Scattering from a rough layer of a random medium

This paper deals with scattering from a random-medium layer with rough boundaries. The fluctuations of the surface heights and medium permittivity are assumed to be small and smooth. All random quantities are assumed to be stationary and independent of each other. After the introduction of approximate boundary conditions, the system of partial differential equations is transformed into an integral equation where the fluctuations of the problem are represented as a zero-mean random operator. Employing smoothing, integral equations for the coherent fields are obtained. Use of the Helmholtz operator leads to solution for the coherent propagation constant while the boundary operators lead to coherent Fresnel coefficients. The characteristics of the results are illustrated by considering several examples.     

三维Helmholtz方程外问题的自然积分方程及其数值解

用文[2,3]提出的自然边界归化方法来处理三维Helmholtz方程的外边值问题。在简要介绍如何用球谐展开的方法得到Helmholtz问题在外球域上的自然积分方程后,给出求解该自然积分方程的一种数值方法及相应的数值算例。     

标量波传播问题的双互易精细积分法

为避免使用计算多种特征频率下的声场响应,采用双互易方法将边界积分方程中时间二次导数项的域积分转化为边界积分.首先,将计算场点配置在边界上并考虑边界条件,可以获得由内部节点上声压量线性表示的边界节点上的物理量;其次,将计算场点配置于域内离散节点上,将所得边界积分方程组中关于边界物理量用内部节点的声压量线性表示,获得关于声压量的二阶常微分方程组;第三,引入声压变化速度作为未知量,将二阶常微分方程组转化为一阶常微分方程组;最后,采用精细积分法精确求解常微分方程组.数值算例验证了双互易精细积分法的正确性和稳定性.     

Homology theory of lattice fermion doubling

I give a general discussion of the phenomenon of spectrum degeneracy in transcribing continuum field equations to the lattice, using concepts of homology theory. This leads to a topological understanding of the problems in transcribing the Dirac equation and a unified treatment of the many lattice fermion schemes in the literature. The connection between spectrum degeneracy and chiral symmetry is explained geometrically without appealing to quantum effects such as anomalies.     

多缝衍射反演光谱学的稳定性理论研究── 一种第一类Fredholm积分方程数值解之稳定性分析

作者在前文^1,2中通过计算机模拟实验给出了由多缝衍射反演光谱学的积分方程─一种第一类Fredholm积分方程─获得稳定线性方程组的条件,本文将进一步从数学上利用积分方程的本征值理论分析计算机的实验结果,阐明这些条件使线性方程组稳定的机制.     

Plasmon resonances in nanoparticles,their applications to magnetics and relation to the Riemann hypothesis

The review of the mathematical treatment of plasmon resonances as an eigenvalue problem for specific boundary integral equations is presented and general properties of plasmon spectrum are outlined. Promising applications of plasmon resonances to magnetics are described. Interesting relation of eigenvalue treatment of plasmon resonances to the Riemann hypothesis is discussed.     

On the modeling of narrow gaps using the standard boundary element method

Numerical methods based on the Helmholtz integral equation are well suited for solving acoustic scattering and diffraction problems at relatively low frequencies. However, it is well known that the standard method becomes degenerate if the objects that disturb the sound field are very thin. This paper makes use of a standard axisymmetric Helmholtz integral equation formulation and its boundary element method (BEM) implementation to study the behavior of the method on two test cases: a thin rigid disk of variable thickness and two rigid cylinders separated by a gap of variable width. Both problems give rise to the same kind of degeneracy in the method, and modified formulations have been proposed to overcome this difficulty. However, such techniques are better suited for the so-called thin-body problem than for the reciprocal narrow-gap problem, and only the first is usually dealt with in the literature. A simple integration technique that can extend the range of thicknesses/widths tractable by the otherwise unmodified standard formulation is presented and tested. This technique is valid for both cases. The modeling of acoustic transducers like sound intensity probes and condenser microphones has motivated this work, although the proposed technique has a wider range of applications.     

