GPU Gems 3: Chapter 30 Real-Time Simulation and Rendering of ...

GPU Gems 3: Chapter 30 Real-Time Simulation and Rendering of Fluids Crane, Llamas, Tariq ME290R Presentation by Brian Kazian 1

Outline • • • •

Existing Fluid Simulation Techniques Fluid Equations GPU Implementation Modeling Different Types of Fluids

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Fluids Simulation Techniques • Grid Based (Eulerian) – Stable fluids over time steps – Particle level set (combines Lagrangian & Eulerian) • Particle Based (Lagrangian) – Guarantees conservation of mass – Easier to have free surface interactions (empty space) • Height Field – FFT (Tessendorf)

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2D Waves • Used 2D wave equation – Useful for shallow water waves – Explicit solution – First implemented in 2003

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2D Fluid Flow • 2D Navier Stokes Solver – Implicit solutions

• Model smoke, liquid interactions

CFD Example

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GPU Particle Systems • Model gas/liquid as interacting particles • Pros: – Computation only where necessary – Not constrained to a finite grid

• Cons: – Large number of particles needed

Particle Fluid

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3D Fluids • Solve the 3D Navier Stokes equations • Handles free surfaces of liquids – Boundary between liquid and air

• Uses tiled 2D textures • Interaction with models

Box of Smoke

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Uses of Fluid Simulations • Aerospace applications • Medicine – Blood flow in the body

• Computer Games / Films

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Navier-Stokes Equations

∇⋅u=0 ut =

2 k∇ u

Incompressibility

–(u⋅∇)u – ∇p + f

Diffusion

Pressure Advection

Change in Velocity

Body Forces 9

Discretization of Equation • All values live on regular grids • Initially uses an Eulerian discretization – Frame of reference is fixed to fluid elements

• • • •

Need scalar and vector fields Scalar fields: amount of smoke or dye Vector fields: fluid velocity Subtract adjacent quantities to approximate derivatives (finite differences) 10

Navier-Stokes Equations

∇⋅u=0 ut =

2 k∇ u

Incompressibility

–(u⋅∇)u – ∇p + f

Diffusion

Pressure Advection

Change in Velocity

Body Forces 11

What is Diffusion? • Property that some fluids are “thicker” than others – Honey vs. Water

• This is viscosity – Measure of how resistive a fluid is to flowing • Diffuses the momentum

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Scalar Field (Smoke, Dye) 1.2

3.7

5.1

…

cij

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Diffusion

cij

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Diffusion

1

∇2c

ct = k

change in value

value relative to neighbors

1

-4

1

1 cijnew = cij + k t (ci-1j + ci+1j + cij-1 + cij+1 - 4cij) 15

Diffusion = Blurring

Original

Some Diffusion

More Diffusion 16

Vector Field Diffusion

ut =

2 k∇ u viscosity

Two separate diffusions:

uxt = k∇2ux y 2 y u t = k∇ u … blur the x-velocity and the y-velocity 17

Effect of Viscosity Low

Medium

High

Very High

• Each one is ten times higher viscosity than the last “Melting and Flowing” Mark Carlson, Peter J. Mucha, Greg Turk Symposium on Computer Animation 2002

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Variable Viscosity • Viscosity can vary based on position • Viscosity field k can change with temperature • Need implicit solver for high viscosity

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Wax

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Navier-Stokes Equations

∇⋅u=0 ut =

2 k∇ u

Incompressibility

–(u⋅∇)u – ∇p + f

Diffusion

Pressure Advection

Change in Velocity

Body Forces 21

What is Advection? • Measure of the velocity of a fluid field – Field “advects” both itself and other particles through the field

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Advection = Pushing Stuff

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Advection

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Advection

0.3

0.3 2.7

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Scalar Field Advection

ct = –(u⋅∇)c change in value

advection

current values

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Vector Field Advection

ut = –(u⋅∇)u Two separate advections:

x u t=

x –(u⋅∇)u

uyt = –(u⋅∇)uy … push around x-velocity and y-velocity 27

Advection • Easy to code • Method stable even at large time steps • Problem: numerical inaccuracy diffuses flow

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Navier-Stokes Equations

∇⋅u=0 ut =

2 k∇ u

Incompressibility

–(u⋅∇)u – ∇p + f

Diffusion

Pressure Advection

Change in Velocity

Body Forces 29

Divergence

High divergence

Low divergence

Zero divergence 30

Enforcing Incompressibility • • • •

First do velocity diffusion and advection Find “closest” divergence-free vector field Need to calculate divergence Need to find and use pressure

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Measuring Divergence

∇⋅u=?

uyij+1

-uxi-1j

? uxi+1j

-uyij-1

∇⋅uij=(uxi+1j - uxi-1j) + (uyij+1-uyij-1) 32

Pressure Term new u =

u – ∇p

Take divergence of both sides…

∇⋅

new u =

∇⋅ u – ∇⋅∇p

zero

∇⋅ u = ∇2p

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Pressure Term

∇⋅ u = known

pnew

2 ∇p

= p + (∇⋅ u -

Let dij = ∇⋅ uij

1

unknown

∇2p)

1

-4

1

1

pnewij = pij + (dij - (pi-1j + pi+1j + pij-1 + pij+1 - 4pij)) 34

Pressure Term new u =

u – ∇p

…and velocity is now divergence-free

Found “nearest” divergence-free vector field to original.

