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# Related fields

There are quite some important fields related to flow visualization. Similar techniques, for example, are used in the area of tensor field visualization. A brief overview is given below:

Tensor field visualization -
beside flow data-sets also tensor data is examined (tensor fields provide multi-dimensional data usually represented by the use of matrices). Stress propagation within certain objects like engines, turbines, etc., produce tensor data. Simulation techniques that are similar to methods known from CFD are used to compute dense data-sets of volume tensors.

One way to intuitively describe a matrix in 3D is to represent it in terms of its eigenvectors and eigenvalues. Depending on whether the eigenvalues are real or complex, all different from each other or not, the eigenvectors build up either three characteristic directions, or one surface of rotational dynamics additional to one characteristic direction.

Hyper stream lines by Delmarcelle and Hesselink  are a visualization concept for tensor data based on the decomposition described above. Certain characteristic curves are integrated like stream lines for flow data. Such curves follow, for example, the direction of maximum stress propagation. See Fig. 2.11 of a typical image in this area. Other related fields are:

Computational fluid dynamics (CFD) -
as data to be visualized often originates from flow simulations, techniques for simulating dynamics are strongly related to the field of flow visualization. Usually the domain of flow is subdivided into a grid of many small cells. Then, the equations of pressure, motion, etc., are solved locally. Various grid structures are used, e.g., regular grids, curvilinear grids, etc.

Mathematics / ordinary differential equations (ODEs) -
Lots of mathematical theory is available for the analysis of dynamical systems. The extraction of the topology of behavior is just one example. Finding critical points usually is simple compared to finding cycles or characteristic sub-sets of dimension one or higher. Advanced techniques like trapping regions must be used.

Numerics -
simulating the dynamics of flow requires careful computations and advanced numerical techniques. Especially numerical integration and numerical derivation of flow characteristics are crucial components within flow visualization techniques.

Sampling and reconstruction -
often flow data is given as a huge set of samples. Reconstructing the continuous solution from the discrete data is usually non-trivial and must be done carefully. Advanced interpolation and approximation techniques, for example, working on arbitrary grids, are necessary.
After some notes on terms and the local analysis of dynamical systems (Chapter 3), in Chapters 4, 5, 6, and 7 four approaches to the visualization of dynamical systems are described in detail. In each case a set of specific goals has to be met and thus rather different approaches emerged.    Next: Notes on the local Up: State of the art Previous: Flow visualization
Helwig Löffelmann, November 1998,
mailto:helwig@cg.tuwien.ac.at.