A Uniaxial Optimal Perfectly Matched Layer Method for Time-harmonic Scattering Problems

We develop a uniaxial optimal perfectly matched layer （opt PML） method for solving the time-harmonic scattering problems by choosing a particular absorbing function with unbounded integral in a rectangular domain. With this choice, the solution of the optimal PML problem not only converges exponentially to the solution of the original scatting problem, but also is insensitive to the thickness of the PML layer for sufficiently small parameter ε0. Numerical experiments are included to illustrate the competitive behavior of the proposed optimal method.

Fast Calculation of Electromagnetic Forces in IPMSMs Under PWM VSI Supply Based on Small-Signal Time-Harmonic Finite Element Method

This paper introduces a novel method for fast calculating the electromagnetic forces in interior permanent magnet synchronous machines(IPMSMs)under pulse width modulation(PWM)voltage source inverter(VSI)supply based on the small-signal time-harmonic finite element analysis(THFEA),which has been successfully utilized for fast calculating the PWMinduced losses in silicon steel sheets and permanent magnets.Based on the small-signal THFEA,the functional relationships between high-frequency harmonic voltages(HFHVs)and corresponding airgap flux densities are established,which are used for calculating the flux density spectra caused by each HFHV in the PWM voltage spectra.Then,the superposition principle is applied for calculating the flux density spectra caused by fundamental currents and all HFHVs,which are converted to the electromagnetic force spectra at last.The relative errors between the force density spectra calculated with the proposed method and those obtained from traditional time-stepping finite element analysis(TSFEA)using PWM voltages as input are within 3.1%,while the proposed method is 24 times faster than the traditional TSFEA.

与Time-harmonic Maxwell方程有关的Cauchy型积分算子的性质

首先介绍了与N算子有关的Cauchy型积分算子,并证明了以下三种情况下的Holder连续性:两点在边界上;一点在边界上,另一个在区域内(在区域外);两点在区域内(在区域外).接下来介绍了与Time-harmonic Maxwell方程有关的Cauchy型积分算子,研究了它与N矩阵算子有关的Cauchy型积分算子的关系.最后利用上述关系,给出了与Time-harmonic Maxwell方程有关的Cauchy型积分算子在上述三种情况下的Holder连续性.

On the Uniqueness Theorem of Time-Harmonic Electromagnetic Fields

The uniqueness theorem of time-harmonic electromagnetic fields, which is the theoretical basis of boundary value problem (BVP) of electromagnetic fields, is reviewed. So far there are many versions of the statements and proofs on the theorem. However, there exist some limitations and lack of strictness in these versions, for instance, the discussion of the uniqueness of solution without considering the existence of solution and the lack of strictness in the case of loss-less medium. In contrast with the traditional statements and proofs, this paper introduces some important conclusions on operator equation from modern theory of partial differential equation (PDE) and attempts to solve the problems on the existence and uniqueness of the solution to operator equation which is derived from Maxwell’s equations of time-harmonic electromagnetic fields. This method provides a novel and rigorous approach to discuss and solve the existence and uniqueness of the solution to time- harmonic fields in the new mathematical framework. Some important conclusions are presented.

Electric Potential in a Dielectric Sphere Head Produced by a Time-Harmonic Equivalent Current Dipole

A time-harmonic equivalent current dipole model is proposed to simulate EEG source which deals with the problem concerning the capacitance effect. The expressions of potentials in both homogeneous infinite dielectric medium and dielectric sphere on the electroquasistatic condition are presented. The potential in a 3-layer inhomogeneous spherical head is computed by using this model. The influences on potential produced by time-harmonic character and permittivity are discussed. The results show that potentials in dielectric sphere are affected by frequency and permittivity.

OPTIMAL ERROR ESTIMATES FOR NEDELEC EDGE ELEMENTS FOR TIME-HARMONIC MAXWELL'S EQUATIONS

In this paper, we obtain optimal error estimates in both L^2-norm and H（curl）-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell＇s equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L^2 error estimates into the L^2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec＇s second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS

In this paper,we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model.The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology.Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.

REALISTIC AND CORRECT MODELS OF IMPRESSED SOURCES FOR TIME-HARMONIC ELECTROMAGNETIC BOUNDARY VALUE PROBLEMS INVOLVING METAMATERIALS

The aim of this work is to analyze the role of the impressed sources in determining the well or ill-posedness of time harmonic electromagnetic boundary value problems involving isotropic effective media.It is shown,in particular,that,even if all interfaces are regular,the class of ill-posed problems can be very large in the presence of general square-integrable impressed sources.However,when a simple and realistic constraint is enforced on these sources,requiring that the support of the sources does not include any interface between a traditional medium and a metamaterial,among the problems here considered just those involving an interface between complementary materials remain ill-posed.These considerations have a very significant impact also on the approximability of the solution of well-posed problems since the numerical noise can introduce small fictitious sources even where the sources to be simulated are not present.These effects on finite element simulators are fully analyzed.Finally,we propose an algorithm that allows to obtain much better approximations of the solutions of the most critical wellposed problems.

