Dependent Maxwell's Equations Research Articles

Virtual elements for Maxwell's equations

We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities associated with de Rahm sequences of nodal, edge, and face virtual element spaces.

Developing a Time-Domain Finite Element Method for the Lorentz Metamaterial Model and Applications

In this paper, we propose a new time-domain finite element method for solving the time dependent Maxwell's equations coupled with the Lorentz metamaterial model. The Lorentz metamaterial Maxwell's equations are much more complicated than the standard Maxwell's equations in free space. Our fully discrete scheme uses edge elements to approximate the unknowns in space, and uses the leap-frog scheme in time discretization. Numerical stability and the optimal error estimate in the $$L^2$$L2 norm are proved for our proposed scheme. Extensive numerical results are presented to confirm the theoretical analysis and applications of our scheme to model many interesting phenomena happened when wave propagates in the Lorentz metamaterials. Examples include the convergence effect happened in the concave lenses formed by the negative refraction index metamatrials, and total reflection and total transmission observed in the zero index metamaterials.

A Polynomial Chaos Method for Dispersive Electromagnetics

AbstractElectromagnetic wave propagation in complex dispersive media is governed by the time dependent Maxwell's equations coupled to equations that describe the evolution of the induced macroscopic polarization. We consider “polydispersive” materials represented by distributions of dielectric parameters in a polarization model. The work focuses on a novel computational framework for such problems involving Polynomial Chaos Expansions as a method to improve the modeling accuracy of the Debye model and allow for easy simulation using the Finite Difference Time Domain (FDTD) method. Stability and dispersion analyzes are performed for the approach utilizing the second order Yee scheme in two spatial dimensions.

A new efficient 3D Discontinuous Galerkin Time Domain (DGTD) method for large and multiscale electromagnetic simulations

A new Discontinuous Galerkin Time Domain (DGTD) method for solving the 3D time dependent Maxwell's equations via the electric field intensity E and magnetic flux density B fields is proposed for the first time. It uses curl-conforming and divergence-conforming basis functions for E and B, respectively, with the same order of interpolation. In this way, higher accuracy is achieved at lower memory consumption than the conventional approach based on the field variables E and H. The centered flux and Riemann solver are both used to treat interfaces with non-conforming meshes, and both explicit Runge–Kutta method and implicit Crank–Nicholson method are implemented for time integration. Numerical examples for realistic cases will be presented to verify that the proposed method is a non-spurious and efficient DGTD scheme.

An MPI/GPU parallelization of an interior penalty discontinuous Galerkin time domain method for Maxwell's equations

In this paper we discuss our approach to the MPI/GPU implementation of an Interior Penalty Discontinuous Galerkin Time domain (IPDGTD) method to solve the time dependent Maxwell's equations. In our approach, we exploit the inherent DGTD parallelism and describe a combined MPI/GPU and local time stepping implementation. This combination is aimed at increasing efficiency and reducing computational time, especially for multiscale applications. The CUDA programming model was used, together with non‐blocking MPI calls to overlap communications across the network. A 10× speedup compared to CPU clusters is observed for double precision arithmetic. Finally, for p = 1 basis functions, a good scalability with parallelization efficiency of 85% for up to 40 GPUs and 80% for up to 160 CPU cores was achieved on the Ohio Supercomputer Center's Glenn cluster.

Unified Analysis of Leap-Frog Methods for Solving Time-Domain Maxwell’s Equations in Dispersive Media

In this paper, we consider the time dependent Maxwell's equations resulting from dispersive medium models. First, the stability and Gauss's law are proved for all three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Then leap-frog mixed finite element methods are developed for these three models. Optimal error estimates are proved for all three models solved by the lowest-order Raviart-Thomas-Nedelec spaces. Extensions to multiple pole dispersive media are presented also. Numerical results confirming the analysis are presented.

Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier Singular Complement Method

We present an efficient method for computing numerically the solution to the time- dependent Maxwell equations in an axisymmetric domain, with arbitrary (not necessarily ax- isymmetric) data. The method is an extension of those introduced in (20) for Poisson's equation, and in (4) for Maxwell's equations in the fully axisymmetric setting (i.e., when the data is also axisymmetric). It is based on a Fourier expansion in the azimuthal direction, and on an improved variant of the Singular Complement Method in the meridian section. When solving Maxwell's equations, this method relies on continuous approximations of the fields, and it is both H(curl)- and H(div)-conforming. Also, it can take into account the lack of regularity of the solution when the domain features non-convex edges or vertices. Moreover, it can handle noisy or ap- proximate data which fail to satisfy the continuity equation, by using either an elliptic correction method or a mixed formulation. We give complete convergence analyses for both mixed and non-mixed formulations. Neither refinements near the reentrant edges or vertices of the domain, nor cutoff functions are required to achieve the desired convergence order in terms of the mesh size, the time step and the number of Fourier modes used.

Premetric Electrodynamics

Classical electrodynamics can be divided into two parts. In the first one, with the use of a plenty of directed quantities, namely multivectors and differential forms, no scalar product is necessary. It is called premetric electrodynamics. In this part, principal laws of the theory can be tackled, among them the two observer-independent Maxwell's equations. The second part concerns specific media and requires establishing of a scalar product and, consequently a metric. We present an axiomatic approach to electrodynamics in which the metric enters as late as possible. Also a line of research is mentioned in which the notion of non-inertial observer is studied and its influence on observer- dependent Maxwell equations.

Error analysis of finite element methods for 3-D Maxwell's equations in dispersive media

We consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. A semi-discrete standard finite element method and a semi-discrete mixed finite element method are developed and error estimates are provided. We believe this is the first finite element error analysis carried out for Maxwell's equations in dispersive media.

PARALLELIZATION AND PERFORMANCE OF A 3-D ELECTROMAGNETIC PLASMA PIC CODE

International Journal of Computational Engineering ScienceVol. 03, No. 02, pp. 117-127 (2002) No AccessPARALLELIZATION AND PERFORMANCE OF A 3-D ELECTROMAGNETIC PLASMA PIC CODESAIKAT SAHASAIKAT SAHADepartment of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Search for more papers by this author https://doi.org/10.1142/S1465876302000587Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractA fully 3 dimensional parallel explicit electromagnetic plasma Particle-in-Cell code is developed and is tested on 64 processors of SGI-Origin 2000 supercomputer at JPL / Caltech. The SGI-Origin 2000 allows one to run simulations as large as 256 × 128 × 32, i.e. approximately 106 grid cells and 18 × 106 particles of each species (electrons and ions) with 18 particles per cell. The particles in this code are pushed using the standard leapfrog scheme and the self-consistent electromagnetic fields are updated by solving the time dependent Maxwell's equations using the Finite-Difference-Time-Domain technique. We have adopted the one dimensional domain decomposition technique to parallelize the sequential electromagnetic particle-in-cell code. Testing and performance analyses of this code were done by generating a narrow current sheet in the system as seen in magnetic reconnection phenomenon.Keywords:Parallel3-DElectromagneticPlasmaPIC FiguresReferencesRelatedDetails Recommended Vol. 03, No. 02 Metrics History KeywordsParallel3-DElectromagneticPlasmaPICPDF download

Application of biorthogonal interpolating wavelets to the Galerkin scheme of time dependent Maxwell's equations

A family of biorthogonal interpolating wavelets has been applied to time-domain electromagnetic field modeling through the wavelet-Galerkin scheme. The scaling functions are the Deslauriers-Dubuc interpolating functions and the wavelets are the shifted and contracted version of the scaling functions. This set of bases yields a simple algorithm for the solution of Maxwell's equations in time domain due to their interpolation properties. The derivation of the algorithm is presented in this paper, followed by a series of numerical verifications on some resonant structures.

High‐order, leapfrog methodology for the temporally dependent Maxwell's equations

A high‐order, leapfrog integrator is presented and analyzed for Maxwell's time domain equations. The integrator is shown to be quite robust in terms of its efficiency and memory demands. Moreover, the integrator retains the original simplicity of the traditional second‐order, leapfrog integrator. Rigorous Fourier analyses are presented to quantify the dispersion, dissipation, and stability properties of the integrator. To limit the scope of the presentation, the 2×2, 2×4, 4×2, and 4×4 schemes are examined and compared. Here the first digit denotes temporal accuracy; the second digit denotes spatial accuracy. Numerical demonstrations, using the three‐dimensional rectangular waveguide as the object under test, are provided to further substantiate the theoretical analysis.

Hybrid finite element solutions of time dependent Maxwell's curl equations

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>