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Kam, Hyeong Ryeol
Feedback Control of Cumuliform Cloud Formation
based on Computational Fluid Dynamics
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AbstractIntroductionRelated WorkSimulation of Cumuliform Cloud FormationOur Control Method for Cloud FormationControlling Cloud SimulationResultsDiscussion and Future WorkConclusionAppendix
Contents
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Cloud play an important role for creating re-alistic images
The atmospheric fluid dynamics already ex-istsBut difficult to adjust parameters and the ini-
tial statusSo, we focus on
Controlling cumuliform cloud formationThe user specifies the shape of the cloudsAutomatically adjusts parameters to form that
shape
Abstract
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Although CFD(Computational fluid dynamics) methods can generate realistic shapes and motion, it is difficult for the user to control the simulation result and impossible to adjust the parameters manually.
In previous methods, the motion of smoke and water such as letters and animals has been calculated.
In this paper, we focus on controlling cloud formation.
1. Introduction
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We only focus on cumuliform clouds( 적운 ).Our method generates realistic clouds.The user specifies the contours of the clouds.Previous approach did not produce convinc-
ing results because there are several physical processes: phase transition from water vapor to water
droplets We developed a new method that controls
the physical parameters affecting the cloud formation process.
1. Introduction
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Fluid ControlAfter the pioneering work by Treuille et al. [2003]
Control smoke and water by using the adjoint methodMcNamara et al [2004]
Control smoke by adding external forcesFattal and Lischinski [2004]
Calculate the motion of smoke by using a potential field Hong and Kim [2004]
Feedback control mechanism Shi and Yu [2005], Kim et al [2007]
SIMILAR, but NOT directly APPLICABLE to CLOUDSSince physical phenomena involved in cloud formation
is not concerned.
2. Related Work
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Fluid ControlMore recently,
Control the motion of water by using control parti-clesThurey et al. [2006]
Make smoke move along a user-specified pathKim et al. [2006]
There are NO methods for controlling the for-mation of clouds based on computational fluid dynamics
2. Related Work
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Cloud Simulation1. procedural techniques
Generate the density distribution of clouds using the idea of fractals.Ebert et al [2002]
Modeling without generating 3d density distribu-tionsTrembilski and Brobler [2002], Bouthors and Neyret
[2002], Neyret [1997], Gardner [1985]
Although the cost is very low and it’s possible to generate the desired cloud shapes, a trial and error process is required.
2. Related Work
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Cloud Simulation2. physical simulation of formation process
Generate clouds by numerical simulationDobashi et al [2000], Kajiya and Herzen [1984],
Miyazaki et al [2001], Miyazaki et al [2002], Harris et al [2003]
Although these methods have the potential to gen-erate realistic clouds, many physical parameters have to be adjusted to generate convincing results.
Adjusting the parameters MANUALLY is almost impossible.
2. Related Work
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The numerical simulation of cumuliform cloud formation
(b) the temperature of the rising air currents de-creases due to adiabatic cooling.
(c) the latent heat is liberated, which creates addi-tional buoyancy forces and promotes further growth of the clouds.
Our method mainly controls the amount of latent heat.
3. Simulation of Cumuliform Cloud Formation
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The density of air is assumed to be constant The atmosphere is assumed to be an incom-
pressible and inviscid fluid media.
