Applying 2D Tensor Field Topology to Solid Mechanics Simulations

Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

There has been much work in the topological analysis of symmetric tensor fields, both in 2D and 3D. However, there has been relatively little work in the physical interpretations of the topological analysis, such as why wedges and trisectors appear in stress and strain tensors. In this chapter, we explore the physical meanings of degenerate points and describe some results made during our initial investigation.

References

  1. 1.
    Alliez, P., Cohen-Steiner, D., Devillers, O., Lévy, B., Desbrun, M.: Anisotropic polygonal remeshing. ACM Trans. Graph. (SIGGRAPH 2003) 22(3), 485–493 (2003)Google Scholar
  2. 2.
    Alliez, P., Meyer, M., Desbrun, M.: Interactive geometry remeshing. In: Proceedings of the 29th annual conference on Computer graphics and interactive techniques, SIGGRAPH ’02, pp. 347–354. ACM, New York (2002). doi:10.1145/566570.566588. http://doi.acm.org/10.1145/566570.566588
  3. 3.
    Basser, P.J., Pierpaoli, C.: Microstructural and physiological features of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. Reson. B 111(3), 209–219 (1996)CrossRefGoogle Scholar
  4. 4.
    Cabral, B., Leedom, L.C.: Imaging vector fields using line integral convolution. In: Poceedings of ACM SIGGRAPH 1993, Annual Conference Series, pp. 263–272 (1993)Google Scholar
  5. 5.
    Cammoun, L., Castano-Moraga, C.A., Munoz-Moreno, E., Sosa-Cabrera, D., Acar, B., Rodriguez-Florido, M., Brun, A., Knutsson, H., Thiran, J., Aja-Fernandez, S., de Luis Garcia, R., Tao, D., Li, X.: Tensors in Image Processing and Computer vision. Advances in Pattern Recognition. Springer, London (2009). http://www.springer.com/computer/computer+imaging/book/978-1-84882-298-6 Google Scholar
  6. 6.
    Delmarcelle, T., Hesselink, L.: Visualizing second-order tensor fields with hyperstream lines. IEEE Comput. Graph. Appl. 13(4), 25–33 (1993)CrossRefGoogle Scholar
  7. 7.
    Delmarcelle, T., Hesselink, L.: The topology of symmetric, second-order tensor fields. In: Proceedings IEEE Visualization ’94 (1994)Google Scholar
  8. 8.
    Hesselink, L., Levy, Y., Lavin, Y.: The topology of symmetric, second-order 3D tensor fields. IEEE Trans. Vis. Comput. Graph. 3(1), 1–11 (1997)CrossRefGoogle Scholar
  9. 9.
    Hotz, I., Feng, L., Hagen, H., Hamann, B., Joy, K., Jeremic, B.: Physically based methods for tensor field visualization. In: Proceedings of the Conference on Visualization ’04, VIS ’04, pp. 123–130. IEEE Computer Society, Washington (2004). doi:10.1109/VISUAL.2004.80. http://dx.doi.org/10.1109/VISUAL.2004.80
  10. 10.
    Kratz, A., Auer, C., Stommel, M., Hotz, I.: Visualization and analysis of second-order tensors: Moving beyond the symmetric positive-definite case. Comput. Graph. Forum 32(1), 49–74 (2013). http://dblp.uni-trier.de/db/journals/cgf/cgf32.html#KratzASH13 CrossRefGoogle Scholar
  11. 11.
    Kratz, A., Meyer, B., Hotz, I.: A visual approach to analysis of stress tensor fields. In: Scientific Visualization: Interactions, Features, Metaphors, pp. 188–211 (2011). Dagstuhl Publishing, Saarbrücken. doi: 10.4230/DFU.Vol2.SciViz.2011.188. http://dx.doi.org/10.4230/DFU.Vol2.SciViz.2011.188
  12. 12.
