fluid & rigid body interaction

University of North Carolina - Chapel HillUniversity of North Carolina - Chapel Hill

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Fluid & Rigid Body Interaction

Fluid & Rigid Body Interaction

Comp 259 - Physical ModelingCraig BennettsApril 25, 2006

Comp 259 - Physical ModelingCraig BennettsApril 25, 2006

University of North Carolina - Chapel Hill


MotivationMotivation

Fluid/solid interactions are ubiquitous in our environment

Realistic fluid/solid interaction is complex not feasible through manual animation

Fluid/solid interactions are ubiquitous in our environment

Realistic fluid/solid interaction is complex not feasible through manual animation



Types of CouplingTypes of Coupling

One-way solid-to-fluid reaction

One-way fluid-to-solid reaction

Two-way coupled interaction

One-way solid-to-fluid reaction

One-way fluid-to-solid reaction

Two-way coupled interaction



Solid-to-Fluid Reaction

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

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



Fluid-to-Solid Reaction

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

The fluid moves the solid without the solid affecting the fluid

Solids are treated as massless particles Foster and Metaxas,1996



One-Way InadequacyOne-Way Inadequacy

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

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



Two-Way Interaction Methods

Two-Way Interaction Methods

Volume Of Fluid and Cubic Interpolated Propagation (VOFCIP)

Arbitrary Lagrangian-Eulerian (ALE) Distributed Lagrange Multiplier (DLM) Rigid Fluid

Volume Of Fluid and Cubic Interpolated Propagation (VOFCIP)

Arbitrary Lagrangian-Eulerian (ALE) Distributed Lagrange Multiplier (DLM) Rigid Fluid



VOFCIP methodVOFCIP 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

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



ALE methodALE 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

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



DLM methodDLM 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

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



DLM Method (cont.)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

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



Prior Two-Way Limitations

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

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



Rigid Fluid MethodRigid 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

Carlson, 2004 Extends the DLM method

except uses finite differences

Uses a Marker-And-Cell (MAC) technique

Pressure projection ensures the incompressibility of fluid



Rigid Fluid Method (cont.)

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

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




Semi-Lagrangian MethodSemi-Lagrangian Method

Advantage: simple to use

Disadvantage: additional numerical dampening to the

advection process


Advantage: simple to use

Disadvantage: additional numerical dampening to the

advection process




Computational DomainsComputational Domains Distinct computational domains for fluid

(F) and rigid solids (R) within the entire domain (C):

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

€

C = F ∪R


are needed to see this picture.



Marker-And-Cell Technique

Marker-And-Cell Technique

Harlow and Welch (1965) Harlow and Welch (1965)


are needed to see this picture.€

u = u,v,w( )



MAC Technique (cont.)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

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



MAC Boundary Conditions

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)

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)



Fluid DynamicsFluid Dynamics

Navier-Stokes Equations Incompressible fluids

Conservation of mass:

Conservation of momentum:

Navier-Stokes Equations Incompressible fluids

Conservation of mass:

Conservation of momentum:

€

∇⋅u = 0

€

ut = − u ⋅∇( )u +∇ ⋅ ν∇u( ) −1

ρ∇p + f



Simplifying AssumptionSimplifying Assumption

For fluids of uniform viscosity More familiar momentum diffusion form

For fluids of uniform viscosity More familiar momentum diffusion form

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f

€

ut = − u ⋅∇( )u +∇ ⋅ ν∇u( ) −1

ρ∇p + f



NotationNotation

Fluid velocity:

Time derivative:

Kinematic viscosity:

Fluid density:

Scalar pressure field:

Fluid velocity:

Time derivative:

Kinematic viscosity:

Fluid density:

Scalar pressure field:€

ut = ∂u∂t

€

u = u,v,w( )

€

ν

€

ρ

€

p

€

∇⋅u = 0

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f



Differential OperatorsDifferential Operators

Gradient:

Divergence:

Vector Laplacian:

Gradient:

Divergence:

Vector Laplacian:€

∇⋅u = 0

€

∇p = px, py, pz( )

€

∇⋅u = ux +vy + wz

€

∇2u =∇ ∇ ⋅u( ) −∇ × ∇ × u( )€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f

€

∇×u =

ˆ x ˆ y ˆ z ∂∂x

∂∂y

∂∂z

u v w

Curl:Curl:



Conservation of MassConservation of Mass

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

the amount leaving the cell

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

the amount leaving the cell

€

∇⋅u = 0



Conservation of MomentumConservation of Momentum

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

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

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f




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

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

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f




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

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

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f




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

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

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f



Overview of Fluid Steps

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

1. Numerically solve for the best guess velocity without accounting for pressure gradient

2. Pressure projection to re-enforce the incompressibility constraint



1. Best Guess Velocity1. Best Guess Velocity

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ∇p + f

€

˜ u t = − u ⋅∇( )u + ν∇ 2u + f

€

˜ u = u + ˜ u tΔt

€

˜ u = u + − u ⋅∇( )u + ν∇ 2u + f[ ]Δt



2. Pressure Projection2. Pressure Projection

€

un +1 = ˜ u −1

ρ∇p

⎡

⎣ ⎢

⎤

⎦ ⎥Δt

€

∇⋅un +1 = 0

€

∇⋅˜ u −Δt

ρ∇ ⋅ ∇p( ) = 0

€

Δt∇ 2 p = ρ∇ ⋅ ˜ u

Solve for p and plug back in to find un+1



Rigid Body DynamicsRigid 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

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



Conservation of Rigidity

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)

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)

€

D u[ ] = 12 ∇u +∇uT[ ]

€

D u[ ] = 0



Conservation of Momentum

Conservation of Momentum

For fluid:

For rigid body:

is implicitly defined as an extra part of the deformation stress

For fluid:

For rigid body:

is implicitly defined as an extra part of the deformation stress

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ f

∇p + f

€

ut = − u ⋅∇( )u +∇ ⋅Π −1

ρ r

∇p + f



Governing EquationsGoverning Equations

For fluid (F):

For rigid body (R):

For fluid (F):

For rigid body (R):

€

ut = − u ⋅∇( )u + ν∇ 2u −1

ρ f

∇p + f

€

ut = − u ⋅∇( )u +∇ ⋅Π −1

ρ r

∇p + f

€

D u[ ] = 0€

∇⋅u = 0



ImplementationImplementation

1. Solve Navier-Stokes equations2. Calculate rigid body forces3. Enforce rigid motion

1. Solve Navier-Stokes equations2. Calculate rigid body forces3. Enforce rigid motion



1. Solve Navier-Stokes1. 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

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

€

u → ˜ u



2. Calculate Rigid Body Forces

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:

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:

€

ρr

ρ f

=>1,

<1,

⎧ ⎨ ⎩

sinks

rises and floats

€

˜ u → ˆ u



3. Enforce Rigid Motion

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 :)

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 :)

€

ˆ u → ut +Δt

€

D ut +Δt[ ] = D ˆ u +

Δt

ρ r

R ⎡

⎣ ⎢

⎤

⎦ ⎥= 0



Rigid Fluid AdvantagesRigid 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

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

fluid & rigid body interaction

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