2D Asymmetric Tensor Field Analysis and Visualization

1. 2D Asymmetric Tensor Field Analysis and Visualization Eugene Zhang, Darrel Palke, Harry Yeh, Zhongzang Lin, Guoning Chen, Robert S. Laramee

2. Introduction • Flow visualization has a wide range of applications in areo- and hydro-dynamics: • Weather prediction • Aircraft and missile design • Natural disaster modeling: • Tsunami, hurricane, and tornado • Water quality study

3. Introduction • Existing techniques often focus on velocity [Laramee et al. 2004, Laramee et al. 2007] • Good for visualizing particle movement Trajectories Vector magnitudes Topology

4. Introduction • Basic types of non-translational motions Rotation (+/-) Expansion = Positive Isotropic Scaling Contraction = Negative Isotropic Scaling Anisotropic stretching

5. Introduction • Given a vector field , the local linearization at is: translation Velocity gradient: rotation, isotropic scaling, anisotropic stretching

6. Introduction • Rotation • Isotropic scaling • Anisotropic scaling

7. Introduction • Flow motions and physical meanings: [Batchelor 1967, Fischer et al. 1979, Ottino 1989, Sherman 1990] • Rotation: • Vorticity • Isotropic scaling: • volume change and/or stretching in the third dimension • Anisotropic stretching: • rate of angular deformation, related to energy dissipation and rate of fluid mixing

8. Introduction • Velocity gradient tensor has been used in vector field visualization • Singularity classification (source, sink, saddle, etc) [Helman and Hesselink 1991] • Periodic orbit extraction [Chen et al. 2007] • Attachment and separation detection [Kenwright 1998] • Vortex core identification [Sujudi and Haimes 1995, Jeong and Hussain 1995, Peikert and Roth 1999, Sadarjoen and Post 2000]

9. Introduction • However, tensor field structures were not investigated and applied to flow visualization • Velocity gradient is asymmetric • Past work in tensor field visualization focus on symmetric tensors [Delmarcelle and Hesselink 1994, Hesselink et al. 1997, Tricoche et al. 2001, Tricoche et al. 2003, Hotz et al. 2004, Zheng and Pang 2004, Zheng et al. 2005, Zhang et al. 2007]

10. Introduction • Symmetric tensors • have two families of mutually perpendicular real-valued eigenvectors • Asymmetric tensors • do not always have real-valued eigenvectors • when they do, the eigenvectors are not always mutually perpendicular

11. Introduction • Zheng and Pang provide analysis on 2D asymmetric tensor fields Circular points Complex domains (major dual-eigenvectors) Degenerate curves Real domains (major eigenvectors) Image courtesy: Xiaoqiang Zheng and Alex Pang

12. Introduction • Interesting questions • Orientation of flow rotation • Circular point • Dual-eigenvector computation • What roles do eigenvalues play? • What can these concepts tell us about the flow? • And how to present such information?

13. Introduction • Our contributions • Present geometric construction of dual-eigenvectors • Enable circular point classification • Incorporate flow orientation • Provide two techniques to better illustrate tensor structures inside complex domains • Pseudo-eigenvectors • Glyph packing • Describe analysis based on eigenvalues • Provide physical interpretation of our analysis

14. Outline • Review tensors • Eigenvector analysis • Eigenvalue analysis • Applications

15. Tensor Review • A 2x2 tensor is matrix • Typically asymmetric • Symmetric tensor • Anti-symmetric tensor • A tensor field is a continuous tensor-valued function

16. Tensor Review • Symmetric tensor • Eigenvalues • Eigenvectors: major and minor • Hyperstreamlines • Degenerate points • Tensor index

17. Tensor Review • Asymmetric tensors • Real domains and complex domains • Degenerate curves • Dual-eigenvectors (bisector) • Circular points

18. Our Analysis • Tensor reparameterization • Eigenvalue and eigenvector manifolds

19. Decomposition • Isotropic scaling: • Rotation: • Anisotropic stretching:

20. Decomposition • The set of 2x2 tensors can be parameterized by , , , and , such that • , and • This is a four-dimensional space • Can we focus on configuration spaces with lower-dimensions?

21. Decomposition • Eigenvalues only depend on , , and • Eigenvectors are dependent on , , and • Define eigenvector and eigenvalue manifolds

22. Eigenvector Manifold • Eigenvectors of is the same as is the same as which can rewritten as

23. Eigenvector Manifold Image credit: http://math.etsu.edu/MultiCalc/Chap3/Chap3-4/sphere1.gif

24. Eigenvector Manifold • Eigenvalues are constant along any latitude

25. Eigenvector Manifold • Along any latitude • Angular component of the eigenvectors and dual-eigenvectors linearly depend on

26. We can focus on any longitude and understand how eigenvectors/dual-eigenvectors change Eigenvector Manifold

27. Eigenvector Manifold • We can focus on any longitude and understand how eigenvectors/dual-eigenvectors change

28. Eigenvector Manifold

29. Eigenvector Manifold • Dual-eigenvectors never change along any longitude • Major and minor dual-eigenvectors swap roles when , i.e., symmetric tensors • Major and minor dual-eigenvectors are the major and minor eigenvectors of the following symmetric matrix:

30. Eigenvector Manifold • Dual-eigenvectors of

31. Eigenvector Manifold • Degenerate (circular) points of • Number, location, index, orientation Major Eigenvectors of Symmetric Component Major Dual-Eigenvectors

32. Eigenvector Manifold • Degenerate (circular) points of • It matters which side of the Equator the point is in Major Eigenvectors of Symmetric Component Major Dual-Eigenvectors

33. Eigenvector Manifold • Poincaré-Hopf theorem (asymmetric tensors): • Given a continuous asymmetric tensor field defined on a closed surface S such that has only isolated degenerate points , then

34. Eigenvector Manifold • Which side of the Equator matters • Incorporate the Equator into asymmetric tensor topology

35. Visualization Black curves: White curves: Blue curves: Degenerate points The Equator Degenerate curves Eigenvector Manifold

36. Eigenvector Manifold

37. Eigenvector Manifold

38. Eigenvector Manifold

39. Complex domains? Show ellipses Eigenvector Manifold

40. Eigenvector Manifold • How can we visualize the ellipses? • It is not enough to show or even and • We introduce pseudo-eigenvectors

41. Eigenvector Manifold • Visualization Blue = Dual-Eigenvectors Blue = Pseudo-Eigenvectors

42. Eigenvector Manifold Visualization Blue = Dual-Eigenvectors Blue = Pseudo-Eigenvectors

43. Eigenvector Manifold Visualization Blue = Dual-Eigenvectors Blue = Pseudo-Eigenvectors

44. Eigenvector Manifold Visualization Vector Field Tensor Field

45. Decomposition Isotropic scaling: Rotation: Anisotropic stretching:

46. Eigenvalue Manifold • Eigenvalue depend on , , and • We are interested in relatively strengths among the three components

47. Eigenvalue Manifold

48. Eigenvalue Manifold

49. Eigenvalue Manifold Dominant component

50. Eigenvalue Manifold