A new alternating positive semidefinite splitting preconditioner for saddle point problems from time-harmonic eddy current models

Based on the special positive semidefinite splittings of the saddle point matrix, we propose a new Mternating positive semidefinite splitting （APSS） iteration method for the saddle point problem arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current problem. We prove that the new APSS iteration method is unconditionally convergent for both cases of the simple topology and the general topology. The new APSS matrix can be used as a preconditioner to accelerate the convergence rate of Krylov subspace methods. Numerical results show that the new APSS preconditioner is superior to the existing preconditioners.

A Two-Scale Asymptotic Analysis of a Time-Harmonic Scattering Problem with a Multi Layered Thin Periodic Domain

The scope of this paper is to show how a two-scale asymptotic analysis,based on a superposition principle,allows us to derive high order approximate boundary conditions for a scattering problem of a time-harmonic wave by a thin and tangentially periodic multi-layered domain.The periods are assumed of the same order of the thickness.New terms like memory effect and variance-covariance ones are observed contrarily to the laminar case.As a result,optimal error estimates are obtained.

Dynamic behavior of rectangular crack in three-dimensional orthotropic elastic medium by means of non-local theory

The dynamic behavior of a rectangular crack in a three-dimensional （3D） orthotropic elastic medium is investigated under a harmonic stress wave based on the non-local theory. The two-dimensional （2D） Fourier transform is applied, and the mixed- boundary value problems are converted into three pairs of dual integral equations with the unknown variables being the displacement jumps across the crack surfaces. The effects of the geometric shape of the rectangular crack, the circular frequency of the incident waves, and the lattice parameter of the orthotropic elastic medium on the dynamic stress field near the crack edges are analyzed. The present solution exhibits no stress singularity at the rectangular crack edges, and the dynamic stress field near the rectangular crack edges is finite.

Rotor End Factors for 2-D FEA of Induction Motors with Smooth or Slitted Solid Rotor

Modifying the equivalent rotor resistivity with rotor end factor in 2-dimension(2-D)finite element analysis(FEA)is an effective way to analyze the 3-dimension(3-D)solid rotor problems.For the smooth solid rotor,five different rotor end factors are discussed and compared with each other.It is theoretically clarified that the resistivity of rotor in 2-D FEA should be multiplied by the square of rotor end factors to take the 3-D end effect of solid rotor into account.For the slitted solid rotor,an improved rotor end factor is proposed based on the equivalent area algorithm of eddy currents in rotor,since the end factors of smooth solid rotor are not applicable.Finally,the time-harmonic finite element method(FEM)combined with the rotor end factor is applied to analyze the performance of solid rotor induction motor.The tested and computed results are in good agreement,which proves the effectiveness of rotor end factor for the simplication of 3-D solid rotor problems.

Dynamic Stress Intensity Factors for Three Parallel Cracks in an Infinite Plate Subject to Harmonic Stress Waves

Dynamic stresses around three parallel cracks in an infinite elastic plate that is subjected to incident time-harmonic stress waves normal to the cracks have been solved. Using the Fourier transform technique, the boundary conditions are reduced to six simultaneous integral equations. To solve these equations, the differences of displacements inside the cracks are expanded in a series. The unknown coefficients in those series are solved using the Schmidt method such that the conditions inside the cracks are satisfied. Numerical calculations are carried out for some crack configurations.

An hp Adaptive Uniaxial Perfectly Matched Layer Method for Helmholtz Scattering Problems

We propose an adaptive strategy for solving high frequency Helmholtz scattering problems.The method is based on the uniaxial PML method to truncate the scattering problem which is defined in the unbounded domain into the bounded domain.The parameters in the uniaxial PML method are determined by sharp a posteriori error estimates developed by Chen and Wu[8].An hp-adaptive finite element strategy is proposed to solve the uniaxial PML equation.Numerical experiments are included which indicate the desirable exponential decay property of the error.

THE FACTORIZATION METHOD FOR A MIXED SCATTERING PROBLEM FROM A BOUNDED OBSTACLE AND AN OPEN ARC

In this paper,we consider the scattering problem of time-harmonic electromagnetic waves from an infinite cylinder having an open arc Γ and a bounded domain D in R^2 as cross section.We focus on the inverse scattering problem,that is,to reconstruct the shape of Γ and D from the far-field pattern by using the factorization method.Through establishing a mixed reciprocity relation,we prove that the scatters Γ and D can be uniquely determined by the far-field pattern.Furthermore,the mathematical basis is given to explain that the factorization method is feasible to our problem.At the end of this paper,we give some numerical examples to show the efficaciousness of the algorithms.

Thin Layer Models for Electromagnetism

We present a review on the accuracy of asymptotic models for the scattering problem of electromagnetic waves in domains with thin layer.These models appear as first order approximations of the electromagnetic field.They are obtained thanks to a multiscale expansion of the exact solution with respect to the thickness of the thin layer,that makes possible to replace the thin layer by approximate conditions.We present the advantages and the drawbacks of several approximations together with numerical validations and simulations.The main motivation of this work concerns the computation of electromagnetic field in biological cells.The main difficulty to compute the local electric field lies in the thinness of the membrane and in the high contrast between the electrical conductivities of the cytoplasm and of the membrane,which provides a specific behavior of the electromagnetic field at low frequencies.