Modeling Falling and Accumulating Snow (by Moeslund, Madsen, Aagaard and Lerche) K. H. Ko Department of Mechatronics Gwangju Institute of Science and Technology November 5, 2008

Introduction Snow is arguably one of the world’s most complex naturally occurring substances. Snow is arguably one of the world’s most complex naturally occurring substances. Accurate simulation is still a significant challenge. Accurate simulation is still a significant challenge. Motivations for investigating snowfall Motivations for investigating snowfall –Without an automatic model of fallen snow, animators have so far relied upon intuition to produce snow covered surfaces. It is an extremely tedious, time-consuming and potentially inaccurate task. It is an extremely tedious, time-consuming and potentially inaccurate task. –Snow transforms commonplace scenes into fantastic wonderlands, greatly changing the appearance and mood of the landscape, allowing us to see familiar sights in a fresh, exciting way.

Introduction

Introduction

Introduction When making a model based on physics a balance between the complexity and the visual result has to be found. When making a model based on physics a balance between the complexity and the visual result has to be found. –It is not necessary to make a model that calculates the exact physics if no significant visual improvement is achieved. –Assumptions are often introduced in order to simplify the physical model.

Modeling a Snowflake The formation of a snowflake is a highly complicated process. The formation of a snowflake is a highly complicated process. It is known that a snowflake consists of ice crystals which can occur when five conditions are met. It is known that a snowflake consists of ice crystals which can occur when five conditions are met. –The air in the atmosphere has to be saturated with water. –The presence of condensation nuclei is necessary. Ex) dust, bacteria, on whose surfaces the water can condensate to produce small droplets. –Enough water has to be present in the cloud. –The temperature has to be below zero degrees Celsius in the cloud so that the water in the air will freeze onto the nuclei. –The nuclei has to be of specific shapes so that the water molecules line up correctly to form the basis ice crystals.

Modeling a Snowflake How ice crystals merge into actual snowflakes depends on How ice crystals merge into actual snowflakes depends on –the level of saturation –the temperature in the area between the cloud and the ground. The result is a virtually endless number of different snowflakes. The result is a virtually endless number of different snowflakes.

Modeling a Snowflake Snowflakes, however, follow some general shapes. Snowflakes, however, follow some general shapes.

Modeling the Size and Density The main characteristics to be modeled The main characteristics to be modeled –The shape –The size –The density They are controlled by the level of saturation in the air and the temperature. They are controlled by the level of saturation in the air and the temperature. –Modeling the saturation of the air is not necessary for rendering a scene. –It is necessary to model the temperature This can also affect other aspects of the scene. This can also affect other aspects of the scene. It is useful to find a way of controlling the density and size of a snowflake using the temperature. It is useful to find a way of controlling the density and size of a snowflake using the temperature.

Modeling the Size and Density The diameter of snowflakes has been measured as a function of temperature. The diameter of snowflakes has been measured as a function of temperature. –D: the diameter in meters –T: the temperature in degree Celsius The uncertainties are up to ±50%. The uncertainties are up to ±50%. –Add a random number to the diameter.

Modeling the Size and Density The density of snowflakes is inversely proportional to the diameter of the snowflakes: The density of snowflakes is inversely proportional to the diameter of the snowflakes: –The proportionality constant C kg/m 2 for dry snow kg/m 2 for dry snow kg/m 2 for wet snow kg/m 2 for wet snow

Modeling the Shape Real snowflakes are primarily constructed by ice crystals colliding. Real snowflakes are primarily constructed by ice crystals colliding. –We can model a snowflake by combining triangular polygons in a random manner. A number of concentric spheres are used to construct a snowflake. A number of concentric spheres are used to construct a snowflake. –Each sphere is considered a layer. –The same number of polygons is used for each layer. The number of layers directly gives the size of the snowflake and is controlled by the temperature. The number of layers directly gives the size of the snowflake and is controlled by the temperature. By fixing the number of polygons per layer, the density is also controlled by the temperature. By fixing the number of polygons per layer, the density is also controlled by the temperature.

Modeling the Shape When a new triangle is added to a layer one of its three corner coordinates must be connected with one of the triangles in the immediate inner layer, denoted the reference triangle. When a new triangle is added to a layer one of its three corner coordinates must be connected with one of the triangles in the immediate inner layer, denoted the reference triangle.

Modeling the Shape The effect of the density change due to the temperature. The effect of the density change due to the temperature. The wet ones are more compact as they have four times higher density. The wet ones are more compact as they have four times higher density.

Movement of a Snowflake The movement of a snowflake is caused by the four forces: The movement of a snowflake is caused by the four forces: –Constant forces: F gravity, F buoyant –Variable according to the wind direction: F lift, F drag F gravity : the gravitational force F gravity : the gravitational force F buoyant : the force that represents the up-drift by the surrounding air. F buoyant : the force that represents the up-drift by the surrounding air.

Movement of a Snowflake F gravity is always opposite F buoyant. F gravity is always opposite F buoyant. –F buoyant is relatively small compared to F gravity. It can be ignored as it has no impact on the visual movement result. F lift : the force that makes the snowflake move in circular and irregular patterns caused by the aerodynamic shape of the snowflake and the turbulence created behind the snowflake. F lift : the force that makes the snowflake move in circular and irregular patterns caused by the aerodynamic shape of the snowflake and the turbulence created behind the snowflake. –At high wind speed, the force is insignificant. –But in calm weather it cannot be ignored. Ex: When observing a snowflake in calm weather one would observe that the snowflake follows the path of a helix towards the ground while rotating around its center of mass. Ex: When observing a snowflake in calm weather one would observe that the snowflake follows the path of a helix towards the ground while rotating around its center of mass.

