A numerical meshless particle method in solving the magnetoencephalography forward problem

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this paper, a difierential wavelet-based operator deflned on an interval is presented and used in evaluating the electromagnetic fleld described by Maxwell's curl equations, in time domain. The wavelet operator has been generated by using Daubechies wavelets with boundary functions. A spatial difierential scheme has been performed and it has been applied in studying electromagnetic phenomena in a lossless medium. The proposed approach has been successfully tested on a bounded axial-symmetric cylindrical domain.

9 - Computational Electromagnetics: The Finite-Difference Time-Domain Method

This contribution studies a particular solution of Maxwell wave equations known as Hopfions by making use of the well-known Finite Difference Time Domain (FDTD) method. Hopfions are a family of solutions, in which the field lines are closed, forming different knot topologies that are preserved during its time evolution. These solutions have been studied in different analytical ways (e.g. [1]), but which to the best of our knowledge they have never been simulated before. Their simulation can be of interest because they are demonstrated to exist, not from purely analytical arguments, but from the direct numerical resolutions of the Maxwell's curl equations. This numerical solution would ease the study of how these fields could interact with other structures made of different materials, with other Hopfions, and on how they could be possibly be generated with antenna arrays.

The finite-difference time domain (FDTD) method is a numerical technique based on the finite difference concept. It is employed to solve Maxwell's equations for the electric and magnetic field distributions in both the time and spatial domains. The FDTD method utilizes the central difference approximation to discretize two of Maxwell's curl equations, namely, Faraday's and Ampere's laws, in both the time and spatial domains, and then solve the resulting equations numerically to derive the electric and magnetic field distributions at each time step and spatial point using the explicit leap-frog scheme. The FDTD solution, thus derived, is second-order accurate, although the difference formulation is first order, and is stable if the time step size is chosen to satisfy the special criterion.

Summary form only given. We present our recent results on the calculation of the distribution of the light scattered by red blood cells in the full range of scattering angle between 0 to 180 degrees using an accurate quantitative finite-difference-time-domain (FDTD) method. The FDTD algorithm allows computation of scattering from cells of any shape and with any index of refraction profile. The algorithm discretizes Maxwell's curl equations in space and time, resulting in a set of explicit finite-difference equations.

In this paper an efficient algorithm is presented for the development of compact and passive macro‐models of electromagnetic devices through the systematic reduction of the order of discrete models for these devices obtained through the use of finite elements. The proposed methodology is founded on a new finite element formulation that casts Maxwell's curl equations in a state‐space form. Such state‐space representations are very compatible with a class of robust model order reduction techniques based on Krylov subspaces. However, the advantage of this compatibility appears to be hindered by the fact that the state matrix of the discretized Maxwellian system is of dimension almost twice that obtained from the finite element approximation of the electromagnetic vector wave equation. It is shown in this paper that the apparent penalty in both memory and computation efficiency can be avoided by a proper selection of the finite element expansion functions used for the discretization of the electromagnetic fields. More specifically, it is shown that the proper selection of expansion functions renders the state‐space form of the Maxwellian system equivalent to the discrete problem obtained from the approximation of the vector wave equation using tangentially continuous vector finite elements. This equivalence is then used to effect Krylov‐based model order reduction directly from the finite element approximation of the vector wave equation. In particular, a passive model order reduction algorithm is used for this purpose. The proposed reduced order macro‐modelling algorithm is demonstrated through its application to a variety of microwave passive components. Copyright © 2002 John Wiley &amp; Sons, Ltd.

The time-harmonic Discontinuous Galerkin Method (DGM) is a flexible forward solver for inverse scattering problems in electromagnetics. Standard DGM discretizations of Maxwell's curl equations, simultaneously solving for both electric and magnetic fields, are generally more accurate but less efficient than the DGM applied to the electric vector wave equation. A natural extension is to mix DGM formulations such that either the curl equation formulation or the wave equation formulation is locally adopted to preserve accuracy. Herein we present such a mixed DGM formulation for time-harmonic scattering problems, where the formulation choice is locally based on the presence of electric or magnetic constitutive parameters that differ from the background medium.

A numerical method for predicting the sinusoidal steady-state electromagnetic fields penetrating an arbitrary dielectric or conducting body is described here. The method employs the finite-difference time-domain (FD-TD) solution of Maxwell's curl equations implemented on a cubic-unit-cell space lattice. Small air-dielectric loss factors are introduced to improve the lattice truncation conditions and to accelerate convergence of cavity interior fields to the sinusoidal steady state. This method is evaluated with comparison to classical theory, method-of-moment frequency-domain numerical theory, and experimental results via application to a dielectric sphere and acylindrical metal cavity with an aperture. Results are also given for a missile-like cavity with two different types of apertures illuminated by an axial-incidence plane wave.

In this paper, a multi-sphere particle method is built-up in order to estimate the solution of the Poisson's equation with Neumann boundary conditions describing the neuronal human brain&#13;\nactivity. The partial differential equations governing the relationships&#13;\nbetween neural current sources and the data produced by neuroimaging&#13;\ntechnique, are able to compute the scalp potential and magnetic field&#13;\ndistributions generated by the neural activity. A numerical approach is proposed with current dipoles as current sources and going on in the&#13;\ncomputation by avoiding the mesh construction. The current dipoles&#13;\nare into an homogeneous spherical domain modeling the head and the&#13;\ncomputational approach is extended to multilayered con ̄guration with&#13;\ndifferent conductivities. A good agreement of the numerical results is&#13;\nshown and, for the first time compared with the analytical ones.