3.1 Simulation of CCF- Governing Equations
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The motion of the atmosphere is expressed by Navier-Stokes equations ,
B : buoyancy force / f : an external forces (such as wind)
T : the temperature / z=(0,0,1) ↑ / kb : the buoyancy coefficient
Tamb : the ambient temperature, inverse-proportion to the height
1st term : proportional to the difference between the temperature of the rising air parcel and the surrounding air
2nd term : the drag force due to the falling water droplets
3.1 Simulation of CCF- Governing Equations
fBuuu
pt
1)( 0 u
zzB camb
ambb gq
T
TTk
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The phase transition between the water vapor and the clouds , ,
Cc : the amount of clouds generated by the phase tran-sition
Sv : the amount of water vapor / α : the phase transi-tion ratio
qs : the saturation vapor content
If Cc < 0 then, the amount of clouds qc is reducedIt means : the evaporation of water droplets
3.1 Simulation of CCF- Governing Equations
ccc Cqt
q
)(u vcvv SCqt
q
)(u
)( svc qqC ))),/(exp(max( cvs qqCTBAq
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The temperature
Γd : the adiabatic lapse rate ω : the z component of the fluid velocityQ : the coefficient of the latent heatST : the heat supplied from the ground
1st term : the advection by the flow field2nd term : the adiabatic cooling of rising air3rd term : the latent heat
The amount of the latent heat is assumed to be propor-tional to the amount of cloud generated by the phase transition
3.1 Simulation of CCF- Governing Equations
Tcd SQCwTt
)(uT
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T0(the initial temperature)=Tamb(the ambient temp)
qv,0(the initial water vapor)Decrease exponentially from the bottom of sim.
spaceThe initial temperature and water vapor
Are constant in the horizontal directionqc,0(the initial cloud density) = zero
u0(the initial velocity) = zeroFor the boundary conditions,
A periodic boundary condition is used in the horizontal
A fixed boundary condition(u=0) is used on the bottom and top of the simulation space
3.2 Simulation of CCF- Initial Status and Boundary Conditions
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pink curve = the desired shape
The contour line is projected onto a plane perpendicular to the xy component of the vector connecting the view-point and the center of the simulation space
3d shape(target shape) is generated from the contour line
4. Our Control Method for Cloud Formation
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Our method controls the simulation so that the difference between the target shape
and the simulated clouds becomes zeroThe effect of wind is not concerned
Since it doesn’t contribute very much to the ccf
The convection due to the buoyancy force is the main driving force for the cumuliform clouds.
4. Our Control Method for Cloud Formation
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To measure the difference, we use the height ratio R of the top of the simulated clouds to the top of the target shape.
4. Our Control Method for Cloud Formation
),...,1 ;,...,1(
),,(/),(),(
yx
targetc
NjNi
jiHjiHjiR
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Key features in our control method1. Feedback control
adjusts the vertical extent of the clouds
2. Geometric potential fieldadjusts the horizontal extent of the clouds
4. Our Control Method for Cloud Formation
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1. Feedback controllerPromotes the cloud growth until the clouds
reach the top of the target shape (R(i,j)=1.0)2. Geometric potential field
When the target shape is not a height field, the clouds may grow outside of the target shapeSo, geometric potential field is used
External forces preventing the clouds from growing outside
We use only the horizontal components of the external forces.
4. Our Control Method for Cloud Formation
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1. Feedback controllerCloud growth is promoted by controlling the la-
tent heat and by supplying additional water va-por2 functions
The latent heat controller controls the coefficient for the latent heat Q (old : constant). - updates Qc(i,j) that was for the grid point (i,j,0) as a control
variable - the coefficient for latent heat for grid point (i,j,k) = Qc(i,j)
The water vapor supplier adds water vapor where clouds does not reach the top of the
target shape (old: at the bottom of the simulation space + top of the clouds)
- determines the amount of water vapor, Sv,c(i,j), to be supplied
There are Nx*Ny*2 control variablesTo adjust these parameters, we employ PID controller
PID=proportional-integral-derivative
4. Our Control Method for Cloud Formation
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2. Geometric potential fieldis generated in a preprocessing step so that the
potential value becomes large inside the target shape
The external forces are generated in propor-tion to the gradient of the potential field.
As a result, the external forces prevent clouds from growing outside the target shape.
4. Our Control Method for Cloud Formation
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How our control mechanism works?1. Compute the height ratio R(i,j)
Ratio is sent to latent heat controller, the water va-por supplier
2. the water vapor supplier determines the amount of the water vapor, Sv,c
and adds this at the grid points corresponding to the top of the simulated clouds.
