Ronald Fedkiw

Stanford UniversityIndustrial Light + Magic

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?• There is no safe place to start...There is no safe place to start...

• One cannot even assume that the One cannot even assume that the correctcorrect answer answer is plausibleis plausible– suppose a character threw a ball forward at the ground and suppose a character threw a ball forward at the ground and

it bounced back into their hand like a boomerangit bounced back into their hand like a boomerang

– this is a backward scattering of the ball, and does have this is a backward scattering of the ball, and does have some small probability of occurringsome small probability of occurring

– unfortunately, it is not very likely and the audience will be unfortunately, it is not very likely and the audience will be hard pressed to accept ithard pressed to accept it

– in fact, one would expect the characters to be rather in fact, one would expect the characters to be rather surprised at the outcomesurprised at the outcome

– that is, character emotion is needed to aid plausibility in that is, character emotion is needed to aid plausibility in this casethis case

• There is no safe place to start...There is no safe place to start...

• One cannot even assume that the One cannot even assume that the correctcorrect answer answer is plausibleis plausible– suppose a character threw a ball forward at the ground and suppose a character threw a ball forward at the ground and

it bounced back into their hand like a boomerangit bounced back into their hand like a boomerang

– this is a backward scattering of the ball, and does have this is a backward scattering of the ball, and does have some small probability of occurringsome small probability of occurring

– unfortunately, it is not very likely and the audience will be unfortunately, it is not very likely and the audience will be hard pressed to accept ithard pressed to accept it

– in fact, one would expect the characters to be rather in fact, one would expect the characters to be rather surprised at the outcomesurprised at the outcome

– that is, character emotion is needed to aid plausibility in that is, character emotion is needed to aid plausibility in this casethis case

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?• If 10,000 balls are thrown forward at the ground, If 10,000 balls are thrown forward at the ground,

and 1 bounces back that might be okand 1 bounces back that might be ok– the collision result of the ball with the ground obeys a the collision result of the ball with the ground obeys a

“BRDF-like” probability for the scattering direction“BRDF-like” probability for the scattering direction

– one might believe that 1 out of 10,000 balls backscattersone might believe that 1 out of 10,000 balls backscatters

– in fact, one might expect some backscatterin fact, one might expect some backscatter

– in this instance the audience witnesses a reasonable in this instance the audience witnesses a reasonable probability distribution first hand (lots of balls)probability distribution first hand (lots of balls)

– but when only one ball is thrown, the audience expects the but when only one ball is thrown, the audience expects the most likely solution, and insists that the characters expect most likely solution, and insists that the characters expect this toothis too

– however, this can appear too sterile (e.g. mirror reflection)however, this can appear too sterile (e.g. mirror reflection)

• If 10,000 balls are thrown forward at the ground, If 10,000 balls are thrown forward at the ground, and 1 bounces back that might be okand 1 bounces back that might be ok– the collision result of the ball with the ground obeys a the collision result of the ball with the ground obeys a

“BRDF-like” probability for the scattering direction“BRDF-like” probability for the scattering direction

– one might believe that 1 out of 10,000 balls backscattersone might believe that 1 out of 10,000 balls backscatters

– in fact, one might expect some backscatterin fact, one might expect some backscatter

– in this instance the audience witnesses a reasonable in this instance the audience witnesses a reasonable probability distribution first hand (lots of balls)probability distribution first hand (lots of balls)

– but when only one ball is thrown, the audience expects the but when only one ball is thrown, the audience expects the most likely solution, and insists that the characters expect most likely solution, and insists that the characters expect this toothis too

– however, this can appear too sterile (e.g. mirror reflection)however, this can appear too sterile (e.g. mirror reflection)

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?• When rendering, many rays are cast per pixel, When rendering, many rays are cast per pixel,

and both sampling and averaging occurs until a and both sampling and averaging occurs until a single solution is converged uponsingle solution is converged upon– rendering algorithms tend to pick a solution near the rendering algorithms tend to pick a solution near the

average valueaverage value

– in dynamics it is not always sensical to compute and in dynamics it is not always sensical to compute and display an average solutiondisplay an average solution

– can only depict a single solution (unless we have more can only depict a single solution (unless we have more balls)balls)

– so it never makes sense to show something improbable so it never makes sense to show something improbable unless the characters will act surprised by the eventunless the characters will act surprised by the event

– plausible simulation seems somehow connected with plausible simulation seems somehow connected with story telling, imagination, etc.story telling, imagination, etc.