Electromagnetic wave propagation in magnetized ferrites is modeled by solving Maxwell's time-dependent curl equations coupled with the equation of motion of the magnetization vector. A discretization approach based on the rotated Richtmyer finite-difference scheme is proposed. The new approach has been used to calculate the phase constants of transversally magnetized ferrite-loaded waveguides. The numerical dispersion equation for TE/sub n0/ modes is derived. The results obtained with this approach for a ferrite-filled and a ferrite-slab loaded waveguide are compared with those obtained with Yee's scheme extended for the treatment of ferrites and with the exact results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The time-explicit finite-difference time-domain method is used to simulate wave interaction in optical waveguide crossings at large angles. The wave propagation at the intersecting structure is simulated by time stepping the discretized form of the Maxwell's time dependent curl equations. The power distribution characteristics of the intersections are obtained by extracting the guided-mode amplitudes from these simulated total field data. A physical picture of power flow in the intersection is also obtained from the total field solution; this provides insights into the switching behavior and the origin of the radiations.

A posteriori error estimates for the problem of electrostatics with a dipole source

The purpose of the neuromagnetic inverse problem is to reconstruct primary neural current from measured MEG data. It is known that this inverse problem is ill-posed: uniqueness of the solution to the inverse problem is not guaranteed without a priori assumptions on the current source (Fokas et al. (2004)), and, even when using the source model that can be uniquely reconstructed, the obtained solution changes very sensitively depending on the noise contained in MEG data. Thus, employment of a stable algorithm is highly required in order that MEG becomes a non-invasive brain monitoring tool with not only high temporal resolution but also high spatial resolution. Basically, conventional methods are categorized into two groups: parametric approaches and imaging approaches. See the detailed list of references in Baillet et al. (2001). The former methods assume that the current source can be represented by a relatively small number of equivalent current dipoles (ECDs). This source model is shown to be uniquely reconstructed from radial MEG data, except the radial component of the dipole moment, when the head is assumed to consist of concentric spheres. The usual algorithm for this source model is the non-linear least-squares method that minimizes the squared error of data and the forward solution. An advantage of this algorithm is that the parametric description allows us the accurate estimation of the center position of the activated region. However, the problems of this algorithm are: 1) it requires an initial solution, 2) it requires an iterative computing of forward solution, 3) it is often trapped by the local minimum of the cost function, which gives a solution far from the true one, 4) estimation of the number of ECDs is difficult, and 5) spatial extent of the source is not considered. The secondmethods, imaging approaches, assume that there exist current dipoles at the nodes of artificial meshes on the cerebral surface, and solve a system of linear equations for the dipole moments at the fixed positions. An advantage of this method is that it can describe the spatial distribution of the neural current. However, the problems of this algorithm are: 1) the solution is not unique, 2) adding a regularization term often gives a unique but over-smoothed solution, 3) choice of the regularization parameters strongly affects the solution. Recently, a method with the multipolar representation of the source has been developed that incorporates some of the advantages of the above twomethods, and has attracted considerable attention (Irimia et al. (2009); Jerbi et al. (2002; 2004); Nara (2008a); Nolte et al. (1997; 2000)). In this model, instead of expressing the current source by an equivalent current dipole, 6

Introduction and problem formulation Elucidating how the human brain is structured and how it functions is a fundamental aim of human neuroscience. To achieve such an aim, the activity of the human brain has been measured using noninvasive neuroimaging techniques, the most popular of which is functional magnetic resonance imaging (fMRI) (Ogawa et al . 1990). The fMRI signals are obtained at a spatial resolution of typically 3mm and measure changes of blood flow and blood oxygen consumption whose temporal dynamics are slower than that of neuronal electrical activities, resulting in a poor temporal resolution of the order of seconds. In contrast, magnetoencephalography (MEG) and electroencephalography (EEG) can detect changes of neuronal activities by the millisecond measurement of magnetic and electric fields, respectively, outside the skull (Hamalainen et al . 1993; Nunez & Srinivasan 2006). The high temporal resolution of MEG (and EEG) is useful, especially for studying the dynamic integration of functionally specialized brain regions, which is a subject of growing interest in human neuroscience (de Pasquale et al . 2012). The major problem of MEG is that spatial brain activity patterns are not easily understandable from sensor measurements. This is because the magnetic fields produced by neuronal current sources are superimposed to form rather uninterpretable spatial patterns of signals on sensors. Estimating the position and intensity of these current sources from the sensor measurements is called source reconstruction, or source localization. Solving the source reconstruction problem allows the mapping of temporally dynamic electrical activities in the human brain (Baillet et al . 2001). Since how brain regions are dynamically integrated to produce a variety of functions is of great interest in human neuroscience research, the mission of MEG source reconstruction is not only to localize position of the current sources, but also to identify directed interactions between these sources. A possible approach to this involves constructing a dynamic model of brain electrical activities, as well as developing an estimation algorithm for the source positions and interactions that are parametrized in this model.