Promotes the phase transition from water vapor to clouds.
4. Our Control Method for Cloud Formation
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How our control mechanism works?3. the latent heat controller
Increases Qc where the clouds have not reached the top of the target shape (R(i,j)<1.0)
Qc increases the temperature when the clouds are generated due to the phase transition
The increase in temperature results in an increase in the buoyancy force
This promotes the cloud growth to higher regions.4. During cloud growth
The external forces due to the geometric potential field push clouds inwards the inside the target shape
4. Our Control Method for Cloud Formation
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How our control mechanism works?5. As the clouds approach the top of the target
shapeThe water vapor supplier decreases the amount of
additional water vapor . The latent heat controller also stops increasing the
latent heat coefficient6. So, user-specified clouds are generated au-
tomatically
The simulation is terminated manually by the userAfter done, the details might be reduced because
the control forces could prevent vortices and detail from developing
4. Our Control Method for Cloud Formation
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1. The generation of the target shape2. The geometric potential field3. The feedback control of the simulation4. The water vapor supplier
5. Controlling Cloud Simulation
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The target shape is generated from the user specified contours of the desired cloud shape.1. the user specifies the generating region
The center of the simulation space is placed at the center of the specified region
2. the user draws the contours on the screen3. the contours are projected onto a plane4. the target shape is generated
Thickens the 2d sketch by extracting its medial axis5. a bounding box of the target shape is gener-
ated and is subdivided into a 3d gridGrid is used to compute the geometric potential
field and to simulate the cloud formation process
5.1 Controlling Cloud Simulation- Generation of Target Shape
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1. An initial potential field is generated1 : inside grid points0 : outside grid points
2. Applying a 3d Gaussian filter to the initial field in order to create a smooth and continu-ous potential fieldThe result field is the geometric potential field.
5.2 Controlling Cloud Simulation- Geometric Potential Field
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During simulation,F, the external force due to the geometric po-
tential filed is calculated at each grid pointBecause of using only the horizontal compo-
nent of F,
qc : the density of clouds / ψ : the potential fieldSince the external force is proportional to the gra-
dient of the potential field, it only works near the boundary of the target shape.
No external forces are generated at the grid points where no clouds exist since F is also proportional to qc
5.2 Controlling Cloud Simulation- Geometric Potential Field
000
010
001
),( cc qqF
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updates the coefficient for the latent heat, Qc(i,j)Using PID control mechanism -> PI control
∆H(i,j) : the normalized height difference between the top of the clouds and the target shape
5.3 Controlling Cloud Simulation- Latent Heat Controller
D
HIHPc djijiKjijiKjiQ0
),,(),( ),(),(),(
j)(i,H
j)(i,Hj)(i,HjiRji
target
ctargetH
),(0.1),(
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1st term : proportional controller
KP : proportional gainWhen only using this, small gaps between the simu-
lated clouds and the target shape are left (So, 2nd term is needed)Not counting when clouds grow near the top of the
target shape, Because ∆H becomes very small
2nd term : integral controller Contributes when the accumulated difference be-
comes largeD : Duration for the accumulation / KI : integral
gainRemoves the gaps and updates Qc until ∆H be-
comes zero
5.3 Controlling Cloud Simulation- Latent Heat Controller
D
HIHPc djijiKjijiKjiQ0
),,(),( ),(),(),(
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The control parameters need to be specified.We determine KP and KI experimentally
It is difficult to determine all these parameterFrom experiments, we found that larger values are re-
quired for KP and KI where the top of the target shape is highBecause the cloud growth in such regions has to be pro-
moted further than the regions where the top of the target shape is low.
We assume that KP and KI are proportional to the height of the target shape.
кP,кI : proportionality coefficientsĤtarget : the height of the target shape divided by the height
of the simulation space.
As a result, кP,кI are specified by the user.
