1. Ronny Peikert peikert@inf.ethz.ch Over Two Decades of Integration-Based, Geometric Vector Field Visualization Part III: Curve based seeding Planar based seeding Ronny Peikert ETH Zurich 1 http://graphics.ethz.ch/~peikert

2. Ronny Peikert peikert@inf.ethz.ch Overview  Curve-based seeding objects  steady flow  stream surfaces  unsteady flow  "path surfaces"  "streak surfaces"  Planar-based seeding objects  steady flow  unsteady flow  Orthogonal surfaces of a vector field  Discussion, future research opportunities 2 http://graphics.ethz.ch/~peikert

3. Ronny Peikert peikert@inf.ethz.ch Stream surfaces 3 http://graphics.ethz.ch/~peikert  Definition  A stream surface is the union of the stream lines seeded at all points of a curve (the seed curve).  Motivation  separates (steady) flow, flow cannot cross the surface  surfaces offer more rendering options than lines (perception!)

4. Ronny Peikert peikert@inf.ethz.ch Stream surfaces 4 http://graphics.ethz.ch/~peikert  First stream surface computation  done before SciVis existed!  Early use in flow visualization (Helman and Hesselink 1990) for flow separation Image: Ying et al.

5. Ronny Peikert peikert@inf.ethz.ch Stream surface integration  Problem: naïve algorithm fails if streamlines diverge or grow at largely different speeds.  Example of failure: seed curve which extends to no-slip boundary: 5 http://graphics.ethz.ch/~peikert fixed time steps slightly better: fixed spatial steps wall (u = 0) streamlines

6. Ronny Peikert peikert@inf.ethz.ch Hultquist's algorithm  Hultquist's algorithm (Hultquist 1992) does optimized triangulation:  Of two possible connections choose the one which is closer to orthogonal to both streamlines. 6 http://graphics.ethz.ch/~peikert systematic triangulation optimized triangulation streamlines

7. Ronny Peikert peikert@inf.ethz.ch Hultquist's algorithm (2)  The problem of divergence or convergence is solved by inserting or terminating streamlines. 7 http://graphics.ethz.ch/~peikert inserted streamline terminated streamline

8. Ronny Peikert peikert@inf.ethz.ch  Curvature of front curve controlled by limiting the dihedral angle of the mesh (Garth et al. 2004)  Adaptive refinement  Intricate structure of vortex breakdown bubble Refined Hultquist methods 8 http://graphics.ethz.ch/~peikert

9. Ronny Peikert peikert@inf.ethz.ch Refined Hultquist methods (2)  Cubic Hermite interpolation along the front curves. Runge-Kutta used to propagate front and its covariant derivatives (Schneider et al. 2009) 9 http://graphics.ethz.ch/~peikert

10. Ronny Peikert peikert@inf.ethz.ch Analytical methods  In a tetrahedral cell the vector field is linearly interpolated:  Streamline has equation  Stream surface seeded on straight line (entry curve) is a ruled surface.  Exit curve is computed analytically respecting boundary switch curves 10 http://graphics.ethz.ch/~peikert HultquistScheuermann (Scheuermann et al., 2001).

11. Ronny Peikert peikert@inf.ethz.ch Implicit methods  A stream function is a special *) solution of the PDE [don't confuse with a potential which has PDE ] 11 http://graphics.ethz.ch/~peikert streamline stream surfaces *) mass flux =  (b-a)(d-c)

12. Ronny Peikert peikert@inf.ethz.ch Implicit methods (2)  Stream functions  exist for divergence-free vector fields (= incompressible flow)  … and for compressible flow, if there are no sinks/sources  are computed by solving a PDE (with appropriate boundary conditions)  yield stream surfaces by isosurface extraction  Advantage of stream function method (Kenwright and Mallinson, 1992, van Wijk, 1993): conservation of mass! 12 http://graphics.ethz.ch/~peikert

13. Ronny Peikert peikert@inf.ethz.ch Implicit methods (3)  Computational space method, implicit method per cell, respects conservation of mass (van Gelder, 2001) 13 http://graphics.ethz.ch/~peikert Delta wing. Stream surface close to boundary. Flow separation and attachment.

