Current induced along horizontal wire above an imperfectly conducting half-space

The boundary element/exponential approximation technique for calculating the loaded straight wire horizontally located above a dissipative half-space is presented. The influence of a lossy ground is taken into account via Sommerfeld integrals appearing within the kernel of the electric field integral equation for thin wire. These integrals are computed by means of the exponential approximation technique. The resulting integral equation for loaded wire above an imperfect earth is solved by the boundary element method. Numerical results are obtained for current distribution along a resistively loaded dipole antenna and along a transmission line of a finite length.     

New numerical approach in the analysis of a thin wire radiating over a lossy half-space

The finite element formulation for a half-space electric field integral equation is described. Sommerfeld integrals, appearing in the kernel of the integral equation are calculated by means of exponential approximations. This approach shows advantages over the usual techniques. Obtained results are compared with other results available.     

On the use of piecewise linear wavelets for fast construction of sparsified moment matrices in solving the thin-wire EFIE

Multiresolution wavelet expansion technique has been successfully used in the method of moments (MoM), and sparse matrix equations have been attained. Solving boundary integral equations arising in electromagnetic (EM) problems by the wavelet-based moment method (WMM) involves a time-consuming double numerical integration for each entry of the resultant matrix which in turn can outweigh the advantages of achieving a sparse matrix. The paper presents an alternative computational model to speed up the WMM by excluding double numerical integrations in the evaluation of matrix elements. In this regard, pieces of linear wavelet bases are replaced by proper sinusoidal functions for which closed-form analytical expressions are available. In addition, by introducing approximate closed-form expressions for radiating EM fields of wavelet current elements, the thresholding procedure is modified so that one can compute only the matrix elements of interest. To demonstrate the effectiveness of the proposed method, the thin-wire electric field integral equation (EFIE) is numerically solved by non-orthogonal linear spline wavelet bases.     

On the spectrum of the electric field integral equation and the convergence of the moment method

Existing convergence estimates for numerical scattering methods based on boundary integral equations are asymptotic in the limit of vanishing discretization length, and break down as the electrical size of the problem grows. In order to analyse the efficiency and accuracy of numerical methods for the large scattering problems of interest in computational electromagnetics, we study the spectrum of the electric field integral equation (EFIE) for an infinite, conducting strip for both the TM (weakly singular kernel) and TE polarizations (hypersingular kernel). Due to the self‐coupling of surface wave modes, the condition number of the discretized integral equation increases as the square root of the electrical size of the strip for both polarizations. From the spectrum of the EFIE, the solution error introduced by discretization of the integral equation can also be estimated. Away from the edge singularities of the solution, the error is second order in the discretization length for low‐order bases with exact integration of matrix elements, and is first order if an approximate quadrature rule is employed. Comparison with numerical results demonstrates the validity of these condition number and solution error estimates. The spectral theory offers insights into the behaviour of numerical methods commonly observed in computational electromagnetics. Copyright © 2001 John Wiley & Sons, Ltd.     

Improving the accuracy of the magnetic field integral equation with the use of a new impedance matrix element formulation

The electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) are widely used in conjunction with method of moments for electromagnetic scattering analysis of three-dimensional conducting objects with closed surfaces. However, the MFIE suffers from an accuracy problem compared with the EFIE with the use of the Rao?Wilton?Glisson (RWG) basis function. This accuracy problem is more serious for objects with sharp edges or corners. To solve this problem, a new technique to compute the impedance matrix elements (IME) of the MFIE using an RWG basis function is presented here. Details to compute the IME and the advantage of this new formulation are displayed. In addition, the relationship between this new IME formulation and the formulation using the low-order curl-conforming basis function for the MFIE is given. Through the computation of the RCS of several relatively small sharp-edged conducting objects, it is shown that the accuracy of the MFIE can be greatly improved by the use of the new IME formulation.     

