[PPT] - Vector Visualiza0on Applica0ons Climate Modeling Computational PowerPoint Presentation

SLIDE 1

Vector Field Visualiza0on

Han-Wei Shen The Ohio State University

(Some images courtesy of David Kao, NASA Ames Research Center)

SLIDE 2

Vector Visualiza0on Applica0ons

Computational Fluid Dynamics Climate Modeling Medical Applications Astronomy

SLIDE 3

Vector field: F(u) = v

u: posi0on in the domain (x,y) in 2D (x,y,z) in 3D v: the vector (u,v) or (u,v,w) at u

Vector Field Visualiza0on

SLIDE 4

Vector field: F(u) = v

u: posi0on in the domain (x,y) in 2D (x,y,z) in 3D v: the vector (u,v) or (u,v,w) at u

Vectors are oPen defined at discrete points as aQributes

at different 0me steps

Vector Field Visualiza0on

…

SLIDE 5

Methods for Experimental Flow Visualiza0on - Adding Foreign Material

Streaklines: dye injected from a fixed posi0on. By injec0ng the

dye for a period of 0me, a line of dye in the fluid is visible

Timelines: a row of small par0cles (hydrogen bubbles) released

at right angles to flow. The mo0on of the par0cles shows the fluid behavior

Pathlines: small par0cles (magnesium powder in liquid; oil

drops in gas) are added to the fluid. Velocity is measured by photographing the mo0on of the par0cles with a known exposure 0me

SLIDE 6

Experimental Flow Visualization

SLIDE 7

Numerical Flow Visualiza0on

Though numerical flow visualiza0on is not able to totally replicate the results from experimental flow visualiza0on, it has been widely accepted as an effec0ve mean to obtain accurate representa0on of the CFD flow solu0ons.

SLIDE 8

Numerical Flow Visualization – Basic Methods

SLIDE 9

Numerical Flow Visualization – Basic Methods

Velocity Magnitude Contours Vector Arrow Plots Par0cle Traces - Streamlines

SLIDE 10

Visualiza0on Techniques I

Geometry-based methods:

Render primi0ves from par0cle trajectories

1D: streamlines, pathlines,

streaklines

Line varia0ons: tubes,

ribbons

2D: stream surfaces
3D: flow volumes

SLIDE 11

Visualiza0on Techniques II

Texture-based methods: Create

texture paQerns following flow direc0ons

– Line Integral Convolu0on – Texture Splates

SLIDE 12

Visualiza0on Techniques III

Topological-based methods:

Extract features based on flow topology

– Critical points – Vortex cores – Skin-friction lines

SLIDE 13

Par0cle Tracing

Visualizing the flow direc0ons by releasing par0cles and calcula0ng a series of par0cle posi0ons based on the vector field

SLIDE 14

Par0cle Tracing Challenges

Displaying streamlines is a local technique because you can
nly visualize the flow direc0ons ini0ated from one or a few

par0cles

When the number of streamlines is increased, the scene

becomes cluQered

You need to know where to drop the par0cle seeds
Streamline computa0on is not cheap
It is much more difficult to predict the data access paQern

SLIDE 15

Seed Placement

Density and distribu0on

SLIDE 16

Vector Glyphs

Display the en0re flow field in a single picture
Minimal user interven0on
Example: Hedgehogs (global arrow plots)

SLIDE 17

Example of hedgehogs

SLIDE 18

Varia0ons of hedgehogs

SLIDE 19

Line Integral Convolu0on (LIC)

SLIDE 20

Flow Line Computa0on

SLIDE 21

Flow Line Visualiza0on

One of the most fundamental flow visualiza0on

techniques

Type of flow lines:

– Streamline: a field line tangent to the velocity field at an instant in 0me – Pathline: the trajectory of a mass less par0cle released from a seed point over a period of 0me – Streakline: a line joining the posi0ons, at an instant in 0me, of par0cles released from a seed point – Timeline: a line connec0ng a row of par0le that are released simultaneously In a steady flow filed, streamlines, pathlines, and streaklines are iden0cal

SLIDE 22

Streamlines

Streamline: a line that is tangen0al to the instantaneous

velocity direc0on (velocity is a vector, and it has a magnitude and a direc0on)

Release a par0cle into the flow and perform numerical

integra/on to compute the path of the par0cle

SLIDE 23

Pathlines

A pathline shows the trajectory of a single particle

released from a fixed location (seed point)

Experimental method: marking a fluid par0cle and taking

a 0me exposure photo of its mo0on will generate a pathline

This is similar to what you see when you take a long-

exposure photograph of car lights on a freeway at night

SLIDE 24

Streaklines

Streakline: a line joining the posi0ons, at an instant in

0me, of all par0cles that were previously released from a fixed loca0on (seed point)

