A technique for dielectric measurement of cylindrical objects in arectangular waveguide

In this paper, the inverse scattering problem of a homogeneous dielectric post in a rectangular waveguide is considered. A novel inversion algorithm, based on the method of moments and eigen analysis, for computation of the dielectric constant of the post (ϵ) from the measured voltage reflection coefficient is introduced. In this method the integral equation for the polarization current induced in the dielectric post is cast into a matrix equation, and then the contribution of ϵ to the resulting reflection coefficient is expressed explicitly using the eigen analysis. It is shown that the dielectric constant can be obtained from the solution of a complex polynomial function which in turn can be obtained numerically using the conjugate gradient method. Practical aspects of dielectric measurement using this technique are discussed. The HP-8510 network analyzer is used to measure the reflection coefficient of dielectric posts in an X-band waveguide sample holder. Metallic and known dielectric posts are used to determine the accuracy of the dielectric measurement technique     

Time domain modeling of a thin wire in a two-media configuration featuring a simplified reflection/transmission coefficient approach

The paper deals with a transient analysis of a straight thin wire in a presence of a two-media configuration using a simplified reflection coefficient approach. A direct time domain formulation is based on thin wire antenna theory and on the corresponding space–time integral equation for the wire above a dissipative half-space, or buried in a real ground. The effect of a half-space is taken into account via the simplified reflection/transmission coefficient arising from the modified image theory. The resulting space–time integral equation for the wire above and below ground, respectively, is handled via the time domain Galerkin–Bubnov variant of the indirect boundary element method (GB-IBEM) and some illustrative numerical results are presented.The transient response computed via the simplified reflection/transmission coefficient approach is compared, where possible, to the results obtained via the Fresnel coefficients approach.     

Guiding mode expansion of a TE and TM transverse-mode integral equation for dielectric slab waveguides with an abrupt termination.

We propose a rigorous transverse-mode integral equation formulation for analyzing TE and TM electromagnetic radiation fields on the facet of dielectric slab waveguides with an abrupt termination in free space. Both exact waveguide guiding modes and discretized radiation modes are included in the kernels of the integral equation. To reduce the size of the matrix that approximates the exact integral equation, we expand the unknown field at the junctions in terms of guiding modes of a selected waveguide with sufficiently large normalized frequency and core thickness. By direct matrix inversion, we obtain numerical solutions of the scattered fields at the junctions. Our method can be used to study the field distribution as well as the energy reflection and transmission coefficients of dielectric waveguides with multiple step discontinuities.     

Implementation of a symmetric boundary element method in transient heat conduction with semi‐analytical integrations

A time‐convolutive variational hypersingular integral formulation of transient heat conduction over a 2‐D homogeneous domain is considered. The adopted discretization leads to a linear equation system, whose coefficient matrix is symmetric, and is generated by double integrations in space and time. Assuming polynomial shape functions for the boundary unknowns, a set of compact formulae for the analytical time integrations is established. The spatial integrations are performed numerically using very efficient formulae just recently proposed. The competitiveness from the computational point of view of the symmetric boundary integral equation approach proposed herein is investigated on the basis of an original computer implementation. Copyright © 1999 John Wiley & Sons, Ltd.     

Time domain modelling of thin wire arrays in the presence of a two media configuration using the boundary element analysis

Transient response of a multiple wire configuration in the presence of a two-media configuration excited by a voltage source (antenna mode) or illuminated by an incident field is analysed using the boundary element method (BEM). The analysis is based on the solution of the corresponding set of the coupled space-time Hallen integral equation and it is carried out directly in the time domain. The influence of a two media configuration is taken into account via the space time reflection coefficient. The corresponding integral equation set is handled via the time domain variant of the Galerkin–Bubnov indirect boundary element method (GB-IBEM). Some illustrative numerical results for both antenna and scattering mode are presented in the paper.     