Movement of a Snowflake F drag : drag that the air will assert on the snowflake. F drag : drag that the air will assert on the snowflake. –This is the force that makes a snowflake follow the wind direction. –U fluid consists of two components: the wind velocity and the velocity of the air friction. –U maximum : the maximum vertical velocity taking wind resistance into consideration.

The Wind Field When snow is falling under the influence of a wind the snow will fly around obstacles in very distinct patterns that are caused by the wind field. When snow is falling under the influence of a wind the snow will fly around obstacles in very distinct patterns that are caused by the wind field. –These wind patterns are important to ensure that The snow falls correctly and the visual appearance resembles reality. The snow falls correctly and the visual appearance resembles reality.

The Wind Field The wind field can be described by the Navier-Stokes equations. The wind field can be described by the Navier-Stokes equations. The air in a wind field can be assumed to be incompressible, inviscid and has a constant density of one. The air in a wind field can be assumed to be incompressible, inviscid and has a constant density of one. –A simplified Navier-Stokes equation

The Wind Field The term –(u  ∇ )u, called the convection term describes how the velocity of the fluid evolves over time. The term –(u  ∇ )u, called the convection term describes how the velocity of the fluid evolves over time. The term ∇ p is the acceleration of the fluid caused by the pressure gradient. The term ∇ p is the acceleration of the fluid caused by the pressure gradient. Solution of the Navier-Stokes equation Solution of the Navier-Stokes equation –The Semi-Lagrangian scheme for the convection term. –The gradient of the pressure is calculated and used in a projection step where it is ensured that the mass is conserved.

Convection Step This step will ensure that a small change in the air at a specific point will have an influence on the air in the rest of the domain. This step will ensure that a small change in the air at a specific point will have an influence on the air in the rest of the domain. –The propagation of the changes in the air is governed by the term –(u  ∇ )u. –A first order Taylor approximation of –(u  ∇ )u is given by u* is the intermediate velocity field from the S-L method.

Convection Step

Projection Step The purpose of this step is to ensure conservation of mass of the fluid. The purpose of this step is to ensure conservation of mass of the fluid. –It can ensure the creation of plausible wind fields with the correct swirly nature.

Projection Step The implementation of the projection is done using the Helmholtz-Hodge decomposition. The implementation of the projection is done using the Helmholtz-Hodge decomposition. –Any vector field can be decomposed into two parts: w = v + ∇ s w is a vector field without conservation of mass w is a vector field without conservation of mass v is a vector field with conservation of mass v is a vector field with conservation of mass ∇ s is the gradient of a scalar field. ∇ s is the gradient of a scalar field. –The vector field is a velocity for a fluid and the scalar field is equal to the pressure field for the wind. u = u* - ∇ p u = u* - ∇ p

Modeling Accumulated Snow A six step algorithm is used. A six step algorithm is used. –Create edge groups This step divides the scene into larger entities, called the edge groups, having the same orientation. This step divides the scene into larger entities, called the edge groups, having the same orientation. Functions as an initialization and is basically a question of determining possible locations where snow can accumulate. Functions as an initialization and is basically a question of determining possible locations where snow can accumulate. –Emit snow particles Emit snow from the sky and calculate with which triangle in the edge groups there is a collision. Emit snow from the sky and calculate with which triangle in the edge groups there is a collision. To ensure realism the emitting and movement of the snowflakes use the schemes for describing movement of snowflakes. To ensure realism the emitting and movement of the snowflakes use the schemes for describing movement of snowflakes.

Modeling Accumulated Snow A six step algorithm is used. A six step algorithm is used. –Refine edge groups A very detailed snow surface is important when representing the snow around obstacles as the snow height will vary a great deal. A very detailed snow surface is important when representing the snow around obstacles as the snow height will vary a great deal. However, it is not always necessary to have a detailed representation of the snow surface on a large plane surface where no obstacles exist. However, it is not always necessary to have a detailed representation of the snow surface on a large plane surface where no obstacles exist. Divide the triangles into a finer grid of triangles according to the number of particles hitting the triangles. Divide the triangles into a finer grid of triangles according to the number of particles hitting the triangles.

Modeling Accumulated Snow A six step algorithm is used. A six step algorithm is used. –Resolve stability There are two purposes of the step. There are two purposes of the step. –It functions as a smoothing mechanism that makes sure that too abrupt transitions between two levels of accumulated snow do not occur. –It determines if the snow cover is stable enough. Can determine whether or not a snow cover on a rooftop is too high to remain stable or falls down. Can determine whether or not a snow cover on a rooftop is too high to remain stable or falls down. A snowdrift stays stable when leaning against an object. A snowdrift stays stable when leaning against an object.

Modeling Accumulated Snow A six step algorithm is used. A six step algorithm is used. –Resolve stability Redistribute snow from one triangle to its neighbors if the height difference is too steep. Redistribute snow from one triangle to its neighbors if the height difference is too steep. The height difference between neighboring triangles is expressed as an angle. The height difference between neighboring triangles is expressed as an angle. –For snow to be distributed this angle has to be above the angle of repose (AOR)

Modeling Accumulated Snow A six step algorithm is used. A six step algorithm is used. –Smoothing and rendering the surface Smoothing the snow surface Smoothing the snow surface –Average the height of each triangle using its adjacent neighbors and then connecting these averaged centers. The final result is rendered. The final result is rendered.

Examples