a unified lagrangian approach to solid-fluid animation richard keiser, bart adams, dominique gasser,...

A Unified Lagrangian Approach to Solid-Fluid

Animation

A Unified Lagrangian Approach to Solid-Fluid

AnimationRichard 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 changesSimple advectionBoundary conditionsIncompressibility

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 representationImplicit: 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


• Conservation of momentum



Material Derivative in Eulerian setting:



Material Derivative in Eulerian setting:

Material Derivative in Lagrangian setting:



– External force (per volume) due to• Gravitation, surface tension, …




– Internal forces (per volume) due to• Pressure stress




– Internal forces (per volume) due to• Pressure stress• Viscosity stress

Deformable Solids


Deformed configurationReference configuration

u(x)x x+u(x)

Lagrangian Approach

• Deformable Solids

• Fluids

• Conservation of mass

Lagrangian Approach

• Conservation of mass


• Fluids

Lagrangian Approach

• Conservation of massmass moves

with particles


• 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-plasticbehavior

Fluidbehavior

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

flu

id

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


• Smoothing potential

Potentials



• Attracting potential

Potentials



• 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: 58kTimings per frame: physics: 3.1 s, surface: 21 s

Freezing

# particles: 2.4k, avg. # surfels: 3.4kTimings 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-plasticbehavior

Fluidbehavior

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

a unified lagrangian approach to solid-fluid animation richard keiser, bart adams, dominique gasser,...

Documents

surface tension forces

solidssmooth surface

internal forces

lagrangian setting

eulerian setting

meshfree continuummechanics

based model

elastic term