Time-harmonic Solutions Research Articles

Transient calculation of the induced currents inside the brain during magnetic stimulation

PurposeTransient calculation of currents in brain tissue induced during a transcranial magnetic stimulation treatment.Design/methodology/approachBecause of the short pulses used in this technique a time‐harmonic approximation is no longer valid, and transient effects have to be considered. We have performed a Fourier analysis of the induced currents calculated in a high‐resolution model of the brain using the extended scalar potential finite differences (Ex‐SPFD) approach.FindingsThe peak induced currents in the transient development of the pulse are higher by a factor of approximately seven than the time harmonic solutions at the fundamental frequency. Furthermore, an analysis of the impact of the conductivity dispersion revealed an increase in the peak induced currents by 17.3 percent for white matter and by 20.8 percent for gray matter.Originality/valueUsing the numerically efficient Ex‐SPFD approach, along with a high performance cluster, the current densities inside the brain can be calculated incorporating more details than previous calculations of this type.

Boundary Element Simulation of Thermal Waves

In this paper we survey computational techniques based on boundary integral formulations for the simulation of thermal waves. Time-harmonic solutions to diffusion problems appear in many physical situations of interest and give rise to many interesting problems related to material characterization, parameter assessment or detection of defects. We review the main direct, indirect and mixed integral numerical methods for a model of scattering of thermal waves by many obstacles and discuss how multiple scattering techniques and other physical tools can be understood as iterative methods or used as preconditioners. We also deal with some transient problems that can be solved with boundary element methods using the Laplace transform and with coupled finite and boundary element schemes for non-homogeneous obstacles

A finite-element analysis of reflection coefficient variability due to interface roughness

The effect of sea floor roughness on the reflection of sound diverging from a spherical source is investigated. An ensemble of water‐sediment interface realizations are modeled, each with height statistics matching those of a rough surface prepared in a laboratory experiment. Using the finite‐element method to solve the Helmholtz equation, time harmonic solutions from 40 to 60 kHz are calculated. From these frequency domain solutions, the band limited impulse response time domain solution is constructed via Fourier synthesis. The first two moments of the spherical wave reflection coefficient are determined as a function of grazing angle, and results are compared to measurements from the laboratory experiment. [Work supported by ONR, Ocean Acoustics.]

Asymptotics of Time Harmonic Solutions to a Thin Ferroelectric Model

We introduce new model equations to describe the dynamics of the electric polarization in a ferroelectric material. We consider a thin cylinder representing the material with thicknessɛand discuss the asymptotic behavior of the time harmonic solutions to the model whenɛtends to0. We obtain a reduced model settled in the cross-section of the cylinder describing the dynamics of the plane components of the polarization and electric fields.

Mathematical modelling of indirect measurements in scatterometry

In this work, we illustrate the benefits and problems of mathematical modelling and effective numerical algorithms to determine the diffraction of light by periodic grating structures. Such models are required for reconstruction of the grating structure from the light diffraction patterns. With decreasing structure dimensions on lithography masks, increasing demands on suitable metrology techniques arise. Methods like scatterometry as a non-imaging indirect optical method offer access to the geometrical parameters of periodic structures including pitch, side-wall angles, line heights, top and bottom widths. The mathematical model for scatterometry is based on the Helmholtz equation derived as a time-harmonic solution of the Maxwell equations. It determines the incident and scattered electric and magnetic fields, which fully specify the light propagation in a periodic two-dimensional grating structure. For numerical simulations of the diffraction patterns, a standard finite element method (FEM) or a generalized finite element method (GFEM) is used for solving the elliptic Helmholtz equation. In a first step, we performed systematic forward calculations for different varying structure parameters to evaluate the applicability and sensitivity of different scatterometric measurement methods. Furthermore our programs include several iterative optimization methods for reconstructing the geometric parameters of the grating structure by the minimization of a functional. First reconstruction results for different test data sets are presented.

