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Physics-based Fluid Simulations for Computer Graphics Ryoichi Ando Contents Contents Fluid Solver Contents Fluid Solver Surface Tracker Contents Fluid Solver Surface

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Physics-based Fluid Simulations for Computer Graphics

Ryoichi Ando

ＣＧ物理ベース流体シミュレーション

安東 遼一

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Contents

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Contents

Fluid Solver

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Contents

Fluid Solver Surface Tracker

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Contents

Fluid Solver Surface Tracker FLIP Solver

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  • Ph.D student @ Kyushu Univ
  • Visiting Scientist at IST Austria
  • Fluid Simulations
  • SIGRRAPH 2013 Tech. paper with

Chris Wojtan and Nils Thuerey

Short Bio

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Fluid Solver Surface Tracker FLIP Solver

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Fluid Solver

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Navier-Stokes Equations s.t.

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Navier-Stokes Equations s.t.

How velocity changes

  • ver time
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Navier-Stokes Equations s.t.

How velocity changes

  • ver time

Advection term

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Navier-Stokes Equations s.t.

How velocity changes

  • ver time

Advection term Pressure term

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Navier-Stokes Equations s.t.

How velocity changes

  • ver time

Advection term Pressure term Viscosity term

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Navier-Stokes Equations s.t.

How velocity changes

  • ver time

Advection term Pressure term Viscosity term External force term

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Navier-Stokes Equations s.t.

How velocity changes

  • ver time

Advection term Pressure term Viscosity term Incompressiblity Constraint External force term

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Navier-Stokes Equations

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Navier-Stokes Equations Large Matrix

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Navier-Stokes Equations Large Matrix p u

=

RHS

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Navier-Stokes Equations Large Matrix p u

=

RHS

Newton Iterations...

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Navier-Stokes Equations Large Matrix p u

=

RHS

Newton Iterations...

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Operator Splitting

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Operator Splitting

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Operator Splitting

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Operator Splitting

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Operator Splitting

[Stable fluid, Stam99]

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First-order integration

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First-order integration

e.g, advection

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First-order integration

e.g, advection First-order accuracy!

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Simulation Steps

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Simulation Steps

Advection term

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Simulation Steps

Advection term Pressure term

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Simulation Steps

Advection term Pressure term Viscosity term

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Simulation Steps

Advection term Pressure term Viscosity term External force term

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Simulation Steps

Advection term Pressure term

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Simulation Steps

Advection term Pressure term

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Simulation Steps

Advection term Pressure term Simulation Loop...

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Advection term Pressure term

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Advection term Pressure term

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Advection term

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Advection term

Computing

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Advection term

Computing

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Advection term

Computing

Scalar Scalar

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Advection term

Computing

Expand...

Scalar Scalar

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Advection term

Computing

Expand...

Scalar Scalar

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Advection term

Computing

Expand... Key component !

Scalar Scalar

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Advection term

Computing

Velocity field Scalar field

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Advection term

Computing

Velocity field Scalar field

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Advection term

Computing

Velocity field Scalar field

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Advection term

Computing

Velocity field Scalar field

Ground Truth

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Advection term

Computing

How to discretize ?

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Advection term

Computing

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Advection term

Computing

Central Difference

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Advection term

Computing

Central Difference Accurate

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Advection term

Computing

Central Difference Accurate

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Advection term

Computing

Central Difference

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Advection term

Computing

Central Difference

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Advection term

Computing

Upwind Scheme If If

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Advection term

Computing

Upwind Scheme If If

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Advection term

Computing

High-order Upwind Scheme ( WENO5 )

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Advection term

Computing

High-order Upwind Scheme ( WENO5 )

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Advection term

Computing

High-order Upwind Scheme ( WENO5 ) Upwind Scheme

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Advection term

Computing

High-order Upwind Scheme ( WENO5 ) Upwind Scheme

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Advection term

Computing

High-order Upwind Scheme ( WENO5 ) Upwind Scheme

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Advection term

Computing

High-order Upwind Scheme ( WENO5 ) Upwind Scheme

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Advection term Pressure term

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Advection term Pressure term

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Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term

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Computing

Pressure term Pressure Poisson Equation

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Computing

Pressure term Pressure Poisson Equation

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Pressure term Pressure Poisson Equation

