Fluid & Rigid Body Interaction

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

2. Motivation • Fluid/solid interactions are ubiquitous in our environment • Realistic fluid/solid interaction is complex • not feasible through manual animation University of North Carolina - Chapel Hill

3. Types of Coupling • One-way solid-to-fluid reaction • One-way fluid-to-solid reaction • Two-way coupled interaction University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

6. One-Way Inadequacy • Fails to simulate true fluid/solid interaction • Reactive as opposed to interactive University of North Carolina - Chapel Hill

7. Two-Way Interaction Methods • Volume Of FluidandCubic Interpolated Propagation (VOFCIP) • Arbitrary Lagrangian-Eulerian(ALE) • Distributed Lagrange Multiplier(DLM) • Rigid Fluid University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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

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

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

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 University of North Carolina - Chapel Hill

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 thematrix used to solve the system or will be singular (cannot be inverted uniquely) University of North Carolina - Chapel Hill

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

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

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

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

24. Conservation of Mass • Velocity field has zero divergence • amount of fluid entering the cell is equal to the amount leaving the cell University of North Carolina - Chapel Hill

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

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

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

28. Conservation of Momentum • The external forces per unit mass that act globally on the fluid • e.g. gravity, wind, etc. University of North Carolina - Chapel Hill

29. Overview of Fluid Steps • Numerically solve for the best guess velocity without accounting for pressure gradient • Pressure projection to re-enforce the incompressibility constraint University of North Carolina - Chapel Hill

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

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

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 University of North Carolina - Chapel Hill

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) University of North Carolina - Chapel Hill

34. Conservation of Momentum • For fluid: • For rigid body: •  is implicitly defined as an extra part of the deformation stress University of North Carolina - Chapel Hill

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

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

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 University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill

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 :) University of North Carolina - Chapel Hill

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 University of North Carolina - Chapel Hill