Variational Approach to Vector Field Decomposition

Konrad Polthier and Eike Preuß


For the feature analysis of vector fields we decompose a given vector field into three components: a divergence-free, a rotation-free, and a harmonic vector field. This Hodge-type decomposition splits a vector field using a variational approach, and allows to locate sources, sinks, and vortices as extremal points of the potentials of the components. Our method applies to discrete tangential vector fields on surfaces, and is of global nature. Results are presented of applying the method to test cases and a CFD flow.

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Features of vector fields strongly affect the characteristics of flows and their physical behavior. Among the most important features are vortices and pairs of sources and sinks. In many applications, vortices must be avoided to avoid energy losses, or sources and sinks must be located, for example to understand atmospheric behaviors. Although feature analysis is an important area only few technical tools are available for their detection and visualization.

A number of heuristic criteria are currently used for vortex detection []. Physical quantities of the underlying grid such as vortex magnitude and helicity are located at isolated vertices of the grid. Such local characterizations depend on the chosen neighbourhood and have deficiencies in regions with lower flow velocity. A slightly more global approach analyses the behaviour of streamlines and other integral curves. For example, geometric quantities derived from curvature properties of streamlines are used by Sadarjoen and Post [] to find vortex cores, and the polyhedral winding angle of a discrete streamline in is used to detect closed streamlines around a possible vortex core. Topological methods as introduced by Helman and Hesselink [] try to decompose vector fields in different global regions of interest by computing integral curves from critical points found by local linear approximations of the Jacobian. Higher-order approximations yield different decomposition []. For an overview of known methods in vector field visualization see Teitzel [].

In this paper we present a variational approach for the decomposition of a given vector field in different components, a divergence-free, a rotation-free, and a harmonic remainder. Instead of trying to define a discrete version of the Jacobian äx of the discrete vector field and its splitting in a stretching tensor S and a vorticity matrix W, we derive a Hodge type decomposition by minimizing certain energies. This more global point of view reduces the dependency on local measurement inaccuracies and discretization artifacts. Our approach comes along the lines with a definition of discrete differential operators of higher differentiability order on piecewise linear functions and vector fields .

The application of our decomposition is two-fold. First, the derived vector field components have distinguished properties which are mixed in the original vector field. Second, two components are the gradient respectively the co-gradient of potential functions. The existence of potential functions allows to identify features of the original vector field as local extrema of the associated potentials which are easily detectable.