14. Ronny Peikert peikert@inf.ethz.ch Rendering of stream surfaces  Stream arrows (Löffelmann et al. 1997)  Texture advection on stream surfaces (Laramee et al. 2006) 14 http://graphics.ethz.ch/~peikert

15. Ronny Peikert peikert@inf.ethz.ch Rendering of stream surfaces (2) 15 http://graphics.ethz.ch/~peikert

16. Ronny Peikert peikert@inf.ethz.ch Invariant 2D manifolds  Critical points of types saddle and focus saddle (spiral saddle) have a stream surface converging to them.  And so do periodic orbits of types saddle and twisted saddle. 16 http://graphics.ethz.ch/~peikert

17. Ronny Peikert peikert@inf.ethz.ch Invariant 2D manifolds  Saddle connectors (Theisel et al, 2003)  Visualization of topological skeleton of 3D vector fields  Intersection of 2D manifolds of (focus) saddles 17 http://graphics.ethz.ch/~peikert Flow past a cylinder saddle-connector of a pair of focus saddle crit. points

18. Ronny Peikert peikert@inf.ethz.ch  Geodesic circles stream surface algorithm (Krauskopf and Osinga 1999)  Front grows radially (not along stream lines)  by solving a boundary value problem  "immune" against spiraling Invariant 2D manifolds (2) 18 http://graphics.ethz.ch/~peikert

19. Ronny Peikert peikert@inf.ethz.ch Invariant 2D manifolds (3)  Topology-aware stream surface method (Peikert and Sadlo, 2009)  starts at critical point, periodic orbit, or given seed curve  handles convergence to saddle or sink 19 http://graphics.ethz.ch/~peikert

20. Ronny Peikert peikert@inf.ethz.ch Path surfaces  Particle based path surfaces (Schafhitzel et al. 2007)  Density control a la Hultquist  Point splatting  1 st order Euler integration  GPU implementation interactive seeding! 20 http://graphics.ethz.ch/~peikert Path surface of unsteady flow past a cylinder

21. Ronny Peikert peikert@inf.ethz.ch Streak surfaces  Smoke surfaces are a technique based on streak surfaces (von Funck et al. 2008)  advected mesh is not retriangulated, but  size/shape of triangles is mapped to opacity  simplified optical model for smoke 21 http://graphics.ethz.ch/~peikert

22. Ronny Peikert peikert@inf.ethz.ch Planar based seeding  Planar-based seeding for steady flow  Stream polygons (Schroeder et al. 1991)  Flow volumes (Max et al. 1993)  Implicit flow volumes (Xue et al. 2004) 22 http://graphics.ethz.ch/~peikert

23. Ronny Peikert peikert@inf.ethz.ch Planar based seeding (2)  Planar-based seeding for unsteady flow  Extension of flow volume technique to unsteady vector fields (Becker et at. 1995). 23 http://graphics.ethz.ch/~peikert Image: Crawfis, Shen, Max Unsteady flow volume

24. Ronny Peikert peikert@inf.ethz.ch Orthogonal surfaces  Surfaces (approximately) orthogonal to a vector (or eigen- vector) field as a visualization technique (Zhang et al. 2003).  If a vector field is conservative,, its potential can be visualized with a scalar field visualization technique, such as isosurfaces.  Orthogonal surfaces exist also in the slightly more general case of helicity-free vector fields.  However, 3D flow fields usually have helicity. Also eigenvector fields of symmetric 3D tensors.  Consequence: For many applications, orthogonal surfaces are less suitable (discussed by Schultz et al. 2009). 24 http://graphics.ethz.ch/~peikert

25. Ronny Peikert peikert@inf.ethz.ch Discussion, future research  There is no single best flow vis technique!  Most effort spent so far on streamlines  Extension to unsteady flow somewhat lacking behind  Also extension to stream surfaces (and unsteady variants)  Other areas needing more research:  Uncertainty visualization tools for geometric techniques  Comparative visualization tools for geometric techniques  Improved surface and volume construction methods  Automatic seeding for surfaces and volumes 25 http://graphics.ethz.ch/~peikert

26. Ronny Peikert peikert@inf.ethz.ch The End 26 http://graphics.ethz.ch/~peikert  Thank you for your attention! Any questions?  We would like to thank the following: R. Crawfis, W. v. Funck, C. Garth, J.L. Helman, J. Hultquist, H. Loeffelmann, N. Max, H. Osinga, T. Schafhitzel, G. Scheuermann, D. Schneider, W. Schroeder, H.W. Shen, H. Theisel, T. Weinkauf, S.X. Ying, D. Xue  PDF versions of STAR and MPEG movies available at: http://cs.swan.ac.uk/~csbob