Hybrid near-field acoustic holography

Hybrid near-field acoustical holography (NAH) is developed for reconstructing acoustic radiation from an arbitrary object in a cost-effective manner. This hybrid NAH is derived from a modified Helmholtz equation least squares (HELS) formula that expands the acoustic pressure in terms of outgoing and incoming waves. The expansion coefficients are determined by solving an overdetermined linear system of equations obtained by matching the assumed-form solution to measured acoustic pressures through the least squares. Measurements are taken over a conformal surface around a source at close range so that the evanescent waves can be captured. Next, the modified HELS is utilized to regenerate as much acoustic pressures on the conformal surface as necessary and take them as input to the Helmholtz integral formulation implemented numerically by boundary element method (BEM). The acoustic pressures and normal velocities on the source surface are reconstructed by using a modified Tikhnov regularization (TR) with its regularization parameter determined by generalized cross validation (GCV) method. Results demonstrate that this hybrid NAH combines the advantages of HELS and inverse BEM. This is because a majority of the input data are regenerated but not measured, thus the efficiency of reconstruction is greatly enhanced. Meanwhile, the accuracy of reconstruction is ensured by the Helmholtz integral theory and modified TR together with GCV method, provided that HELS converges fast enough on the measurement surface. Numerical examples of reconstructing acoustic quantities on the surface of a simplified engine block are demonstrated. [Work supported by NSF.]     

A fast direct solver for the integral equations of scattering theory on planar curves with corners

We describe an approach to the numerical solution of the integral equations of scattering theory on planar curves with corners. It is rather comprehensive in that it applies to a wide variety of boundary value problems; here, we treat the Neumann and Dirichlet problems as well as the boundary value problem arising from acoustic scattering at the interface of two fluids. It achieves high accuracy, is applicable to large-scale problems and, perhaps most importantly, does not require asymptotic estimates for solutions. Instead, the singularities of solutions are resolved numerically. The approach is efficient, however, only in the low- and mid-frequency regimes. Once the scatterer becomes more than several hundred wavelengths in size, the performance of the algorithm of this paper deteriorates significantly. We illustrate our method with several numerical experiments, including the solution of a Neumann problem for the Helmholtz equation given on a domain with nearly 10000 corner points.     

A rigorous method for solving the problems of diffraction of monochromatic electromagnetic waves by periodic structures

We present a rigorous method for solving the problems of diffraction of monochromatic electromagnetic waves on periodic structures of many types. The method is based on transformation of the Helmholtz equation and the boundary conditions to a certain system of integral equations and a subsequent rigorous solution of this system.     

The generalized non-linear Schrödinger model on the interval

The generalized (1+1)-D$(1+1)\text{-}\mathrm{D}$ non-linear Schrödinger (NLS) theory with particular integrable boundary conditions is considered. More precisely, two distinct types of boundary conditions, known as soliton preserving (SP) and soliton non-preserving (SNP), are implemented into the classical glN$g{l}_N$ NLS model. Based on this choice of boundaries the relevant conserved quantities are computed and the corresponding equations of motion are derived. A suitable quantum lattice version of the boundary generalized NLS model is also investigated. The first non-trivial local integral of motion is explicitly computed, and the spectrum and Bethe ansatz equations are derived for the soliton non-preserving boundary conditions.     

Radiative temperature dissipation in a finite atmosphere—I. The homogeneous case

We have developed a new approach toward solving problems of linear radiative relaxation of LTE temperature perturbations in a plane-parallel atmosphere of finite extent. We show that the mathematical problem is one of solving an integral eigenvalue equation, for which non-trivial solutions exist only for discrete values of the radiative relaxation time. The solutions for the spatial part of the perturbation constitute a complete and orthogonal set of basis functions, making it possible to solve more general problems of temperature relaxation. In applying this method to radiative relaxation in the middle atmosphere of earth, we show how the additional influences of photochemical coupling, advection by winds, and eddy diffusion by small-scale turbulence may be easily included using matrix perturbation techniques. We have solved the homogeneous integral equation for a wide variety of vertical thicknesses in an idealized homogeneous slab medium. Adopting a number of different analytic line profiles (rectangular, Doupler, Voigt, and Lorentz) we have obtained numerical solutions using an exponential-kernel method for solving the integral equation. The discrete eigenvalue “spectrum” is presented for vertical optical depths (0–10³) at line-center, and is used in solving several initial-value problems for a decaying temperature perturbation. We find that the eigenvalue spectrum is bounded from above by the lowest-order eigenvalue, and bounded from below by the familiar transparent approximation. The dependence of the lowest even eigenvalue on optical depth and the relative separation of the higher eigenvalues are found to depend sensitively on the line profile.     

