Kam, Hyeong Ryeol

Abstract Introduction Related Work Simulation of Cumuliform Cloud Formation Our Control Method for Cloud Formation Controlling Cloud Simulation Results Discussion and Future Work Conclusion Appendix

Cloud play an important role for creating realistic images The atmospheric fluid dynamics already exists But difficult to adjust parameters and the initial status So, we focus on Controlling cumuliform cloud formation The user specifies the shape of the clouds Automatically adjusts parameters to form that shape

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.

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 convincing 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.

Fluid Control After the pioneering work by Treuille et al. [2003] Control smoke and water by using the adjoint method McNamara et al [2004] Control smoke by adding external forces Fattal 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 CLOUDS Since physical phenomena involved in cloud formation is not concerned.

Fluid Control More recently, Control the motion of water by using control particles Thurey et al. [2006] Make smoke move along a user-specified path Kim et al. [2006] There are NO methods for controlling the formation of clouds based on computational fluid dynamics

Cloud Simulation 1. procedural techniques Generate the density distribution of clouds using the idea of fractals. Ebert et al [2002] Modeling without generating 3d density distributions Trembilski 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.

Cloud Simulation 2. physical simulation of formation process Generate clouds by numerical simulation Dobashi 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 generate realistic clouds, many physical parameters have to be adjusted to generate convincing results. Adjusting the parameters MANUALLY is almost impossible.

The numerical simulation of cumuliform cloud formation (b) the temperature of the rising air currents decreases due to adiabatic cooling. (c) the latent heat is liberated, which creates additional buoyancy forces and promotes further growth of the clouds. Our method mainly controls the amount of latent heat.

The density of air is assumed to be constant The atmosphere is assumed to be an incompressible and inviscid fluid media.

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) ↑ / k b : the buoyancy coefficient T amb : the ambient temperature, inverse-proportion to the height 1 st term : proportional to the difference between the temperature of the rising air parcel and the surrounding air 2 nd term : the drag force due to the falling water droplets

The phase transition between the water vapor and the clouds,, C c : the amount of clouds generated by the phase transition S v : the amount of water vapor / α : the phase transition ratio q s : the saturation vapor content If C c < 0 then, the amount of clouds q c is reduced It means : the evaporation of water droplets

The temperature Γ d : the adiabatic lapse rate ω : the z component of the fluid velocity Q : the coefficient of the latent heat S T : the heat supplied from the ground 1 st term : the advection by the flow field 2 nd term : the adiabatic cooling of rising air 3 rd term : the latent heat The amount of the latent heat is assumed to be proportional to the amount of cloud generated by the phase transition

T 0 (the initial temperature)=T amb (the ambient temp) q v,0 (the initial water vapor) Decrease exponentially from the bottom of sim. space The initial temperature and water vapor Are constant in the horizontal direction q c,0 (the initial cloud density) = zero u 0 (the initial velocity) = zero For 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

pink curve = the desired shape The contour line is projected onto a plane perpendicular to the xy component of the vector connecting the viewpoint and the center of the simulation space 3d shape(target shape) is generated from the contour line

Our method controls the simulation so that the difference between the target shape and the simulated clouds becomes zero The 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.

To measure the difference, we use the height ratio R of the top of the simulated clouds to the top of the target shape.

Key features in our control method 1. Feedback control adjusts the vertical extent of the clouds 2. Geometric potential field adjusts the horizontal extent of the clouds

1. Feedback controller Promotes 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 shape So, geometric potential field is used External forces preventing the clouds from growing outside We use only the horizontal components of the external forces.

1. Feedback controller Cloud growth is promoted by controlling the latent heat and by supplying additional water vapor 2 functions The latent heat controller controls the coefficient for the latent heat Q (old : constant). - updates Q c (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) = Q c (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, S v,c (i,j), to be supplied There are N x *N y *2 control variables To adjust these parameters, we employ PID controller PID=proportional-integral-derivative

2. Geometric potential field is generated in a preprocessing step so that the potential value becomes large inside the target shape The external forces are generated in proportion to the gradient of the potential field. As a result, the external forces prevent clouds from growing outside the target shape.

