## The Finite Integration TechniqueThe Finite Integration Technique (FIT) is a consistent formulation for the discrete representation of Maxwell's equations on spatial grids. First proposed by Weiland [1] in 1977, the finite integration technique can be viewed as a generalization of the FDTD method. It is also similar to the finite element method. Weiland [1, 2] proposed exact algebraic analogues to Maxwell's equations that guarantee physical properties of computed fields and lead to a unique solution. By discretizing the integral form of Maxwell's equations on a pair of dual interlaced discretization grids, the finite integration technique generates so-called Maxwell's Grid Equations (MGEs) that guarantee the physical properties of computed fields and lead to a unique solution. (1) (2) (3) (4) where
Figure 3: Allocation of the voltage and flux components in the mesh. Due to the consistent transformation, the analytical properties of the fields are maintained resulting in corresponding discrete topological operators on the staggered grid duplet. The topology matrices , , and correspond to the curl- and the div- operators. The tilde means that the operator is performed on the dual grid. After discretization, the material property relations become (5) (6) (7) where M
and _{μ}M are matrices
describing the material properties. The relations in (1) - (4) are exact
on a given mesh, however, the material matrices contain the unavoidable
approximations of any numerical procedure. In addition, these matrices have diagonal
form [78]._{k}Employing a so-called leap-frog scheme which samples
values of (8) (9) The recursion is stable if the time step inside an equidistant grid is restricted by the Courant criterion to (10) The calculation of each further time step only requires one matrix-vector multiplication. Thus it has the advantage being an explicit algorithm. The FIT can be applied to different mesh types [3, 4]. On Cartesian grids, the time-domain FIT is equivalent to FDTD. ## References[1] T. Weiland,
"A discretization method for the solution of Maxwell's equations for
six-component fields", [2] T. Weiland,
"Time domain electromagnetic field computation with finite difference methods",
[3] B. Krietenstein, R. Schuhmann, P. Thoma, and T. Weiland,
"The perfect boundary approximation technique facing the challenge of high precision field
computation", in [4] M. Walter, I. Munteanu,
"FIT for EMC," |