A Unified Lagrangian Approach to Solid-Fluid Animation Richard Keiser, Bart Adams, Dominique Gasser, Paolo Bazzi, Philip Dutré, Markus Gross

Motivation Increasing importance of realistic animation of physics phenomena –Deformable solids and fluids –Phase transitions, melting and freezing User interaction –Animations in interactive time

Motivation Solving the continuum mechanics equations using –Eulerian methods –Lagrangian methods Meshfree particle methods have become popular Implicit handling of topological changes Simple advection Boundary conditions Incompressibility Müller et al., SCA 2005

Motivation Challenge: Surface reconstruction –Represent fine detail for solids –Smooth surface for fluids –Handle topological changes Explicit/implicit surface? Explicit: Detail representation Implicit: Topological changes

Related Work Carlson et al. [02] –Model different materials by varying the viscosity Müller et al. [04] –Mesh-free continuum- mechanics-based model for animating elasto- plastic objects Goktekin et al. [04] –Viscoelastic fluids by adding an elastic term to the Navier- Stokes equations

Overview Governing Equations Lagrangian Approach for Solid-Fluid Simulations Melting & Freezing Hybrid Explicit-Implicit Surface Implicit Surface Deformation Results Conclusions

Navier-Stokes Equations Momentum equation Continuity equation

Navier-Stokes Equations Conservation of momentum

Navier-Stokes Equations Conservation of momentum Material Derivative in Eulerian setting:

Navier-Stokes Equations Conservation of momentum Material Derivative in Eulerian setting: Material Derivative in Lagrangian setting:

Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, …

Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress

Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

Navier-Stokes Equations Conservation of momentum –External force (per volume) due to Gravitation, surface tension, … –Internal forces (per volume) due to Pressure stress Viscosity stress

Deformable Solids Conservation of momentum Deformed configurationReference configuration u(x)u(x) xx+u(x)

Lagrangian Approach Deformable Solids Fluids Conservation of mass

Lagrangian Approach Conservation of mass Deformable Solids Fluids

Lagrangian Approach Conservation of mass Deformable Solids Fluids

Lagrangian Approach Conservation of mass Deformable Solids Fluids

Lagrangian Approach Conservation of mass Deformable Solids Fluids

Lagrangian Approach Conservation of mass mass moves with particles Deformable Solids Fluids

Lagrangian Approach Merged Equation Elastic, pressure and viscous stress Body force f –Gravity, surface tension, …

Forces Viscous, pressure and surface tension forces are derived using Smoothed Particle Hydrodynamics (SPH) Derive elastic body forces via strain energy Explicit integration using leap-frog

Material Properties Animation control: –Stiffness (Young’s Modulus E ) –Compressibility (Poisson’s ratio) –Plasticity –Viscosity (µ) –Cohesion / surface tension Elasto-plastic behavior Fluid behavior

Viscoelastic Materials Fluid: No elastic forces ( E = 0 ) Solid: No viscosity (μ = 0 ) and surface tension Viscoelastic materials: couple parameters to scalar a elastic solid fluid

Demo

Melting and Freezing Define properties per particle Change properties depending on a scalar T (called temperature) Heat transfer between particles –Solve heat equation using SPH:

Surface Solid surface –Highly detailed Fluid surface –Smooth surface due to surface tension –Inherent topological changes Local changes from solid to fluid surfaces for melting and freezing

Hybrid Surface Point-sampled surface –wrapped around the particles Hybrid implicit-explicit –Explicit representation for solids Exploit displacement field –Implicit representation for fluids defined as iso-value from particle density field –Blend locally between implicit / explicit surfaces for melting and freezing Depending on temperature T

Implicit Surface Problems of implicit surface defined by particles: –“blobby” surface –Surface with large offset to particles Control surface by defining energy potentials

Potentials Implicit potential

Potentials Implicit potential Smoothing potential

Potentials Implicit potential Smoothing potential Attracting potential

Potentials Implicit potential Smoothing potential Attracting potential Repulsion potential

Forces Potential energy of a surfel is the weighted sum of the potentials Derive forces which minimize potential energy: –Apply implicit, attraction and smoothing force in new normal direction –Apply repulsion force in tangential direction

Melting # particles: 3.9k, avg. # surfels: 58k Timings per frame: physics: 3.1 s, surface: 21 s

Freezing # particles: 2.4k, avg. # surfels: 3.4k Timings per frame: physics: 0.4 s, surface: 1.2 s

Conclusion Lagrangian approach for physics –Wide range of materials from stiff solids, elasto-plastic and visco-elastic objects, to fluids –Stable and efficient –Simple to program Lagrangian approach for surface –Hybrid implicit-explicit approach allows both detailed and smooth surfaces undergoing rapid topological changes –Potentials for better surface control

Discussion

Fluid Forces Viscous, pressure and surface tension forces are derived using Smoothed Particle Hydrodynamics (SPH):

Elastic Force Derive elastic body forces via strain energy Green-Saint-Venant strain tensor Hookean Material

Integration Elastic, pressure, viscosity, surface tension and external forces Explicit integration using Leap-frog Animation control: –Stiffness (Young’s Modulus E ) –Compressibility (Poisson’s ratio) –Plasticity –Viscosity (µ) –Cohesion / surface tension Elasto-plastic behavior Fluid behavior

Constraints Restrict position and movement of surface Implicit constraint –Restrict surfel to be within a minimal iso- level –Enforces automatic splitting External constraint –For adapting to a contact surface –Potentials prevent discontinuities

Contributions Framework for animation of both solids and fluids, and phase transitions Lagrangian approach for both physics and surface Hybrid implicit-explicit surface generation Surface control by defining potentials and geometric constraints