Alternative Integral Equation Research Articles

A 3-D Tensorial Integral Formulation of Scattering Containing Intriguing Relations

We investigate the role of the electric field and its normal derivative in 3-D electromagnetic scattering theory. We present an alternative integral equation formulation that uses the electric field and its normal derivative as the boundary unknowns. In particular, we extend a traditional formulation that is used in 2-D scattering theory to three-dimensions. We uncover several intriguing relationships involving closed surface integrals of the field and/or its derivative. In order not to obscure the physical/geometric awareness, the derivation is made from a tensor calculus perspective.

Size-dependent resonance frequencies of cantilevered and bridged nanosensors

This paper studies transverse vibration of nanoscale cantilevered and bridged sensors carrying a nanoparticle. The nanoscale sensors are modelled as Euler–Bernoulli beams with surface effect and nanoparticle as a concentrated mass. Frequency equations of cantilevered and bridged beam-mass system are derived and exact resonance frequencies are calculated. An alternative Fredholm integral equation method is used to obtain an approximate explicit expression for the fundamental frequency for both cases. A comparison between the approximate and analytical results is made and the approximation accuracy is satisfactory. The influences of the residual surface stress, surface elasticity, and attached mass on the resonance frequencies and mode shapes are discussed. These results are useful to illustrate the surface phenomena and are helpful to design micro-/nano-mechanical sensors.

Alternative Surface Integral Equation Formulations for Characteristic Modes of Dielectric and Magnetic Bodies

In this paper, the physical meanings and various computing methods of characteristic modes (CMs) of dielectric and magnetic bodies are investigated. Two types of surface integral equation (SIE) formulations based on the dependent relationships between the equivalent surface electric and magnetic currents are proposed. Following the two types of SIE formulations, two generalized eigenvalue equations are established to determine the CMs of dielectric or magnetic bodies. This is in contrast to the approach using induced volume currents. It is found that the eigenvalues possess clear physical meanings similar to those of perfectly electric conducting bodies. The CMs can be used to describe the resonant features, including resonant frequencies and resonant field distributions. Numerical results are presented to validate the proposed formulations.

Solutions of plasmonic structures using the multilevel fast multipole algorithm

We consider accurate full-wave solutions of plasmonic problems using the multilevel fast multipole algorithm (MLFMA). Metallic structures at optical frequencies are modeled by using the Lorentz-Drude model, formulated with surface integral equations, and analyzed iteratively via MLFMA. Among alternative choices, the electric and magnetic current combined-field integral equation (JMCFIE) and the combined tangential formulation (CTF), which are popular integral-equation formulations for penetrable objects, are discretized with the conventional Rao-Wilton-Glisson functions and used to model plasmonic structures. We discuss electromagnetic interactions in plasmonic media and show how far-field interactions may be omitted for improving the efficiency without sacrificing the accuracy of results. © 2016 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:335–341, 2016.

Coherent scattering by a collection of randomly located obstacles – An alternative integral equation formulation

Scattering of electromagnetic waves by discrete, randomly distributed objects is addressed. In general, the non-intersecting scattering objects can be of arbitrary form, material and shape. The main aim of this paper is to calculate the coherent reflection and transmission characteristics of a slab containing discrete, randomly distributed scatterers. The integral representation of the solution of the deterministic problem constitutes the underlying framework of the stochastic problem. Conditional averaging and the employment of the Quasi Crystalline Approximation lead to a system of integral equations in the unknown expansion coefficients. Of special interest is the slab geometry, which implies a system of integral equations in the depth variable. Explicit solutions for tenuous media and low frequency approximations can be obtained for spherical obstacles.

Transverse vibration of free–free beams carrying two unequal end masses

Transverse vibration of free–free slender beams carrying two tip masses is studied. An exact frequency equation is derived and the natural frequencies are calculated. An alternative Fredholm integral equation approach is presented to determine resonant frequencies of a vibrating mass–beam system. Simple approximate expressions for fundamental frequencies with high accuracy are obtained. Analytical and approximate results of the frequencies are computed and compared. The effects of concentrated masses at two ends on the natural frequencies and mode shapes are discussed. The results are helpful to design micromechanical sensors and energy harvesters.

Analysis of High Intensity Radiated Field Coupling into Aircraft Using the Method of Moments

In this paper, electromagnetic effects arising from high intensity radiated fields (HIRF) within realistic aircraft structures are investigated using the method of moments (MoM) and experimental tests. This study focuses on simulation-to-measurement correlation, where two potential error sources are investigated: Accuracy of the numerical techniques and the generation of numerical models. It is shown that the conventional mixed potential electric field integral equation may lead to inaccurate results regarding the field penetration through apertures. An alternative integral equation is introduced which splits the computational domain into an interior and an exterior domain, improving the results for aperture problems. To analyze the accuracy and the range of applicability for these integral equations, a generic aircraft structure is investigated, both numerically and experimentally. Thereby, various modeling aspects required for a meaningful simulation of aircraft are discussed. Following these observations, a real aircraft is analyzed in a broad frequency range from 10 kHz to 1 GHz. It is demonstrated that accurate numerical techniques are as important as modeling and appropriate simplification of the considered structures.