统一非对称光波导横向耦合模理论分析

光波导横向耦合模理论包含正交耦合模理论和非正交耦合模理论两大类, 为了追求形式上的完美, 两类理论没有统一的解析解, 并且都没有对非匹配耦合系统中耦合功率的非对称性进行深入研究. 本文一方面由Helmholtz方程出发推导出了一种新型的非正交耦合模方程, 并对两类理论中的耦合模方程进行了统一处理和求解, 得到了一种统一的解析解; 另一方面根据所得到的统一的解析解对非匹配耦合系统中耦合功率的非对称性进行了详细研究, 计算结果表明非匹配耦合系统中的最大互耦合功率和最小自耦合功率均可用统一的解析解进行近似计算. 关键词： 光波导 耦合模理论 耦合器 最大耦合功率     

On the Boundary Integral Equations for a Two-Dimensional Slowly Rotating Highly Viscous Fluid Flow

In this paper, the two-dimensional slowly rotating highly viscous fluid flow in small cavities is modelled by the triharmonic equation for the streamfunction. The Dirichlet problem for this triharmonic equation is recast as a set of three boundary integral equations which however, do not have a unique solution for three exceptional geometries of the boundary curve surrounding the planar solution domain. This defect can be removed either by using modified fundamental solutions or by adding two supplementary boundary integral conditions which the solution of the boundary integral equations must satisfy. The analysis is further generalized to polyharmonic equations.     

ON THE DISCRETIZATION OF AN ELASTIC ROD WITH DISTRIBUTED SLIDING FRICTION

A one-dimensional elastic system with distributed contact under fixed boundary conditions is investigated in order to study dynamic behavior under sliding friction. A partial differential equation of motion is established and its exact solution is presented. Due to the friction the eigenvalue problem is non-self-adjoint. Mathematical methods for handling the non-self-adjoint system, such as the non-self-adjoint eigenvalue problem and the eigenvalue problem with a proper inner product, are reviewed and applied. The exact solution showed that the undamped elastic system under fixed boundary conditions is neutrally stable when the coefficient of friction is a constant. The assumed mode approximation and the lumped-parameter discretization method are evaluated and their solutions are compared with the exact solution. As a cautionary example the assumed modes approximation leads to false conclusions about stability. The lumped-parameter discretization algorithm generates reliable results.     

Computation of discrete transparent boundary conditions for the 2D Helmholtz equation

We present a family of non-local transparent boundary conditions for the 2D Helmholtz equation. The whole domain, on which the Helmholtz equation is defined, is decomposed into an interior and an exterior domain. The corresponding interior Helmholtz problem is formulated as a variational problem in a standard manner, representing a boundary value problem, whereas the exterior problem is posed as an initial value problem in the radial variable. This problem is then solved approximately by means of the Laplace transformation. The derived boundary conditions are asymptotically correct, model inhomogeneous exterior domains and are simple to implement.     

Data completion method for the characterization of sound sources

This paper presents a different approach to solve the inverse acoustic problem. This problem is an "ill-posed" problem since the solution is very sensitive to measurement precision. A classical way to solve this problem consists in inversing a propagation operator which relates structure quantities (acoustic pressures or gradients) to near-field quantities (acoustic pressures or gradients). This can be achieved by using near-field acoustical holography (NAH) in separable coordinate systems. In order to overcome this limitation, the inverse boundary element method (IBEM) can be implemented to recover all acoustic quantities in a three-dimensional space and on an arbitrary three-dimensional source surface. In this paper, the data completion method (DCM) is developed: the acoustic gradients and pressures are known on a surface surrounding the source, but are unknown on its structure. The solution is given by the resolution of the Helmholtz formulation applied on the empty domain between the two boundaries made by the measurements quantities and the structure of the source. The conventional method applies directly the integral formulation for the empty domain. Another way of solving this Helmholtz formulation can be achieved by splitting it in two well-posed subproblems in a Steklov-Poincare?'s formulation. The data completion method allows one to solve the problem with acoustic perturbations due to sources on the exterior domain, or due to a confined domain, without altering the results.     

Sound field separating on arbitrary surfaces enclosing a sound scatterer based on combined integral equations

To eliminate the limitations of the conventional sound field separation methods which are only applicable to regular surfaces, a sound field separation method based on combined integral equations is proposed to separate sound fields directly in the spatial domain. In virtue of the Helmholtz integral equations for the incident and scattering fields outside a sound scatterer, combined integral equations are derived for sound field separation, which build the quantitative relationship between the sound fields on two arbitrary separation surfaces enclosing the sound scatterer. Through boundary element discretization of the two surfaces, corresponding systems of linear equations are obtained for practical application. Numerical simulations are performed for sound field separation on different shaped surfaces. The influences induced by the aspect ratio of the separation surfaces and the signal noise in the measurement data are also investigated. The separated incident and scattering sound fields agree well with the original corresponding fields described by analytical expressions, which validates the effectiveness and accuracy of the combined integral equations based separation method.