Simulating Liquid SoundPart I: Fluid Simulation for Sound RenderingLiquid SimulationSlide 4Sound GenerationFluid Simulation TechniquesGrid Based MethodsSlide 8Slide 9Slide 10Particle Based MethodsParticle InteractionsSlide 13Smoothed Particle HydrodynamicsEquations of MotionBubblesSlide 17Shallow Water EquationsSlide 19Small Bubbles?HeuristicsTexture SynthesisReferencesPart II: Bubble SoundSpherical BubbleFree OscillationRayleigh-Plesset EquationLinearization of R-P eq.DampingSlide 30Shifted Resonant FrequencyPressure RadiationExperimentsNonspherical Bubble OscillationsBurstMore IssueSlide 37Simulating Liquid SoundWill MossHengchin YehPart I: Fluid Simulation forSound RenderingSolve the Navier-Stokes equationswhere v is the flow velocity, ρ is the fluid density, p is the pressure, T is the (deviatoric) stress tensor, and f represents body forcesLiquid SimulationGenerally, graphics people assume the fluid is incompressible and inviscid (no viscosity)Looks fine for water and other liquids.Cannot handle shockwaves or acoustic wavesFor these, wee work by Jason or NikunjLiquid SimulationSound GenerationMore detail in the second halfSound is generated by bubblesOur fluid simulator must be able to handle bubblesFluid Simulation TechniquesGrid Based (Eulerian)Accurate to within the grid resolutionSlowParticle Based (Lagrangian)FasterCan look a little strangeOthersShallow water equationsCoupled shallow water and particle basedGrid Based MethodsSplit the inviscid, incompressible Navier-Stokes equations into the three partsAdvectionForcePressureCorrect within a factor of O(Δt)Grid Based MethodsConsiders a constant grid and observes what moves into an out of a cellStagger the grid points to avoid problemsMeasure the pressure at the center of a grid cellMeasure the velocity at the faces between the grid pointsuxGrid Based Methodsui 12, jui 12, jvi, j 12pi, jpi 1, jpi, j  1pi, j 1pi  1, jvi, j 12Grid Based MethodsNaturally handle bubblesJust grid cells that are empty with liquid surrounding themMust take rendering into accountUsed in boiling simulations (Kim, et al)DemosEarly Foster and FedkiwFluid-fluid interactionsBoilingParticle Based MethodsParticles are created by an emitter and exist for a certain length of timeStore mass, position, velocity, external forces and their lifetimeNo particle interactionsBased on smoothed particle hydrodynamics [CITE]Particle InteractionsNo particle interactionsFast, system is decoupledCan only simulate splashing and sprayingParticle InteractionsTheoretically n2 interactionsDefine a cutoff distance outside of which particles do not interactAllows for puddles, pools, etc.Particle InteractionsInteractions of liquids look something likeMathematically we model this with:Smoothed Particle HydrodynamicsNavier-Stokes equations operate on continuous fields, but we have particlesAssume each particle induces a smooth local fieldThe global fluid field is simply the sum of all the local fieldsEquations of MotionSimple particle equations:Reformulate Navier-Stokes equations in terms of forcesEach particle feels a force due to pressure, viscosity and any external forcesBubblesBubbles are not inherently handled (like in Eulerian approaches)Add an air particle to the systemCreate air particles at the surface, so they can be incorporated into the fluidAdd a interaction force and a surface tension force to the particlesSmoothed Particle HydrodynamicsDemosSimple SPH DemoAdding air particlesBoilingPouringShallow Water EquationsReduce the problem to 2DAt each x and y in the grid, store the height of the fluidDrastically reduces the complexity of the Navier-Stokes equationsRuns in real timeShallow Water EquationsOne value for each grid cell means no bubbles or breaking wavesExtension to the method by Thuerey, et. AlSimulate the bubbles as particles interacting with the fluidCan also simulate foam on the surface with SPH particlesVideoSmall Bubbles?What about small scale bubbles?Increase the resolutionComputationally expensiveUse finer grid sizes near the surfaceComplicated, still expensiveUse a heuristic near the surfaceInaccurate, but fasterWe have seen before, sounds can be inaccurate and still portray the necessary feelingHeuristicsAssume bubbles and foam form at regions of the surface where measureable quantities exceed a thresholdCould use curvature, divergence, Jacobian, etc.Generate bubble profiles for those regions heuristically based on the physical propertiesOther heuristics possibleTexture SynthesisUsed at UNC for generating realistic textures for dynamic fluidsVideoReferencesThürey, N., Sadlo, F., Schirm, S., Müller-Fischer, M., and Gross, M. 2007. “Real-time simulations of bubbles and foam within a shallow water framework”. In Proceedings of the 2007 ACM Siggraph/Eurographics Symposium on Computer AnimationMüller, M., Solenthaler, B., Keiser, R., and Gross, M. 2005. “Particle-based fluid-fluid interaction”. In Proceedings of the 2005 ACM Siggraph/Eurographics Symposium on Computer AnimationBridson, R. and Müller-Fischer, M. 2007. Fluid simulation: SIGGRAPH 2007 course notes Narain, R., Kwatra, V., Lee, H.P., Kim, T., Carlson, M., and Lin, M.C., Feature-Guided Dynamic Texture Synthesis on Continuous Flows, Eurographics Symposium on Rendering 2007.Foster, N. and Fedkiw, R. 2001. Practical animation of liquids. In Proceedings of the 28th Annual Conference on Computer Graphics and interactive Techniques SIGGRAPH '01Part II: Bubble SoundCavitation InceptionTensile Strength Cavitation NucleiInsideVacuumGasVaporSpherical Bubblepi=pg+pvpspLRp0Hydrostatic pressureFree Oscillation=0ps + pL > pipi=0RmaxRminR0R0piContractingStart from wall speed =0 ps + pL > piInternal pressure builds up as air is compressedadiabatically (PV = const. )isothermally (PV=nRT)Expandingwall speed =0ps + pL < piInternal pressure decreasesRayleigh-Plesset EquationR-P eq.Work done by pressure difference =Kinetic Energy (Speed of wall)+ Viscosity damping μ+ (Acoustic radiation)+ (Thermal damping)Linearization of R-P eq.R-P eq. is non-linearLinearization for R = R0+rSolution without dampingMinnaert Resonance