A finite integration method for conformal, structured-grid, electromagnetic simulation
We describe a numerical scheme for solving Maxwells equations in the frequency domain on a conformal, structured, non-orthogonal, multi-block mesh. By considering Maxwells equations in a volume parameterized by dimensionless curvilinear coordinates, we obtain a set of tensor equations that are a continuum analogue of common circuit equations, and that separate the metrical and metric-free parts of Maxwells equations and the material constitutive relations. We discretize these equations using a new formulation that treats the electric field and magnetic induction using simple basis-function representations to obtain a discrete form of Faradays law of induction, but that uses finite integral representations for the displacement current and magnetic field to obtain a discrete form of Amperes law, as in the finite integration technique [T. Weiland, A discretization method for the solution of Maxwells equations for six-component fields, Electron. Commun. (AE U) 31 (1977) 116; T. Weiland, Time domain electromagnetic field computation with finite difference methods, Int. J. Numer. Model: Electron. Netw. Dev. Field 9 (1996) 295319]. We thereby derive new projection operators for the discrete tensor material equations and obtain a compact numerical scheme for the discrete differential operators. This scheme is shown to exhibit significantly reduced numerical dispersion when compared to the standard linear finite element method. We take advantage of the mesh structure on a block-by-block basis to implement these numerical operators efficiently, and achieve computational speed with modest memory requirements when compared to explicit sparse matrix storage. Using the JacobiDavidson [G.L.G. Sleijpen, H.A. van der Vorst, A JacobiDavidson iteration method for linear eigenvalue problems, SIAM J. Matrix Anal. Appl. 17 (2) (1996) 401425; S.J. Cooke, B. Levush, Eigenmode solution of 2-D and 3-D electromagnetic cavities containing absorbing materials using the JacobiDavidson algorithm, J. Comput. Phys. 157 (1) (2000) 350370] and quasi-minimal residual [R.W. Freund, N.M. Nachtigal, QMR: a quasi-minimal residual method for non-Hermitian linear systems, Numer. Math. 60 (1991) 315339] iterative matrix solution algorithms, we solve the resulting discrete matrix eigenvalue equations and demonstrate the convergence characteristics of the algorithm. We validate the model for three-dimensional electromagnetic problems, both cavity eigenvalue solutions and a waveguide scattering matrix calculation.
finite integration technique finite element methods eigenvalue problems