Modeling of Granular Materials

Sand SimulationAbhinav GolasCOMP 768 - Physically Based SimulationFinal Project PresentationMay 6, 20091MotivationMovies, games

Engineering design grain silosAvalanches, LandslidesMay 6, 20092

Spiderman 3The Mummywww.stheoutlawtorn.comOverviewWhat are Granular Materials?Proposed ModelActual ProgressMay 6, 20093OverviewWhat are Granular Materials?Proposed ModelActual ProgressMay 6, 20094What are Granular Materials?A granular material is a conglomeration of discrete solid, macroscopic particles characterized by a loss of energy whenever the particles interact (Wikipedia)Size variation from 1m to icebergsFood grains, sand, coal etc.Powders can be suspended in gasMay 6, 20095What are Granular materials?Can exist similar to various forms of matterGas/Liquid powders can be carried by velocity fieldsSandstormsLiquid/Solid similar to liquids embedded with multiple solid objectsAvalanches, landslidesHourglassSimilar to viscous liquidsMay 6, 20096Why the separate classification?Behavior not consistent with any one state of matterCan sustain small shear stresses stable pilesHydrostatic pressure achieves a maximumParticle interactions lose energyCollisions approach inelasticInfinite collisions in finite time inelastic collapseInhomogeneous and anisotropicParticle shape and size inhomogeneousMay 6, 20097Granular solids, liquids, and gases Jaeger et al.Understanding the behavior - StressStress

At equilibrium matrix is symmetric 6 degrees of freedomPressure for fluids tr()/ 3

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yzyzShearNormalStressDifferent matrix for different basis need invariantsPressure! I0Deviatoric invariants Invariants based on J1,J2Eigen values? called principle stressesMay 6, 20099

Understanding the behaviorWhy can sand sustain shear stress?Friction between particlesWhen does it yield? yield surface/conditionMay 6, 200910

Yield surfaceMany surfaces suitable for different materialsMohr Coulomb surface with Von-Mises equivalent stress f(I0, J1)

Condition for stability/rigidity:

sin coefficient of friction

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So why is it difficult to simulate?Scale - >10M particlesNonlinear behavior yield surfaceRepresentation discrete or continuum?May 6, 200912SimulationDepends on what scenario to simulateDiscrete particles Particle-Based Simulation of Granular Materials, Bell et al.Continuum Animating Sand as a Fluid, Zhu et al.May 6, 200913Particle-Based Simulation of Granular MaterialsUse a particle system with collision handlingDefine objects in terms of spheresNeed to define per sphere pair interaction forcesCollision system based on Molecular DynamicsAllow minor spatial overlap between objects

May 6, 200914Sphere pair interactionDefine overlap(), relative velocity(V), contact normal(N), normal and tangential velocities(Vn, Vt), rate of change of overlap(V.N)Normal forces

kd : dissipation during collisions, kr : particle stiffnessBest choice of coefficients: =1/2, =3/2Given coefficient of restitution , and time of contact tc, we can determine kd and kr

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Sphere pair interactionTangential forces

These forces cannot stop motion require true static frictionSprings between particles with persistent contact?Non-spherical objectsMay 6, 200916

Solid bodiesMap mesh to structure built from spheresGenerate distance field from meshChoose offset from mesh to place spheresBuild iso-surface mesh (Marching Tetrahedra)Sample spheres randomly on trianglesLet them float to desired iso-surface by repulsion forcesD=sphere density, A=triangle area, R=particle radius, place particles, 1 more with fractional probabilityMay 6, 200917

Solid bodiesK interaction kernel, P Position of particle, V velocity of particle, distance fieldRigid body evolutionOverall force = forcesOverall torque = torques around center of massMay 6, 200918

Efficient collision detectionSpatial hashingGrid size = 2 x Maximum particle radiusNeed to look at 27 cells for each particle O(n)Not good enough, insert each particle into not 1, but 27 cells check only one cell for possible collisionsWhy better?Spatial coherenceParticles moving to next grid cell, rare (inelastic collapse)Wonderful for stagnant regionsMay 6, 200919Advantages/DisadvantagesThe GoodFaithful to actual physical behaviorThe Bad and the UglyComputationally intensiveSmall scale scenesScenes with some control particlesMay 6, 200920Animating Sand as a FluidMotivationSand ~ viscous fluids in some casesContinuum simulationBootstrap additions to existing fluid simulatorWhy?Simulation independent of number of particlesBetter numerical stability than rigid body simulatorsMay 6, 200921Fluid simulation? whats that?Discretize 3D region into cuboidal grid3 step process to solve Navier Stokes equationsAdvectAdd body forcesIncompressibility projectionStable and accurate under CFL conditionMay 6, 200922vvuup, Extending our fluid simulatorExtra things we need for sandFriction (internal, boundary)Rigid portions in sandRecallStressYield conditionMay 6, 200923Calculating stressExact calculation infeasibleSmart approximationsDefine strain rate D = d/dt(strain)

