Field Boundary Value Problems Research Articles

Core Energy Capacitance of NiZn Inductors

In the high-frequency (3–30 MHz) range, NiZn cores are commonly used. They have much lower permeability and permittivity than materials typically used at lower frequencies, including MnZn ferrites. Previously used capacitance models rely on the perfect electrical conductor assumption. They are not applicable when NiZn cores are used. We propose a general core energy capacitance model found by solving the electric field boundary value problem. We also propose a simplified model by curve fitting to finite element analysis data. Both are verified by simulation and experimental results in two case studies with rod and pot core structures.

On One Normal Boundary Value Problem of the Field Theory

One normal problem of the field theory on a plane for a system of Poisson equations is studied. This problem is one of implementations of the general approach to formulation of boundary value problems of the field theory, which is based on studying the kernels of trace operators and kernels of trace functionals, which are taken as the main space of desired solutions. A peculiarity of this problem is that one of the boundary conditions imposed on the desired solution is nonlocal in nature. The considered boundary condition nonlocality is due to the fact that the base space in which the solution of the problem is sought is the kernel of the regular trace functional in the Sobolev space. The need to supplement the problem on the one-dimensional cokernel of the above-mentioned functional, which arises in this case, determines the second boundary condition, which is local (pointwise) in nature. That is why, in the framework of the duality of the Sobolev space and its conjugate space, the solution to the problem is the pair (u, с), where с is a constant that compensates for the one-dimensional “defect” of the problem. It should be noted that the theory of boundary value problems for vector fields on a plane differs essentially from the theory of boundary value problems for three-dimensional vector fields. One of the reasons for the difficulties in formulating three-dimensional boundary value problems is that a number of important surfaces cannot be made parallel to each other, which rules out the possibility to consider “tangential” boundary value problems, that is, problems with boundary tangential fields. In this respect, the theory of boundary value problems of plane vector fields differs from the three-dimensional theory for the better. On the other hand, the fact that a formula that would interrelating the Laplace operator with first-order field theory operations is not available on a plane is an obstacle to identifying new non-standard boundary value problems for plane fields. Revealing of this interrelation is a separate issue, the solution of which leads to the concept of the trace of first-order field theory operators as a singular functional above the space of traces of functions from the first-order Sobolev space. Thus, boundary value problems for plane fields are a separate section of the general theory of field boundary value problems. One of the non-standard problems in the theory of plane vector fields, which contains non-local boundary conditions, is considered. A theorem for existence and uniqueness of a weak solution of the problem posed is formulated and proved. The proof of the theorem includes a few steps: using the Galerkin method, the existence of a solution to the equation is established; the passage to the limit is made, and the vector function u(x) is found; the number α is determined, and the uniqueness of the function u and the number α is established. The main result of the work is that the problem posed is correct in nature.

Finite element formulation for implicit magnetostrictive constitutive relations

Magnetostrictive materials that couple mechanical and magnetic domains have been widely explored for use in sensors and actuators. These materials often exhibit a nonlinear material response under applied magnetic fields, which limits the use of linear constitutive models. Furthermore, the nonlinear constitutive relations tend to be implicit in nature. Hence, a finite element scheme that can handle the implicit relationship between the mechanical (stresses and strains) and magnetic (magnetic flux density and magnetic field) quantities is proposed in order to arrive at solutions to boundary value problems. In the proposed scheme, while the physical requirements of equilibrium and strain–displacement relation are satisfied point-wise, the constitutive relations hold in a weak integral sense. A fully coupled magnetostrictive plane stress rectangular element is developed based on the proposed scheme and its efficacy in arriving at solutions to coupled field boundary value problems is illustrated by subjecting the element to standard loading conditions.

A Thrust Force Analysis Method for Permanent Magnet Linear Motor Using Schwarz–Christoffel Mapping and Considering Slotting Effect, End Effect, and Magnet Shape

A Schwarz–Christoffel (SC) mapping-based method is presented for accurately and efficiently predicting the thrust force of permanent magnet linear motors (PMLMs). It accounts for not only the slotting effect but also the end effect and the magnet shape. In this method, according to the magnet shape, the PMs are modeled as equivalent line currents on their surfaces. Through analyzing the invariant of the governing Poissonian field equation of line currents under a conformal mapping, the magnetic field boundary value problem in the air gap is mapped to a rectangle by the SC mapping, which is calculated with the SC Toolbox of MATLAB. Furthermore, the magnetic field solution is constructed with the line currents in a strip under the flux continuity condition. The cogging force due to the slotting effect and the end effect is obtained, respectively, by integrating the Maxwell stress tensor along the lines closely surrounding the slotted and finite-length iron core. The validity of the proposed method is examined by the finite-element method. The influences of the magnet shape on the thrust force pulsation characteristic are investigated for a PMLM equipped with rectangular PM and bread loaf PM.

