Modern numerical software packages have potential as development tools for research and educational purposes. One such package is FEMLAB ® , a finite element based environment with a userdefined, flexible equation solver. This paper describes the implementation and use of a vector Helmholtz solver in FEMLAB ® for educational purposes.Numerical modelling is unquestionably a major part of engineering design and analysis in the 21st century. Over the past ten years in particular, modelling based on finite element, finite volume, and finite difference analyses has progressively transformed the methods that engineers use in prototype design, research, and development. 1-7 Once the exclusive domain of researchers skilled in numerical science, these advanced modelling methods are now available for engineers and educators at all levels through ready-to-use software packages. State of the art commercial computer aided design (CAD) software for high frequency design problems includes field solvers, equivalent circuit solvers and some thermal analysis software. 7 FEMLAB ® by Comsol is mainly promoted as a research tool for solving partial differential equations (PDEs), in science and engineering. 1 It is a finite element based program with in-built mesh generator, solvers and post-processing facilities. Coupled problems can also be tackled using a multi-physics feature, which allows the flexible use of solutions from different physical equations. Unlike many other CAD software packages it is an open environment where the user can define problem-specific variational equations, which are then processed and solved using the built-in finite element routines.FEMLAB's inherently modular and open environment also provides an opportunity for augmenting educational courses on numerical methods with practical applications. Problems can be set which allow the student to start by transforming a boundary value problem into a variational expression, before using FEMLAB ® to solve the problem and to visualise the solutions. The student can examine the effects of boundaries and materials and explore a range of matrix solvers and applications. The purpose of this paper is to demonstrate how this can be done for the straightforward example of the vector Helmholtz equation in electromagnetic wave propagation.by guest on July 25, 2015 ije.sagepub.com Downloaded from 2 L. Y. Tio et al.

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SUMMARYA four-transverse-field formulation for a lossless Hermitian tensor material was introduced previously for an arbitrary direction of applied bias field. This method is extended here to include loss, which is critical in modelling experimental devices and which causes variational functionals for gyrotropic waveguides to become non-Hermitian. The functional is developed from Maxwell's equations before demonstrating its stationary properties at the boundary value problem solution. Using the finite element method, the functional is implemented and then validated against various waveguide structures. For the first time, the E t -H t formulation has been explicitly applied to lossy gyroelectric waveguide cross-sections. This method has important implications for the study and design of future phase shift and control components used in sub-millimeter wave and terahertz systems.