Kernel Of Integral Equation Research Articles

Reversible Exciplex Formation Followed Charge Separation

The reversible exciplex formation followed by its decomposition into an ion pair is considered, taking into account the subsequent geminate and bulk ion recombination to the triplet and singlet products (in excited and ground states). The integral kinetic equations are derived for all state populations, assuming that the spin conversion is performed by the simplest incoherent (rate) mechanism. When the forward and backward electron transfer is in contact as well as all dissociation/association reactions of heavy particles, the kernels of integral equations are specified and expressed through numerous reaction constants and characteristics of encounter diffusion. The solutions of these equations are used to specify the quantum yields of the excited state and exciplex fluorescence induced by pulse or stationary pumping. In the former case, the yields of the free ions and triplet products are also found, while in the latter case their stationary concentrations are obtained.

An Improved Multilevel Green's Function Interpolation Method With Adaptive Phase Compensation

An improved multilevel Green's function interpolation method (MLGFIM) with adaptive phase compensation (APC) is proposed. The difficulty in applying interpolation approaches to the fast varying phase term in the integral equation kernel for full-wave electromagnetic (EM) simulations is eradicated by using the phase compensation and adaptive direction separation (ADS). The multilevel tree structure in MLGFIM keeps the number of direction separation invariant at all levels, attributing to the recursive interpolation with multilevel phase compensation. The proposed MLGFIM-APC in conjunction with the Lagrange-Chebyshev interpolation yield an O(N log N) CPU time and O(N) computer memory requirement for surface scattering problems. By introducing a transition level, the MLGFIM-APC can adaptively incorporate interpolation techniques of conventional interpolation (CI), transition interpolation (TI) and phase-compensation interpolation (PI), corresponding to electromagnetic simulation of problems of small, moderate, and large electrical sizes, respectively. Large-scale microstrip antenna arrays are simulated to illustrate the accuracy and efficiency of the proposed method. It is found that the CPU time scales better than O(N log N) for these co-planar problems.

Stabilization of Integral-Equation Formulations for the Accurate Solution of Scattering Problems Involving Low-Contrast Dielectric Objects

The solution of scattering problems involving low-contrast dielectric objects with three-dimensional arbitrary shapes is considered. Using the traditional forms of the surface integral equations, scattered fields cannot be calculated accurately if the contrast of the object is low. Therefore, we consider the stabilization of the formulations by extracting the nonradiating parts of the equivalent currents. We also investigate various types of stable formulations and show that accuracy can be improved systematically by eliminating the identity terms from the integral-equation kernels. Traditional and stable formulations are compared, not only for small scatterers but also for relatively large problems solved by employing the multilevel fast multipole algorithm. Stable and accurate solutions of dielectric contrasts as low as 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> are demonstrated on problems involving more than 250000 unknowns.

A combined nonlinear contrast image reconstruction algorithm under inexact point-spread function

The problem of image reconstruction is considered for the case when the right-hand side of the 2D integral equation and the point-spread function (the integral equation kernel) are given with random errors. A stable image reconstruction algorithm is proposed. It is a combination of a regularizing algorithm for solving an integral equation (frequency filtering) and a local nonlinear filter (spatial filtering). Characteristics of the 2D point-spread function of the regularizing algorithm are introduced. The regularization parameter is chosen according to the required regularizing algorithm resolution. For eliminating the random reconstruction error, the regularized solution is subjected to nonlinear local filtering that preserves high-frequency information components of the image.

A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending

AbstractIn the paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve bending problems of a circular beam with circular holes. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. After moving the null-field point to the boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Stress concentration is also our concern. Finally, several examples, including two holes and four holes, are given to demonstrate the validity of the proposed method. The results are compared with those of Naghdi and Bird and Steele. Also, the position where the maximum concentration factor occurs is examined. The present formulation can be extended to handle beam problems with arbitrary number and various positions of circular holes by using the developed general-purpose program.

