1. University of North Carolina - Chapel Hill Fluid & Rigid Body Interaction Comp 259 - Physical Modeling Craig Bennetts April 25, 2006 Comp 259 - Physical Modeling Craig Bennetts April 25, 2006

2. University of North Carolina - Chapel Hill Motivation  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

3. University of North Carolina - Chapel Hill Types 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

4. University of North Carolina - Chapel Hill 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

5. University of North Carolina - Chapel Hill 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

6. University of North Carolina - Chapel Hill One-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

7. University of North Carolina - Chapel Hill 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

8. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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

12. University of North Carolina - Chapel Hill 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

13. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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  Uses conjugate-gradient solver

15. University of North Carolina - Chapel Hill Semi-Lagrangian Method  Advantage:  simple to use  Disadvantage:  additional numerical dampening to the advection process  Uses conjugate-gradient solver  Advantage:  simple to use  Disadvantage:  additional numerical dampening to the advection process  Uses conjugate-gradient solver

16. University of North Carolina - Chapel Hill Computational Domains  Distinct computational domains for fluid ( F ) and rigid solids ( R ) within the entire domain ( C ):

17. University of North Carolina - Chapel Hill Marker-And-Cell Technique  Harlow and Welch (1965)

18. University of North Carolina - Chapel Hill 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

19. University of North Carolina - Chapel Hill 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)

20. University of North Carolina - Chapel Hill Fluid Dynamics  Navier-Stokes Equations  Incompressible fluids  Conservation of mass:  Conservation of momentum:  Navier-Stokes Equations  Incompressible fluids  Conservation of mass:  Conservation of momentum:

21. University of North Carolina - Chapel Hill Simplifying Assumption  For fluids of uniform viscosity  More familiar momentum diffusion form  For fluids of uniform viscosity  More familiar momentum diffusion form

22. University of North Carolina - Chapel Hill Notation Fluid velocity: Time derivative: Kinematic viscosity: Fluid density: Scalar pressure field: Fluid velocity: Time derivative: Kinematic viscosity: Fluid density: Scalar pressure field:

23. University of North Carolina - Chapel Hill Differential Operators Gradient: Divergence: Vector Laplacian: Gradient: Divergence: Vector Laplacian: Curl:

24. University of North Carolina - Chapel Hill Conservation 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

25. University of North Carolina - Chapel Hill Conservation of Momentum  The advection term accounts for the direction in which the surrounding fluid pushes a small region of fluid

26. University of North Carolina - Chapel Hill Conservation of Momentum  The momentum diffusion term describes how quickly the fluid damps out variation in the velocity surrounding a given point

27. University of North Carolina - Chapel Hill Conservation of Momentum  The pressure gradient describes how a small parcel of fluid is pushed in a direction from high to low pressure

28. University of North Carolina - Chapel Hill Conservation of Momentum  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.

29. University of North Carolina - Chapel Hill 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

30. University of North Carolina - Chapel Hill 1. Best Guess Velocity

31. University of North Carolina - Chapel Hill 2. Pressure Projection  Solve for p and plug back in to find u n+1

32. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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 )

34. University of North Carolina - Chapel Hill 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

35. University of North Carolina - Chapel Hill Governing Equations  For fluid ( F ):  For rigid body ( R ):  For fluid ( F ):  For rigid body ( R ):

36. University of North Carolina - Chapel Hill Implementation 1.Solve Navier-Stokes equations 2.Calculate rigid body forces 3.Enforce rigid motion 1.Solve Navier-Stokes equations 2.Calculate rigid body forces 3.Enforce rigid motion

37. University of North Carolina - Chapel Hill 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  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. University of North Carolina - Chapel Hill 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: sinks rises and floats

39. University of North Carolina - Chapel Hill 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 :)

40. University of North Carolina - Chapel Hill 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  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