over two decades of integration-based, geometric vector field visualization
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Ronny Peikertpeikert@inf.ethz.ch
Over Two Decades of Integration-Based, Geometric Vector Field Visualization
Part III: Curve based seedingPlanar based seeding
Ronny PeikertETH Zurich
1http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@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
2http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@inf.ethz.ch
Stream surfaces
3http://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!)
Ronny Peikertpeikert@inf.ethz.ch
Stream surfaces
4http://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.
Ronny Peikertpeikert@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:
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fixed time steps slightly better: fixed spatial steps
wall (u = 0)
streamlines
streamlines
Ronny Peikertpeikert@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.
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systematictriangulation
optimizedtriangulation
stre
amlin
es
stre
amlin
es
Ronny Peikertpeikert@inf.ethz.ch
Hultquist's algorithm (2) The problem of divergence or convergence is
solved by inserting or terminating streamlines.
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inserted streamline terminated streamline
Ronny Peikertpeikert@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
8http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@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)
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Ronny Peikertpeikert@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
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( ) u x Ax( ) tt e Αx x0
HultquistScheuermann
(Scheuermann et al., 2001).
Ronny Peikertpeikert@inf.ethz.ch
Implicit methods A stream function is a special*) solution of
the PDE[don't confuse with a potential which has PDE ]
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0x u x x
x u x
streamline
stream surfaces
a x b x
d x
c x
*) mass flux = r(b-a)(d-c)
Ronny Peikertpeikert@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!
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Ronny Peikertpeikert@inf.ethz.ch
Implicit methods (3) Computational space method, implicit
method per cell, respects conservation of mass (van Gelder, 2001)
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Delta wing. Stream surface close to boundary. Flow separation and attachment.
Ronny Peikertpeikert@inf.ethz.ch
Rendering of stream surfaces Stream arrows
(Löffelmann et al. 1997) Texture advection on stream
surfaces (Laramee et al. 2006)
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Ronny Peikertpeikert@inf.ethz.ch
Rendering of stream surfaces (2)
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Ronny Peikertpeikert@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.
16http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@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
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Flow past a cylinder
saddle-connector of a pair of focus saddle crit. points
Ronny Peikertpeikert@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)
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Ronny Peikertpeikert@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
19http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@inf.ethz.ch
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!
20http://graphics.ethz.ch/~peikert
Path surface of unsteady flow past a cylinder
Ronny Peikertpeikert@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
21http://graphics.ethz.ch/~peikert
Ronny Peikertpeikert@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)
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Ronny Peikertpeikert@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).
23http://graphics.ethz.ch/~peikert
Image: Crawfis, Shen, MaxUnsteady flow volume
Ronny Peikertpeikert@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).
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u 0
u u 0
Ronny Peikertpeikert@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
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Ronny Peikertpeikert@inf.ethz.ch
The End
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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
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