Solving exterior Neumann boundary value problems for Beltrami fields through the Beltrami system

Solving exterior Neumann boundary value problems for Beltrami fields through the Beltrami system

Three-dimensional forward modeling for magnetotelluric sounding by finite element method

A finite element algorithm combined with divergence condition was presented for computing three-dimensional(3D) magnetotelluric forward modeling. The finite element equation of three-dimensional magnetotelluric forward modeling was derived from Maxwell’s equations using general variation principle. The divergence condition was added forcedly to the electric field boundary value problem, which made the solution correct. The system of equation of the finite element algorithm was a large sparse, banded, symmetric, ill-conditioned, non-Hermitian complex matrix equation, which can be solved using the Bi-CGSTAB method. In order to prove correctness of the three-dimensional magnetotelluric forward algorithm, the computed results and analytic results of one-dimensional geo-electrical model were compared. In addition, the three-dimensional magnetotelluric forward algorithm is given a further evaluation by computing COMMEMI model. The forward modeling results show that the algorithm is very efficient, and it has a lot of advantages, such as the high precision, the canonical process of solving problem, meeting the internal boundary condition automatically and adapting to all kinds of distribution of multi-substances.

Improved iterative algorithm for 3D edge FEM analysis of electromagnetic field boundary value problems

In this article, an improved iterative arithmetic of the symmetric successive over-relaxation preconditioning biconjugate-gradient algorithm (ISSOR-PBCG) is utilized to solve the 3D edge FEM equations derived from the time-harmonic electromagnetic-field boundary value problems. Several typical structures have been analyzed, and the computation time is compared with that of other algorithms such as biconjugate-gradient (BCG) algorithm and the conventional symmetric successive over-relaxation preconditioning biconjugate- gradient algorithm (SSOR-PBCG). The CPU time saved using the ISSOR-PBCG algorithm is nearly 27% and 65.5%, as compared with that using the SSOR-PBCG and the BCG algorithm. It can be seen that the ISSOR-PBCG algorithm is efficient for edge FEM equation sets derived from large-scale time-harmonic electromagnetic-field problems. © 2006 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2006.

A Field Theoretical Method for Analyzing Microwave Cavity with Arbitrary Cross-Section

In this paper a microwave cavity with an arbitrary cross-section is analyzed using the method of lines, and the correctness of the analytic method is demonstrated. A novel truncated circular microwave cavity is presented and its electromagnetic field distribution and resonances are calculated and analyzed. The analytic model in this paper has some advantages including less computational quantity, easy formulation, and extensive application. It can be used for researching electromagnetic field boundary value problems.

Analysis of Method of Lines for a Novel Open Truncated Circular-Groove Guide

Groove guide is a millimeter wave and sub-millimeter wave transmission line. It has some characteristics that are superior to the other transmission lines such as simple structure, easy manufacture for its large size, low loss, low dispersion and wide bandwidth etc., when operational frequency is higher than 100 GHz. The groove guide with arbitrary cross-section is analyzed by using the method of lines in this paper. The correctness of the analytic model for solving this guide is demonstrated. A novel truncated circular-groove guide is presented and its transmission characteristics are numerically calculated and analyzed. It can be consulted for theoretical researching of groove guide and disposing electromagnetic field boundary value problems. The analytic method in this paper has some virtues such as easy formulation, less computation and universal application etc.

Generalised method of lines for Helmholz equations by using discretisation technique of pseudospectral method

A novel semi-analytical higher-order numerical approach to solve electromagnetic field boundary-value problems is described. This pseudospectral-based method of lines is developed by combining a pseudospectral technique with the method of lines so that its solution is not only analytical along the line direction but also maintains high accuracy in the discrete direction. The pseudospectral-based discretisation strategy, distribution of collocation nodes, the global Fourier series interpolation of differential quadrature, a second-order difference matrix for various homogeneous boundary conditions, and the decoupling of coupled ordinary differential equations are all discussed in detail. Numerical results demonstrate that this pseudospectral method of lines can efficiently solve the boundary-value problems with higher accuracy when compared with the conventional method of lines. This method should be a competitive technique for modelling time-harmonic complex electromagnetic problems.

The generalized method of lines based on the discretization technique of the pseudospectral method

In this paper, a novel semianalytical higher order numerical approach to solve electromagnetic field boundary value problems is described. This pseudospectral-based method of lines is developed by combining the pseudospectral technique and the method of lines so that it not only has a high accuracy of the semianalytical technique in the line direction, but also maintains high accuracy in the discrete direction. The pseudospectral-based discretization technique, distribution of collocation nodes, global interpolation of the differential quadrature, second-order difference matrix for various homogeneous boundary conditions, and decoupling of coupled ordinary differential equations are discussed in detail. Our numerical results demonstrate that this higher order semianalytical method can efficiently solve the boundary value problems with fewer mesh lines when compared to the conventional method of lines. It primarily shows that this method will be a competitive technique to model time-harmonic complex electromagnetic problems. ©1999 John Wiley & Sons, Inc. Microwave Opt Technol Lett 20: 339–342, 1999.

