Generation of Accurate Integral Surfaces in Time-Dependent Vector Fields

Generation of Accurate Integral Surfaces inTime-Dependent Vector Fields

C. Garth, H. Krishnan, K. I. JoyUniversity of California, Davis

X. TricochePurdue University

T. BobachUniversity of Kaiserslautern

Overview

• Introduction: Integral Surfaces

• Approximation Algorithm

• Experiments

• Visualization using Integral Surfaces

Overview

‣ Introduction: Integral Surfaces


• Experiments


Introduction

Modern flow design is strongly based on CFD simulation.

Current generation flow simulations:• large (106 cells) – very large (109 cells)• time-dependent (102 – 105 time steps)• typically complex boundary conditions, flow patterns

Analysis relies heavily on Flow Visualization• exploratory / illustrative techniques are important

Integral Curves

Popular technique in this context: integral curves.

Definition: ,

Intuitive interpretation: path of a massless particle, seeded at point .

Integral Surfaces

• Natural generalization: integral surfaces

Interpretation: surface spanned by a family of integral curves

Integral Surfaces

• Natural generalization: integral surfaces

Interpretation: surface spanned by a family of integral curves

t →

s →

Integral Surfaces

Example

seeding curve

Overview


‣ Approximation Algorithm

• Experiments


Algorithm Outline

Integral surface approximation is performed in two phases:

• Phase 1: Time line approximation

• Phase 2: Surface Triangulation

Integral Surfaces are also a continuum of time lines.

Approximation Algorithm: • Approximate surface by a finite set of time lines (skeleton)• Discretize time lines using a finite set of integral curves• Approximate time line ti+1 from time line ti

Phase 1: Time Line Approximation

t0t1

t2 t3t4


• Step 1: Compute initial approximation, points on t1 are advected from t0

t0

t1


• Step 2: Apply refinement predicate on adjacent point triplesto determine where better resolution is needed

t0

t1


• Step 3: Insert new points next to the insufficient points

t0

t1


• Repeat at Steps 2 and 3 until no further refinement is needed

t0

t1


• Approximate sequence of timelines going from ti to ti+1

t0

t1

t2



t0

t1

t2t3



t0

t1

t2t3

t4


• Result: Surface skeleton of integral curves + time lines

t0

t1

t2t3

t4

Refinement Predicates

Predicates: Thresholds on local curve properties such as derivative and curvature.

Discrete versions easily obtained over adjacent point triples.




point distance

!




point distance

!

"

inter-point angle




point distance

!

triangle area

A

"

inter-point angle

Surface “Discontinuities”

• Integral curves can “get stuck” at critical points or boundaries(or just error out due to numerical difficulties)

• We use a discontinuity predicate to avoid over-refinementin such settings: first / second derivative over point triple

• If predicate triggers, split the timeline (removing the indicated point)

• Problem turns into two subproblems, treated with same methods as before

dx/dsd2x/ds2

Algorithm Outline

Integral surface approximation is performed in two phases:

• Phase 1: Time line approximation

• Phase 2: Surface Triangulation

• Discretize integral curves to obtain a point set on the surface

Phase 2: Surface Triangulation

t0

t1

t2t3

t4

• Discretize integral curves to obtain a point set on the surface

• Time lines are no longer needed and can be forgotten.


t0

t1

t2t3

t4


• Use adjacent integral curves (→ strip) and triangulate heuristically.

t0

t1

t2t3

t4


• Use adjacent integral curves (→ strip) and triangulate heuristically.• Shortest diagonal advancing front works well

t0

t1

t2t3

t4


• Use adjacent integral curves (→ strip) and triangulate heuristically.• Insertions are handled by choosing the triangulation with largest sum of angles

t0

t1

t2t3

t4


• Use adjacent integral curves (→ strip) and triangulate heuristically.• After an insertion point, strips are triangulated recursively

t0

t1

t2t3

t4

A Note on Integral Curves

Integral Curves are advanced from ti to ti+1 using numerical integration.

Traditional adaptive integration schemes (Runge-Kutta, Runge-Kutta-Fehlberg) offer good efficiency, but deliver a polyline representation of integral curves.

A Note on Integral Curves

Integral Curves are advanced from ti to ti+1 using numerical integration.

We use a Dormand-Prince scheme (DOPRI5) that represents the integral curve in piecewise polynomial form.

This allows us to adaptively discretize the integral curves after integration by re-using the previous predicates for max. efficiency.

Overview



‣ Experiments


Experiments

• Delta wing dataset (18e6 cells, 1000 timesteps)

Which refinement criterion?Numerical experiment

• Comprison of adaptive vs. non-adaptive high-res time lines• Distance and area critera work well, curvature does not give good results at all• Time lines are approximated correctly in the limit

1e-06

1e-05

0.0001

0.001

0.01

0.1

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

avg.

err

or

threshold magnitude

area refinementdist. refinement

1e-05

0.0001

0.001

0.01

0.1

1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 0.1 1

max

. err

or

threshold magnitude

area refinementdist. refinement

Overview



• Experiments

‣ Visualization using Integral Surfaces

Visualization

• Integral surfaces have natural 2d-parameterization (s,t)

• Lend themselves well to texture mapping• earlier work on this by Löffelmann et al., 1997• animated textures to convey temporal flow behavior

t →

s →

Visualization

• Example: Car (stationary, 15e6 cells)

Visualization

Visualization

• In the time-dependent case, self-intersections are possible• Explicit depiction can be helpful

Visualization

Contributions

• Novel 2-phase integral surface computation algorithm

• adaptive time-line approximation, with pluggable criteria• straightforward triangulation scheme: crack-free, high quality surfaces• efficiently applicable to time-dependent data (streaming)• near-interactive computation times facilitate dataset exploration• parallelization is feasible

• Enhanced visualization through

• highly illustrative, high-quality rendering• texture mapping• animated textures• explicit depiction of self-intersections

Differences to Previous Work

Hulquist 1992:• combined integration-triangulation (recursive-iterative) algorithm• tight coupling introduces mutual dependencies, complex algorithm• time-dependent data is not feasible

Garth et al. 2004:• based on Hultquist, uses arc-length parameterization• visualization enhancements• time-dependent data still not feasible

Our scheme• clean separation of integration / triangulation allows precise refinement

control, parallelization, time-dependent data, ...

Acknowledgements

• colleagues at IDAV, UC Davis, Univ. of Kaiserslautern

• Markus Rütten, DLR Göttingen

• DOE, contract DE-FC02-06ER-25780Scientific Discovery through Advanced Computing (SciDAC)Visualization and Analytics Center for Enabling Technologies (VACET)

• DFG IRTG 1131Visualization of Large and Unstructured Data Sets

Thank you

Questions?

Why can time lines degenerate?

Generation of Accurate Integral Surfaces in Time-Dependent Vector Fields

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