Elisa Francomano

An advanced numerical model in solving thin-wire integral equations by using semi-orthogonal compactly supported spline wavelets

In this paper, the semi-orthogonal compactly supported spline wavelets are used as basis functions for the efficient solution of the thin-wire electric field integral equation (EFIE) in frequency domain. The method of moments (MoM) is used via the Galerkin procedure. Conventional MoM directly applied to the EFIE, leads to dense matrix which often becomes computationally intractable when large-scale problems are approached. To overcome these difficulties, wavelets can be used as a basis set so obtaining the generation of a sparse matrix; this is due to the local supports and the vanishing moments properties of the wavelets. In the paper, this technique is applied to analyze electromagnetic transients in a lightning protection systems schematized as a thin-wire structure. The study is carried out in frequency domain; a discrete fast Fourier transform algorithm can be used to compute time profiles of the electromagnetic interesting quantities. The unknown longitudinal currents are expressed by using multiscale wavelet expansions. Thus, the thin-wire EFIE is converted into a matrix equation by the Galerkin method. Results for linear spline wavelets along with their comparison with conventional MoM that uses triangular basis functions and the point matching procedure are presented, for two case studies. Good agreement has been reached with a strong reduction of the computational complexity.

SOIL IONIZATION DUE TO HIGH PULSE TRANSIENT CURRENTS LEAKED BY EARTH ELECTRODES

This paper proposes a numerical model of the soil ionization phenomena that can occur when earth electrodes are injected by high pulse transient currents, as the one associated with a direct lightning stroke. Based on finite difference time domain numerical scheme, this model ascribes the electrical breakdown in the soil to the process of discharge in the air. In fact, as soon as the local electric field overcomes the electrical strength, the air in the voids trapped among soil particles is ionized, and the current is conducted by ionized plasma paths locally grown. The dimension of these ionized air channels is strictly dependent upon the local temperature. Thus, a local heat balance is enforced in order to obtain the time variable conductivity profile of the medium. This model can be implemented both for concentrated and extended electrodes, since no hypothesis has to be enforced about the geometric shape of the ionized region. Validation of the proposed model is obtained by comparing simulation results with experimental data found in technical literature.

A smoothed particle interpolation scheme for transient electromagnetic simulation

In this paper, the fundamentals of a mesh-free particle numerical method for electromagnetic transient simulation are presented. The smoothed particle interpolation methodology is used by considering the particles as interpolation points in which the electromagnetic field components are computed. The particles can be arbitrarily placed in the problem domain: No regular grid, nor connectivity laws among the particles, have to be initially stated. Thus, the particles can be thickened only in distinct confined areas, where the electromagnetic field rapidly varies or in those regions in which objects of complex shape have to be simulated. Maxwells equations with the assigned boundary and initial conditions in time domain are numerically solved by means of the proposed method. Validation of the model is carried out by comparing the results with those obtained by the FDTD method for a one-dimensional (1-D) case study in order to easily show the capability of the proposed scheme

A Meshfree Solver for the MEG Forward Problem

Non-invasive estimation of brain activity via magnetoencephalography (MEG) involves an inverse problem whose solution requires an accurate and fast forward solver. To this end, we propose the method of fundamental solutions as a meshfree alternative to the boundary element method (BEM). The solution of the MEG forward problem is obtained, via the method of particular solutions, by numerically solving a boundary value problem for the electric scalar potential, derived from the quasi-stationary approximation of Maxwells equations. The magnetic field is then computed by the Biot-Savart law. Numerical experiments have been carried out in a realistic single-shell head geometry. The proposed solver is compared with a state-of-the-art BEM solver. A good agreement and a reduced computational load show the attractiveness of the meshfree approach.

The method of fundamental solutions in solving coupled boundary value problems for M/EEG

The estimation of neuronal activity in the human brain from electroencephalography (EEG) and magnetoencephalography (MEG) signals is a typical inverse problem whose solution process requires an accurate and fast forward solver. In this paper the method of fundamental solutions is, for the first time, proposed as a meshfree, boundary-type, and easy-to-implement alternative to the boundary element method (BEM) for solving the M/EEG forward problem. The solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three-dimensional Laplace equation. Numerical accuracy, convergence, and computational load are investigated. The proposed method is shown to be a competitive alternative to the state-of-the-art BEM for M/EEG forward solving.

A marching-on in time meshless kernel based solver for full-wave electromagnetic simulation

A meshless particle method based on an unconditionally stable time domain numerical scheme, oriented to electromagnetic transient simulations, is presented. The proposed scheme improves the smoothed particle electromagnetics method, already developed by the authors. The time stepping is approached by using the alternating directions implicit finite difference scheme, in a leapfrog way. The proposed formulation is used in order to efficiently overcome the stability relation constraint of explicit schemes. In fact, due to this constraint, large time steps cannot be used with small space steps and vice-versa. The same stability relation holds when the meshless formulation is applied together with an explicit finite difference scheme accounted for the time stepping. The computational tool is assessed and first simulation results are compared and discussed in order to validate the proposed approach.

Wavelet-based efficient simulation of electromagnetic transients in a lightning protection system

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.

Numerical Investigations of an Implicit Leapfrog Time-Domain Meshless Method

Numerical solution of partial differential equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes in time. A predefined grid in the problem domain and a stability step size restriction need. Recently, the authors have reformulated the meshless framework based on smoothed particle hydrodynamics, in order to be applied for time domain electromagnetic simulation. Despite the good spatial properties, the numerical explicit time integration introduces, also in a meshless context, a severe constraint. In this paper, at first, the stability condition is addressed in a general way by allowing the time step increment get away from the minimum points spacing. Then, an alternating direction implicit leapfrog scheme for time evolution is proposed. The unconditional stability of the method is analytically provided and numerically validated. The stability of the method has been proved by avoiding the algebra developments related to the usually adopted von Neumann analysis. Three case studies are investigated by achieving a satisfactory agreement by comparing both numerical and analytical results.

On the use of a meshless solver for PDEs governing electromagnetic transients

In this paper some key elements of the Smoothed Particle Hydrodynamics methodology suitably reformulated for analyzing electromagnetic transients are investigated. The attention is focused on the interpolating smoothing kernel function which strongly influences the computational results. Some issues are provided by adopting the polynomial reproducing conditions. Validation tests involving Gaussian and cubic B-spline smoothing kernel functions in one and two dimensions are reported.

Unconditionally stable meshless integration of time-domain Maxwell's curl equations

Grid based methods coupled with an explicit approach for the evolution in time are traditionally adopted in solving PDEs in computational electromagnetics. The discretization in space with a grid covering the problem domain and a stability step size restriction, must be accepted. Evidence is given that efforts need for overcoming these heavy constraints. The connectivity laws among the points scattered in the problem domain can be avoided by using meshless methods. Among these, the smoothed particle electromagnetics, gives an interesting answer to the problem, overcoming the limit of the grid generation. In the original formulation an explicit integration scheme is used providing, spatial and time discretization strictly interleaved and mutually conditioned. In this paper a formulation of the alternating direction implicit scheme is proposed into the meshless framework. The developed formulation preserves the leapfrog marching on in time of the explicit integration scheme. Studies on the systems matrices arising at each temporal step, are reported referring to the meshless discretization. The new method, not constrained by a grid in space and unconditionally stable in time, is validated by numerical simulations.