How our control mechanism works? 1. Compute the height ratio R(i,j) Ratio is sent to latent heat controller, the water vapor supplier 2. the water vapor supplier determines the amount of the water vapor, S v,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.

How our control mechanism works? 3. the latent heat controller Increases Q c where the clouds have not reached the top of the target shape (R(i,j)<1.0) Q c 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

How our control mechanism works? 5. As the clouds approach the top of the target shape The water vapor supplier decreases the amount of additional water vapor. The latent heat controller also stops increasing the latent heat coefficient 6. So, user-specified clouds are generated automatically The simulation is terminated manually by the user After done, the details might be reduced because the control forces could prevent vortices and detail from developing

1. The generation of the target shape 2. The geometric potential field 3. The feedback control of the simulation 4. The water vapor supplier

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 screen 3. the contours are projected onto a plane 4. the target shape is generated Thickens the 2d sketch by extracting its medial axis 5. a bounding box of the target shape is generated and is subdivided into a 3d grid Grid is used to compute the geometric potential field and to simulate the cloud formation process

1. An initial potential field is generated 1 : inside grid points 0 : outside grid points 2. Applying a 3d Gaussian filter to the initial field in order to create a smooth and continuous potential field The result field is the geometric potential field.

During simulation, F, the external force due to the geometric potential filed is calculated at each grid point Because of using only the horizontal component of F, q c : the density of clouds / ψ : the potential field Since the external force is proportional to the gradient 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 q c

updates the coefficient for the latent heat, Q c (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

1 st term : proportional controller K P : proportional gain When only using this, small gaps between the simulated clouds and the target shape are left (So, 2 nd term is needed) Not counting when clouds grow near the top of the target shape, Because ∆H becomes very small 2 nd term : integral controller Contributes when the accumulated difference becomes large D : Duration for the accumulation / K I : integral gain Removes the gaps and updates Q c until ∆H becomes zero

The control parameters need to be specified. We determine K P and K I experimentally It is difficult to determine all these parameter From experiments, we found that larger values are required for K P and K I where the top of the target shape is high Because the cloud growth in such regions has to be promoted further than the regions where the top of the target shape is low. We assume that K P and K I are proportional to the height of the target shape. к P,к I : proportionality coefficients Ĥ target : the height of the target shape divided by the height of the simulation space. As a result, к P,к I are specified by the user.

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 C v : a control parameter for the water vapor supplier, specified by the user / q v,0 : the initial water vapor at the beginning (i,j,k top ) : the grid point corresponding to the top of the clouds

Intel Core2 Extreme X9650 with nVidia GeForce 8800 GTX The simulation space : 320 * 80 * 100 grid The average computation time for each time step of the simulation : 7 seconds The additional computational cost due to control mechanism is very low and is less than one percent of the total computation cost

к P =4.95, к I =0.6, c v =0.5 c v is from process of trial and error Once the appropriate values for these parameters have been found, we can generate various shapes of clouds

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.

Case that is not the height field

The target shapes are completely different from height fields.

Generating clouds resembling real clouds in a photo

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 promote 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 generate the realistic clouds.

Our method controls the cloud motion in the vertical direction The horizontal movement of the clouds is not controlled Since the forces due to cloud formation work only in the vertical direction It is difficult to control the horizontal movement 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 horizontal directions.

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 clouds Horizontal external forces are required Our goal for generating realistic clouds forming the desired shape has been achieved. Extending the method to handling multiple target shapes is our next important research direction

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 provide ‘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

Controlling cumuliform cloud simulation Our method can generate clouds with desired shapes Controlled by our feedback controller and external forces calculated by the geometric potential 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 generate realistic clouds with desire shapes

We determine к P,к I based on this method 1. Experimental simulations are carried out several times with the proportional controller only. 2. Carried out with increasing к C and the controller tries to make the clouds reach a target height In experiment, the target height is set to 80% of space When к C is small, the clouds cannot reach As к C increases, the clouds can reach At a certain value of к C, the cloud growth exhibits periodical oscillations  Clouds repeatedly exceed and fall below the target height 2. T C : the period of the oscillation

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