Reflectivity of buried slab waveguides

The scattering properties of an abruptly ended buried slab waveguide for both TE and TM modes are examined by an improved iteration technique that is based on the integral equation method with "accelerating" parameters. The waveguide is considered a symmetrical slab, for which the weakly guiding conditions are invalid, and it is embedded in a different dielectric material. The tangential electric field distribution on the terminal plane, the reflection coefficient of the first TE and TM guided modes, and the far-field radiation pattern are computed. Numerical results are presented for several ended waveguides, while special attention is given to the far-field radiation pattern rotation and the terminal field distributions.     

Micromechanics of fibrous composites subjected to combined shear and thermal loading using a truly meshless method

In this study, a micromechanical model is presented to study the combined normal, shear and thermal loading of unidirectional (UD) fiber reinforced composites. An appropriate truly meshless method based on the integral form of equilibrium equations is also developed. This meshless method formulated for the generalized plane strain assumption and employed for solution of the governing partial differential equations of the problem. The solution domain includes a representative volume element (RVE) consists of a fiber surrounded by corresponding matrix in a square array arrangement. A direct interpolation method is employed to enforce the appropriate periodic boundary conditions for the combined thermal, transverse shear and normal loading. The fully bonded fiber–matrix interface condition is considered and the displacement continuity and traction reciprocity are imposed to the fiber–matrix interface. Predictions show excellent agreement with the available experimental, analytical and finite element studies. Comparison of the CPU time between presented method and the conventional meshless local Petrov–Galerkin (MLPG) shows significant reduction of the computational time. The results of this study also revealed that the presented model could provide highly accurate predictions with relatively small number of nodes and less computational time without the complexity of mesh generation.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

A gradient free integral equation for diffusion–convection equation with variable coefficient and velocity

In this paper a boundary-domain integral diffusion–convection equation has been developed for problems of spatially variable velocity field and spatially variable coefficient. The developed equation does not require a calculation of the gradient of the unknown field function, which gives it an advantage over the other known approaches, where the gradient of the unknown field function is needed and needs to be calculated by means of numerical differentiation. The proposed equation has been discretized by two approaches—a standard boundary element method, which features fully populated system matrix and matrices of integrals and a domain decomposition approach, which yields sparse matrices. Both approaches have been tested on several numerical examples, proving the validity of the proposed integral equation and showing good grid convergence properties. Comparison of both approaches shows similar solution accuracy. Due to nature of sparse matrices, CPU time and storage requirements of the domain decomposition are smaller than those of the standard BEM approach.     

Dielectric constant measurement technique for a dielectric strip using a rectangular waveguide

A dielectric constant measurement technique for dielectric strips is presented in this paper. In the measurement, the strip is placed parallel to the broad walls of a rectangular waveguide, and it is found that the measured reflection coefficient is insensitive to the position of the strip when it is placed around the middle of the waveguide cross section. The new sample placement scheme becomes very convenient, especially when a large number of strips have to be measured. To develop the forward scattering formulation, the reflection coefficient of the strip placed in the waveguide is evaluated using the method of moments. With this forward model, a genetic algorithm is developed to retrieve the dielectric constant of the strip from the measured reflection coefficient. The validity of the calculated reflection coefficient is verified by measuring a Teflon strip in a WR187 waveguide, and the dielectric constant of the Teflon is successfully retrieved from the measurement. For the measurement of many strips, a special sample holder is made to ensure insensitivity of the measured reflection coefficient to the position of the strip.     