Green quasifunction method for vibration of simply-supported thin polygonic plates on Pasternak foundation

A new numerical method—Green quasifunction is proposed. The idea of Green quasifunction method is clarified in detail by considering a vibration problem of simply-supported thin polygonic plates on Pasternak foundation. A Green quasifunction is established by using the fundamental solution and boundary equation of the problem. This function satisfies the homogeneous boundary condition of the problem. The mode shape differential equation of the vibration problem of simply-supported thin plates on Pasternak foundation is reduced to two simultaneous Fredholm integral equations of the second kind by Green formula. There are multiple choices for the normalized boundary equation. Based on a chosen normalized boundary equation, a new normalized boundary equation can be established such that the irregularity of the kernel of integral equations is overcome. Finally, natural frequency is obtained by the condition that there exists a nontrivial solution in the numerically discrete algebraic equations derived from the integral equations. Numerical results show high accuracy of the Green quasifunction method.

Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach

In this paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve the exterior Helmholtz problems with alluvial valleys. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals by locating the null-field point exactly on the real boundary. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. Image concept and technique of decomposition are utilized for half-plane problems. After moving the null-field point to the real boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples including single and successive alluvial valleys are given to test our program. Limiting cases of a single canyon and two successive canyons are also addressed. Amplification of soft basin is also observed in this study. The validity of the semi-analytical method is verified. Our advantages, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence and mesh-free, by using the present method are achieved.

A BEM-based meshless method for plates on biparametric elastic foundation with internal supports

In this paper a BEM-based meshless method is developed for the analysis of plates on a biparametric elastic foundation which, in addition to the boundary supports, are also supported inside the domain on isolated points (a group of plies) and/or line supports (continuous plates). The presented method is achieved using the concept of the analog equation method (AEM) of Katsikadelis. According to this method the original governing differential equation is replaced by an equivalent problem for plates without internal supports not resting on an elastic foundation subjected to an “appropriate” fictitious load in addition to the transverse external loads under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by radial basis functions series. The solution of the actual problem is obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The presented method has all the advantages of the pure BEM. To validate the effectiveness, accuracy as well as the applicability of the proposed method, numerical results of various example problems are presented.

A BEM-BASED MESHLESS METHOD FOR ELASTIC BUCKLING ANALYSIS OF PLATES

In this paper, a BEM-based meshless method is developed for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. The proposed method is based on the concept of the Analog Equation Method (AEM) of Katsikadelis. According to this method, the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent plate bending problem subjected to an appropriate fictitious load under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by using the radial basis functions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The method has all the advantages of the pure BEM. To validate its effectiveness, accuracy as well as applicability of the proposed method, numerical results of various problems are presented.

Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions

Generalized exponential integral functions (GEIF) are encountered in multi-dimensional thermal radiative transfer problems in the integral equation kernels. Several series expansions for the first-order generalized exponential integral function, along with a series expansion for the general nth order GEIF, are derived. The convergence issues of these series expansions are investigated numerically as well as theoretically, and a recurrence relation which does not require derivatives of the GEIF is developed. The exact series expansions of the two dimensional cylindrical and/or two-dimensional planar integral kernels as well as their spatial moments have been explicitly derived and compared with numerical values.

Complex potentials and integral equations for curved crack and curved rigid line problems in plane elasticity

In this paper, a simple layer potential and a double layer potential are suggested to solve the curved crack problem. The complex potentials in the simple layer case are formulated on the distributed dislocation along the curve. Meantime, the complex potentials in the double layer case are formulated on the crack displacement opening distribution. Behaviors of the complex potentials, for example the behaviors of increments of some physical quantities around a large circle, are analyzed in detail. Continuity and discontinuity of some physical quantities in the normal direction of the curve are analyzed, which are key points for formulating the integral equations of the problems. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. The relations between the kernels in different integral equations are addressed. Similarly, a simple layer potential and a double layer potential are suggested to solve the curved rigid line problem. The complex potentials in the simple layer case are formulated on the distributed forces along the curve. Meantime, the complex potentials in the double layer case are formulated on the resultant force function. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. When the resultant forces and moment are applied on the deformable line, the constraint equations are suggested. For more general cases, for example, in the case that the tractions applied on the two crack faces are not same in magnitude and opposite in direction, a singular integral equation is suggested. The equation is obtained by a superposition of two kinds of single layer potentials.