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Fluid Simulator 1) 2) 3) 4) 5) 6)

Diffuse velocity Advect velocity Add body forces (e.g. gravity, obstacles) Pressure projection Diffuse dye/smoke Advect dye/smoke

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GPU Implementation for Fluids • Cell attributes stored in 3D textures • Run kernels (fragment shaders) over grid – Slices

• Split N-S equation into 3 parts – Advection – External Forces – Pressure Projection

• No diffusion 37

Advection Implementation • Semi-Lagrangian vs. MacCormack scheme for advection steps • Lagrangian is unconditionally stable – Can introduce numerical smoothing • MacCormack uses 2 intermediate Lagrangian steps – Not unconditionally stable • Higher order scheme is better on the GPU • Math is cheaper than bandwidth

MacCormack 128x64x64

Semi-Lagrangian 256x128x128

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Fluid – Object Interaction • Fluid interaction with solid objects? – Model a solid object and allow for interaction – Modeling dynamic objects • Volumetric representation of object • Velocity at boundary

Movie

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Dynamic Obstacles • Fluid behavior at dynamic boundary – Handle voxel/fluid faces which serve as boundaries

• Create a no-slip boundary condition at faces – Fluid can move freely along the surface – Same velocities normal to voxels

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Solving Pressure with Objects • Object pressure cells – Replace value with current cell’s value

• Object velocity cells – Replace fluid velocity with object velocity

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Velocity Correction at Boundaries • Need to enforce no-slip boundary condition • Compute normal velocity of object – Replace normal velocity of fluid with value

• For two opposing faces, select one • Large time steps, fluid “leaks” into solid – Add advection constraint to prevent this

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Voxelization • How to determine whether a cell contains a solid object or not – Voxelize solid objects into 2 textures

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Inside – Outside Voxelization • Use a stencil buffer and clipping planes – Increment back faces, decrement front

• Voxels inside have nonzero values • Can distinguish types of cells • Need water-tight mesh

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Velocity Voxelization • Need velocity at each grid cell – Find velocity at each vertex • Store previous and current vertex position

• Need to interpolate velocities – Use stencil buffer procedure – Have intersection with triangles • Segment, triangle, point, or empty

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Velocity Voxelization • Segment intersections – Draw thicker segment with a quad • Offset width = diagonal length of a texel • Offset direction = projected normal of triangle

– Linearly interpolate velocities • Ultimately assign to grid points

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Velocity Texture

» Bright blue – next to boundary » Dark blue – outside boundary » Red/Green – velocity components

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Optimizing Voxelization • Optimize with stream output – Can cache transformed vertices • Don’t have to recompute transformations at each slice

• Instancing to voxelize all slices in one call – Instance ID used to specify target slice

• Use low level detail mesh for obstacle – Simulation grid is coarse

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Smoke • Vector field is boring by itself – Inject some smoke! – Need to track density & temperature

• Keep track of temperature for buoyant force • Inject 3-D Gaussian splat for smoke

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Fire • Similar to smoke, but also need a reaction coordinate – Reaction coordinated advected through flow

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Water • Mainly worry about the surface interactions • Level set method used – Need scalar value at each grid cell – Cells record shortest signed distance to surface – Don’t need simulation outside of fluid surface • Mask computation if level set is above 0

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Performance Considerations • Many arithmetic kernels for each frame – Lots of transfer to & from texture memory – Little discernable difference in 16 & 32 bit textures • Arithmetic operations still in 32 bit

• No benefit from packing data into textures – No interleaving of data accesses

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Numerical Issues • Solving pressure-Poisson system is expensive – Solving exactly enforces incompressibility and thus fluid volume – Use other solvers rather than Jacobi – For smoke/fire, not concerned with losing volume

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Numerical Issues • Water is a different matter – Need information to travel • Can’t solve for pressure exactly in real-time simulation

– Without it, fluid disappears

• Defines surface after extended period – A is advection operator – Beta controls the damping • Lessens number of solver iterations needed

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Volume Rendering • Have 3D texture, but no built-in display call – Use ray-marching pixel shader

• Render six simulation volume quad faces – Shoot rays through to gather densities of fluid

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Volume Ray Casting • Need to know information about rays – Where it enters, direction, etc.

• Create a RayData texture with this information – Store depth of faces in alpha channel – RGB channel stores fragment coords. of front

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Rendering Volume • Shader uses RayData for entry & distance – Number of samples determined by marching dist. • Each step, sample values and blend front to back • Can terminate early if color saturates

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Limitations - Occlusion • Rays do not stop for objects inside fluid – Use minimum of back-face & scene distances

• Objects may occlude part of fluid – Compare front-face & scene distances • If front-face distance larger, use negative value in RayData

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Limitations - Clipping • Camera may be inside fluid volume – No information about where rays enter volume – Incorrect volume depth values

• Mark pixels where no front face rendered – Use near plane instead

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Limitations - Banding • Integer number of equally spaced samples – Take one extra sample – Weigh against distance and sample width

• Other solutions – Increasing sampling frequency – Jittering samples – Higher-order filters

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Off Screen Ray Marching • Low resolution simulation grid w.r.t. screen resolution – Render fluid in smaller off-screen target – Composite into final scene – Does not work well for discontinuities in model

• Run edge detector to reflect problem geometry – Render in higher resolution at these points

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Rendering Fire • Similar to rendering smoke – Accumulate values according to reaction coordinate – Map these coordinates to 1D fire texture

• Lighting effects – Sample at several locations, treat as light source • Flickering, inconsistent behavior

– Downsample reaction coords to low resolution 62

Render Liquids • March and look at values from level set – Find first value along ray where f=0

• Refraction needed for water surface – Render rear objects into background texture

• Determine ray intersections with water surface – Hit points shaded with refraction shader

• Background pixel used offset – Offset proportional to surface normal 63

Conclusions • Simulation of fluids has many application • Rendering fluids on GPU is attractive – Cheap mathematics, grid-based

• Advanced hardware leads to better simulations

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