volume | PIER Journals

Abstract

A desire to unify the mathematical description of many physical theories, such as electrodynamics, mechanics, thermal conduction, has led us to understand that global (=integral) physical variables of eachth eory can be associated to spatial geometrical elements suchas points, lines, surfaces, volumes and temporal geometrical elements suchas instants and intervals. This association has led us to build a space-time classification diagram of variables and equations for each theory. Moreover, the possibility to express physical laws directly in a finite rather than differential form has led to the development of a computational methodology called Cell Method [12]. This paper discusses some practical aspects of this method and how it may overcome some of the main limitations of FDTD method in electrodynamics. Moreover, we will provide some numerical examples to compare the two methodologies.

1. Pauli, W., Elettrodinamica, (translation of Vorlesung Elektrodynamik), Boringhieri (ed.), 1964.

2. Yee, K. S., "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Transactions on Antennas and Propagation, Vol. 14, 302-307, 1966.
doi:10.1109/TAP.1966.1138693

3. Carver, K. R. and J. W. Mink, "Microstrip antenna technology," IEEE Transactions on Antennas and Propagation, Vol. 29, 2-24, 1981.
doi:10.1109/TAP.1981.1142523

4. Holland, R., "Finite difference solutions of Maxwell’s equations in generalized nonorthogonal coordinates," IEEE Transactions on Nuclear Science, Vol. 30, No. 6, 4689-4591, 1983.

5. Fusco, M., "FDTD algorithm in curvilinear coordinates," IEEE Transactions on Antennas and Propagation, Vol. 38, No. 1, 76-89, 1990.
doi:10.1109/8.43592

6. Lee, J. F., R. Palandech, and R. Mittra, "Modeling threedimensional discontinuites in waveguides using nonorthogonal FDTD algorithm," IEEE Transactions on Microwave and Techniques, Vol. 40, No. 2, 346-352, 1992.
doi:10.1109/22.120108

7. Lee, C. F., B. J. McCartin, R. T. Shin, and J. A. Kong, "A triangular-grid finite-difference time-domain method for electromagnetic scattering problems," Journal of Electromagnetics Waves and Applications, Vol. 8, No. 4, 449-470, 1994.

8. Madsen, N. K., "Divergence preserving discrete surface integral methods for Maxwell’s curl equations using non-orthogonal unstructured grids," Journal of Computational Physics, Vol. 119, 34-45, 1995.
doi:10.1006/jcph.1995.1114

9. Hano, M. and T. Itoh, "Three-dimensional time-domain method for solving Maxwell’s equations based on circumcenters of elements," IEEE Transactions on Magnetics, Vol. 32, No. 3, 946-949, 1996.
doi:10.1109/20.497398

10. Schuhmann, R. and T. Weiland, "FDTD on nonorthogonal grids withtriangular fillings," IEEE Transactions on Magnetics, Vol. 35, No. 3, 1470-1473, 1999.
doi:10.1109/20.767244

11. Taflove, A., Computational Electrodynamics:The Finite- Difference Time-Domain Method, Artech House, 1995.

12. Tonti, E., "Finite formulation of the electromagnetic field,", this volume.

Dr. M. Marrone