Stream Lines and Surfaces Lecture 9 February 20, 2020 Outline - - PowerPoint PPT Presentation

CS53000 - Spring 2020

Introduction to Scientific Visualization

Lecture

Stream Lines and Surfaces

February 20, 2020

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Outline

Introduction Overview of basic techniques Mathematical / numerical foundations

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Vector Field Visualization

General Goal:

• Display the field’s information
• direction
• norm
• Convey patterns
• Capture overall structure

Domain Specific:

• Detect certain features:
• vortex cores, attachment, shock

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Data set is given by a vector component and its

magnitude Often results from study of fluid flow or by looking at derivatives (rate of change) of some scalar quantity (e.g., bioelectric field) No visual intuition for what a vector field should look like Many visualization techniques proposed

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Vector Field Visualization

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Origins of Flow Visualization

Rich field of Fluid Flow Visualization (experimental) Hundreds of years old!! Computational fluid dynamics (CFD) simulations require powerful computational flow visualization techniques

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Experimental Flow Vis

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Streaklines in the wake behind a cylinder (from Batchelor 1967)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Flow Visualization in SciVis

Direct visualization Integral lines and surfaces

Texture-based and dense methods Topology Feature extraction Coherent structures Extension to tensor data

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Direct Visualization

Glyphs: visual encoding (geometry + color) of local data attributes

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Peikert and Roth, ETH Zurich Pieter W. Groen, Universiteit Amsterdam

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamline-based Techniques

Stream-{lines | tubes | ribbons | polygons |…}

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Hans-Georg Pagendarm, DLR Göttingen Mat Ward WPI Gerris Flow Solver

Stream ribbons

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamline seeding Streamlines on surfaces Illuminated streamlines

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Greg Turk, Georgia Tech Konrad Polthier, Zuse Institut Berlin Mallo et al., ETH Zurich

Streamline-based Techniques

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Stream Surfaces

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Christoph Garth, University of Kaiserslautern Markus Ruetten, DLR Goettingen

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Mathematical Framework

Differential equation associated with smooth vector field with : trajectory (streamline, orbit) Steady: does not depend on time vs. unsteady (“time-dependent”, “transient”) Initial conditions:

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⌃ v : I ⊆ I R × Mn → TMn

x : I R → Mn

x(0) = x0

dx dt = ⇧ v(t, x)

⇥ v

x(0) = x0

⌅ v : I ⊆ I R × I Rn → I Rn

x : I R → I Rn

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Mathematical Framework

Flow of a vector field tells us where a particle ends up after a certain time given a particular starting point. Existence and uniqueness if is smooth (“Lipschitz continuity”)

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⌅⇥(x, t) ⌅t

t=τ = ⌥

v(x, )

⇥ v

⌅⇥(x, t) ⌅t

t=τ = ⌥

v(x, )

⇥ v

φ : U ⊆ I Rn × I ⊆ I R → I Rn

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamline Computation

Requires solution (integration) of an ordinary differential equation (ODE): Closed form solutions (analytic) do not exist in general (linear vector fields are a special case) Numerical ODE solvers

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x(t) = x0 + Z t ~ v(x(u))du

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Euler Method

Simplest method Fast, inaccurate, unstable

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(x0, t0) (x1, t1) (x2, t2) (x3, t3) (x4, t4)

xn+1 = xn + h⌃ v(xn, tn) + O(h2)

(x0, t0) (x1, t1) (x2, t2) (x3, t3) (x4, t4)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 2 (RK2)

Use Euler as trial step

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⌃ k1 = h⌃ v(xn, tn) ⌃ k2 = h⌃ v(1 2 ⌃ k1, tn + 1 2h) xn+1 = xn + ⌅ k2 + O(h3)

(x0, t0) (x1, t1) (x2, t2) (x3, t3)

⌃ k1 = h⌃ v(xn, tn) xn+1 = xn + ⌅ k2 + O(h3)

(x0, t0) (x1, t1) (x2, t2) (x3, t3)

~ k2 = h~ v(xn + 1 2 ~ k1, tn + 1 2h)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 4 (RK4)

Additional Euler steps to increase order

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⌃ k3 = h⌃ v(xn + 1 2 ⌃ k2, tn + 1 2h) ⌃ k3 = h⌃ v(xn + 1 2 ⌃ k2, tn + 1 2h) ⌃ k1 = h⌃ v(xn, tn)

⌃ k3 = h⌃ v(xn + 1 2 ⌃ k2, tn + 1 2h)

⌃ k1 = h⌃ v(xn, tn)

⇧ xn+1 = ⇧ xn + 1 6 ⇧ k1 + 1 3 ⇧ k2 + 1 3 ⇧ k3 + 1 6 ⇧ k4 + O(h5) ~ k2 = h~ v(xn + 1 2 ~ k1, tn + 1 2h)

~ k4 = h~ v(xn + ~ k3, tn + h)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 4

Increased order does not imply increased accuracy!

