Weiying Zheng

An Adaptive Finite Element Method for the H- Ψ Formulation of Time-dependent Eddy Current Problems

In this paper, we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the H − ψ formulation of eddy current problems with multiply connected conductors. Multiply connected domains are considered by making “cuts”. The competitive performance of the method is demonstrated by an engineering benchmark problem, Team Workshop Problem 7, and a singular problem with analytic solution.

An Adaptive Multilevel Method for Time-Harmonic Maxwell Equations with Singularities

We develop an adaptive edge finite element method based on reliable and efficient residual-based a posteriori error estimates for low-frequency time-harmonic Maxwell equations with singularities. The resulting discrete problem is solved by the multigrid preconditioned MINRES (minimum residual) iteration algorithm. We demonstrate the efficiency and robustness of the proposed method by extensive numerical experiments for cavity problems with singular solutions which include, in particular, scattering over screens.

Electromagnetic Scattering by Unbounded Rough Surfaces

This paper is concerned with the analysis of electromagnetic wave scattering in inhomogeneous medium with infinite rough surfaces. Consider a time-harmonic electromagnetic field generated by either a magnetic dipole or an electric dipole incident on an infinite rough surface. The dielectric permittivity is assumed to have a positive imaginary part which accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Maxwell equations, with transparent boundary conditions proposed on plane surfaces with inhomogeneity in between. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. The perfectly matched layer (PML) method is investigated to truncate the unbounded rough surface electromagnetic scattering problem in the direction away from the rough surfaces. It is shown that the truncated PML problem attains a unique solution. An explicit error estimate is given between the solution of the sc...

Convergence of the Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems in Two-Layered Media

In this paper, we propose a uniaxial perfectly matched layer (PML) method for solving the time-harmonic scattering problems in two-layered media. The exterior region of the scatterer is divided into two half spaces by an infinite plane, on two sides of which the wave number takes different values. We surround the computational domain where the scattering field is interested by a PML with the uniaxial medium property. By imposing homogeneous boundary condition on the outer boundary of the PML, we show that the solution of the PML problem converges exponentially to the solution of the original scattering problem in the computational domain as either the PML absorbing coefficient or the thickness of the PML tends to infinity.

An Efficient Eddy Current Model for Nonlinear Maxwell Equations with Laminated Conductors

In this paper, we propose a new eddy current model for the nonlinear Maxwell equations with laminated conductors. Direct simulation of three-dimensional (3D) eddy currents in grain-oriented (GO) silicon steel laminations is very challenging since the coating film over each lamination is only several microns thick and the magnetic reluctivity is nonlinear and anisotropic. The system of GO silicon steel laminations has multiple sizes, and the ratio of the largest scale to the smallest scale can amount to

Finite Element Approximations to the Discrete Spectrum of the Schrödinger Operator with the Coulomb Potential

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Homogenization of Quasi-static Maxwell's Equations

. The new model omits coating films and thus reduces the scale ratio by 2--3 orders of magnitude. It avoids very fine or very anisotropic mesh in coating films and can reduce computations greatly in computing 3D eddy currents. We establish the well-posedness of the new model and prove the convergence of the solution of the original problem to the solution of the new model as the thickness of coating films tends to zero. The new model is validated by finite element computations of an engineering benchm...

A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

In the present paper, the authors consider the Schrodinger operator H with the Coulomb potential defined in R3m, where m is a positive integer. Both bounded domain approximations to multielectron systems and finite element approximations to the helium system are analyzed. The spectrum of H becomes completely discrete when confined to bounded domains. The error estimate of the bounded domain approximation to the discrete spectrum of H is obtained. Since numerical solution is difficult for a higher-dimensional problem of dimension more than three, the finite element analyses in this paper are restricted to the S-state of the helium atom. The authors transform the six-dimensional Schrodinger equation of the helium S-state into a three-dimensional form. Optimal error estimates for the finite element approximation to the three-dimensional equation, for all eigenvalues and eigenfunctions of the three-dimensional equation, are obtained by means of local regularization. Numerical results are shown in the last section.

Numerical solutions of the Schrödinger equation for the ground lithium by the finite element method

This paper studies the homogenization of quasi-static and nonlinear Maxwells equations in grain-oriented (GO) silicon steel laminations. GO silicon steel laminations have multiple scales, and the ratio of the largest scale to the smallest scale can be up to

An Adaptive FEM for a Maxwell Interface Problem

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