Con0nuously inject par0cles into the flow at each 0me

step and track the paths of the par0cles

In a steady flow field, streamlines, pathlines, and

streaklines are iden0cal. However, they can be very different in unsteady flows

SLIDE 25

Timelines

Timeline: a line connec0ng a row of par0cles that

released simultaneously

Timelines are generated by injec0ng rows of

par0cles at some fixed 0me interval

SLIDE 26

[Lane ‘96]e

SLIDE 27

Compu0ng Flow Lines by Par0cle Tracing

The path of a massless par0cle at posi0on p at 0me t

can be described by the following ordinary differen0al equa0on:

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steady flow

r

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unsteady flow

SLIDE 28

Compu0ng Flow Lines - Par0cle Tracing

The path of a massless par0cle at posi0on p at 0me t

can be described by the following ordinary differen0al equa0on:

And the posi0ons of the par0cle can be computed by

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r

The image part with relationship ID rId5 was not found in the file.

steady flow unsteady flow Solved by numerical integra0on

SLIDE 29

Compu0ng Flow Lines - Par0cle Tracing

The path of a massless par0cle at posi0on p at 0me t

can be described by the following ordinary differen0al equa0on:

And the posi0ons of the par0cle can be computed by

The image part with relationship ID rId3 was not found in the file.

r

The image part with relationship ID rId5 was not found in the file.

steady flow unsteady flow Solved by numerical integra0on

SLIDE 30

Numerical Integra0on

Discrete approxima0on of the con0nuous

integra0on result

t

v v(t)

SLIDE 31

Numerical Integra0on

Discrete approxima0on of the con0nuous

integra0on result

Calculate area under curve

for curve y(t)

– Trapezoidal approxima0on

t

v v(t)

t

v v(t)

SLIDE 32

Numerical Integra0on

Discrete approxima0on of the con0nuous

integra0on result

Calculate area under curve

for curve y(t)

– Trapezoidal approxima0on

Error related to the step

size used

Below we describe a few popular methods of

numerical integra0on for par0cle tracing

t

v v(t)

t

v v(t)

SLIDE 33

Euler’s Method

Simple but lower accuracy

pk+1 = pk + v(pk) x Δt

Approxima0on Ground True

SLIDE 34

Euler’s Method

Simple but lower accuracy

pk+1 = pk + v(pk) x δt

The equa0on can be derived from Taylor series expansion

where y is one component of the P posi0on, and y’ is the one component of the velocity, h is the step size (Δt) Error

Approxima0on Ground True

SLIDE 35

2nd Order Runge-KuQa (RK2)

Improved accuracy, more commonly used

p* = pk + v(pk) x δt

SLIDE 36

2nd Order Runge-KuQa (RK2)

Improved accuracy, more commonly used

p* = pk + v(pk) x δt pk+1 = pk + (v(pk) + v(p*) ) x δt/2

Can also be derived from Taylor’s series (next slide)

SLIDE 37

2nd Order Runge-KuQa (RK2)

Improved accuracy from Euler for ODE
Differen0a0ng the above equa0on
From Taylor series expansion:

V(Pk) V(P*)

SLIDE 38

4th Order Runge-KuQa (RK4)

BeQer accuracy, recommended

a = 2δt v(pk), b = 2δt v(pk+a/2), c = 2δt v(pk+b/2) d = 2δt v(pk+c/2), pk+1 = pk + (a+2b+2c+d)/6

SLIDE 39

Put it All Together Par0cle Tracing Algorithm

1. Specify a seed posi0on p(0), t = 0
2. Perform cell search to locate the cell that contains

the p(t)

3. Interpolate the velocity field to determine the

velocity at p(t)

4. Advance the par0cle from p(t) to p(t+Δt) using a

numerical integra0on method

5. Repeat from step 2 un0l the par0cle moves a

certain distance or goes out of bound

SLIDE 40

Line Integral Convolu0on

SLIDE 41

Line Integral Convolution

SLIDE 42

Motivation

To visualize steady 2D vector fields,
r flows near the body of the model

(surface flows)

To create images similar to

experimental oil-flow visualization

To overcome the drawback of

traditional flow visualization techniques

SLIDE 43

Streamline Problems

Seed placement is non trivial
Trial and error
Need prior knowledge
Unsatisfactory visualization
Grid spacing may vary
Difficult to control streamline lengths
Discontinued flow lines

SLIDE 44

Streamlines

Uneven streamline density and lengths

SLIDE 45

Streamline and LIC Comparison

Streamlines LIC

SLIDE 46