Transient performance and loss analysis in solid wound rotor pulsed power generators

High rotation speed is the key to enhanced power density in rotating machines. High speed, shock, and construction considerations argue for solid rotors. However, output transient voltage requirements cause concern that changes in the field current will not be registered quickly in the stator. This paper examines different techniques for estimating the machine time constant, including an analytic solution of a smooth air gap machine and a transfer relation solution based on time harmonic solutions. It discusses an empirical solution and an equivalent surface current approach for predicting the loss in the rotor. These solutions compare favorably to rotational transient finite-element solutions.

Time harmonic solutions to a model of ferroelectric materials

We discuss the model equations of ferroelectric media, represented by a bounded and regular domain introduced by Greenberg in the case of time harmonic dependency of the solutions. The system reduces to a couple of nonlinear Maxwell's equations satisfied by the electromagnetic field ( H, E ) and the electric polarization vector P. By classical methods we prove the existence and uniqueness of the solutions for frequencies which are far from 0. We obtain the -regularity of the solutions when the polarization satisfies the boundary condition P× n = 0. We also consider the particular 2-D case deduced from the full model by using the transverse electric (TE) and transverse magnetic (TM) approximations. For the coupled nonlinear Hemholtz system we establish a general existence theory of the solutions for all frequencies and characterize their low frequency behaviour.

Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems

We will derive the fundamental generalized displacement solution, using the Radon transform, and present the direct formulation of the time-harmonic boundary element method (BEM) for the two-dimensional general piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM, which is formulated following the physical interpretation of Somigliana’s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the Stroh–Lekhnitskii (SL) formalism. The time-harmonic BEM is obtained by adding the boundary integrals for the dynamic regular part which, from the original double integral representation over the boundary element and the unit circle, are reduced to simple line integrals along the unit circle. The BEM will be applied to the determination of the eigen frequencies of piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. A comparative study will be made of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. It is found that the IRAM is faster and has more control over the solution procedure than the QZ algorithm. The use of the time-harmonic fundamental solution provides a clean boundary only formulation of the BEM and, when applied to the eigenvalue problems with IRAM, provides eigen frequencies accurate enough to be used for industrial applications. It supersedes the dual reciprocity BEM and challenges to replace the FEM designed for the eigenvalue problems for piezoelectricity.

Rapid calculations of time-harmonic nearfield pressures produced by rectangular pistons

A rapid method for calculating the nearfield pressure distribution generated by a rectangular piston is derived for time-harmonic excitations. This rapid approach improves the numerical performance relative to the impulse response with an equivalent integral expression that removes the numerical singularities caused by inverse trigonometric functions. The resulting errors are demonstrated in pressure field calculations using the time-harmonic impulse response solution for a rectangular source 5 wavelengths wide by 7.5 wavelengths high. Simulations using this source geometry show that the rapid method eliminates the singularities introduced by the impulse response. The results of pressure field computations are then evaluated in terms of relative errors and computational speeds. The results show that, when the same number of Gauss abscissas are applied to both approaches for time-harmonic pressure field calculations, the rapid method is consistently faster than the impulse response, and the rapid method consistently produces smaller maximum errors than the impulse response. For specified maximum error values of 10% and 1%, the rapid method is 2.6 times faster than the impulse response for pressure field calculations performed on a 61 by 101 point grid. The rapid approach achieves even greater reductions in the computation time for smaller errors and larger grids.

Hard transition to chaotic dynamics in Alfvén wave fronts

The derivative nonlinear Schrödinger (DNLS) equation, describing propagation of circularly polarized Alfvén waves of finite amplitude in a cold plasma, is truncated to explore the coherent, weakly nonlinear, cubic coupling of three waves near resonance, one wave being linearly unstable and the other waves damped. In a reduced three-wave model (equal dampings of daughter waves, three-dimensional flow for two wave amplitudes and one relative phase), no matter how small the growth rate of the unstable wave there exists a parametric domain with the flow exhibiting chaotic relaxation oscillations that are absent for zero growth rate. This hard transition in phase-space behavior occurs for left-hand (LH) polarized waves, paralleling the known fact that only LH time-harmonic solutions of the DNLS equation are modulationally unstable, with damping less than about (unstable wave frequency)2/4×ion cyclotron frequency. The structural stability of the transition was explored by going into a fully 3-wave model (different dampings of daughter waves, four-dimensional flow); both models differ in significant phase-space features but keep common features essential for the transition.