• When rendering, many rays are cast per pixel, When rendering, many rays are cast per pixel, and both sampling and averaging occurs until a and both sampling and averaging occurs until a single solution is converged uponsingle solution is converged upon– rendering algorithms tend to pick a solution near the rendering algorithms tend to pick a solution near the

average valueaverage value

– in dynamics it is not always sensical to compute and in dynamics it is not always sensical to compute and display an average solutiondisplay an average solution

– can only depict a single solution (unless we have more can only depict a single solution (unless we have more balls)balls)

– so it never makes sense to show something improbable so it never makes sense to show something improbable unless the characters will act surprised by the eventunless the characters will act surprised by the event

– plausible simulation seems somehow connected with plausible simulation seems somehow connected with story telling, imagination, etc.story telling, imagination, etc.

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for turbulence we have no idea what the correct for turbulence we have no idea what the correct

equations even areequations even are

– engineers use turbulence models with the goal of engineers use turbulence models with the goal of reproducing some aspect of the physicsreproducing some aspect of the physics

– they tune parameters to get what they wantthey tune parameters to get what they want

– i.e. they work to match experiments for lift, drag, etc.i.e. they work to match experiments for lift, drag, etc.

– however, they could not care less about the visual normhowever, they could not care less about the visual norm

– these models are mostly useless for visually plausible these models are mostly useless for visually plausible simulation simulation

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for turbulence we have no idea what the correct for turbulence we have no idea what the correct

equations even areequations even are

– engineers use turbulence models with the goal of engineers use turbulence models with the goal of reproducing some aspect of the physicsreproducing some aspect of the physics

– they tune parameters to get what they wantthey tune parameters to get what they want

– i.e. they work to match experiments for lift, drag, etc.i.e. they work to match experiments for lift, drag, etc.

– however, they could not care less about the visual normhowever, they could not care less about the visual norm

– these models are mostly useless for visually plausible these models are mostly useless for visually plausible simulation simulation

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for collisions the result depends on the microstructure for collisions the result depends on the microstructure

of the materialof the material

– the microstructure occurs on a microscopic scale that is the microstructure occurs on a microscopic scale that is difficult to measure and even more difficult to modeldifficult to measure and even more difficult to model

– since we cannot see the microstructure, we cannot since we cannot see the microstructure, we cannot expect the audience to believe all high probability expect the audience to believe all high probability microstructure eventsmicrostructure events

– the audience will instead assume a microstructure and the audience will instead assume a microstructure and we need to be faithful to thatwe need to be faithful to that

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for collisions the result depends on the microstructure for collisions the result depends on the microstructure

of the materialof the material

– the microstructure occurs on a microscopic scale that is the microstructure occurs on a microscopic scale that is difficult to measure and even more difficult to modeldifficult to measure and even more difficult to model

– since we cannot see the microstructure, we cannot since we cannot see the microstructure, we cannot expect the audience to believe all high probability expect the audience to believe all high probability microstructure eventsmicrostructure events

– the audience will instead assume a microstructure and the audience will instead assume a microstructure and we need to be faithful to thatwe need to be faithful to that

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for collisions the result depends on the approach for collisions the result depends on the approach

velocity, angular velocity, internal response, etc. velocity, angular velocity, internal response, etc.

– these are all difficult to measure and model exactlythese are all difficult to measure and model exactly

– the flight conditions of a thrown ball depend on the the flight conditions of a thrown ball depend on the texture of the skin on the fingers, how it was released, texture of the skin on the fingers, how it was released, the firing of muscles in the hand, fingers and arm, etc.the firing of muscles in the hand, fingers and arm, etc.

– quite a lot is unknown, unmodeled, etc. quite a lot is unknown, unmodeled, etc.

• Many times, one cannot compute the correct Many times, one cannot compute the correct solution anywaysolution anyway– for collisions the result depends on the approach for collisions the result depends on the approach

velocity, angular velocity, internal response, etc. velocity, angular velocity, internal response, etc.