Discontinuity relations for the AdS4/CFT3 correspondence

We study in detail the analytic properties of the Thermodynamic Bethe Ansatz (TBA) equations for the anomalous dimensions of composite operators in the planar limit of the 3D N=6 superconformal Chern–Simons gauge theory and derive functional relations for the jump discontinuities across the branch cuts in the complex rapidity plane. These relations encode the analytic structure of the Y functions and are extremely similar to the ones obtained for the previously-studied AdS5/CFT4 case. Together with the Y-system and more basic analyticity conditions, they are completely equivalent to the TBA equations. We expect these results to be useful to derive alternative nonlinear integral equations for the AdS4/CFT3 spectrum.

ALTERNATIVE TIME DOMAIN BOUNDARY INTEGRAL EQUATIONS FOR THE SCALAR WAVE EQUATION USING DIVERGENCE-FREE REGULARIZATION TERMS

In this short note alternative time domain boundary integral equations (TDBIE) for the scalar wave equation are formulated on a surface enclosing a volume. The technique used follows the traditional approach of subtracting and adding back relevant Taylor expansion terms of the field variable, but does not restrict this to the surface patches that contain the singularity only. From the divergence-free property of the added-back integrands, together with an application of Stokes' theorem, it follows that the added-back terms can be evaluated using line integrals defined on a cut between the surface and a sphere whose radius increases with time. Moreover, after a certain time, the line integrals may be evaluated directly. The results provide additional insight into the theoretical formulations, and might be used to improve numerical implementations in terms of stability and accuracy.

Weak penetration and radiation through apertures in conducting bodies of revolution

The simplest way to solve for penetration through small apertures in conducting surfaces is to treat the body as a scatterer and determine the interior field as the sum of the incident field and the scattered field contributed by the current induced on the surface of the body. However, it is well known that, if the aperture is very small or if the penetrated field is very weak this method yields very inaccurate results, which, in turn, prohibits proper design of electronic systems, especially for electromagnetic compatibility and interference. Previously, alternative integral equations formulations were proposed to remedy this problem and applied to two-dimensionalconducting cylinders with slots [1--2]. Application of these alternative techniques to three-dimensional conducting bodies of revolution (BOR) is studied in this work. In addition, the reciprocity principle is used to recast a weak penetration problem into a weak radiation problem, and it is shown that the alternative formulations useful for weak penetration are also useful for weak radiation. The important features and relative accuracies of each formulation together with numerical results are detailed for mock missile-shape structures.

Alternative integral equations for the iterative solution of acoustic scattering problems

This paper addresses the first step of the derivation of well-conditioned integral formulations for the iterative solution of exterior acoustic boundary-value problems. These new formulations are designed to be implemented in a fast multipole method coupled to a Krylov subspace iterative algorithm. Their construction is based on the on-surface radiation condition formalism. Both theoretical developments and numerical aspects are examined in detail. This approach can be viewed as a generalization of the usual Burton–Miller integral equations.

TBA equations for excited states in the sine–Gordon model

We propose thermodynamic Bethe ansatz (TBA) integral equations for multi-particle soliton (fermion) states in the sine–Gordon (massive Thirring) model. This is based on T-system and Y-system equations, which follow from the Bethe ansatz solution in the light-cone lattice formulation of the model. Even and odd charge sectors are treated on an equal footing, corresponding to periodic and twisted boundary conditions, respectively. The analytic properties of the Y-system functions are conjectured on the basis of the large volume solution of the system, which we find explicitly. A simple relation between the TBA Y-functions and the counting function variable of the alternative non-linear integral equation (Destri–deVega equation) description of the model is given. At the special value β2 = 6π of the sine–Gordon coupling, exact expressions for energy and momentum eigenvalues of one-particle states are found.

Seismic soil-structure interaction analysis by direct boundary element methods

Upon establishing the mathematical framework necessary for a proper understanding of the analytical theory, a regularized form of the conventional direct boundary integral equation formulation for three-dimensional elastodynamics is presented for a general anisotropic medium. Founded on the basis of a full decomposition of the Greens functions into regular and singular parts, the alternative boundary integral equation format is compact and without demand ingmathematical and numerical complexities such as Cauchy principal values. Extended to deal with general seismic soil-structure interaction problems in semi-infinite media, the formulation is implemented computationally together with a rigorous treatment of singular dynamic multi-layered viscoelastic half-space Greens functions and interfacial boundary tractions arising in typical soil-structure-foundation configurations. A set of new benchmark numerical results are included.