RK4 requires 4 evaluations of RK2 requires only 2 RK4 more “accurate” if same accuracy can be achieved with double step size (at least)

In most practical cases RK4 outperforms RK2

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⇧ v(t, ⇧ x)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Adaptive Step Size

Objectives

Reduce computational effort Increase accuracy

Adapt step size h to achieve required accuracy Compare several integration steps to estimate actual accuracy

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Adaptive Step Size

Step doubling method (with RK4)

1 step with double step size 2h 2 steps with size h Error estimate Increased order

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⌥ x(t + 2h) = ⌥ k1 + (2h)5⌥ + O(h6) ⌥ x(t + 2h) = ⌥ k2 + 2(h)5⌥ + O(h6) ⇥ ∆ = ⇥ k2 − ⇥ k1 ⌃ x(t + 2h) = ⌃ k2 + ⌃ ∆ 15 + O(h6)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 4/5 (RK45)

Embedded Runge-Kutta formulas

Different combinations of same 6 function evaluations yield 4th and 5th order formulas

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x∗

n+1 = xn + c∗ 1k1 + c∗ 2k2 + c∗ 3k3 + c∗ 4k4 + c∗ 5k5 + c∗ 6k6 + O(h5)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 4/5

Embedded Runge-Kutta formulas

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge-Kutta 4/5

Dormand/Prince (DOPRI45): dense output

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Runge Kutta 4/5

Embedded Runge-Kutta formulas

Error evaluation For a desired accuracy , required step size Used to adjust step size

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∆ = xn+1 − x∗

n+1 = 6

• i=1

(ci − c∗

i )ki = O(h5)

∆0

h0 = h1

• ∆0

∆1

• 1

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Definitions

Streamline

trajectory of massless particle in a steady flow or path of a particle in a frozen unsteady field (non-physical)

Pathline

trajectory of massless particle in an unsteady flow

Timeline

connect particles released simultaneously at discrete time-steps

Streakline

continuously inject particles at a fixed point in the flow, connect consecutive particles

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamlines

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• Basic idea: visualizing the flow directions by

releasing particles and calculating a series of particle positions based on the vector field

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamlines

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• Displaying streamlines is a local technique because you

can only visualize the flow directions initiated from one

• r a few particles (sparse / discrete flow sampling)
• When the number of streamlines is increased, the scene

becomes cluttered

• User needs to know where to drop the

seed particles (seeding strategy)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamlines

Many streamlines can give an overall / global view of the vector field

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Rakes = multiple streamlines

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Rake: geometric primitive (typically a line) along which multiple streamlines are seeded at regular interval.

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Pathlines

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• Extension of streamlines for time-varying data

(unsteady flows) Pathlines:

T=1 T=2 T=3 T=4 T=5

Pathline: Trajectory of a given particle. This is generated by injecting a dye into the fluid and following its path by photography or other means

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Timelines

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• Extension of streamlines for time-varying data

(unsteady flows)

T = 1 T = 2 T = 3 timeline

Timeline is generated by drawing a line through adjacent particles in flow at any instant of time

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streaklines

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b.t. =5 b.t. =4 b.t. =3 b.t. =2 b.t. =1

Streakline concentrates on fluid particles that have gone through a fixed station or point. At some instant of time the position of all these particles are marked and a line is drawn through them. Such a line is called a streakline.

• Extension of streamlines for time-varying data

(unsteady flows)

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advection methods comparison

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Stream-ribbon

Basic idea:

Need a means of visualizing swirling flow behavior (vortices) A point primitive or an icon can hardly convey this Idea: advect neighboring particles and connect them with polygons Shade those polygons appropriately and one will detect twists

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Stream-ribbon

Problem - when flow diverges Solution: Just trace one streamline and a constant size vector with it:

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Ribbons

Ribbons can be constructed using 2 or 3 particle advections per segment.

