Quasi-Dynamic Green’s Functions for Efficient Full-Wave Integral Formulations for Microstrip Interconnects

Integral formulations are widely used for full-wave analysis of microstrip interconnects. A weak point of these formulations is the inclusion of the proper planar-layered Green’s Functions (GFs), because of their computational cost. To overcome this problem, usually the GFs are decomposed into a quasi-dynamic term and a dynamic one. Under suitable approximations, the ?rst may be given in closed form, whereas the second is approximated. Starting from a general criterion for this decomposition, in this paper we derive some simple criteria for using the closed-form quasi-dynamic GFs instead of the complete GFs, with reference to the problem of evaluating the full-wave current distribution along microstrips. These criteria are based on simple relations between frequency, line length, dielectric thickness and permittivity. The layered GFs have been embedded into a full-wave transmission line model and the results are ?rst benchmarked with respect to a full-wave numerical 3D tool, then used to assess the proposed criteria.

Study on Characteristics of 3-D Translating-Pulsating Source Green Function of Deep-Water Havelock Form and Its Fast Integration Method

The singularities, oscillatory performances and the contributing factors to the 3-＇D translating-pulsating source Green function of deep-water Havelock form which consists of a local disturbance part and a far-field wave-like part, are analyzed systematically. Relative numerical integral methods about the two parts are presented in this paper. An improved method based on LOBATTO rule is used to eliminate singularities caused respectively by infinite discontinuity and jump discontinuous node from the local disturbance part function, which makes the improvement of calculation efficiency and accuracy possible. And variable substitution is applied to remove the singularity existing at the end of the integral interval of the far-field wave-like part function. Two auxiliary techniques such as valid interval calculation and local refinement of integral steps technique in narrow zones near false singularities are applied so as to avoid unnecessary integration of invalid interval and improve integral accordance. Numerical test results have proved the efficiency and accuracy in these integral methods that thus can be applied to calculate hydrodynamic performance of floating structures moving in waves.

The discrete variational principle and the first integrals of Birkhoff systems＊

This paper shows that first integrals of discrete equation of motion for Birkhoff systems can be determined explicitly by investigating the invariance properties of the discrete Pfaffian. The result obtained is a discrete analogue of theorem of Noether in the calculus of variations. An example is given to illustrate the application of the results.

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS WITH NON-SEPARATED TYPE INTEGRAL BOUNDARY CONDITIONS

In this paper, we study a boundary value problem of nonlinear fractional dif- ferential equations of order q （1 〈 q 〈 2） with non-separated integral boundary conditions. Some new existence and uniqueness results are obtained by using some standard fixed point theorems and Leray-Schauder degree theory. Some illustrative examples are also presented. We extend previous results even in the integer case q = 2.

Boundedness of singular integrals along surfaces on Lebesgue spaces

In the paper, we establish the LP（Rn＋1）-boundedness for a class of singular integral operators associated to surfaces of revolution {（y,γ（｜y｜）, y ∈ Rn} with rough kernels. We also give several applications of this inequality.

Curve integral with path independent in orthogonal curvilinear coordinate system

It is explored that the line integral is a path independent in two or three arbitrary dimensional orthogonal curvilinear coordinate systems, which is based on the integral condition with the path independent in two or three dimensional rectangular coordinate systems. Firstly, according to the coordinate transformation, the condition that the line integral is the path independent in the polar coordinate system is obtained easily from the Green's theorem in two-dimensional rectangular coordinate system and the condition is extended to arbitrary two-dimension orthogonal curvilinear coordinates. Secondly, through the coordinate transformation relationship and the area projection method, the Stokes formula in three-dimensional rectangular coordinate system is promoted to the spherical coordinate system and cylindrical coordinate system, and the condition that the line integral is a path independent is obtained. Furthermore, the condition is extended to arbitrary three-dimension orthogonal curvilinear coordinates. Lastly, the conclusions are made.

THE GAUSS–GREEN THEOREM IN CLIFFORD ANALYSIS AND ITS APPLICATIONS

In this article, we establish the Gauss Green type theorems for Clifford-valued functions in Clifford analysis. The boundary conditions in theorems obtained are very general by using the geometric measure theoretic method. The Cauchy-Pompeiu formula for Clifford-valued functions under the weak condition will be derived as their simple application. Furthermore, Cauchy formula for monogenic functions under the weak condition is derived directly from the Cauchy-Pompeiu formula.

Revisiting the Evaluation of a Multidimensional Gaussian Integral

The evaluation of Gaussian functional integrals is essential on the application to statistical physics and the general calculation of path integrals of stochastic processes. In this work, we present an elementary extension of an usual result of the literature as well as an alternative new derivation.

Solution of Singular Integrals in Mathematical Model of Mode I Crack near Strength Mismatched Interface

Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.

METHOD OF GREEN’S FUNCTION OF CORRUGATED SHELLS

By using the fundamental equations of axisymmetric shallow shells of revolution, the nonlinear bending of a shallow corrugated shell with taper under arbitrary load has been investigated. The nonlinear boundary value problem of the corrugated shell was reduced to the nonlinear integral equations by using the method of Green＇s function. To solve the integral equations, expansion method was used to obtain Green＇s function. Then the integral equations were reduced to the form with degenerate core by expanding Green＇s function as series of characteristic function. Therefore, the integral equations become nonlinear algebraic equations. Newton＇ s iterative method was utilized to solve the nonlinear algebraic equations. To guarantee the convergence of the iterative method, deflection at center was taken as control parameter. Corresponding loads were obtained by increasing deflection one by one. As a numerical example,elastic characteristic of shallow corrugated shells with spherical taper was studied.Calculation results show that characteristic of corrugated shells changes remarkably. The snapping instability which is analogous to shallow spherical shells occurs with increasing load if the taper is relatively large. The solution is close to the experimental results.

