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Over Two Decades of Integration-Based, Geometric Vector Field Visualization

Over Two Decades of Integration-Based, Geometric Vector Field Visualization. Part III: Curve based seeding Planar based seeding Ronny Peikert ETH Zurich. Overview. Curve-based seeding objects steady flow stream surfaces unsteady flow "path surfaces" "streak surfaces"

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Over Two Decades of Integration-Based, Geometric Vector Field Visualization

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

  2. 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. Stream surfaces • 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!) 3 http://graphics.ethz.ch/~peikert

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

  5. 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: fixed time steps slightly better: fixed spatial steps streamlines streamlines wall (u= 0) 5 http://graphics.ethz.ch/~peikert

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

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

  8. Refined Hultquist methods • Curvature of front curve controlled by limiting the dihedral angle of the mesh (Garth et al. 2004) • Adaptive refinement • Intricate structure of vortexbreakdown bubble 8 http://graphics.ethz.ch/~peikert

  9. 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. 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 • (Scheuermann et al., 2001). Scheuermann Hultquist 10 http://graphics.ethz.ch/~peikert

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

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

  14. Rendering of stream surfaces • Stream arrows (Löffelmann et al. 1997) • Textureadvection on streamsurfaces(Laramee et al. 2006) 14 http://graphics.ethz.ch/~peikert

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

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

  18. Invariant 2D manifolds (2) • Geodesiccirclesstreamsurfacealgorithm(Krauskopf and Osinga 1999) • Front growsradially (notalongstreamlines) • bysolving a boundaryvalueproblem • "immune" againstspiraling 18 http://graphics.ethz.ch/~peikert

  19. Invariant 2D manifolds (3) • Topology-awarestreamsurfacemethod(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. Path surfaces • Particle based path surfaces (Schafhitzel et al. 2007) • Density control a la Hultquist • Point splatting • 1st order Euler integration • GPU implementation interactive seeding! Path surface of unsteady flow past a cylinder 20 http://graphics.ethz.ch/~peikert

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

  24. 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. 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. The End • 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 26 http://graphics.ethz.ch/~peikert

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