Eugene Zhang Oregon State University

Transcription

1 2D Asymmetric Tensor Field Analysis and Visualization Eugene Zhang Oregon State University

2 Introduction Asymmetric tensors can model the gradient of a vector field velocity gradient in fluid dynamics deformation gradient in solid mechanics

3 Introduction Flow visualization has a wide range of applications in areo- and hydro-dynamics: y Climatology Oceanography and limnology Hydraulic engineering Aircraft and undersea vehicle design

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

5 Introduction Basic types of non-translational motions Rotation (+/-) Expansion Pure shear Contraction

6 Introduction Given a vector field, the local linearization at is: Velocity vector field: translation Velocity gradient tensor field: yg rotation, isotropic scaling (expansion and contraction), and pure shear.

7 Introduction Rotation: Isotropic scaling: Anisotropic stretching (aka pure shear):

8 Introduction Flow motions and physical meanings: [Batchelor 1967, Lighthill 1986, Ottino 1989, Sherman 1990] Rotation: Vorticity Isotropic scaling: volume change and/or stretching t in the third dimensioni Anisotropic stretching: rate of angular deformation, related to energy dissipation and rate of fluid mixing

9 Introduction Velocity gradient tensor has been used in vector field visualization Singularity classification [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]

10 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]

11 Introduction Symmetric tensors two real eigenvalues two mutually perpendicular eigenvectors when not degenerate Asymmetric tensors can have complex eigenvalues eigenvectors not always mutually perpendicularp

12 Introduction Questions: What are features in an asymmetric tensor field? How to visualize these features? Wh t l b t th fl f th What can we learn about the flow from these features?

13 Tensor Decomposition Isotropic scaling: Rotation: Pure shear:

14 Tensor 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?

15 Tensor Decomposition Eigenvalues only depend on,, and Eigenvectors are dependent on,, and Define eigenvector and eigenvalue manifolds

16 Eigenvector Manifold Eigenvectors of are same as are same as can be rewritten as

17 Eigenvector Manifold Image credit:

18 Eigenvector Manifold Eigenvalues are constant along any latitude

19 Eigenvector Manifold Real domains and complex domains Degenerate curves

20 Eigenvector Manifold We can focus on any longitude and understand how eigenvectors change

21 Eigenvector Manifold

22 Eigenvector Manifold

23 Eigenvector Manifold

24 Eigenvector Manifold

25 Eigenvector Manifold

26 Eigenvector Manifold

27 Eigenvector Manifold

28 Eigenvector Manifold

29 Eigenvector Manifold

30 Eigenvector Manifold Bisectors never change along any longitude They are dual-eigenvectors (Zheng and Pang 2005) Dual-eigenvectors are eigenvectors of

31 Eigenvector Manifold Dual-eigenvectors of

32 Eigenvector Manifold Dual-eigenvectors of

33 Eigenvector Manifold Which side of the Equator matters Incorporate the Equator into asymmetric Incorporate the Equator into asymmetric tensor topology

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

35 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

36 Eigenvector Manifold Visualization Black curves: White curves: Blue Bl curves: Degenerate points The Equator Degenerate curves

37 Eigenvector Manifold

38 Eigenvalue Manifold Eigenvalues depend on,, and We are interested in relatively strengths We are interested in relatively strengths among the three components

39 Eigenvalue Manifold

40 Eigenvalue Manifold

41 Eigenvalue Manifold Dominant component

42 Eigenvalue Manifold

43 Eigenvalue Manifold All components

44 Eigenvalue Manifold Tensor Magnitude

45 Combining Eigenvector and Eigenvalue Manifolds CCW-dominant region (red) must be in the northern hemisphere (red) CW-dominant region (green) must be in the southern hemisphere (red)

46 Combining Eigenvector and Eigenvalue Manifolds

47 Applications Sullivan flow (a tornado model) [Sullivan:1959]

48 Applications Sullivan flow (a tornado model) Vector field topology Dominant eigenvalue + major eigenvector and major dual-eigenvector Tensor magnitude

49 Applications Sullivan flow (a tornado model) Vector Field Topology Dominant Eigenvalue + Major Eigenvector and Major Dual-Eigenvector Tensor Magnitude

50 Applications Sullivan flow (a tornado model) Vector Field Topology Dominant Eigenvalue + Major Eigenvector and Major Dual-Eigenvector Tensor Magnitude

51 Applications Sullivan flow (a tornado model) Vector field topology Dominant eigenvalue + major eigenvector and major dual-eigenvector Tensor magnitude

52 Applications Diesel engine

53 Applications Diesel engine

54 Open Questions What does tensor index tell us about the flow, other than zero stretching? Can we generate a graph representation for asymmetric tensors, much like the vector field topology?

55 Open Questions How do we integrate information from vector and tensor field analysis?

56 Open Questions How does the analysis carry over to 3D? Axis of rotation is not always aligned with any of the eigenvector directions, which means all of them could be moving when going from real domains into complex domains How to deal with 3D anisotropic stretching? What do the eigenvalue and eigenvector manifolds look like?

57 Open Questions What is the topology of higher-order tensor fields, symmetric or not, 2D or 3D, static or time-varying? And why do we care?

58 Acknowledgement Collaborators Dr. Harry Yeh, Professor in fluid mechanics, Oregon State University Darrel Palke, Intel Zhongzang Lin, Ph.D. student at Oregon State University Dr. Guoning Chen, postdoctoral researcher at University of Utah Dr. Robert S. Laramee, Lecturer (Assistant Professor) at Swansea University, UK

59 Acknowledgement Inspirations from: Dr. Xiaoqiang Zheng, Nvidia Dr. Alex Pang, Professor in computer science, UC Santa Cruz The pioneers in vector and tensor field analysis and The pioneers in vector and tensor field analysis and visualization

60 References Xiaoqiang Zheng and Alex Pang, 2D Asymmetric Tensor Analysis, IEEE Vis 2005 ( Eugene Zhang, Harry Yeh, Zhongzang Lin, and Robert S. Laramee, Asymmetric Tensor Analysis for Flow Visualization, IEEE Trans. on Visualization and Computer Graphics ( org/portal/web/csdl/doi/ /TVCG Darrel Palke, Guoning Chen, Zhongzang Lin, Harry Yeh, Robert S. Laramee, and Eugene Zhang, Asymmetric Tensor Visualization with Glyph and Hyperstreamline Placement on 2D Manifolds, Tech Report, Oregon State University (

61 Questions?