Modeling light scattered from and transmitted through dielectric periodic structures on a substrate

Light scattering and transmission by rough surfaces are of considerable interest in a variety of applications including remote sensing and characterization of surfaces. In this work, the finite-difference time-domain technique is applied to calculate the scattered and transmitted electromagnetic fields of an infinite periodic rough surface. The elements of the Mueller matrix for scattered light are calculated by an integral of the near fields over a significant number of periods of the surface. The normalized Mueller matrix elements of the scattered light and the spatial distribution of the transmitted flux for a monolayer of micrometer-sized dielectric spheres on a silicon substrate are presented. The numerical results show that the nonzero Mueller matrix elements for scattering from a surface consisting of a monolayer of dielectric spheres on a silicon substrate have specific maxima at some scattering angles. These maxima may be used in the characterization of features of the surface. For light transmitted through the monolayer of spheres, our results show that the transmitted energy focuses around the ray passing through centers of the spheres. At other locations, the transmitted flux is very small. Therefore, micrometer-sized dielectric spheres might be placed on a semiconductor surface to burn nanometer-sized holes in a layer using laser pulses. The method may also be useful in the assembly of periodic microstructures on surfaces.     

Uncertainties in the determination of dielectric properties in theinfinite sample method

In the infinite-sample method, the sample permittivity is found from the measured input reflection coefficient. The reflection coefficient domain of a traveling-wave system is transformed into the complex permittivity domain by two subsequent mappings which include the inverse bilinear and Schwarz-Christoffel transformations. The error bounds in permittivity measurements are derived. The dielectric properties and respective uncertainties are presented for a water sample in the form of the Cole-Cole diagram. The case presented represents the first step leading to a more general presentation of uncertainties in dielectric measurements     

Effect of the number of sublayers on axial optics of anisotropic helical structures

The transition from the continuously rotating dielectric ellipsoid approach to the discrete periodic collection of wave plates is investigated by comparing an exact analytic solution found using a Bloch-Lyaponov transformation on the 4x4 matrix form of Maxwell's equations and the numerical solution. The validity of the analytic solution depends only on the number of sublayers within the unit cell and not on the number of periods in the sample. The circularly polarized Bragg-type selective reflection peak is shown to exist with helices containing as few as N(l)=3 sublayers in a single period, but due to the appearance of higher Fourier harmonics in the dielectric tensor, additional selective reflection peaks appear. The case of N(l)=4 corresponds to the folded Solc-type filter case with a twist angle of pi/4, but it acts as a reflection filter for unpolarized light. Hence using this fact with liquid crystalline structures, polarization insensitive tunable filters can be built.     

Applications of dual MRM for determining the natural frequencies and natural modes of an Euler–Bernoulli beam using the singular value decomposition method

In this paper, a dual multiple reciprocity method (MRM) is employed to solve the natural frequencies and natural modes for an Euler–Bernoulli beam. It is found that the conventional MRM using an essential integral equation results in spurious eigenvalues and modes. By using the natural integral equation of dual MRM, the spurious eigendata can be filtered out. Four numerical examples are given to verify the validity of the present formulation. In one of these four examples, fixed–fixed supported beam, it is found that the boundary eigenvector cannot be determined by either the essential or natural integral equation alone since the rank of the corresponding leading coefficient matrix is insufficient. The singular value decomposition method is then used to solve the eigenproblem after combining the essential and natural integral equations. This method can avoid the spurious eigenvalue problem and possible indeterminancy of boundary eigenvectors at the same time.     

Independent electromagnetic optimization of the two coating thicknesses of a dielectric layer on the facets of an echelle grating in Littrow mount

Abstract

Electromagnetic investigations using the integral equation system method with parametrization (IESMP) show that the two-coating thicknesses of a dielectric layer on the facets of an echelle grating in a Littrow mount have to be independently optimized. While the optimal coating thickness on the blaze facet is the same for maximal efficiency and minimal absorption in both polarizations, this is not the case for the anti-blaze facet. Therefore, it is only possible to optimize the two-coating thicknesses for one of the purposes. On the blaze facet, a simple formula based on thin-film optical considerations describes the optimal thickness very well. Additionally, we found that resonance anomalies can significantly reduce efficiency if the wrong coating thickness is used on the anti-blaze facet. The coating thickness creating the resonance anomaly can be deduced by investigating the poles of the reflection coefficient of a dielectric coated metallic mirror in grazing incidence. This value can be used to optimize the layer for maximal efficiency. Consequently, we are generally able to describe the optimal coating thicknesses for minimal absorption as well as for maximum efficiency in both, TE- and TM-polarization, using only thin-film optical considerations without any further rigorous calculation.     