Modelling Rupture Dynamics of a Planar Fault in 3-D Half Space by Boundary Integral Equation Method: An Overview

In this article, we first reviewed the method of boundary integral equation (BIEM) for modelling rupture dynamics of a planar fault embedded in a 3-D elastic half space developed recently (ZHANG and CHEN, 2005a,b). By incorporating the half-space Green's function, we successfully extended the BIEM, which is a powerful tool to study earthquake rupture dynamics on complicated fault systems but limited to full-space model to date, to half-space model. In order to effectively compute the singular integrals in the kernels of the fundamental boundary integral equation, we proposed a regularization procedure consisting of the generalized Apsel-Luco correction and the Karami-Derakhshan algorithm to remove all the singularities, and developed an adaptive integration scheme to efficiently deal with those nonsingular while slowly convergent integrals. The new BIEM provides a powerful tool for investigating the physics of earthquake dynamics. We then applied the new BIEM to investigate the influences of geometrical and physical parameters, such as the dip angle (δ) and depth (h) of the fault, radius of the nucleation region (R asp), slip-weakening distance (D c ), and stress inside (T i ) and outside (T e ) the nucleation region, on the dynamic rupture processes on the fault embedded in a 3-D half space, and found that (1) overall pattern of the rupture depends on whether the fault runs up to the free surface or not, especially for strike-slip, (2) although final slip distribution is influenced by the dip angle of the fault, the dip angle plays a less important role in the major feature of the rupture progress, (3) different value of h, δ, R asp, T e , T i and D c may influence the balance of energy and thus the acceleration time of the rupture, but the final rupture speed is not controlled by these parameters.

WIRE ANTENNA MODEL FOR TRANSIENT ANALYSIS OF SIMPLE GROUNDING SYSTEMS, PART II: THE HORIZONTAL GROUNDING ELECTRODE

The paper deals with the transient impedance calculation for simple grounding systems. The mathematical modelmodel is based on the thin wire antenna theory. The formulation of the problem is posed in the frequency domain, while the corresponding transient response of the grounding system is obtained by means of the inverse Fourier transform. The current distribution induced along the grounding system due to an injected current is governed by the corresponding frequency domain Pocklington integro-differential equation. The influence of a dissipative half-space is taken into account via the reflection coefficient (RC) appearing within the integral equation kernel. The principal advantage of the RC approach versus rigorous Sommerfeld integral approach is simplicity of the formulation and significantly less computational cost. The Pocklington integral equation is solved by the Galerkin Bubnov indirect boundary element procedure thus providing the current distribution flowing along the grounding system. The outline of the Galerkin Bubnov indirect boundary element method is presented in Part II of this work. Expressing the electric field in terms of the current distribution along the electrodes the feed point voltage is obtained by integrating the normal field component from infinity to the electrode surface. The frequency dependent input impedance is then obtained as a ratio of feed-point voltage and the value of the injected lightning current. The frequency response of the grounding electrode is obtained multiplying the input impedance spectrum with Fourier transform of the injected current waveform. Finally, the transient impedance of the grounding system is calculated by means of the inverse Fourier transform. The vertical and horizontal grounding electrodes, as simple grounding systems, are analyzed in this work. The Part I of this work is related to the vertical

An Integral-equation technique for solving thick irises in rectangular waveguides

This paper presents an efficient integral-equation (IE) technique for analysis of thick irises inside multilayered rectangular waveguides. The IEs remain the same as in the case of zero-thickness iris and the thickness is accounted for only as a correction term in the IE kernel. This technique halves the number of unknowns on the iris, thus leading to computational effort and simulation times comparable to the zero-thickness case. In this paper, two efficient ways for computing the correction term are introduced and the accuracy of the approach is discussed on several waveguide structures and filters.

Modeling of the scattering of electromagnetic waves by a thin dielectric coating on a cylinder

Double-sided boundary conditions containing only tangential components of a diffracted field are used to model the interaction of electromagnetic waves with a cylinder of arbitrary cross section covered with a thin dielectric layer. The obtained boundary-value problem is reduced to a system of two singular integral equations of the second kind with kernels whose structure is similar to the kernels of integral equations of the first kind for a perfectly conducting scatterer. The numerical solution of the integral equations of the problem is obtained by the method of mechanical quadratures. The scattering properties of an elliptic cylinder with different dielectric coatings are studied in the superhigh-frequency band. It is shown that the coating strongly affects the diffraction properties of the cylinder.