Perturbed incomplete ILU preconditioner for efficient solution of electric field integral equations

To efficiently solve large, dense, complex linear systems that arise in the electric field integral equation (EFIE) formulation of electromagnetic scattering problems, a new modified incomplete LU (ILU) preconditioner is developed and used in the context of the generalised minimal residual iterative method accelerated with the multilevel fast multipole method. The key idea is to perturb the near-field impedance matrix of EFIE with the principle value term of the magnetic field integral equation operator before constructing ILU preconditioners. Numerical experiments indicate that this new perturbation technique is very effective with the ILU preconditioner and the resulted ILU preconditioner can reduce both the iteration number and the computational time substantially.     

Analytic preconditioners for the electric field integral equation

Since the advent of the fast multipole method, large‐scale electromagnetic scattering problems based on the electric field integral equation (EFIE) formulation are generally solved by a Krylov iterative solver. A well‐known fact is that the dense complex non‐hermitian linear system associated to the EFIE becomes ill‐conditioned especially in the high‐frequency regime. As a consequence, this slows down the convergence rate of Krylov subspace iterative solvers. In this work, a new analytic preconditioner based on the combination of a finite element method with a local absorbing boundary condition is proposed to improve the convergence of the iterative solver for an open boundary. Some numerical tests precise the behaviour of the new preconditioner. Moreover, comparisons are performed with the analytic preconditioner based on the Calderòn's relations for integral equations for several kinds of scatterers. Copyright © 2004 John Wiley & Sons, Ltd.     

Using MBPE technique to accelerate solving the thin-wire EFIE used in numerical simulation of lightning

The antenna theory (AT) model is widely used to numerically simulate the propagation of current wave along lightning return-stroke channels and compute the radiated electromagnetic fields. In this model, the return stroke channel is considered as a vertical monopole antenna above perfectly conducting ground for which the numerical solution of the governing electric field integral equation (EFIE) in the frequency domain by the conventional method of moment (MoM) is prohibitively slow. In this paper, a model-based parameter estimation (MBPE) technique is proposed to reduce the number of frequency-domain calculation points required for the evaluation of space-time current distribution along a lightning return stroke channel. In applying this technique to a rational function model for the channel current distribution, a uniform-like sampling strategy is investigated. In order to accelerate the building of the moment impedance matrix, the reciprocal closed-form mutual impedance of sinusoidal electric dipoles and the symmetry of the model are used. The proposed technique is validated against the conventional inverse fast Fourier transform algorithm which uses a MoM solution for all frequencies within the channel base current spectrum. It is shown that considerable computation efficiency is achieved in terms of CPU time without losing accuracy.     

A remark on the numerical solution of singular integral equations and the determination of stress-intensity factors

Summary As is well-known, an efficient numerical technique for the solution of Cauchy-type singular integral equations along an open interval consists in approximating the integrals by using appropriate numerical integration rules and appropriately selected collocation points. Without any alterations in this technique, it is proposed that the estimation of the unknown function of the integral equation is further achieved by using the Hermite interpolation formula instead of the Lagrange interpolation formula. Alternatively, the unknown function can be estimated from the error term of the numerical integration rule used for Cauchy-type integrals. Both these techniques permit a significant increase in the accuracy of the numerical results obtained with an insignificant increase in the additional computations required and no change in the system of linear equations solved. Finally, the Gauss-Chebyshev method is considered in its original and modified form and applied to two crack problems in plane isotropic elasticity. The numerical results obtained illustrate the powerfulness of the method.     

Application of the complete multiple reciprocity method for 3D impedance extraction with multiple frequency points

A complete multiple reciprocity method (CMRM), usually employed for the eigenvalue analysis of Helmholtz equation, is applied to impedance calculation of 3D electric structures for multiple frequency points. Based on a recently proposed boundary integral formulations for impedance calculation, the CMRM is used to separate the boundary integrals into the frequency-dependent and frequency-independent portions so as to accelerate the computation for multiple frequency points. A set of approaches is proposed to handle the severe numerical problems induced by the large varieties of distance r and frequency-dependent k, when applying the CMRM to impedance calculation. As a result, the near-field integrals are calculated with the inner product of a frequency-independent sequence and a frequency-dependent sequence, while the far-field integrals are calculated with an efficient approximate formula. Since the majority of the calculation for generating the overall linear equation system becomes reusable, the impedance extraction with multiple frequency points is greatly accelerated. Several typical structures of interconnects are calculated with the boundary element method combined with CMRM. Numerical results verify the accuracy and efficiency of the proposed methods.     