Boundary Conditions of

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

Divergence Theorem

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing MAC Grid

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

1 2 3 4 p

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Pressure term

Discretizing

Solid wall

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Pressure term

Discretizing

Solid wall

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Pressure term

Discretizing

Solid wall 1 2

=

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Pressure term

Discretizing

Empty Matrix

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Pressure term

Discretizing

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Pressure term

Discretizing

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Pressure term

Discretizing

p u*

=

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Pressure term

Discretizing

p u*

=

(Semi-)Positive Definite System

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Pressure term

Discretizing

p u*

=

(Semi-)Positive Definite System

  • 1
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Pressure term

Discretizing

p u*

=

(Semi-)Positive Definite System

  • 1

Preconditioned Conjugate Gradient Solver

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Pressure term

Discretizing

p

=

(Semi-)Positive Definite System Solution

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Pressure term

Discretizing

Solution

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Pressure term

Discretizing

Solution

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Pressure term

Discretizing

Solution

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Advection term Pressure term

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Advection term Pressure term

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Advection term Pressure term

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Examples

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Examples

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Short Break

Next: Liquid Simulations...

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Fluid Solver Surface Tracker FLIP Solver

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Fluid Solver Surface Tracker FLIP Solver

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Surface Tracker

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Liquid Simulation

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Liquid Simulation

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Liquid Simulation Surface Tracker

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Surface Tracker

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Surface Tracker

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Surface Tracker

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Surface Tracker

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Surface Tracker

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Surface Tracker

  • Water tight
  • Topology change
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Surface Tracker

  • Water tight
  • Topology change
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Surface Tracker

  • Water tight
  • Topology change
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Surface Tracker

  • Water tight
  • Topology change
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Surface Tracker

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Surface Tracker

Levelset Method

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Surface Tracker

Levelset Method Mesh-based Method

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Surface Tracker

Levelset Method Mesh-based Method Particle-based Method

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Surface Tracker

Levelset Method

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Levelset Method

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φ0

Levelset Method

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φ0 φ1

Levelset Method

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φ0 φ1 φ2

Levelset Method

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φ < 0 φ > 0

Levelset Method

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φ < 0 φ > 0

Levelset Method

∂φ ∂t = u · rφ

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Levelset Method

φ < 0 φ < 0 φ > 0

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Levelset Method

φ < 0 φ < 0 φ > 0

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Levelset Method

φ < 0 φ < 0 φ > 0

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Levelset Method

φ < 0 φ < 0 φ > 0

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Levelset Method

φ > 0 φ < 0

Topology change

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Levelset Method

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Levelset Method

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p = 0

Levelset Method

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Levelset Method

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Levelset Method

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Fluid Solver Surface Tracker FLIP Solver

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Fluid Solver Surface Tracker FLIP Solver

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FLIP Solver

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FLIP Solver

Particles Levelset Velocity Pressure

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FLIP Solver

Particles Levelset Velocity Pressure

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FLIP Solver

Particles Levelset Velocity Pressure

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FLIP Solver

Particles Levelset Velocity Pressure

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FLIP Solver

Particles Levelset Velocity Pressure Boundary Condition

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FLIP Solver

Particles Levelset Velocity Pressure Boundary Condition Map Velocity

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FLIP Solver

Particles Levelset Velocity Pressure Boundary Condition Map Velocity

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FLIP Solver

Particles Levelset Velocity Pressure Boundary Condition Map Velocity Solve Pressure

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FLIP Solver

Particles Levelset Velocity Pressure Boundary Condition Map Velocity Solve Pressure

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FLIP Solver

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FLIP Solver

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FLIP Solver

  • Easy advection
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FLIP Solver

  • Easy advection
  • Fast pressure solver
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FLIP Solver

  • Easy advection
  • Fast pressure solver
  • Splash !
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FLIP Solver

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FLIP Solver

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FLIP Solver

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FLIP Solver

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FLIP Solver

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FLIP Solver

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Summary

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Summary

Fluid Solver

  • Pressure Solver
  • Advection
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Summary

Fluid Solver Surface Tracker

  • Pressure Solver
  • Advection
  • Levelset
  • Topology Change
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Summary

Fluid Solver Surface Tracker FLIP Solver

  • Pressure Solver
  • Advection
  • Levelset
  • Topology Change
  • FLIP Steps