A versatile integral equation technique for magnetic modelling

A requirement currently exists in both mineral exploration and environmental or engineering geophysics for a technique to model the magnetic fields caused by bodies with large to extreme susceptibilities in which both induced and remanent magnetizations are significant. It is well known that modelling such magnetic fields is not amenable to any known approximation. It is a significantly difficult task that requires the solution of a magnetostatic boundary value problem. Analytical solutions to the problem are extremely useful for providing insight but generally of limited application in practical interpretation due to the geometrical complexity of real situations. Available numerical solutions include both volume and surface integral equation formulations. However neither of these are particularly efficient for the purpose. An alternative surface integral equation formulation is presented here which represents the required magnetic field in terms of a double layer over the surface of the body. The technique accommodates both remanent and induced magnetization and is generally applicable to any 3D body in a magnetic environment for which the Green's function is available. The present technique has significant advantages over other integral equation solutions in the geophysical literature. It is particularly economic in terms of the density of the surface discretization and consequently the computational effort. Moreover, it is extremely robust. It is found to yield accurate solutions for the type of thin bodies that cause numerical instability with other surface integral equation approaches.

Identification and control of distributed parameter systems by means of the singular value decomposition

In this paper the basic properties of the singular value decomposition (SVD) for integral equation models of distributed parameter systems (DPS) are presented in the context of process identification and model-based control. In addition, new methods of analysis and computation are described in which SVD-based techniques are used to provide a practical solution for the DPS identification and control problem. Once developed, these novel procedures are applied to a class of linear DPS which are conventionally described by parabolic partial differential equation (PDE) models. An alternative integral equation representation for these processes is used to provide a general and readily identifiable pseudomodal input/output model for a wide variety of such DPS, including both spatially self-adjoint (diffusive) and non-self-adjoint (convective/diffusive) systems It is demonstrated that this SVD-based internal equation model of DPS is a sound and convenient basis for pseudo-modal feedback control system designs. A later paper will extend these methods to allow for model-predictive controller designs and nonlinear DPS

Axisymmetric elastic postbuckling of circular plates

This paper deals with the elastic postbuckling behaviour of axisymmetric thin plates. The analysis is based on an alternative boundary integral equation formulation which has been applied successfully in connection with the large deflection analysis of plates and shells. Numerical results are obtained and compared with corresponding ones available in the literature. Further applications of the proposed formulation to more advanced problems are also discussed.

Modelling spontaneous mineralization potentials with a new integral equation - a reply

A requirement currently exists in both mineral exploration and environmental or engineering geophysics for a technique to model the magnetic fields caused by bodies with large to extreme susceptibilities in which both induced and remanent magnetizations are significant. It is well known that modelling such magnetic fields is not amenable to any known approximation. It is a significantly difficult task that requires the solution of a magnetostatic boundary value problem. Analytical solutions to the problem are extremely useful for providing insight but generally of limited application in practical interpretation due to the geometrical complexity of real situations. Available numerical solutions include both volume and surface integral equation formulations. However neither of these are particularly efficient for the purpose. An alternative surface integral equation formulation is presented here which represents the required magnetic field in terms of a double layer over the surface of the body. The technique accommodates both remanent and induced magnetization and is generally applicable to any 3D body in a magnetic environment for which the Green's function is available. The present technique has significant advantages over other integral equation solutions in the geophysical literature. It is particularly economic in terms of the density of the surface discretization and consequently the computational effort. Moreover, it is extremely robust. It is found to yield accurate solutions for the type of thin bodies that cause numerical instability with other surface integral equation approaches.

An alternative fredholm integral equation approach for multiple crack problem and multiple rigid line problem in plane elasticity

An alternative Fredholm integral equation approach for the multiple crack problem and the multiple rigid line problem in plane elasticity is suggested in this paper. After using some operators on both sides of the singular integral equation for the relevant problem, an alternative Fredholm integral equation is obtainable. The solutions of the suggested integral equations satisfy the single-valuedness condition of displacement (for the multiple crack problem) and the zero resultant force condition (for the multiple rigid line problem), respectively. The obtained integral equation is compact in form and easy for computation. A technique for evaluating the rotation in the multiple rigid line problem is also proposed. Numerical examples are given to demonstrate the efficiency of the proposed approach.

Alternative field representations and integral equations for modeling inhomogeneous dielectrics

New volume and volume-surface integral equations are presented for modeling inhomogeneous dielectric regions. In particular, it is shown that materials with nontrivial permeability and permittivity can be modeled using a single unknown equivalent current or field component. The integral equations result in more efficient numerical implementations and should be useful in a variety of electromagnetic applications. >

On Two Alternative Field Solutions for Second-Order Wave Diffraction

The present paper set out a nonlinear boundary value problem involving the interaction of surface waves with large submerged structures in two dimensions. The problem is solved to second order by a time-stepping procedure on the basis of two alternative methods. Both of these involve the application of Green’s theorem, a Taylor series expansion of variables at the free surface and a perturbation series representation of the pertinent variables. Differences between the two solutions are associated with alternative integral equations applicable to second-order terms. The two second-order solutions are compared with each other and with previous theoretical and experimental results, and their applicability is assessed.