Estimating ∇u in terms of divu, curlu, either (ν,u) or ν×u and the topology

In the present paper we prove Cα-estimates for ∇u using components of boundary values of u, div u, curl u and quantities given by components of boundary values of u as well as boundary values of elements belonging to de Rhams cohomology modules. The vector field u is defined on a bounded set G¯⊂ℝ3, meanwhile the cohomology group will be defined with regard to ℝ3−G. Our inequalities turn out to be a priori estimates concerning well-known boundary value problems for vector fields. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Mode coupling between ICRF waves propagating outside the B- Delta B plane

The conductivity tensor is obtained directly from the perturbed Vlasov equation for a plasma in an inhomogeneous magnetic field. The conductivity tensor elements are obtained consistently to all orders in the 'perpendicular wavenumbers' and to first order in the equilibrium magnetic field gradient. Conditions on the parallel wavenumber and the magnetic field gradient for which such a method is valid are given. The wave differential operator is obtained from the conductivity tensor using Maxwell's equations. The coupled differential equations are then truncated to second order to model to case of minority and second harmonic ion cyclotron heating. The forms of the electromagnetic and kinetic power flows and the minority and majority cyclotron damping are obtained simply. The differential equations are solved numerically as a linear combination of boundary value problems for the perturbing electric fields using standard NAG library routines, and the transmission, reflection and mode conversion are calculated for a range of ky and kz values. On setting ky to zero, excellent agreement is found between the results obtained from the authors' code and published results from codes by other authors.

A new method for solving Dyadic Green’s Function of electromagnetic field

A new method for solving electromagnetic field boundary value problem is given. By using this method, the boundary value problem of the vector wave equation can be transformed into the independent boundary value problem of scalar wave equations and the two additional vector differential operations. All the dyadic Green’s functions got by eigenfunction expansion of the dyadic Green’s function can be got by this method easily and some of the dyadic Green’s functions for complex systems which are very difficult to get by the ordinary method have been got by this new method. The dyadic Green’s function for a dielectric loaded cavity is one of the given examples.

Generating fields from data onH ? ?H +andH ? ?I +

There are series solutions for characteristic boundary value problems for fields on black hole backgrounds that converge when the data are given on ℋ=ℋ− ∪ ℋ+, or on ℐ=ℐ− ∪ ℐ+, but may not converge when the data are given on ℋ− ∪ ℐ−, or on ℐ+ ∪ ℐ+. We specialize to oscillatory data of frequency ω and calculate approximate reflection and transmission coefficientsR(ω) andT(ω), using a field generated by data on ℋ=ℋ− ∪ ℋ+, and again, using a field generated by data on ℋ− ∪ ℐ−. The first calculation gives qualitatively good results at all frequencies at each order of approximation, and quantitatively better results at higher orders of approximation; the second calculation, using the series which may not converge, gives bad results except at very high frequencies. Thus for the physically unnatural case of a field that vanishes on ℋ− and goes toe iωv on ℋ+ we have a series that is convergent, and uniformly so with respect to frequency, while for the natural case of a field that vanishes on ℋ− and goes toe iωv on ℐ− we are limited to high frequencies. It is argued that a frequency-dependent renormalization of a series of the first type provides an approximation scheme that is convergent, and uniformly so with respect to frequency, for the physically important problems of the second type. The difficulties posed by the ω-dependent renormalization for the study of incident pulses are discussed.

Anwendung eines “elementen-methode” -programmes zur losung von temperaturverteilungsproblemen

A computer program has been developed which can solve 2- or 3-dimensional structural problems and general boundary value problems for scalar and vector fields, based on the finite element techniques. As an example of its application to scalar fields, heat conduction problems for isotropic and anisotropic bodies are treated. It is also shown that the finite element method does not require the existence of a variational principle and can be applied to non-symmetric problems like temperature distributions in slowly moving fluids or to nonlinear problems like temperature distributions in a solid with temperature dependent thermal conductivity.

Three-dimensional nonlinear magnetic field boundary value problem and its numerical solution

An accurate solution to the nonlinear magnetic field boundary value problem which results when ferromagnetic materials are present is generally very difficult to obtain. Several computer codes based on a magnetic vector potential have been published in the literature. While those codes can be used to solve 2-dimensional problems, difficulties arise in their application to 3-dimensional problems. One difficulty is the complexity of the resulting differential equations which must be solved. This paper treats the more complex 3-dimensional solutions as derived from the concept of a scalar magnetic potential. The most desirable feature of this approach is that the magnetic material characteristics are incorporated directly into one second order nonlinear partial differential equation which can be solved on a digital computer using standard finite difference schemes. In order to add credence to the validity of the technique presented, a simple 3-dimensional magnetic field boundary value problem is posed and solved.

Nonlinear magnetic field solutions based on a vector magnetic potential

The earlier successful techniques for numerical solution of magnetic field problems required defining a field quantity and magnetic permeability over the problem grid model. Later improved but limited techniques were introduced whereby it was necessary to define only a field quantity, usually a potentia function, over the grid because the magnetic material characteristics were incorporated directly into the equations. The work presented in this paper represents a culmination of these latter efforts in that technique is presented whereby the general three-dimensional magnetic field boundary value problem with current regions can be solved numerically.

Solution of the Hall field boundary value problem by Fourier series

The paper describes a method for obtaining solutions of the Hall field in Fourier series. The method making use of functions of the complex variable is illustrated by the example of the Hall field on a region near to the front of a semi-strip.