Whitney elements time domain (WETD) methods

Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2004

Maxwell's curl equations in the time domain are solved using an explicit linear finite-element approach implemented on unstructured tetrahedral meshes. For the simulation of scattering problems, a perfectly matched layer is added at the artificial far-field boundary, created by the truncation of the physical domain prior to the numerical solution. The complete solution procedure is parallelized. The computational challenges that are encountered when attempting simulations at higher frequencies suggest that the implementation of a hybrid algorithm could have certain advantages. The hybrid approach adopted uses a combination of the finite-element procedure and the well-known low operation count/low storage finite-difference timedomain method. Examples are included to demonstrate the numerical performance of the techniques that are described.

IEEE Transactions on Magnetics, 1998

A mixed finite-differencewhitney-elements timedomain (FDwE-TD) method is proposed for the analysis of transient electromagnetic field problems. The method consists in discretizing the spatial region in two parts composed respectively by structured and unstructured elements. In the structured mesh Maxwell's curl equations are numerically solved by Uee's algorithm while in the unstructured mesh the wave equation is solved in terms of electric field by the WETD method. The proposed method leads to an explicit-implicit solution scheme which is convenient to model curved boundaries and complex configurations without a significant increase of the computational cost with respect to the basic FDTD method. The FD/WE-TD method is a valid alternative of the subgridding algorithms and nonuniform grid models used in the FDTD calculations. The nonphysical wave reflection introduced by the finite elements i s calculated for simple canonical configurations. Index terms-FDTD, WETD, FEM, time domain numerical techniques, hybrid techniques.

Progress In Electromagnetics Research B, 2012

In this paper, we present a new model using a Fourdimensional (4D) Element-Oriented physical concepts based on a topological approach in electromagnetism. Its general finite formulation on dual staggered grids reveals a flexible Finite-Difference Time-Domain (FDTD) method with reasonable local approximating functions. This flexible FDTD method is developed without recourse to the traditional Taylor based forms of the individual differential operators. This new formulation generalizes both the standard FDTD (S-FDTD) and the nonstandard FDTD (NS-FDTD) methods. Moreover, it can be used to generate new numerical methods. As proof, we deduce a new nonstandard scheme more accurate than the S-FDTD and the known nonstandard NS-FDTD methods. Through some numerical examples, we validate this proposal, and we show the power and the advantage of this Element-Oriented Model.

International Journal of Numerical Modelling Electronic Networks Devices and Fields

This paper presents a 3D body-conforming finite element solution of the time-dependent vector wave equation. The method uses edge elements on tetrahedra for the electric field interpolation. This kind of element is suited to model Maxwell's equations since it only enforces tangential continuity of vector fields. For the discretization of time derivatives we use the Newmark method, which allows obtaining an unconditionally stable scheme with second-order accuracy. The Silver–Müller absorbing boundary condition is employed for the domain truncation in unbounded problems. Numerical results for some examples are provided to validate the presented method. Copyright © 2000 John Wiley & Sons, Ltd.

SEG Technical Program Expanded Abstracts 2009, 2009

We present a 3-D finite-element time-domain (FETD) algorithm for the simulation of electromagnetic (EM) diffusion phenomena. The algorithm simulates transient electric fields and time derivatives of the magnetic fields for a general anisotropic earth. In order to compute transient fields, the electric field wave equation is transformed into a system of ordinary differential equations (ODE) via a Galerkin method with Dirichlet boundary conditions. To ensure both numerical stability and an efficient time step size, the system of ODE is discretized in time using the implicit backward Euler scheme. The resultant FETD matrix-vector equation is solved using a sparse direct solver with a fill-in reducing algorithm. When advancing the solution in time, the algorithm adjusts the tine step by examining if or not a current step size can be doubled without affecting the accuracy of the solution. Instead of directly solving another FETD matrix-vector equation for transient magnetic fields, Faraday's law is employed to compute time-derivatives of magnetic fields only at receiver positions. The accuracy and efficiency of the FETD algorithm are demonstrated using time-domain controlled source EM (TD-CSEM) simulations.

IEEE Transactions on Magnetics, 2004

2013

We construct a parallel and explicit finite-element time-domain (FETD) algorithm for Maxwell equations in simplicial meshes based on a mixed E-B discretization and a sparse approximation for the inverse mass matrix. The sparsity pattern of the approximate inverse is obtained from edge adjacency information, which is naturally encoded by the sparsity pattern of successive powers of the mass matrix. Each column of the approximate inverse is computed independently, allowing for different processors to be used with no communication costs and hence linear (ideal) speedup in parallel processors. The convergence of the approximate inverse matrix to the actual inverse (full) matrix is investigated numerically and shown to exhibit exponential convergence versus the density of the approximate inverse matrix. The resulting FETD time-stepping is explicit is the sense that it does not require a linear solve at every time step, akin to the finite-difference time-domain (FDTD) method.

2007

IEEE Transactions on Antennas and Propagation

A discontinuous Galerkin time-domain (DGTD) method based on dynamically adaptive Cartesian meshes (ACM) is developed for a full-wave analysis of electromagnetic fields in dispersive media. Hierarchical Cartesian grids offer simplicity close to that of structured grids and the flexibility of unstructured grids while being highly suited for adaptive mesh refinement (AMR). The developed DGTD-ACM achieves a desired accuracy by refining non-conformal meshes near material interfaces to reduce stair-casing errors without sacrificing the high efficiency afforded with uniform Cartesian meshes. Moreover, DGTD-ACM can dynamically refine the mesh to resolve the local variation of the fields during propagation of electromagnetic pulses. A local time-stepping scheme is adopted to alleviate the constraint on the time-step size due to the stability condition of the explicit time integration. Simulations of electromagnetic wave diffraction over conducting and dielectric cylinders and spheres demonstrate that the proposed method can achieve a good numerical accuracy at a reduced computational cost compared with uniform meshes. For simulations of dispersive media, the auxiliary differential equation (ADE) and recursive convolution (RC) methods are implemented for a local Drude model and tested for a cold plasma slab and a plasmonic rod. With further advances of the charge transport models, the DGTD-ACM method is expected to provide a powerful tool for computations of electromagnetic fields in complex geometries for applications to high-frequency electronic devices, plasmonic THz technologies, as well as laser-induced and microwave plasmas.

IEEE Transactions on Magnetics, 2000

We report on results concerning a discontinuous Galerkin time domain (DGTD) method for the solution of Maxwell equations. This DGTD method is formulated on unstructured simplicial meshes (triangles in 2-D and tetrahedra in 3-D). Within each mesh element, the electromagnetic field components are approximated by an arbitrarily high order nodal polynomial while, in the original formulation of the method, time integration is achieved by a second order Leap-Frog scheme. Here, we discuss about several recent developments aiming at improving the accuracy and the computational efficiency of this DGTD method in view of the simulation of problems involving general domains and heterogeneous media.