Identifying vector field singularities using a discrete Hodge decomposition. (English) Zbl 1065.37018

Polthier, Konrad; Preuß, Eike

Hege, Hans-Christian (ed.) et al., Visualization and mathematics III. Outgrowth of the 3rd international workshop, Berlin, Germany, May 22–25, 2002. Berlin: Springer (ISBN 3-540-01295-8/hbk). 113-134, 441-442 (2003).

Summary: We derive a Hodge decomposition of discrete vector fields on polyhedral surfaces, and apply it to the identification of vector field singularities. This novel approach allows us to easily detect and analyze singularities as critical points of corresponding potentials. Our method uses a global variational approach to independently compute two potentials whose gradient respectively co-gradient are rotation-free respectively divergence-free components of the vector field. The sinks and sources respectively vortices are then automatically identified as the critical points of the corresponding scalar-valued potentials. The global nature of the decomposition avoids the approximation problem of the Jacobian and higher-order tensors used in local methods, while the two potentials plus a harmonic flow component are an exact decomposition of the vector field containing all information.
For the entire collection see [Zbl 1014.00012].

MSC:

37C10	Dynamics induced by flows and semiflows
37N10	Dynamical systems in fluid mechanics, oceanography and meteorology
65D18	Numerical aspects of computer graphics, image analysis, and computational geometry
58K45	Singularities of vector fields, topological aspects
76B99	Incompressible inviscid fluids
76M10	Finite element methods applied to problems in fluid mechanics

Keywords:

discrete rotation; divergence potential function; discrete differential forms