# Finite Integration with nonorthogonal grids

** Fundamentals:**

The Finite Integration method (FIT) is already in its classical formulation applicable to a large number of different grid types. Successful implementations are available, among others, for two- and three-dimensional Cartesian grids, grids in cylinder coordinates and special two-dimensional triangular grids. An important requirement for the use of the classical FI method is however the **orthogonality of primary and dual grids **i.e., the edges of one grid and the facets of the second one intersect at right angles. (Unlike other methods, there is however no restriction as to the angles between the edges of one grid.)

The goal of the research is to enhance the mesh-generation flexibility of the method by allowing nonorthogonal grid systems. Until now, the important class of structured nonorthogonal grids (i.e. whose topology corresponds to the one of a Cartesian system) was exhaustively studied.

**The nonorthogonal Finite Integration Technique (NFIT):**

When using such structured nonorthogonal grid systems, the matrix operators S and C of the FIT, which describe only the grid topology, can be used in unmodified form. It is olny the material matrices that need to be adapted: a one-to-one relation between a grid voltage and its corresponding grid flux is not possible anymore, due to the different directions of the grid edge and of the dual facet normal.

Instead, a new material operator is used, which first interpolates the neighbouring flux quantities and then, by using a local discrete metric, projects them on the new direction. In this process, it is absolutely necessary to ensure that this projection scheme leads to a symmetric material operator, as this is the only one that guarantees the stability of the whole method. To this aim, a local discretization scheme for the discrete metric was introduced.

This approach leads to a non-diagonal material matrix in block band format, with up to 8 extra-diagonals. These matrices can be used instead of the usual diagonal matrices in explicit time domain algorithms (HF), in 2D- and 3D eigenvalues problems, as well in quasi-static calculations.

**Grid generation:**

In order to be able to apply the nonorthogonal FI method to practical problems, a method for the generation of adapted nonorthogonal structured grids must be developed and implemented.

A **projection method** starts from a Cartesian basic grid and locally displaces the grid points on the structure's surface. The most important decision hereby is which points to move, since degenerated cells (with angles around 180°) can be avoided only through an appropriate criterion. A high precision of the geometric modelling can be ensured by using a generalized fill pattern for the cells, in which pyramid-shaped fillings are allowed, combined with an appropriate smoothing algorithm.

** Examples:**

2D-grid (part of a three-dimensional simulation) for modelling a coaxial waveguide. With very few grid points and a very smooth grid, a very good approximation of the conductor can be achieved. (The inner conductor has a square form in the problem description.)

Electric field on the surface of a skew waveguide. The discretization with a nonorthogonal grid almost perfectly models the structure's geometry. At equal desired precision (e.g. for the reflection parameter), the use of NFIT leads to computational time savings of several orders of magnitude in comparison with the classical method.

Electric field in a cylinder-symmetric cell of the TESLA accelerating structure (2D cut through an automatically generated 3D nonorthogonal grid). Again, the high-precision approximation of the geometry is evident, while the produced grid is almost equidistant and uses triangle-shaped partial fillings.

**Publications on this theme: **

- R. Schuhmann, T. Weiland: Stability of the FDTD Algorithm on Nonorthogonal Grids Related to the Spatial Interpolation Scheme. IEEE Transactions on Magnetics, Vol. 34,5, S. 2751-2754, 1998.
- R. Schuhmann, T. Weiland: A Stable Interpolation Technique for FDTD on Nonorthogonal Grids. Int. J. of Num. Modelling, Vol 11, No. 6, 1998, pp. 299-306.
- M. Hilgner, R. Schuhmann, T. Weiland: Advanced Generation of Structured Hexahedral Grids for Electromagnetic Field Computations with the Finite Integration Technique. ACES '00.