over two decades of integration-based, geometric vector field visualization

Ronny [email protected]

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


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



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!)


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.


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

wall (u = 0)

streamlines

streamlines


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.


systematictriangulation

optimizedtriangulation

stre

amlin

es

stre

amlin

es


Hultquist's algorithm (2) The problem of divergence or convergence is

solved by inserting or terminating streamlines.


inserted streamline terminated streamline


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



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)



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


( ) u x Ax( ) tt e Αx x0

HultquistScheuermann

(Scheuermann et al., 2001).


Implicit methods A stream function is a special*) solution of

the PDE[don't confuse with a potential which has PDE ]


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)


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!



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.


Rendering of stream surfaces Stream arrows

(Löffelmann et al. 1997) Texture advection on stream

surfaces (Laramee et al. 2006)



Rendering of stream surfaces (2)



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.



Invariant 2D manifolds Saddle connectors (Theisel et al, 2003)

Visualization of topological skeleton of 3D vector fields

Intersection of 2D manifolds of (focus) saddles


Flow past a cylinder

saddle-connector of a pair of focus saddle crit. points


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)



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



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


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



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)



Planar based seeding (2) Planar-based seeding for unsteady flow

Extension of flow volume technique to unsteady vector fields(Becker et at. 1995).


Image: Crawfis, Shen, MaxUnsteady flow volume


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).


u 0

u u 0


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



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

over two decades of integration-based, geometric vector field visualization

Documents

peikert ronny

seed curve

exit curve

peikertfirst stream

equation stream surface

stream lines

incompressible flow

stream functionsexist