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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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:

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

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:

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

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

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

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

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

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.

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

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

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

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

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 )

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

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

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

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

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

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 :)

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