– these are all difficult to measure and model exactlythese are all difficult to measure and model exactly

– the flight conditions of a thrown ball depend on the the flight conditions of a thrown ball depend on the texture of the skin on the fingers, how it was released, texture of the skin on the fingers, how it was released, the firing of muscles in the hand, fingers and arm, etc.the firing of muscles in the hand, fingers and arm, etc.

– quite a lot is unknown, unmodeled, etc. quite a lot is unknown, unmodeled, etc.

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?

• SummarySummary– the correct physics may seem implausiblethe correct physics may seem implausible

– the correct physics may be impossible to computethe correct physics may be impossible to compute

– often, engineers only care about macroscopic integral often, engineers only care about macroscopic integral quantities like lift and drag, not high frequency visual quantities like lift and drag, not high frequency visual featuresfeatures

– characters probably need to react a certain way in order characters probably need to react a certain way in order to make the correct physics seem plausibleto make the correct physics seem plausible

• SummarySummary– the correct physics may seem implausiblethe correct physics may seem implausible

– the correct physics may be impossible to computethe correct physics may be impossible to compute

– often, engineers only care about macroscopic integral often, engineers only care about macroscopic integral quantities like lift and drag, not high frequency visual quantities like lift and drag, not high frequency visual featuresfeatures

– characters probably need to react a certain way in order characters probably need to react a certain way in order to make the correct physics seem plausibleto make the correct physics seem plausible

What is plausible What is plausible simulation?simulation?What is plausible What is plausible simulation?simulation?

• I have no ideaI have no idea

• But, let me tell you how to take advantage of But, let me tell you how to take advantage of it...it...

• I have no ideaI have no idea

• But, let me tell you how to take advantage of But, let me tell you how to take advantage of it...it...

Mathematical EquationsMathematical EquationsMathematical EquationsMathematical Equations

• Physicists, chemists, engineers, etc. are all Physicists, chemists, engineers, etc. are all interested in mathematical descriptions of the interested in mathematical descriptions of the world around usworld around us– these are models!these are models!

– they are intended to capture some aspect of the they are intended to capture some aspect of the problemproblem

– they are not intended to be exact, true, or delivered they are not intended to be exact, true, or delivered from your favorite deityfrom your favorite deity

– that does not mean that they are not usefulthat does not mean that they are not useful

– e.g., even though electrons do not orbit atoms like e.g., even though electrons do not orbit atoms like planets, that model does account for certain planets, that model does account for certain phenomena, and can be quite usefulphenomena, and can be quite useful

• Physicists, chemists, engineers, etc. are all Physicists, chemists, engineers, etc. are all interested in mathematical descriptions of the interested in mathematical descriptions of the world around usworld around us– these are models!these are models!

– they are intended to capture some aspect of the they are intended to capture some aspect of the problemproblem

– they are not intended to be exact, true, or delivered they are not intended to be exact, true, or delivered from your favorite deityfrom your favorite deity

– that does not mean that they are not usefulthat does not mean that they are not useful

– e.g., even though electrons do not orbit atoms like e.g., even though electrons do not orbit atoms like planets, that model does account for certain planets, that model does account for certain phenomena, and can be quite usefulphenomena, and can be quite useful

Numerical AlgorithmsNumerical AlgorithmsNumerical AlgorithmsNumerical Algorithms

• Since the equations are not necessarily correct, Since the equations are not necessarily correct, there is no reason to force the numerical there is no reason to force the numerical algorithms to exactly mimic the equationsalgorithms to exactly mimic the equations– consistency is over-ratedconsistency is over-rated

• if your algorithm does not solve the given equations, if your algorithm does not solve the given equations, it may solve some other set of equations that are it may solve some other set of equations that are interesting in their own rightinteresting in their own right

– accuracy is over-ratedaccuracy is over-rated

• there is no reason to strive for 8 digits of accuracy there is no reason to strive for 8 digits of accuracy when solving a set of equations that is inconsistent when solving a set of equations that is inconsistent with the physics in the second or third decimal pace with the physics in the second or third decimal pace

• Since the equations are not necessarily correct, Since the equations are not necessarily correct, there is no reason to force the numerical there is no reason to force the numerical algorithms to exactly mimic the equationsalgorithms to exactly mimic the equations– consistency is over-ratedconsistency is over-rated

