflectance models based on the radiative transfer equation handle shoot level clumping by correcting the radiation attenuation coefficient with a clumping index. The clumping index causes a reduction in the interception of radiation by the canopy at a fixed leaf area index (LAI). In this study, we show how within-shoot multiple scattering is related to shoot scale clumping and derive a similar, but wavelength dependent, correction to the scattering coefficient. The results provide a method for integrating shoot structure into current radiative transfer equation based forest reflectance models. The method was applied to explore the effect of shoot scale clumping on canopy spectral reflectance using simple model canopies with a homogeneous higher level structure. The clumping of needles into shoots caused a wavelength dependent reduction in canopy reflectance, as compared to that of a leaf canopy with similar interception. This is proposed to be one reason why coniferous and broad-leaved canopies occupy different regions in the spectral space and exhibit different dependency of spectral vegetation indices on LAI.     

Solution of Electromagnetic Scattering Problems Involving Three-Dimensional Homogeneous Dielectric Objects by the Single Integral Equation Method

The problem of electromagnetic scattering by a homogeneous dielectric object is usually formulated as a pair of coupled integral equations involving two unknown currents on the surface S of the object. In this paper, however, the problem is formulated as a single integral equation involving one unknown current on S. Unique solution at resonance is obtained by using a combined field integral equation. The single integral equation is solved by the method of moments using a Galerkin test procedure. Numerical results for a dielectric sphere are in good agreement with the exact results. Furthermore, the single integral equation method is shown to have superior convergence speed of iterative solution compared with the coupled integral equations method.     

Exact solutions of a KdV equation hierarchy with variable coefficients

In this paper, a nonisospectral and variable-coefficient KdV equation hierarchy with self-consistent sources is derived from the related linear spectral problem. Exact solutions of the KdV equation hierarchy are obtained through the inverse scattering transformation (IST). It is shown that the IST is an effective mathematical tool for solving the whole hierarchy of nonisospectral nonlinear partial differential equations with self-consistent sources.     

Numerical solution of the nonlinear Schrödinger equation,starting from the scattering data

Starting from the scattering data, the initial-value problem for the focusing nonlinear Schrödinger equation is solved numerically by following the path of the inverse scattering transform. The numerical results of an extensive experimentation suggest that: (a) our method is very effective, whenever the scattering data are analytically known; (b) the split-step Fourier method is not really effective if the exact solution is not smooth enough.     

Nonreflecting boundary condition for the Helmholtz equation

To solve the Helmholtz equation in an infinite three-dimensional domain a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of (primary) nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. An approximate nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The accuracy of the boundary condition is determined using a point source, and the computational results are presented for the scattering of a wave from a sphere.     

Momentum-time flux conservation method for one-dimensional wave equations

We present a conservation element and solution element method in time and momentum space. Several paradigmatic wave problems including simple wave equation, convection-diffusion equation, driven harmonic oscillating charge and nonlinear Korteweg-de Vries (KdV) equation are solved with this method and calibrated with known solutions to demonstrate its use. With this method, time marching scheme is explicit, and the nonreflecting boundary condition is automatically fulfilled. Compared to other solution methods in coordinate space, this method preserves the complete information of the wave during time evolution which is an useful feature especially for scattering problems.     

基于Bi-Mimics模型的GNSS-R农作物生物量监测理论研究

GNSS-R(GNSS-Reflectometry)遥感从机理上讲属于双站雷达,在一阶辐射传输方程Mimics(Michigan Microwave Canopy Scattering)模型的基础上,将其修改为双站散射模型Bi-Mimics(Bistatic\|Mimics);将模型中的树干层去掉,保留树冠层和地表间的散射机制,发展了适用于农作物的Bi-Mimics模型。利用该模型,模拟分析在GNSS-R工作的L波段农作物的散射特性;根据GNSS-R设置模拟分析了镜像散射系数与农作物生物量之间的关系;并根据双站雷达理论公式,模拟了农作物生物量与接收机信号之间的关系。结果表明GNSS-R从理论上用来研究和监测农作物的生物量存在可行性,但研究工作有待进一步深入。     

A scattering model for perfectly conducting random surfaces I. Model development

Abstract

The standard integral equation for the surface current is solved iterativcly to obtain an estimate of the surface current on a perfectly conducting randomly rough surface. The far-zone scattered fields and the backscattering coefficients for vertical, horizontal and cross-polarizations are then computed using this current estimate. The polarized backscattering coefficients are explicit functions of the surface parameters and reduce to the Kirchhoff solution in the high-frequency region and to the first-order perturbation solution in the low-frequency region. The cross-polarized scattering coefficient reduces to the second-order perturbation result in the low-frequency region and to zero in the high-frequency limit. A comparison is made with scattering measurements taken under laboratory conditions on a random surface with ka equal to 0-44 and kl equal to 3-25 ( l is the correlation length) It is found that better agreement is obtained with the current model than with the first-order perturbation model in predicting polarized scattering. It is also shown that the separation between VV and HH polarizations decreases gradually with frequency and approaches zero in the high-frequency limit     

Accelerating subsurface scattering using Cholesky factorization

In this paper we present a simplified subsurface scattering model that exploits a diffusion mechanism to provide a simpler solution to the transport equation. Our model is based on numerical analysis techniques that are amenable to Cholesky factorization. We treat the factorization as a precomputed scattering quantity which can be used to significantly speed up multiple scattering calculations as the global light source changes. On low resolution meshes, we have been able to achieve real-time solutions of the subsurface scattering while still maintaining good visual quality of the solution.     

Application of the finite-element method for the forward and inverse models in optical tomography

The development of an optical tomographic imaging system for biological tissue based on time-resolved near-infrared transillumination has received considerable interest recently. The reconstruction problem is ill posed because of scatter-dominated photon propagation, and hence it requires both an accurate and fast transport model and a robust solution convergence scheme. The iterative image recovery algorithm described in this paper uses a numerical finite-element solution to the diffusion equation as the photon propagation model. The model itself is used to compare the influence of absorbing and scattering inhomogeneities embedded in a homogeneous tissue sample on boundary measurements to estimate the possibility of separating absorption and scattering images. Images of absorbers and scatterers reconstructed from both mean-time-of-flight and logarithmic intensity data are presented. It is found that mean-time-of-flight data offer increased resolution for reconstructing the scattering coefficient, whereas intensity data are favorable for reconstructing absorption.