Combinatorial Vector Field Topology in Three Dimensions

  • Wieland ReichEmail author
  • Dominic Schneider
  • Christian Heine
  • Alexander Wiebel
  • Guoning Chen
  • Gerik Scheuermann
Part of the Mathematics and Visualization book series (MATHVISUAL)


In this paper, we present two combinatorial methods to process 3-D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are a straightforward extension of an existing 2-D technique to 3-D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. We evaluate our method on a CFD-simulation of a gas furnace chamber. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow.


Periodic Orbit Outer Approximation Quotient Graph Graph Layout Conley Index 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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We want to thank Tomasz Kaczynski, Matthias Schwarz, the people from the Krakau research group for “Computer Assisted Proofs in Dynamics” and the reviewers for many valuable hints and comments. We also thank the DFG for funding the project by grant SCHE 663/3-8.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Wieland Reich
    • 1
    Email author
  • Dominic Schneider
    • 1
  • Christian Heine
    • 1
  • Alexander Wiebel
    • 2
  • Guoning Chen
    • 3
  • Gerik Scheuermann
    • 1
  1. 1.University of LeipzigLeipzigGermany
  2. 2.Max-Planck-Institut for Human Cognitive and Brian Sciences, LeipzigLeipzigGermany
  3. 3.University of UtahSalt Lake CityUSA

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