• if your algorithm does not solve the given equations, if your algorithm does not solve the given equations, it may solve some other set of equations that are it may solve some other set of equations that are interesting in their own rightinteresting in their own right

– accuracy is over-ratedaccuracy is over-rated

• there is no reason to strive for 8 digits of accuracy there is no reason to strive for 8 digits of accuracy when solving a set of equations that is inconsistent when solving a set of equations that is inconsistent with the physics in the second or third decimal pace with the physics in the second or third decimal pace

ExamplesExamplesExamplesExamples

• The Navier-Stokes equations do not allow 2 water The Navier-Stokes equations do not allow 2 water drops to merge into a single dropdrops to merge into a single drop

• The Navier-Stokes equations do not allow a The Navier-Stokes equations do not allow a falling drop to actually hit the groundfalling drop to actually hit the ground

• the numerous models for frictional contact and the numerous models for frictional contact and collision do not accurately account for collision do not accurately account for microstructuremicrostructure

• For large deformations, accurate stress strain For large deformations, accurate stress strain relationships are unknown (a nonlinear Green relationships are unknown (a nonlinear Green strain coupled to linear elasticity is useless)strain coupled to linear elasticity is useless)

• bending of shells and cloth, plasticity, fracture, bending of shells and cloth, plasticity, fracture, damping, etc... damping, etc...

• The Navier-Stokes equations do not allow 2 water The Navier-Stokes equations do not allow 2 water drops to merge into a single dropdrops to merge into a single drop

• The Navier-Stokes equations do not allow a The Navier-Stokes equations do not allow a falling drop to actually hit the groundfalling drop to actually hit the ground

• the numerous models for frictional contact and the numerous models for frictional contact and collision do not accurately account for collision do not accurately account for microstructuremicrostructure

• For large deformations, accurate stress strain For large deformations, accurate stress strain relationships are unknown (a nonlinear Green relationships are unknown (a nonlinear Green strain coupled to linear elasticity is useless)strain coupled to linear elasticity is useless)

• bending of shells and cloth, plasticity, fracture, bending of shells and cloth, plasticity, fracture, damping, etc... damping, etc...

What do numerical What do numerical analysts do?analysts do?What do numerical What do numerical analysts do?analysts do?

• They sit in their office waiting for some scientist They sit in their office waiting for some scientist or engineer to walk in with a set of equations or engineer to walk in with a set of equations and say “simulate this”and say “simulate this”– a numerical analyst does not know the underlying a numerical analyst does not know the underlying

physicsphysics

– thus, the best they can do is to faithfully and thus, the best they can do is to faithfully and consistently solve the equations they are givenconsistently solve the equations they are given

– if they’re algorithm is not consistent with the equations, if they’re algorithm is not consistent with the equations, they are essentially constructing a new modelthey are essentially constructing a new model

– this is less than ideal since they do not know the physics this is less than ideal since they do not know the physics of the problem at handof the problem at hand

– once they have consistency, they strive for accuracy in once they have consistency, they strive for accuracy in order to get a more efficient algorithm order to get a more efficient algorithm

• They sit in their office waiting for some scientist They sit in their office waiting for some scientist or engineer to walk in with a set of equations or engineer to walk in with a set of equations and say “simulate this”and say “simulate this”– a numerical analyst does not know the underlying a numerical analyst does not know the underlying

physicsphysics

– thus, the best they can do is to faithfully and thus, the best they can do is to faithfully and consistently solve the equations they are givenconsistently solve the equations they are given

– if they’re algorithm is not consistent with the equations, if they’re algorithm is not consistent with the equations, they are essentially constructing a new modelthey are essentially constructing a new model

– this is less than ideal since they do not know the physics this is less than ideal since they do not know the physics of the problem at handof the problem at hand

– once they have consistency, they strive for accuracy in once they have consistency, they strive for accuracy in order to get a more efficient algorithm order to get a more efficient algorithm

We are the “scientists”We are the “scientists”We are the “scientists”We are the “scientists”

• We understand the underlying problem and we We understand the underlying problem and we know what we want... plausible simulationknow what we want... plausible simulation– I don’t know what it is, but I know that I want itI don’t know what it is, but I know that I want it

