Efcient Morse Decompositions of Vector Fields

Vector eld analysis plays a crucial role in many engineering appli-cations, such as weather prediction, tsunami and hurricane study, and airplane and automotive design. Existing vector eld analy-sis techniques focus on individual trajectories such as xed points, periodic orbits and separatrices which are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector eld analysis unsuitable for rigorous interpretations. In this paper, we advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of the vector eld in the form of a directed graph, called a Morse decomposition connection graph (MCG). While an MCG exists for every vector eld, it need not be unique. We develop the idea of a τ map, which decouples the MCG construction process and the conguration of the underlying mesh. This, in general, re-sults in ner MCGs than mesh-dependent approaches. To compute MCGs effectively, we present an adaptive approach in constructing better approximations of the images of triangles in the meshes used for simulation. These techniques result in fast and efcient MCG construction. We demonstrate the efcacy of our technique on various examples in planar elds and on surfaces in-cluding engine simulation data