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Illuminated Streamlines

Remember Phong lighting model:

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θ θ

γ

L R V N I = Ia + Id + Is = ka + kd L · N + ks (V · R)n

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Illuminated Streamlines

Model requires normal vector but curves have an entire normal plane

Rewrite with Hence Similarly

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T

• L

L = LT + LN

LT = L · T

L · N = |LN| = p 1 − |LT |2 = p 1 − (L · T)2

V · R = (L · T)(V · T)−

p 1 − (L · T)2p 1 − (V · T)2

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Illuminated Streamlines

These quantities can be computed efficiently by the graphics hardware using texture mapping.

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Interactive visualization of 3D-vector fields using illuminated stream lines. Zöckler, Stalling, Hege, IEEE Vis 1996

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Illuminated Streamlines

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Illuminated Streamlines Revisited. Mallo, Peikert, Sigg, Sadlo, IEEE Vis 2005

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamline Seeding

Isolated streamlines provide poor picture of the flow continuum Large number of streamlines create visual clutter Seeding strategy is important to effectively use streamlines in visualization

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Turk & Banks ‘96

Mimic hand-drawn flows Use multi-scale analysis to determine position/ density Minimize energy function Rules for adding/removing/moving streamlines

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Hand Drawn Flows

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Automated Results

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Approach: Image-based

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Comparison

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Evenly sampled “Optimized”

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streamline Seeding

Streamline Placement

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Farthest Point Seeding

Motivations

Speed up Turk and Banks algorithm (x200!) Achieve better results by mitigating issue of short streamlines

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Mebarki et al., Farthest point seeding for efficient placement of streamlines. IEEE Visualization 2005

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Farthest Point Seeding

Seed next streamline at point furthest away from existing streamlines

Streamlines are approximated by discrete set of points Points are Delaunay triangulated Empty circumscribed circle of triangles used to determine largest empty regions

54 Mebarki et al., Farthest point seeding for efficient placement of streamlines. IEEE Visualization 2005

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Farthest Point Seeding

Seed next streamline at point furthest away from existing streamlines

55 Mebarki et al., Farthest point seeding for efficient placement of streamlines. IEEE Visualization 2005

streamlines are approximated by discrete set of points points are Delaunay triangulated Empty circumscribed circle of triangles used to determine largest empty regions

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Stream Surface

Surface spanned by union of infinitely many streamlines integrated from a curve

Rake, polygonal line, closed curve, … Remark: circle ⇒ “stream tube”

Everywhere tangential to the flow “Natural” extension of streamlines in 3D

Capture volumes, shading, depth cues

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Stream Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Definition

Natural parameterization [s x t]

Starting curve (→ front) Along the flow (integration parameter, time)

Constant s: streamline Constant t: time line

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t s

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

(Naïve) Implementation

Sparse set of streamlines integrated independently Ribbons connecting neighboring streamlines Corresponds to a triangulation in parameter space Extremely inaccurate and/or inefficient

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front (Hultquist)

Advance a discrete front iteratively Triangulate during integration

Ensure well-shaped triangles Local heuristic Handle shearing effects

Dynamically adapt front resolution to account for divergence / convergence

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Front is stored as a double-linked list of streamlines

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i i-1 i+1

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

A streamline on the front is locally advanced w.r.t. two neighboring streamlines

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Objective: maintain front locally

• rthogonal to flow direction

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i i-1 i+1

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Objective: maintain front locally orthogonal to flow direction Triangulation heuristic creates shortest diagonal (greedy algo.)

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i i-1 i+1

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

i i-1 i+1

Advancing Front

Triangulation implicitly determines which streamlines must be advanced next

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Triangulation locally determines which streamline must be advanced next

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i i-1 i+1

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Triangulation locally determines which streamline must be advanced next

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i i-1 i+1

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Once a streamline has been advanced, its neighbors must be advanced too, if required Local decision propagates along the front to ensure orthogonality to the flow Active point moves back and forth Iteration complete when all neighboring streamlines have roughly the same size

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Front resolution (=distance between neighboring streamlines) varies during integration (convergence/ divergence) Ensure constant resolution by inserting or removing additional streamlines when necessary Requires to advance all streamlines simultaneously

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Hultquist’s criterion for splitting: relative width / length of current quadrilateral exceeds factor 2

