Fluid

1
Fluid Rigid Body Interaction

• Comp 259 - Physical Modeling
• Craig Bennetts
• April 25, 2006

2
Motivation

• Fluid/solid interactions are ubiquitous in our
  environment
• Realistic fluid/solid interaction is complex
• not feasible through manual animation

3
Types of Coupling

• One-way solid-to-fluid reaction
• One-way fluid-to-solid reaction
• Two-way coupled interaction

4
Solid-to-Fluid Reaction

• The solid moves the fluid without the fluid
  affecting the solid
• Rigid bodies are treated as boundary conditions
  with set velocities
• Foster and Metaxas, 1997
• Foster and Fedkiw, 2001
• Enright et al., 2002b

5
Fluid-to-Solid Reaction

• The fluid moves the solid without the solid
  affecting the fluid
• Solids are treated as massless particles
• Foster and Metaxas,1996

6
One-Way Inadequacy

• Fails to simulate true fluid/solid interaction
• Reactive as opposed to interactive

7
Two-Way Interaction Methods

• Volume Of Fluid and Cubic Interpolated
  Propagation (VOFCIP)
• Arbitrary Lagrangian-Eulerian (ALE)
• Distributed Lagrange Multiplier (DLM)
• Rigid Fluid

8
VOFCIP method

• Takahashi et al. (2002,2003)
• Models forces due to hydrostatic pressure
• neglects dynamic forces and torques due to the
  fluid momentum
• Only approximates the solid-to-fluid coupling

9
ALE method

• Originally used in the computational physics
  community Hirt et al. (1974)
• Finite element technique
• Drawbacks
• computational grid must be re-meshed when it
  becomes overly distortion
• at least 2 layers of cell elements are required
  to separate solids as they approach

10
DLM method

• Originally used to study particulate suspension
  flows Glowinski et al. 1999
• Finite element technique
• Does not require grid re-meshing
• Ensures realistic motion for both fluid and solid

11
DLM Method (cont.)

• Does not account for torques
• Restricted to spherical solids
• Surfaces restricted to be at least 1.5 times the
  velocity element size apart
• requires application of repulsive force

12
Prior Two-Way Limitations

• Solids simulated as fluids with high viscosity
• ultimately results in solid deformation, which is
  undesirable in modeling rigid bodies
• Do not account for torque on solids
• Boundary proximity restrictions

13
Rigid Fluid Method

• Carlson, 2004
• Extends the DLM method
• except uses finite differences
• Uses a Marker-And-Cell (MAC) technique
• Pressure projection ensures the incompressibility
  of fluid

14
Rigid Fluid Method (cont.)

• Treats the rigid objects as fluids
• Ensures rigidity through rigid-body-motion
  velocity constraints within the object
• Avoids need to directly enforce boundary
  conditions between rigid bodies and fluid
• approximately captured by the projection
  techniques
• Uses conjugate-gradient solver

15
Semi-Lagrangian Method

• Advantage
• simple to use
• Disadvantage
• additional numerical dampening to the advection
  process
• Uses conjugate-gradient solver

16
Computational Domains

• Distinct computational domains for fluid (F) and
  rigid solids (R) within the entire domain (C)

17
Marker-And-Cell Technique

• Harlow and Welch (1965)

18
MAC Technique (cont.)

• Well suited to simulate fluids with relatively
  low viscosity
• Permits surface ripples, waves, and full 3D
  splashes
• Disadvantage
• cannot simulate high viscosity fluids (with free
  surfaces) without reducing time step significantly

19
MAC Boundary Conditions

• Can use any combination of Dirichlet or Neumann
  boundary conditions between fluid and air
• there must be at least one empty air cell
  represented in the matrix used to solve the
  system or will be singular (cannot be inverted
  uniquely)

20
Fluid Dynamics

• Navier-Stokes Equations
• Incompressible fluids
• Conservation of mass
• Conservation of momentum

21
Simplifying Assumption

• For fluids of uniform viscosity
• More familiar momentum diffusion form

22
Notation

• Fluid velocity
• Time derivative
• Kinematic viscosity
• Fluid density
• Scalar pressure field

23
Differential Operators

• Gradient
• Divergence
• Vector Laplacian

Curl
24
Conservation of Mass

• Velocity field has zero divergence
• amount of fluid entering the cell is equal to the
  amount leaving the cell

25
Conservation of Momentum

• The advection term accounts for the direction in
  which the surrounding fluid pushes a small region
  of fluid

26
Conservation of Momentum

• The momentum diffusion term describes how quickly
  the fluid damps out variation in the velocity
  surrounding a given point

27
Conservation of Momentum

• The pressure gradient describes how a small
  parcel of fluid is pushed in a direction from
  high to low pressure

28
Conservation of Momentum

• The external forces per unit mass that act
  globally on the fluid
• e.g. gravity, wind, etc.

29
Overview of Fluid Steps

1. Numerically solve for the best guess velocity
   without accounting for pressure gradient
2. Pressure projection to re-enforce the
   incompressibility constraint

30
1. Best Guess Velocity

31
2. Pressure Projection

?
Solve for p and plug back in to find un1
32
Rigid Body Dynamics

• Typical rigid body solver
• rigidity is implicitly enforced due to the nature
  of affine transformations (translation and
  rotation about center of mass)
• Rigid fluid solver
• rigid body motion is determined using the
  Navier-Stokes equations
• requires a motion constraint to ensure rigidity
  of the solid

33
Conservation of Rigidity

• Similar to the incompressibility constraint
  presented for fluids, but more strict
• The rigidity constraint is not only divergence
  free, but deformation free
• The deformation operator (D) for a vector
  velocity field (u) is
• Rigid body constraint is (in
  R)

34
Conservation of Momentum

• For fluid
• For rigid body
• ? is implicitly defined as an extra part of the
  deformation stress

35
Governing Equations

• For fluid (F)
• For rigid body (R)

36
Implementation

1. Solve Navier-Stokes equations
2. Calculate rigid body forces
3. Enforce rigid motion

37
1. Solve Navier-Stokes

• Solve fluid equations for the entire
  computational domain C F ? R
• Rigid objects are treated exactly as if they were
  fluids
• Perform two steps as described in fluid dynamics
  section
• Result
• divergence-free intermediate velocity field
• collision and relative density forces of the
  rigid bodies are not yet accounted for

38
2. Calculate Rigid Body Forces

• Rigid body solver applies collision forces to the
  solid objects as it updates their positions
• These forces are included in the velocity field
  to properly transfer momentum between the solid
  and fluid domains
• Account for forces due to relative density
  differences between rigid body and fluid

sinks
rises and floats
39
3. Enforce Rigid Motion

• Use conservation of rigidity and solve for the
  rigid body forces, R
• similar to the pressure projection step in the
  fluid dynamics solution ( but crazier )

40
Rigid Fluid Advantages

• Relatively straightforward to implement
• Low computational overhead
• scales linearly with the number of rigid bodies
• Can couple independent fluid and rigid body
  solvers
• Permits variable object densities and fluid
  viscosities
• Allows dynamic forces and torques