– it makes lot’s of $$ at the box officeit makes lot’s of $$ at the box office

– it makes graduate students workit makes graduate students work

– it makes our friends think we are coolit makes our friends think we are cool

• We understand the underlying problem and we We understand the underlying problem and we know what we want... plausible simulationknow what we want... plausible simulation– I don’t know what it is, but I know that I want itI don’t know what it is, but I know that I want it

– it makes lot’s of $$ at the box officeit makes lot’s of $$ at the box office

– it makes graduate students workit makes graduate students work

– it makes our friends think we are coolit makes our friends think we are cool

We are the “scientists”We are the “scientists”We are the “scientists”We are the “scientists”

• We can write down a set of equationsWe can write down a set of equations

• We understand the visual normWe understand the visual norm

• We can consult numerical textbooks to solve the We can consult numerical textbooks to solve the equationsequations– ode and pde solversode and pde solvers

– linear algebra routineslinear algebra routines

– optimizationoptimization

– solid and fluid mechanicssolid and fluid mechanics

– statisticsstatistics

• We can consult with the numerical analysts We can consult with the numerical analysts directly directly

• We can write down a set of equationsWe can write down a set of equations

• We understand the visual normWe understand the visual norm

• We can consult numerical textbooks to solve the We can consult numerical textbooks to solve the equationsequations– ode and pde solversode and pde solvers

– linear algebra routineslinear algebra routines

– optimizationoptimization

– solid and fluid mechanicssolid and fluid mechanics

– statisticsstatistics

• We can consult with the numerical analysts We can consult with the numerical analysts directly directly

CautionCautionCautionCaution

• Be careful borrowing equations and numerical Be careful borrowing equations and numerical algorithms from other fieldsalgorithms from other fields– most likely, the equations were not written down to most likely, the equations were not written down to

respect the visual normrespect the visual norm

– most likely, the numerical methods were not concerned most likely, the numerical methods were not concerned with the visual normwith the visual norm

– simple import of existing technology has little valuesimple import of existing technology has little value

– instead, novel contributions are needed to get instead, novel contributions are needed to get equations and algorithms that respect and flourish equations and algorithms that respect and flourish under the visual norm under the visual norm

• Be careful borrowing equations and numerical Be careful borrowing equations and numerical algorithms from other fieldsalgorithms from other fields– most likely, the equations were not written down to most likely, the equations were not written down to

respect the visual normrespect the visual norm

– most likely, the numerical methods were not concerned most likely, the numerical methods were not concerned with the visual normwith the visual norm

– simple import of existing technology has little valuesimple import of existing technology has little value

– instead, novel contributions are needed to get instead, novel contributions are needed to get equations and algorithms that respect and flourish equations and algorithms that respect and flourish under the visual norm under the visual norm

One way to exploit One way to exploit plausible simulationplausible simulationOne way to exploit One way to exploit plausible simulationplausible simulation

• There are many examples of visually rich There are many examples of visually rich phenomena that are nearly impossible or phenomena that are nearly impossible or completely impossible to simulatecompletely impossible to simulate

• We can choose or derive a set of equations and We can choose or derive a set of equations and a numerical method that is faithful to our needs, a numerical method that is faithful to our needs, the visual normthe visual norm

• This is no worse than a turbulence model that This is no worse than a turbulence model that gets the correct lift and drag but is inaccurate gets the correct lift and drag but is inaccurate on the small scale eddieson the small scale eddies

• This is no worse than any number of friction This is no worse than any number of friction modelsmodels

• There are many examples of visually rich There are many examples of visually rich phenomena that are nearly impossible or phenomena that are nearly impossible or completely impossible to simulatecompletely impossible to simulate

• We can choose or derive a set of equations and We can choose or derive a set of equations and a numerical method that is faithful to our needs, a numerical method that is faithful to our needs, the visual normthe visual norm

• This is no worse than a turbulence model that This is no worse than a turbulence model that gets the correct lift and drag but is inaccurate gets the correct lift and drag but is inaccurate on the small scale eddieson the small scale eddies

• This is no worse than any number of friction This is no worse than any number of friction modelsmodels

One way to exploit One way to exploit plausible simulationplausible simulationOne way to exploit One way to exploit plausible simulationplausible simulation