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Hultquist’s criterion for merging: 3 neighboring points are coplanar and close Required to speed up computation and decrease complexity of generated surface Both merging and splitting criteria must be evaluated at each step

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Ripping is necessary if stream surface encounters an obstacle

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Improvements of original method

Higher integration accuracy (RK 45) Integration step controlled by arc length

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integration parameter arc length

• C. Garth et al., Surface Techniques for Vortex Visualization

Joint Eurographics - IEEE TCVG Symposium on Visualization

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Possible improvements of original method

Curvature control in refinement criterion

Avoids creases on surface Properly handle folding behavior (vortex)

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segment size only inter-segment angle

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Possible improvements of original method

Topological information

Front resolution control Ripping criterion

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Examples

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Examples

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Advancing Front

Examples

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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• C. Garth et al., Generation of Accurate Integral Surfaces

in Time-Dependent Vector Fields. IEEE Visualization 2008

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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

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t → s →

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Path Surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Texture Mapping on Path

Vortex formation behind an ellipsoid (2.6e6 cells, 400 timesteps)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Texture Mapping on Path

Vortex formation behind an ellipsoid (2.6e6 cells, 400 timesteps)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Texture Mapping

Vortex formation behind an ellipsoid (2.6e6 cells, 400 timesteps)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Texture Mapping

Vortex formation behind an ellipsoid (2.6e6 cells, 400 timesteps)

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Texture Mapping

Vortex formation behind an ellipsoid (2.6e6 cells, 400 timesteps)

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• C. Garth et al., Generation of Accurate Integral Surfaces

in Time-Dependent Vector Fields. IEEE Visualization 2008

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020 87

• C. Garth et al., Generation of Accurate Integral Surfaces

in Time-Dependent Vector Fields. IEEE Visualization 2008

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020 87

• C. Garth et al., Generation of Accurate Integral Surfaces

in Time-Dependent Vector Fields. IEEE Visualization 2008

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Streak and Time

Similar adaptive front refinement principle

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• H. Krishnan et al., Time and Streak Surfaces for Flow Visualization

in Large Time-Varying Data Sets. IEEE Visualization 2009

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Integration must be carried out on smooth

surfaces (tangent bundle of manifold)

• Problems
• polygonal surfaces are not smooth
• Tangent plane undefined at edges and vertices
• Global parameterization (i.e. invertible mapping from the

plane to the surface) of surface may not be available or not exist

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Quick and dirty
• integrate in surrounding (3D) flow
• Control distance to surface
• Successive projections for a posteriori

corrections

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Cell-wise integration
• Use well-defined (smooth) cell-wise

tangent plane to carry out integration over projected vector field in each cell

• Use analytic formulas (triangles) or RK
• Iterative computation from edge to edge

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Cell-wise integration
• Problem: possible inconsistencies of cell-

wise vector field projections at edges

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Use discrete geodesics defined on

polygonal surfaces

• Geodesics in continuous setting
• Extension of straight lines to curved surfaces
• Shortest path between two points
• Smooth curve is geodesics iff it is not curved within

the surface

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CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Geodesics on polygonal surfaces

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• K. Polthier, M. Schmies, Straightest Geodesics on Polyhedral Surfaces,

Mathematical Visualization, H.C. Hege, K. Polthier (eds.), Springer Verlag, 1998

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Parallel transport of vectors

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• K. Polthier, M. Schmies, Straightest Geodesics on Polyhedral Surfaces,

Mathematical Visualization, H.C. Hege, K. Polthier (eds.), Springer Verlag, 1998

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

: Parallel transport

along

Integration on Surfaces

• Geodesic Runge-Kutta method

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: Geodesic path

• K. Polthier, M. Schmies, Straightest Geodesics on Polyhedral Surfaces,

Mathematical Visualization, H.C. Hege, K. Polthier (eds.), Springer Verlag, 1998

⇧ yi+1 = (h, yi, ⇧ yi)

δ π|δ

δ

CS530 / Spring 2020 : Introduction to Scientific Visualization. Streamlines and Stream Surfaces Feb 20, 2020

Integration on Surfaces

• Geodesic Runge-Kutta method

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: Parallel transport

along : Geodesic path

⇧ yi+1 = (h, yi, ⇧ yi)

δ π|δ

δ

• K. Polthier, M. Schmies, Straightest Geodesics on Polyhedral Surfaces,

Mathematical Visualization, H.C. Hege, K. Polthier (eds.), Springer Verlag, 1998