    Maries, A., Haque, M.A., Yilmaz, S.L., Nik, M.B., Marai, G.: Interactive exploration of stress tensors used in computational turbulent combustion. In: Laidlaw, D., Villanova, A. (eds.) New Developments in the Visualization and Processing of Tensor Fields. Springer, Heidelberg (2012)Google Scholar
  13. 13.
    Neeman, A., Jeremic, B., Pang, A.: Visualizing tensor fields in geomechanics. In: IEEE Visualization, p. 5 (2005)Google Scholar
  14. 14.
    Nieser, M., Palacios, J., Polthier, K., Zhang, E.: Hexagonal global parameterization of arbitrary surfaces. IEEE Trans. Vis. Comput. Graph. 18(6), 865–878 (2012). doi: 10.1109/TVCG.2011.118. http://dx.doi.org/10.1109/TVCG.2011.118 CrossRefGoogle Scholar
  15. 15.
    Schultz, T., Kindlmann, G.L.: Superquadric glyphs for symmetric second-order tensors. IEEE Trans. Vis. Comput. Graph. 16(6), 1595–1604 (2010)CrossRefGoogle Scholar
  16. 16.
    Tricoche, X., Kindlmann, G., Westin, C.F.: Invariant crease lines for topological and structural analysis of tensor fields. IEEE Trans. Vis. Comput. Graph. 14(6), 1627–1634 (2008). doi:http://doi.ieeecomputersociety.org/10.1109/TVCG.2008.148
  17. 17.
    Tricoche, X., Scheuermann, G.: Topology simplification of symmetric, second-order 2D tensor fields. In: Geometric Modeling Methods in Scientific Visualization (2003)MATHGoogle Scholar
  18. 18.
    Westin, C.F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: ISMRM ’97, p. 1742. Vancouver, Canada (1997)Google Scholar
  19. 19.
    Wiebel, A., Koch, S., Scheuermann, G.: Glyphs for Non-Linear Vector Field Singularities, pp. 177–190. Springer, Berlin (2012). doi:10.1007/978-3-642-23175-9_12. http://dx.doi.org/10.1007/978-3-642-23175-9_12
  20. 20.
    Zhang, E., Hays, J., Turk, G.: Interactive tensor field design and visualization on surfaces. IEEE Trans. Vis. Comput. Graph. 13(1), 94–107 (2007)CrossRefGoogle Scholar
  21. 21.
    Zhang, E., Mischaikow, K., Turk, G.: Vector field design on surfaces. ACM Trans. Graph. 25(4), 1294–1326 (2006)CrossRefGoogle Scholar
  22. 22.
    Zhang, S., Kindlmann, G., Laidlaw, D.H.: Diffusion tensor MRI visualization. In: Visualization Handbook. Academic Press, London (2004). http://www.cs.brown.edu/research/vis/docs/pdf/Zhang-2004-DTM.pdf Google Scholar
  23. 23.
    Zheng, X., Pang, A.: Volume deformation for tensor visualization. In: IEEE Visualization, pp. 379–386 (2002)Google Scholar
  24. 24.
    Zheng, X., Pang, A.: Hyperlic. In: Proceeding IEEE Visualization, pp. 249–256 (2003)Google Scholar
  25. 25.
    Zheng, X., Pang, A.: Topological lines in 3D tensor fields. In: Proceedings IEEE Visualization 2004, VIS ’04, pp. 313–320. IEEE Computer Society, Washington (2004). doi:10.1109/VISUAL.2004.105. http://dx.doi.org/10.1109/VISUAL.2004.105
  26. 26.
    Zheng, X., Parlett, B., Pang, A.: Topological structures of 3D tensor fields. In: Proceedings IEEE Visualization 2005, pp. 551–558 (2005)Google Scholar
  27. 27.
    Zheng, X., Parlett, B.N., Pang, A.: Topological lines in 3D tensor fields and discriminant hessian factorization. IEEE Trans. Vis. Comput. Graph. 11(4), 395–407 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of Electrical Engineering and Computer Science, 3117 Kelley Engineering CenterOregon State UniversityCorvallisUSA
  2. 2.School of Electrical Engineering and Computer Science, 1148 Kelley Engineering CenterOregon State UniversityCorvallisUSA
  3. 3.School of Electrical Engineering and Computer Science, 2111 Kelley Engineering CenterOregon State UniversityCorvallisUSA

Personalised recommendations