A tool for visualizing the topology of three-dimensional vector fields

Abstract

We describe a software system, TOPO, that numerically analyzes and graphically displays topological aspects of a three dimensional vector field, v, to produce a single relatively simple picture that characterizes v. The topology of v that we consider consists of its critical points (where v = 0), their invariant manifolds, and the integral curves connecting these invariant manifolds. Many of the interesting features of v are associated with its critical points. The field in the neighborhood of each critical point is approximated by the Taylor expansion. The coefficients of the first non-zero term of the Taylor expansion around a critical point are the 3x3 matrix dv. Critical points are classified by examining dv's eigenvalues. The eigenvectors of dv span the invariant manifolds of the linearized field around a critical point. Curves integrated from initial points on the eigenvectors a small distance from a critical point connect with other critical points (or the boundary) to complete the topology. In addition, one class of critical surfaces important in computational fluid dynamics is analyzed.TOPO is implemented as a module in the FAST [1] visualization environment. FAST is general purpose visualization software with modules for isosurface generation, particle tracing, etc. TOPO operates on curvilinear, structured grids, including large multi-zone grids. We have used TOPO to visualize a number of computational fluid dynamics (CFD) data sets. The results agree well with other topology software and hand generated topologies. TOPO has proved useful in finding surface topology, flow attachment and separation points, vortex cores, scalar field local extrema, and generally interesting regions of v. We believe there may be other interesting applications yet to be discovered. This paper, along with the references, contains most of the information needed for a scientific programmer to code a topology module in another environment.