A Unified Lagrangian Approach to SolidFluid Animation Richard

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 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 Reference configuration x Deformed configuration u(x) x+u(x)

Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

Lagrangian Approach • Deformable Solids • Fluids • Conservation of mass

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

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

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. 9 k, avg. # surfels: 58 k Timings per frame: physics: 3. 1 s, surface: 21 s

Freezing # particles: 2. 4 k, avg. # surfels: 3. 4 k 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