Approximate stressesRigidFluidMay 6, 200924

The algorithm in a nutshellCalculate strain rateFind rigid stress for cellCell satisfies yield condition?Yes mark rigid, store rigid stressNo mark fluid, store fluid stressFor each rigid connected componentAccumulate forces and torquesFor fluid cells, subtract friction forceMay 6, 200925Yield conditionRecap

Can add a cohesive force for sticky materialsMay 6, 200926

Rigid componentsAll velocities must lie in allowed space of rigid motion (D=0)Find connected components graph searchAccumulate momentum and angular momentum

Ri solid region, u velocity, density, I moment of inertiaMay 6, 200927

Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid, Carlson et al.Friction in fluid cellsUpdate cell velocityBoundary conditionsNormal velocity: Tangential velocity:

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RepresentationDefining regions of sandLevel setsParticlesAllow improved advectionHybrid simulationPIC Particle In CellFLIP FLuid Implicit ParticleMay 6, 200929Advantages/DisadvantagesAdvantagesFast & stableIndependent of number of particles large scale scenes possibleDisadvantagesNot completely true to actual behaviorDetail issues smoothing in simulation, surface reconstructionMay 6, 200930OverviewWhat are Granular Materials?Proposed ModelActual ProgressMay 6, 200931Proposed ModelMinimization problemInelasticity -> stress tries to minimize kinetic energyConstraintsFriction, yield conditionBoundary conditionsUnilateral incompressibilityMay 6, 200932Proposed ModelFriction

Nice, but not linear Frobenius NormInfinity/1 Norm linearUnilateral Incompressibility

Boundary conditionsMay 6, 200933

ProblemsFriction not orthogonal Goodbye LCPKKT solver slowIterative solversLCP for unilateral incompressibility, boundary conditionsFriction checking after thatRecurse till convergenceMay 6, 200934OverviewWhat are Granular Materials?Proposed ModelActual ProgressMay 6, 200935Actual progressBoundary cases!90% of all effort in writing fluid solverMinor detailsParticle reseeding / preventing clumpingContinuity of stress fieldParameter tuningMay 6, 200936Actual progressRunning implementation of Animating Sand as a Fluid, Zhu et al.3D real-time albeit with simple renderingTakes care of frictionRigid, fluid casesBoundary cases tangential contact frictionVariational formulation Variational Fluids, Bridson and BattyMay 6, 200937ImplementationGrids coupled with particlesParticles dictate fluid densityOne way velocity mapping need ghost fluids for proper 2 way mappingGrid based advectionVariational model better interaction handling with non-axis-aligned objectsRigid projectionMay 6, 200938Actual progressThe nitty-grittyImplemented 3D fluid solver from scratchParticle System reused from previous assignmentsUsed Bridsons CG solverRigid projection Rigid FluidsVariational pressure solve equality constraintsMay 6, 200939Pluses/Minuses?IssuesBugs known and unknown remainLCP solver couldnt be completed in time no unilateral incompressibility, improved contactIterative testing couldnt be donePluses3D fluid simulator workingWith minor fixes should be perfectly functionalMay 6, 200940Future workFix remaining bugsAdd LCP solver to do iterative testingLook for alternatives for full KKT solver, to allow for coupled solveAdd better cube/plane intersection codesMay 6, 200941ReferencesGranular Solids, Liquids, and Gases Jaeger et al.Review of Modern Physics 96Particle-Based Simulations of Granular Materials Bell et al., Eurographics 05Animating Sand as a Fluid Zhu et al.SIGGRAPH 05Rigid Fluid: Animating the Interplay Between Rigid Bodies and Fluid Carlson et al.SIGGRAPH 04May 6, 200942ReferencesA Fast Variational Framework for Accurate Solid-Fluid Coupling Batty et al.SIGGRAPH 2007May 6, 200943