Influence of a Magnetically Permeable Surface Layer on Transient Fields for a Thin Circular Loop Antenna

The transient fields, in the time-domain, of a thin circular loop antenna, on a two-layered earth’s model are reexamined when the usually neglected magnetic permeability contrast is considered. It is shown that for a two-layered earth model, where the upper layer is permeable, the transient fields are modified over the nonpermeable case. The fields in the time domain are obtained as the inverse Laplace transforms of derived full wave time-harmonic solution. These time-domain solutions are obtained as a summation of waveguide modes plus contributions from branch cuts in the complex plane of the longitudinal wave number. The results should be useful for interpreting airborne electromagnetic systems and in cases where super-paramagnetic mineral constant is present.

Working Nonlinear Transient Eddy Current Problems With Time Harmonic Solutions

The classical approach to solving transient problems is a finite difference-based time-stepping procedure. A powerful alternative when motion is not involved is to use time harmonic solutions to build the transfer function of the system, appropriate for a spectrum of source excitations. This paper shows how nonlinear problems are approached with this method, and how the transfer function is derived using variable metrics and multivariate splines.

The Concurrent Complementary Operators Method Applied to Two-Dimensional Time-Harmonic Radiation and Scattering Problems

The concurrent complementary operators method (C-COM) has recently been applied to time-harmonic solution of Maxwell equations in two-dimensional space. In this paper, we present the application of the C-COM to the problem of electromagnetic scattering by conducting cylindrical objects in two-dimensional spaces. We show that the C-COM can effectively annihilate surface waves that impinge on the boundary at or near grazing incidence. The effectiveness of the C-COM is due to the fact that its wave absorption mechanism is independent of the wave number. To demonstrate the effectiveness and robustness of the C-COM in time-harmonic simulation, we present several numerical experiments. A strong agreement is obtained between the C-COM solution and the reference solutions even when the computational boundary is positioned only a fraction of a wavelength from the scatterer's body. Furthermore, we present a numerical experiment that shows the behavior of C-COM when waves impinge on the outer boundary at oblique incidence.

Oscillating Streaming Potential and Electro-osmosis of Multilayer Membranes

Artificial membranes consist of multilayers that have different physical or chemical properties. They are often considered as an equivalent single-layer membrane without taking into account the detail inside. Based on the Debye−Hückel approximation, we have provided analytical solutions of oscillating electrokinetic flow in multilayer membranes. Both pressure-driven flow (streaming potential) and electric-field-driven flow (electro-osmosis) were studied. The pressure and electric-field distributions in each layer can be obtained from our analytical solutions. This allows a better understanding of electrokinetic flow in multilayer membranes and benefits the design and selection of artificial membranes. The derived analytical solutions are useful for more-general time-dependent problems through a superposition of time-harmonic solutions weighted by appropriate Fourier coefficients. The properties of each membrane layer are reflected as complex quantities; therefore, a method is proposed to determine the electrokinetic properties (such as the zeta potential) of each layer by applying a high-frequency alternating electric field or pressure.

Characterization of porous membranes by zeta-potential under an ac electric field: analytical treatment of time-dependent electroosmostic flow

Electroosmosis has been used to determine zeta-potentials of porous membranes with nano-pores. In order to eliminate the influence of electrode polarization and concentration polarization on the membrane surface, one should employ an alternating electric field instead of a direct electric field. However, theoretical study of time-dependent electroosmosis flow is limited. In this work, we obtained an analytical solution of oscillating electroosmosis flow that is based on the Debye–Hückel approximation. The derived analytical solutions are useful for more general time-dependent problems through superposition of time-harmonic solutions weighted by the appropriate Fourier coefficients. We have provided a new formula for the determination of zeta-potentials for porous membranes under an alternating electric field. Our results suggest that the use of a steady-state equation to interpret zeta-potentials of nano-pores by means of the amplitude or plateau value of an oscillating sinusoidal and step function electric field can be misleading only when the frequency of oscillation is as high as 10 8 Hz.