• If we choose the equations and the numerical If we choose the equations and the numerical method wisely, impossible problems may method wisely, impossible problems may become practical or even trivialbecome practical or even trivial– e.g. it would be quite costly to simulate all the waves in e.g. it would be quite costly to simulate all the waves in

the ocean with the nonlinear three dimensional Navier-the ocean with the nonlinear three dimensional Navier-Stokes equationsStokes equations

– instead the equations can be both linearized and reduced instead the equations can be both linearized and reduced to two dimensions based on a height fieldto two dimensions based on a height field

– this made the Titanic sink, a perfect storm, and 300 this made the Titanic sink, a perfect storm, and 300 million dollars for Pixar million dollars for Pixar

– that’s plausible enough for methat’s plausible enough for me

• If we choose the equations and the numerical If we choose the equations and the numerical method wisely, impossible problems may method wisely, impossible problems may become practical or even trivialbecome practical or even trivial– e.g. it would be quite costly to simulate all the waves in e.g. it would be quite costly to simulate all the waves in

the ocean with the nonlinear three dimensional Navier-the ocean with the nonlinear three dimensional Navier-Stokes equationsStokes equations

– instead the equations can be both linearized and reduced instead the equations can be both linearized and reduced to two dimensions based on a height fieldto two dimensions based on a height field

– this made the Titanic sink, a perfect storm, and 300 this made the Titanic sink, a perfect storm, and 300 million dollars for Pixar million dollars for Pixar

– that’s plausible enough for methat’s plausible enough for me

One Way Wave Equation 0v vt x

1 12

v vi ivx h

upwindcentral 1v vi ivx h

24

( )6

hv v v O ht x xxx

3( )

2

hv v v O ht x xx

initial data exact solution

central upwind

Higher Order Approximation

upwindcentral2

4( )

6

h rv v v h G O ht x xxx

3( )

2

h rv v v h G O ht x xx

- add a new rh G term to cancel out the leading order error terms

Thoughts...

- in traditional CFD, the results of numerical calculations are only meaningful when the computed solution is well-resolved

- well-resolved computations are within the convergent asymptotic regime where the numerical errors are proportional to the mesh spacing

- the only sensical

rh G terms are those that accelerate convergence

in the asymptotic regime, i.e. high order methods that cancel error

• What happens in very complex flow fields where one cannot possibly use enough grid points to resolve all the important features?

In general, one can claim very little about under resolved calculationson relatively coarse grids.

true solution

high order method

new method?

asymptotic regime

x

- scale the force so that it vanishes for consistency, but still gives a good answer on a coarse grid

- calculate the magnitude and direction of the force that the vorticity should exert

Vorticity Confinement – coarse grid fix

2( )

pu u u u x Nt

u

N

N

- vorticity needs help to overcome coarse grid dissipation

- locate the vorticity with

x N

Smoke

Simulation of Large Simulation of Large Scale PhenomenaScale PhenomenaSimulation of Large Simulation of Large Scale PhenomenaScale Phenomena

• The computational cost of 3D simulations The computational cost of 3D simulations can be overwhelmingcan be overwhelming

– use 2D fluid simulations to obtain the desired use 2D fluid simulations to obtain the desired detail and synthesize rich motiondetail and synthesize rich motion

– ““define” a 3D velocity field from 2D fluid define” a 3D velocity field from 2D fluid simulations and interpolationsimulations and interpolation

– advect particles with the virtual 3D velocity field advect particles with the virtual 3D velocity field and a spatially tiled Komolgorov spectrumand a spatially tiled Komolgorov spectrum

Velocity Field Velocity Field ConstructionConstructionVelocity Field Velocity Field ConstructionConstruction

2D Simulation

2D Simulation

2D Simulation

Interpolation

KolmogorovSpectrum

3D VelocityField

Velocity Field Velocity Field ConstructionConstructionVelocity Field Velocity Field ConstructionConstruction

Velocity Field Velocity Field ConstructionConstructionVelocity Field Velocity Field ConstructionConstruction

Simulation of Large Simulation of Large Scale PhenomenaScale PhenomenaSimulation of Large Simulation of Large Scale PhenomenaScale Phenomena

NIH Visible HumanNIH Visible HumanNIH Visible HumanNIH Visible Human

NIH Visible HumanNIH Visible HumanNIH Visible HumanNIH Visible Human