Neilen Marais

Numerical Evaluation of High-Order Finite Element Time Domain Formulations in Electromagnetics

This paper compares three full-wave finite-element time-domain (FETD) formulations. The first is based on the vector wave equation; the others on Maxwells equations, viz. the EBHD formulation that discretizes E rarr, B rarr, H rarr and Drarr and the EB formulation that discretizes only E rarr and B rarr. The latter two formulations use a combination of 1- and 2-form discretization to avoid an auxiliary mesh. A novel method for making the EBHD formulation operational is presented. Conditions for finite-difference time-domain (FDTD)-like explicit operation are discussed. The formulations are compared numerically by solving a three-dimensional cavity and a rectangular waveguide using high-order field representations up to mixed fourth order. The error balance between time integration and field representation is investigated. Difficulties in making the EBHD formulation operational which have not previously been addressed in the literature are discussed and worked around. Novel numerical results show that the EBHD formulation has serious performance limitations.

Numerical evaluation of hierarchical vector finite elements on curvilinear domains in 2-D

When using the finite element method, computational efficiency may be improved by using higher order elements. Typical 2-D triangular rectilinear elements cannot exactly model curved geometries, negating the advantage of higher order elements when such geometries are modeled. Curvilinear elements can be applied to such problems, and have significantly better computational efficiency than rectilinear elements when higher order elements are used. Results are shown for elemental basis orders ranging from CT/LN through full fifth order in rectilinear and curvilinear domains.

Efficient High-Order Time Domain Hybrid Implicit/Explicit Finite Element Methods for Microwave Electromagnetics

Abstract An efficient high-order time domain hybrid implicit/explicit finite element method scheme is addressed. An earlier low-order implicit/explicit (finite element time domain–finite difference time domain hybrid is reviewed, and the need for a higher-order hybrid is motivated. The necessary components of a higher-order hybrid are described, including formulations, diagonalizable hexahedral elements, the block structure of the matrix and required groupings of degrees of freedom, and the hybrid hexahedral/tetrahedral mesh. Results presented using the full hybrid demonstrate excellent performance on a challenging waveguide filter problem.

Near-field to far-field transforms using PDE based methods

When approximating an electromagnetic scattering problem using a volume based finite element, finite difference or finite volume scheme, the far-field pattern is often calculated using a near-field to far-field transform. The focus of this paper is the implementation of both a surface integral and variational method in a locally developed FEniCS environment called SUCEM:FEM. The results obtained by these methods are then compared to an analytical far-field expression for an infinitesimal dipole.

High order explicit convolution free time-domain finite element PML implementation

This paper presents a simple, convolution free, uni-axial perfectly matched layer (UPML) implementation applicable to high-order, explicit finite element time-domain (FETD) solvers. While implementing the UPML for the general FETD case is fairly complex, a simple FDTD-inspired implementation can be derived for the special case of diagonalised Cartesian hexahedra (also called Lobatto-cells). The FDTD description is not directly applicable to Lobatto-cells; analysis in a discrete differential-forms framework results in a suitable FEM description. The resulting semi-discrete form is discretised in time using the leapfrog central-difference method, although an approximation (also present in the FDTD implementation) is made to avoid the need for time-convolution.

Validating a high-order time domain hybrid implicit/explicit FEM implementation

This paper discusses aspects of the verification and validation of a high-order time domain hybrid implicit/explicit FEM scheme, which also incorporates hybrid meshes. Methods to test both the spatial and temporal discretizations, as well as the hybrid mesh, are discussed.

A Comparison of Some Finite Element Time Domain Formulations in Electromagnetics

This paper compares three full-wave finite element time domain (FETD) formulations. The first is based on the vector wave equation; the others on Maxwells equations, viz. the EBHD formulation that discretises Eoarr, Boarr, Hoarr and Doarr and the EB formulation that discretises only Eoarr and Boarr. The latter two formulations use a combination of 1-form and 2-form discretisation to avoid an auxiliary mesh. A method for making the EBHD formulation operational is presented and conditions for finite difference time domain (FDTD)-like explicit operation are discussed. Numerical results for a three dimensional cavity and a coaxial transmission line show that the EBHD formulation has serious performance limitations.