Vector potential integral formulation for eddy-current proberesponse to cracks

A boundary integral vector potential formulation has been developed to evaluate eddy-current interactions with three-dimensional finite cracks in conductors. The approach is compared with an electric field integral equation method also used for solving crack problems in eddy-current nondestructive evaluation. An important advantage of the vector potential integral formulation is that the kernel has a weak singularity, but a drawback is that two unknown functions must be found on the crack surface. One of these functions, the current dipole density, represents the effect of the crack in terms of an induced source, and the other function is a solution of the two-dimensional Laplace equation. By contrast, the source density alone is needed for a complete solution of the electric field integral equation. In order to determine the surface Laplacian for finite cracks of arbitrary shape, a general numerical solution utilizing the boundary element technique is introduced. Numerical predictions of the eddy-current probe response to a crack give good agreement with experimental measurements, supporting the validity of the formulation     

An electromagnetic model of lightning return stroke channel using electric field integral equation in time domain

We use an electromagnetic approach based on antenna theory (AT) to evaluate the lightning return stroke current as a function of time and height. The lightning channel is modeled as a lossy, straight, and vertical monopole antenna above a perfectly conducting ground, which is excited by a source voltage at the base of the channel. This voltage source is a function of the current assumed at the ground level and the input impedance of the monopole antenna. An electric field integral equation (EFIE) is employed to describe the electromagnetic behavior of the antenna. The numerical solution of EFIE by the method of moments in time domain provides the time-space distribution of the current along the lightning channel. This AT model with specified current at the channel base requires only two adjustable parameters, namely the return-stroke propagation speed and the channel resistance per unit length. To demonstrate the accuracy of the proposed model, we compare it to the most commonly used models in terms of the temporal and spatial distributions of channel current and predicted electromagnetic fields. We also present results to show the effectiveness of the model in the analysis of lightning-related problems dealing with complex structures. In this regard, the lightning induced overvoltages on the neighboring overhead lines and the lightning strikes to tall structures are investigated.     

A fast wavelet-based moment method for solving thin-wire EFIE

Solving the thin-wire electric field integral equation (EFIE) by the multiresolution wavelet expansion method involves a time-consuming double numerical integration for each nonzero element of the moment matrix which in turn can outweigh the advantages of achieving a sparse matrix. To speed up the matrix fill process in wavelet-based moment method codes, first, the triangular scaling functions of a nonorthogonal piecewise liner wavelet at the finest spatial resolution are appropriately replaced by sinusoidal dipoles for which mutual impedances are available in closed-form analytical expressions. The fast wavelet bases transform is then exploited to effectively transfer the resultant matrix equation to multiresolution wavelet domain. Numerical results obtained by the compactly supported semi-orthogonal linear B-spline wavelet demonstrate dramatic reduction of the overall solution time without any degradation in the accuracy of the final solution.     

An Interaction Integral Method for Computing Fracture Parameters in Functionally Graded Magnetoelectroelastic Composites

A contour integral method is developed for the computation of stress intensity, electric and magnetic intensity factors for cracks in continuously nonhomogeneous magnetoelectroelastic solids under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium are the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for the computation of the physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

Numerical treatment of finite part integrals in 2-D boundary element analysis with application infracture mechanics

For the solution of problems in fracture mechanics by the boundary element method usually the subregion technique is employed to decouple the crack surfaces. In this paper a different procedure is presented. By using the displacement boundary integral equation on one side of the crack surface and the hypersingular traction boundary integral equation on the opposite side, one can renounce the subregion technique.An essential point when applying the traction boundary integral equation is the treatment of the thus arising hypersingular integrals. Two methods for their numerical computation are presented, both based on the finite part concept. One may either scale the integrals properly and use a specific quadrature rule, or one may apply the definition formula for finite part integrals and transform the resulting regular integrals into the usual element coordinate system afterwards. While the former method is restricted to linear or circular approximations of the boundary geometry, the latter one allows for arbitrary curved (e.g. isoparametric) elements. Two numerical examples are enclosed to demonstrate the accuracy of the two boundary integral equations technique compared with the subregion technique.     