THE DYADIC GREEN'S FUNCTION IN A LOADED RECTANGULAR WAVEGUIDE AND ITS APPLICATION

This paper presents a method to derive the Dyadic Green’s Function(DGF)ofa loaded rectangular waveguide by using the image method.In the calculation of the DGF,we use the integral transformation and replace the multi-infinite summation by a single one;thus it greatly simplifies the calculation and saves computer time.As an example of the DGF’sapplication,we give the moment method’s scattering field calculation of a metal sphere resting onthe broad wall of the loaded rectangular waveguide.Results of our calculations well agree withboth data of experiments performed in our laboratory and those are published.It is easy to seethat the method used in this paper can be expanded to other related waveguide problems.

Solution of a Green's function for a saturated porous medium in a half-space subjected to a torsional force

Based on the solutions of the Green＇s function for a saturated porous medium obtained by the authors, and using transformation of axisymmetric coordinates, Sommerfeld integrals and superposition of the influence field on a free surface, the authors have obtained displacement solutions of a saturated porous medium subjected to a torsional force in a half-space. The relationship curves of the displacement solutions and various parameters （permeability, frequency, etc.） under action of a unit of torque are also given in this paper. The results are consistent with previous Reissner＇s solutions, where a two-phase medium decays to a single-phase medium. The solution is useful in solving relevant dynamic problems of a two- phase saturated medium in engineering.

Quasi-Green’s function method for free vibration of clamped thin plates on Winkler foundation

The quasi-Green＇s function method is used to solve the free vibration problem of clamped thin plates on the Winkler foundation. Quasi-Green＇s function is established by the fundamental solution and the boundary equation of the problem. The function satisfies the homogeneous boundary condition of tile problem. The mode-shape differential equation of the free vibration problem of clamped thin plates on the Winkler foundation is reduced to the Fredholm integral equation of the second kind by Green＇s formula. The irregularity of the kernel of the integral equation is overcome by choosing a suitable form of the normalized boundary equation. The numerical results show the high accuracy of the proposed method.

Green’s Function Solution for the Dual-Phase-Lag Heat Equation

The present work is devoted to define a generalized Green’s function solution for the dual-phase-lag model in homogeneous materials in a unified manner .The high-order mixed derivative with respect to space and time which reflect the lagging behavior is treated in special manner in the dual-phase-lag heat equation in order to construct a general solution applicable to wide range of dual-phase-lag heat transfer problems of general initial-boundary conditions using Green’s function solution method. Also, the Green’s function for a finite medium subjected to arbitrary heat source and arbitrary initial and boundary conditions is constructed. Finally, four examples of different physical situations are analyzed in order to illustrate the accuracy and potentialities of the proposed unified method. The obtained results show good agreement with works of [1-4].

利用Green公式计算二重积分

二重积分的计算一直都是高等数学教学的核心内容之一.基于被积函数齐次性的特点,结合Green公式,给出了将闭区域上二重积分转化为边界上曲线积分的计算公式以及其特殊情形.将二重积分转化为一重积分,降维使得计算更加地便捷简化;另一方面,该结论又提供了一种计算二重积分的新途径,特别是可以用来求解部分利用常规方法难于求解或计算繁杂的二重积分,这使得该方法具有更大的优势.最后,通过一些典型性算例证实理论的有效性.

Flat Currents of Green-Schwarz Superstring in AdS2 × S^2 Background

<正> From the k symmetric action of IIB string in AdS_2×S~2 background given by Zhou,we derive theequations of motion.By using the twisted dual transformation which was introduced by Hou,we construct the flatcurrents,conserving non-local charge with one free parameter,for the superstring in AdS_2×S_2.

BI-GreenNet:Learning Green’s Functions by Boundary Integral Network

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations(PDEs).However,in most cases,Green’s function is difficult to compute.The troubles arise in the following threefold.Firstly,compared with the original PDE,the dimension of Green’s function is doubled,making it impossible to be handled by traditional mesh-based methods.Secondly,Green’s function usually contains singularities which increase in the difficulty to get a good approximation.Lastly,the computational domain may be very complex or even unbounded.To override these problems,we develop a new framework for computing Green’s function leveraging the fundamental solution,boundary integral method and neural networkswith reasonably high accuracy in this paper.We focus on Green’s function of Poisson and Helmholtz equations in bounded domains,unbounded domains.We also consider Poisson equation and Helmholtz equations in domains with interfaces.Extensive numerical experiments illustrate the efficiency and accuracy of our method for solving Green’s function.In addition,Green’s function provides the operator from the source term and boundary condition to thePDEsolution.We applyGreen’s function to solve PDEs with different sources,and obtain reasonably high-precision solutions,which shows the good generalization ability of our method.However,the requirements for explicit fundamental solutions to remove the singularity of Green’s function hinder the application of our method in more complex PDEs,such as variable coefficient equations,which will be investigated in our future work.

一道曲线积分计算题引出的思考

通过对一道曲线积分计算题的引深、扩展,启发学生思考,掌握有奇点的曲线积分的计算.

Green公式及其证明

首先指出数学分析中反映二重积分和第二类平面线积分联系的Green公式可以看成是散度定理的一种特殊情形,然后利用散度定理给出广义分部积分公式、第一Green公式和第二Green公式的证明.

S积分的收敛定理

讨论了S积分与极限的次序交换问题,所获结果概括了一些非绝对积分的收敛定理.