5.3 Controlling Cloud Simulation- Latent Heat Controller
),(),( & ),(),( argarg jiHjiKjiHjiK ettIIettPP
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PICP T85.0
,45.0
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Adds water vapor if
Indicates that the water vapor is supplied at the grid points where the ratio R(i,j) < the average of the ratios
The water vapor supplier tries to make the ratio of the cloud growth the same for all grid points The top of the cloud at all grid point reaches the top
of the target shape almost simultaneously.The amount of additional water vapor
Cv : a control parameter for the water vapor supplier,
specified by the user / qv,0 : the initial water vapor at the beginning
(i,j,ktop) : the grid point corresponding to the top of the clouds
5.4 Controlling Cloud Simulation- Water Vapor Supplier
x yN
n
N
myx
mlRNN
jiR1 1
),(1
),(
),,(),( 0,, topvvcv kjiqcjiS
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Intel Core2 Extreme X9650 with nVidia GeForce 8800 GTX
The simulation space : 320 * 80 * 100 gridThe average computation time for each time
step of the simulation : 7 secondsThe additional computational cost due to con-
trol mechanism is very low and is less than one percent of the total computation cost
6. Results
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кP=4.95, кI=0.6, cv=0.5cv is from process of trial and errorOnce the appropriate values for these parame-
ters have been found, we can generate various shapes of clouds
6. Results
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Comparing to Fattal’s method
In Fattal’s method, clouds are generated where they should not be
Our method can generate realistic clouds while their shapes closely match the target shapes.
6. Results
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Case that is not the height field
6. Results
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The target shapes are completely different from height fields.
6. Results
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Generating clouds resembling real clouds in a photo
6. Results
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Our control mechanism is fairly indirect.Feedback controllers cannot guarantee that the
clouds will completely form the desired shape.The indirect control makes it possible to pro-
mote the cloud growth as naturally as possible (realistic-looking)
The indirect control allows the user to specify the rough shape of the clouds and finally gen-erate the realistic clouds.
7. Discussion and Future Work
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Our method controls the cloud motion in the vertical directionThe horizontal movement of the clouds is not
controlledSince the forces due to cloud formation work
only in the vertical directionIt is difficult to control the horizontal move-
ment by controlling the cloud formation process
The previous fluid control methods might be suitable for horizontal control it is expected that the combination of methods to
control the cloud motion in both vertical and hori-zontal directions.
7. Discussion and Future Work
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Our method can handle a desired shape that is not a height field.However, it is still difficult to handle a shape
that is very different from a height field the cumuliform cloud formation process affects
the vertical movement of the cloudsHorizontal external forces are required
Our goal for generating realistic clouds form-ing the desired shape has been achieved.Extending the method to handling multiple tar-
get shapes is our next important research di-rection
7. Discussion and Future Work
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The user specifies the shape of clouds viewed from only a single direction. solution : to specify the multiple target
shapes viewed from multiple viewpoints.Ziegler-Nichols method doesn’t always pro-
vide ‘good’ parameters. A trial and error process is still required.
Extension of our method to other types of clouds such as stratus is also an interesting area for future work
7. Discussion and Future Work
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Controlling cumuliform cloud simulationOur method can generate clouds with desired
shapes Controlled by our feedback controller and ex-
ternal forces calculated by the geometric po-tential field.
For feedback control, we developed a latent heat controller and a water vapor supplier.
By controlling the amounts of latent heat and additional water vapor, clouds grow naturally and converge into the desired shape
Our method provides a simple way to gener-ate realistic clouds with desire shapes
8. Conclusion
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We determine кP,кI based on this method1. Experimental simulations are carried out several
times with the proportional controller only.2. Carried out with increasing кC and the controller
tries to make the clouds reach a target heightIn experiment, the target height is set to 80% of space
When кC is small, the clouds cannot reachAs кC increases, the clouds can reach
At a certain value of кC, the cloud growth exhibits period-ical oscillations Clouds repeatedly exceed and fall below the target height
2. TC : the period of the oscillation
Appendix A.Ziegler-Nichols Method
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PICP T85.0
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