Orthotropic piezoelectric patches for control of symmetric laminates via an integral equation

Formulation of the problem for the feedback displacement control of a vibrating laminated plate with orthotropic piezoelectric sensors and actuators is given in terms of an integral equation. The objective is to develop a formulation which facilitates the numerical solution to obtain the eigenfrequencies and eigenfunctions of the piezo-controlled plate. The control is carried out via piezoelectric sensors and actuators which are of orthorhombic crystal class mm² with poling in the z direction. The initial formulation of the problem is given in terms of a differential equation which is the conventional formulation most often used in the literature. The conversion to an integral equation formulation is achieved by introducing an explicit Green’s function. Explicit expressions for the kernel of the integral equation are given and the method of solution using the new formulation is outlined. The solution technique involves approximating the integral equation with an infinite system of linear equations and using a finite number of these equations to obtain the numerical results.     

A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains

This paper presents a direct traction boundary integral equation method (TBIEM) for three-dimensional crack problems. The TBIEM is based on the traction boundary integral equation (TBIE). The TBIE is collocated on both the external boundary and one of the crack surfaces. The displacements and tractions are used as unknowns on the external boundary and the relative crack opening displacements (CODs) are introduced as unknowns on the crack surface. In our implementation, all the surfaces of the considered structure are discretized into discontinuous elements to satisfy the continuity requirement for the existence of finite-part integrals, and special crack-front elements are constructed to capture the crack-tip behavior. To calculate the finite-part integrals, an adaptive singular integral technique is proposed. The stress intensity factors (SIFs) are computed through a modified COD extrapolation method. Numerical examples of SIFs computation are presented to demonstrate the accuracy and efficiency of our method.     

Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids

A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity factors (SIF) and electrical displacement intensity factors (EDIF) are discussed by comparison with available analytical or numerical solutions.     

Fast iterative, coupled-integral-equation technique for inhomogeneous profiled and periodic slabs

A fast coupled-integral-equation (CIE) technique is developed to compute the plane-TE-wave scattering by a wide class of periodic 2D inhomogeneous structures with curvilinear boundaries, which includes finite-thickness relief and rod gratings made of homogeneous material as special cases. The CIEs in the spectral domain are derived from the standard volume electric field integral equation. The kernel of the CIEs is of Picard type and offers therefore the possibility of deriving recursions, which allow the computation of the convolution integrals occurring in the CIEs with linear amounts of arithmetic complexity and memory. To utilize this advantage, the CIEs are solved iteratively. We apply the biconjugate gradient stabilized method. To make the iterative solution process faster, an efficient preconditioning operator (PO) is proposed that is based on a formal analytical inversion of the CIEs. The application of the PO also takes only linear complexity and memory. Numerical studies are carried out to demonstrate the potential and flexibility of the CIE technique proposed. Though the best efficiency and accuracy are observed at either low permittivity contrast or high conductivity, the technique can be used in a wide range of variation of material parameters of the structures including when they contain components made of both dielectrics with high permittivity and typical metals.     

The radial integration method for evaluation of domain integrals with boundary-only discretization

In this paper, a simple and robust method, called the radial integration method, is presented for transforming domain integrals into equivalent boundary integrals. Any two- or three-dimensional domain integral can be evaluated in a unified way without the need to discretize the domain into internal cells. Domain integrals consisting of known functions can be directly and accurately transformed to the boundary, while for domain integrals including unknown variables, the transformation is accomplished by approximating these variables using radial basis functions. In the proposed method, weak singularities involved in the domain integrals are also explicitly transformed to the boundary integrals, so no singularities exist at internal points. Some analytical and numerical examples are presented to verify the validity of this method.