The Adaptive Cross Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems

This paper presents the adaptive cross approximation (ACA) algorithm to reduce memory and CPU time overhead in the method of moments (MoM) solution of surface integral equations. The present algorithm is purely algebraic; hence, its formulation and implementation are integral equation kernel (Green's function) independent. The algorithm starts with a multilevel partitioning of the computational domain. The interactions of well-separated partitioning clusters are accounted through a rank-revealing LU decomposition. The acceleration and memory savings of ACA come from the partial assembly of the rank-deficient interaction submatrices. It has been demonstrated that the ACA algorithm results in O(NlogN) complexity (where N is the number of unknowns) when applied to static and electrically small electromagnetic problems. In this paper the ACA algorithm is extended to electromagnetic compatibility-related problems of moderate electrical size. Specifically, the ACA algorithm is used to study compact-range ground planes and electromagnetic interference and shielding in vehicles. Through numerical experiments, it is concluded that for moderate electrical size problems the memory and CPU time requirements for the ACA algorithm scale as N/sup 4/3/logN.

Problem-matched basis functions for microstrip coupled slot arrays based on transmission line Green's functions (TLGF)

Problem matched basis functions are proposed for the method of moments analysis of printed slot coupled microstrips. The appropriate equivalent currents of the integral equation kernel are represented in terms of two sets of entire domain basis functions. These functions synthesize on one hand the resonant behavior of slots, microstrips or dipoles and on the other hand the field in proximity of the feeding source and of the discontinuities. In order to define these basis functions, canonical geometries are identified, whose Green's functions have been found in semi-analytical form. The accuracy and the effectiveness of the method in terms of convergence rate and number of unknowns is demonstrated by comparison with a standard fine meshing full-wave analysis. The method is extremely convenient for large arrays, where the subwavelength details should be treated together with large global dimensions. Since the proposed solution is independent of the dimensions of these details, it provides dramatic reduction of the number of unknowns and improvement of condition number.

Impact parameter dependence in the Balitsky–Kovchegov equation

We study an impact parameter dependence of solutions of the Balitsky–Kovchegov (BK) equation. We argue that if the kernel of the BK integral equation is regulated to cutoff infrared singularities, then it can be approximated by an equation without diffusion in impact parameter. For some purposes, when momentum scales large compared to Λ QCD are probed, the kernel may be approximated as massless. In particular, we find that the Froissart bound limit is saturated for physical initial conditions and seem to be independent of the cutoff as long as the cutoff is sufficiently large compared to the momentum scale associated with the large distance falloff of the impact parameter distribution.

MFIE MoM-formulation with curl-conforming basis functions and accurate kernel integration in the analysis of perfectly conducting sharp-edged objects

We present a novel technique to integrate analytically the highest-order terms of the Kernel of a low-order curl-conforming magnetic-field integral equation (MFIE) operator. In the computation of the bistatic RCS of moderately small perfectly conducting sharp-edged examples, we show that this curl-conforming choice yields very similar performance to that of the MoM-EFIE formulation and outperforms a MoM-MFIE formulation based on the RWG basis functions, both with very accurate Kernel integration. © 2005 Wiley Periodicals, Inc. Microwave Opt Technol Lett 44: 354–358, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20633

A near-field preconditioner and its performance in conjunction with the BiCGstab(ell) solver

This paper describes a simple physically-motivated near-field preconditioning scheme that is effective in accelerating convergence of surface, volume, and combined surface/volume integral equations for a broad variety of electromagnetic scattering problems. It can be easily implemented numerically in method of moment (MoM) solvers (both conventional and those employing matrix-compression techniques), irrespective of the analytical form of the integral-equation kernel. It has low memory and CPU requirements, both of which scale linearly with the number of unknowns, and is easily amenable to efficient parallelization. We demonstrate the preconditioner's performance (in conjunction with the BiCGstab(ell) iterative solver) on two representative geometries, and observe a significant reduction in the number of iterations required for convergence.