# The Finite Integration Technique and the MAFIA™ software package

**The Finite Integration Technique (FIT)**

The Finite Integration Technique (FIT) is a consistent formulation for the discrete representation of Maxwell's equations on numeric grids. It was introduced as early as 1977 by T. Weiland and represents ever since one of the most important fundaments for the development of algorithms for electromagnetic field simulation. The matrix equations for the electromagnetic integral quantities obtained by FIT possess the inherent properties of Maxwell's equations with respect to charge and energy conservation and thus ensure an especially favorable stability and convergence behavior in the numerical implementation. Another decisive advantage of this formulation in comparison with other methods is that it represents a comprehensive theory which can be successfully used within the whole spectrum of electromagnetic applications.

The Finite Integration Technique is also the basis for the research and development activities at TEMF. It is an important starting point for the development of new numerical simulation methods. Among them are topical research themes such as the development of higher-order FIT algorithms, subgridding, and “Reduced Order” Models for the system matrices of the FIT discretization.

Some representative publications on the Finite Integration Technique are:

- T. Weiland, “A discretization method for the solution of Maxwell's equations for six-component Fields”, Electronics and Communications AEUE, vol. 31, no. 3, pp. 116-120, 1977.
- U. van Rienen and T. Weiland, “Triangular discretization method for the evaluation of RF-Fields in cylindrically symmetric cavities”, IEEE Transactions on Magnetics, vol. MAG-21, no. 6, pp.2317-2320, 1985.
- T. Weiland, “Time domain electromagnetic Field computation with Finite Difference Methods”, International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, vol.9, pp. 259-319, 1996.
- R. Schuhmann, M. Clemens, P. Thoma, T. Weiland, “Frequency and Time Domain Computations of S-Parameters Using the Finite Integration Technique”, Proc. of the 12th Annual Review of Progress in Applied Computational Electromagnetics (ACES Conference), Monterey, 1996, pp. 1295-1302.
- M. Clemens, R. Schuhmann, T. Weiland, “Algebraic Properties and Conservation Laws in the Discrete Electromagnetism”, FREQUENZ, Band 53 (1999) , Ausg. 11-12, S. 219 – 225.
- R. Schuhmann and T. Weiland, “Conservation of Discrete Energy and Related Laws in the Finite Integration Technique”, Progress in Electromagnetic Research (PIER) Monograph Series: Special Issue on Geometric Methods for Computational Electromagnetics, Vol. 32, 2001, pp. 301-316.

**The MAFIA™ software package**

MAFIA™ is an interactive and generally applicable software package for the numeric simulation of electromagnetic fields. It is based on the FIT formulation of Maxwell's equations on structured grids. Most of the program components were developed over the last 20 years at the TEMF Laboratory. The first official version of MAFIA™ was released in 1983 and contained already a 3-D time domain solver and an eigenmode solver. In the meantime, MAFIA™ has become an internationally recognized simulation program, with more than 500 installations in universities, research institutes and High-Tech companies of over 25 countries.

MAFIA™ has a modular structure: It contains a preprocessor, a postprocessor, several solvers for various application types, as well as a graphical user interface with modern modelling capabilities. The various solvers and their application areas are listed below:

**S**: Electrostatic-magnetostatic problems, stationary currents, stationary heat conduction**T3**: 3D time domain solver in Cartesian coordinates. Offers greatest generality for solving waveguide and open problems.**TS3**: 3D time domain solver in Cartesian coordinates for coupled field-charged particles simulations**TS2**: 2D time domain solver in cylindrical coordinates for coupled field-charged particles simulations**E**: Eigenmode solver**W3**: Frequency domain solver in Cartesian and cylindrical coordinates**T2**: Time domain solver in structures with cylindrical symmetry**OO**: Optimisation module**A3**: 3D acoustics solver**H3**: Solution of the nonstationary heat conduction equations**TL3**: 3D time domain solver for low frequency problems

**© MAFIA™ is a protected trade mark of CST AG, Darmstadt**