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Ronny Peikert Over Two Decades of Integration-Based, Geometric Vector Field Visualization Part III: Curve based seeding Planar based seeding Ronny Peikert ETH Zurich 1

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Ronny Peikert 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

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Ronny Peikert Stream surfaces 3 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!)

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Ronny Peikert Stream surfaces 4 First stream surface computation done before SciVis existed! Early use in flow visualization (Helman and Hesselink 1990) for flow separation Image: Ying et al.

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Ronny Peikert 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 fixed time steps slightly better: fixed spatial steps wall (u = 0) streamlines

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Ronny Peikert 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 systematic triangulation optimized triangulation streamlines

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Ronny Peikert Hultquist's algorithm (2) The problem of divergence or convergence is solved by inserting or terminating streamlines. 7 inserted streamline terminated streamline

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Ronny Peikert 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

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Ronny Peikert 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

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Ronny Peikert 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 HultquistScheuermann (Scheuermann et al., 2001).

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Ronny Peikert Implicit methods A stream function is a special *) solution of the PDE [don't confuse with a potential which has PDE ] 11 streamline stream surfaces *) mass flux = (b-a)(d-c)

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Ronny Peikert 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

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Ronny Peikert Implicit methods (3) Computational space method, implicit method per cell, respects conservation of mass (van Gelder, 2001) 13 Delta wing. Stream surface close to boundary. Flow separation and attachment.

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Ronny Peikert Rendering of stream surfaces Stream arrows (Löffelmann et al. 1997) Texture advection on stream surfaces (Laramee et al. 2006) 14

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Ronny Peikert Rendering of stream surfaces (2) 15

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Ronny Peikert 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

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Ronny Peikert 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 Flow past a cylinder saddle-connector of a pair of focus saddle crit. points

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Ronny Peikert 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

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Ronny Peikert 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

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Ronny Peikert 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 Path surface of unsteady flow past a cylinder

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Ronny Peikert 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

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Ronny Peikert 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

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Ronny Peikert Planar based seeding (2) Planar-based seeding for unsteady flow Extension of flow volume technique to unsteady vector fields (Becker et at. 1995). 23 Image: Crawfis, Shen, Max Unsteady flow volume

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Ronny Peikert 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

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Ronny Peikert 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

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Ronny Peikert The End 26 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:

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