流体-孔隙介质周期性粗糙界面超声波反射与透射特性研究

为了研究超声波在流体-孔隙介质周期性粗糙界面的传播特性,文章基于周期性锯齿粗糙界面的衍射模型分析孔隙介质开孔与闭孔状态下孔隙度对反射与透射的影响。通过孔隙介质比奥特(Biot)理论与光栅方程理论,得到包含各阶反射系数、透射系数的线性方程组,再利用傅里叶变换进行数值计算,得到孔隙度与流体-孔隙介质周期性粗糙界面反射系数、透射系数之间的关系。结果表明,由于界面的周期性,频率对反射与透射系数的影响没有呈现一定的规律。但孔隙度对反射与透射系数有显著影响,且由于孔隙介质状态的差异性,导致反射与透射系数在开孔与闭孔时变化趋势不同。     

渐变截面梁波透射反射系数求解的数值验证

推导任意截面形状的预应力Timoshenko梁波动微分方程,对预应力箱板梁渐变截面段弹性波透(反)射系数矩阵进行计算,为求得小端传播到大端的透射波幅值,将渐变截面梁近似当作具有一系列截面间断的阶梯梁,按层状结构波传播的传递矩阵法求波幅在大小端间的透(反)射系数矩阵,通过基于Maple软件平台编写子程序对梁截面尺寸及渐变段分段数目对透(反)射系数矩阵精确性的影响进行数值分析。在波动微分方程伽辽金近似解基础上,对渐变截面段梁波透(反)射系数矩阵的求解法进行数值验证,验证中以第一类贝塞尔函数组合作为系数具有高次项幂级数函数的高阶常微分方程伽辽金法近似解的试函数,该试函数对这类方程具有普遍的适用性。     

Analytical integration of the moments in the diagonal form fast multipole boundary element method for 3-D acoustic wave problems

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton-Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numerical integration. Numerical examples show that using the analytical moments can reduce the CPU time by a lot as compared with that using the direct numerical integration. The percentage of CPU time reduction largely depends on the proportion of the time used for moments calculation to the overall solution time. Several examples are studied to investigate the effectiveness and efficiency of the developed diagonal fast multipole BEM as compared with earlier p³ fast multipole method BEM, including a scattering problem of a dolphin modeled with 404,422 boundary elements and a radiation problem of a train wheel track modeled with 257,972 elements. These realistic, large-scale BEM models clearly demonstrate the effectiveness, efficiency and potential of the developed diagonal form fast multipole BEM for solving large-scale acoustic wave problems.     

Spectral-domain analysis of multilayer cylindrical–rectangular microstrip antennas

In this paper, the resonance and radiation characteristics of multilayered cylindrical–rectangular microstrip antennas are analysed. The problem is formulated, in the spectral-domain, using an electric field integral equation and the spectral-domain Green's function. An efficient compact method is developed to obtain the Green's function of the electric field due to the current distribution of a patch located in an arbitrary layer of a cylindrical stratified medium. To accomplish this, efficient matrix representations are used to describe the electric and magnetic fields in each layer of the geometry in terms of ABCD matrices characteristic of its material properties. The boundary integral equation for the unknown patch current is solved numerically by applying the boundary element method using three kinds of basis functions. Taking into account the symmetry properties of the elements of the moment method matrix, the CPU time is reduced significantly. The convergence of the method is proven by performing the resonant frequencies for a single layer cylindrical–rectangular microstrip patch vs. the substrate thickness. The computed data are found to be in very good agreement with those found in the literature, using only two basis functions. Once the validity of the method is checked, further results for various antennas structures are presented proving the generality of the method. Finally, the effect of an air gap between the substrate layer and the ground conducting cylinder on far zone radiation field is investigated using the steepest-descent method.     

Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002