Microfluid Flow in Circular Microchannel with Electrokinetic Effect and Navier's Slip Condition

Electrokinetic transport phenomena and slippage of fluid at the surface are often considered independently for description of microchannel flow. Anomalous cases of experimental results disagreeing with theoretical prediction have frequently been reported. By combining the electrokinetic effect and Navier's slip condition, we illustrate a possible explanation and present an analytical solution for oscillating flow in a circular microchannel. The derived analytical solution is useful for more general time-dependent problems through superposition of time-harmonic solutions weighted by appropriate Fourier coefficients. Our parametric results suggest that these two surface phenomena can be important for a better understanding of microfluid flow with hydrophobic channel walls.

A complete lumped equivalent circuit of three-phase squirrel-cage induction motors using two-dimensional finite-elements technique

This paper presents the use of the finite element technique for determining the parameters of a three-phase squirrel-cage induction motor. The common parameters, in addition to the core losses and ratio of number of tums, are obtained from the finite element field solutions. The magnetizing characteristic and core losses curve are used to determine the flux distribution within the motor structure. The linear time harmonic vector potential field solution is utilized for the inductances computation. The accuracy of the finite element application is verified using the available precise results.

Time-domain study of transient fields for a thin circular loop antenna

The transient fields in the time-domain of a thin circular loop antenna on a two-layer conducting earth model are expressed in analytical form. In these expressions, the displacement currents both in the two-layer ground and in the air region are taken into consideration. The closed-form expressions of the time-domain are obtained as the inverse Laplace transform of the derived full-wave time-harmonic solution. These time-domain solutions are obtained as a summation of wave-guide modes plus contributions from branch cuts in the complex plane of the longitudinal wave number. Numerical examples are given to indicate the important features in the wave forms of the surface fields due to step and pulsed current excitation. These features provide the means of detecting the earth's stratification, measuring the overburden height, and determining the ratio of the conductivities of the layers. PACS Nos.: 41.20Jb, 42.25Bs, 42.25Gy, 44.05+e

A Complete Lumped Equivalent Circuit of Three-Phase Squirrel-Cage Induction Motors Using the Two-Dinensional Finite Elements Technique

This paper presents the use of the finite element technique for determining the parameters of a three-phase squirrel-cage induction motor. The common parameters, in addition to the core losses and ratio of number of tums, are obtained from the finite element field solutions. The magnetizing characteristic and core losses curve are used to determine the flux distribution within the motor structure. The linear time harmonic vector potential field solution is utilized for the inductances computation. The accuracy of the finite element application is verified using the available precise results.

Designing localized electromagnetic fields in a source-free space.

An approach to characterizing and designing localized electromagnetic fields, based on the use of differentiable manifolds, differentiable mappings, and the group of rotation, is presented. By way of illustration, novel families of exact time-harmonic solutions to Maxwell's equations in the source-free space--localized fields defined by the rotation group--are obtained. The proposed approach provides a broad spectrum of tools to design localized fields, i.e., to build-in symmetry properties of oscillating electric and magnetic fields, to govern the distributions of their energy densities (both size and form of localization domains), and to set the structure of time-average energy fluxes. It is shown that localized fields can be combined as constructive elements to obtain a complex field structure with desirable properties, such as one-, two-, or three-dimensional field gratings. The proposed approach can be used in designing localized electromagnetic fields to govern motion and state of charged and neutral particles. As an example, motion of relativistic electrons in one-dimensional and three-dimensional field gratings is treated.