An advanced numerical model in solving thin-wire integral equations by using semi-orthogonal compactly supported spline wavelets (original) (raw)
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Wavelet-like bases for thin-wire integral equations in electromagnetics
Journal of Computational and Applied Mathematics, 2005
In this paper, wavelets are used in solving, by the method of moments, a modified version of the thin-wire electric field integral equation, in frequency domain. The time domain electromagnetic quantities, are obtained by using the inverse discrete fast Fourier transform. The retarded scalar electric and vector magnetic potentials are employed in order to obtain the integral formulation. The discretized model generated by applying the direct method of moments via point-matching procedure, results in a linear system with a dense matrix which have to be solved for each frequency of the Fourier spectrum of the time domain impressed source. Therefore, orthogonal wavelet-like basis transform is used to sparsify the moment matrix. In particular, dyadic and M-band wavelet transforms have been adopted, so generating different sparse matrix structures. This leads to an efficient solution in solving the resulting sparse matrix equation. Moreover, a wavelet preconditioner is used to accelerate the convergence rate of the iterative solver employed. These numerical features are used in analyzing the transient behavior of a lightning protection system. In particular, the transient performance of the earth termination system of a lightning protection system or of the earth electrode of an electric power substation, during its operation is focused. The numerical results, obtained by running a complex structure, are discussed and the features of the used method are underlined.
Wavelet-based efficient simulation of electromagnetic transients in a lightning protection system
IEEE Transactions on Magnetics, 2003
In this paper, a wavelet-based efficient simulation of electromagnetic transients in a lightning protection systems (LPS) is presented. The analysis of electromagnetic transients is carried out by employing the thin-wire electric field integral equation in frequency domain. In order to easily handle the boundary conditions of the integral equation, semiorthogonal compactly supported spline wavelets, constructed for the bounded interval [0 1], have been taken into account in expanding the unknown longitudinal currents. The integral equation is then solved by means of the Galerkin method. As a preprocessing stage, a discrete wavelet transform is used in order to efficiently compress the Fourier spectrum of the waveform, used as the current source that directly strikes the LPS. Time profiles of electromagnetic quantities are then obtained by using an inverse discrete fast Fourier transform algorithm. The model has been validated by comparing the results with computed and measured data found in technical literature. A good agreement has been found with a significant computational reduction. Index Terms-Electromagnetic transient analysis, integral equations, wavelet transform.
Modelling of straight thin wires using time-domain electric field integral equations
IEE Proceedings - Microwaves, Antennas and Propagation, 1994
The thin-wire time-domain electric field integral equation is formulated starting from the extended boundary condition theorem and including a first-order approximation for the charges at the end caps of the wire. The equation obtained is solved directly in the time domain by the moment method. The effects of this approximation and of changing the situation of the matching points on the segments of the wire are studied.
This paper deals with the application of wavelets in a computational tool, based on a modified version of the thin-wire electric field integrai equation (EFIE) in frequency domain, devoted to analyse transient performances of electric power substations grounding systems in a homogeneous soil. Time profiles of the electromagnetic quantities can be evaluated via discrete fast Fourier transform. In the numerical scheme, after having discretized the electromagnetic problem by applying the method of moments in a direct formulation, wavelet filter coefficients are directly employed to build a wavelet-matrix transform by using the concept of wavelet decomposition and reconstruction algorithms. The application of the wavelet-matrix transform to the moment matrix generated by the modified EFIE, produces a sparse matrix equation which can be efficiently solved by the conjugate gradient method. Moreover, a preconditioner which strongly reduce the number of iterations of the iterative solver adopted, has been built by using the same wavelet-matrix transform. The computational results obtained by running a complex earthing structure are discussed. I. INTRODUCTION Wavelet theory [1] is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines. Recently, the use of wavelets and wavelet-like basis functions in computational electromagnetics has received considerable attention for the numerical analysis of integrai equations. The integrai equation approaches have the merit of reducing the solution domain to a small finite region containing boundary conditìons implicitly. However, the major drawback of this method is that the integrai equation formulation specifies the global interactions among sources, object and fields; by usually applying the method of moments in a direct formulation, this leads to a dense matrix equation. For small to medium scale problems, the integrai equation approach enables efficient numerical solutions. But, for large scale problems, the dense matrix equation leads to computations which are often prohibitive.
A study of wavelets for the solution of electromagnetic integral equations
IEEE Transactions on Antennas and Propagation, 1995
The use of wavelet basis functions for the e cient solution of electromagnetic integral equations is studied. It has previously been demonstrated that the use of wavelets for expansion and testing functions produces a sparse moment-method matrix. Here, this e ect is examined and analyzed in terms of the radiation/receiving characteristics of the wavelet basis functions. The limitations of wavelets as an e cient solution technique are discussed, and comparison is made to other fast algorithms.
IEEE Journal on Multiscale and Multiphysics Computational Techniques
In this work, a straight forward method of moments (MOM) procedure to solve the time domain integral equation (TDIE) is presented and applied to a wire-grid model of an arbitrarily-shaped conducting body. The conducting body is illuminated by a Gaussian plane wave. Contrary to all the available time domain algorithms, the present procedure does not involve marching in time thus eliminating error accumulation, a major source for late-time instability problem. The procedure presented in this work is conceptually simple, numerically efficient, and handles multiple excitations in a trivial manner, all the while remaining stable. The numerical procedure utilizes pulse functions for space variable and time-shifted Gaussian functions for time variable, respectively. Further, the numerical procedure adopts Galerkin method of solution implying the usage of same time and space functions for both expansion and testing. The numerical results obtained in the time domain are validated by comparing with the data obtained from the frequency domain solution at several frequencies and performing inverse discrete Fourier transform.
An integral equation is developed for determining the time-dependent current distribution on a wire structure excited by an arbitrary time-varying electric field. The sub-sectional collocation form of the method of moments is used to reduce this integral equation to a form that can be evaluated on a digital computer as an initial value problem. A Lagrangian interpolation scheme is introduced so that the dependent variables can be accurately evaluated at any point in the spacetime cone; thus, only mild restrictions on the space and time sample density are required. The integral equation relating present values of the current to previously computed values is presented in a form that can be directly converted into a computer code. Expressions are developed for the computer time and the relative advantages of time-domain and frequency-domain calculations are discussed, providing impetus for analyses in the time domain in certain cases. Part II of this paper will present well-validated numerical results obtained using the technique described.
Journal of Computational Physics, 1973
An integral equation is developed for determining the time-dependent current distribution on a wire structure excited by an arbitrary time-varying electric field. The subsectional collocation form of the method of moments is used to reduce this integral equation to a form that can be evaluated on a digital computer as an initial value problem. A Lagrangian interpolation scheme is introduced so that the dependent variables can be accurately evaluated at any point in the spacetime cone; thus, only mild restrictions on the space and time sample density are required. The integral equation relating present values of the current to previously computed values is presented in a form that can be directly converted into a computer code. Expressions are developed for the computer time and the relative advantages of time-domain and frequency-domain calculations are discussed, providing impetus for analyses in the time domain in certain cases. Part II of this paper will present well-validated numerical results obtained using the technique described.