Animating smoke with dynamic balance Jin-Kyung Hong Chang-Hun Kim 발표 윤종철

컴퓨터 그래픽스 연구실 Contents Abstract Introduction  contribution Related Work Computing Errors in the Advection Term  The Equations of Flow  Errors Compensation Scheme Vortex Advection Based on Vorticity Confinement  Vorticity Confinement  Vortex Advection Scheme Implementation Results and Discussion Conclusion

컴퓨터 그래픽스 연구실 Abstract Numerical method for avoiding dissipation Compensation for the errors induced by semi-Largangian scheme New advection term Vortex advection based on a vorticity confinement force

컴퓨터 그래픽스 연구실 Introduction (1/3) Numerical error caused by discretization is as small as possible fundamentally improve accuracy of the simulation, without additional computation dynamic balance to maintain the coherence of the field

컴퓨터 그래픽스 연구실 Introduction (2/3) Compensation for losses in the energy of the velocity field by an advection step Focus on maintaining the vorticity Separate the vorticity field from main velocity field In new advection step, estimate the error during each time interval and compensate

컴퓨터 그래픽스 연구실 Introduction (3/3) Reduces the numerical dissipation which necessarily results from the linear interpolation of a semi-Lagrangian scheme Unique type of vortex fully Eulerian

컴퓨터 그래픽스 연구실 contribution 1. Improving the method for solving the differential equation for the advection step by using error compensation 2. Allowing the smoke model to remain dynamic near the center of a vortex by the use of vortex advection

컴퓨터 그래픽스 연구실 Related work (1/2) Realistic animation of liquids [Foster & Metaxas 1996] Stable Fluids [Stam 1999] Visual simulation of smoke [Fedkiw 2001]

컴퓨터 그래픽스 연구실 Related work (2/2) Back and forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function [Dupont 2003] A vortex particle method for smoke, water and explosions [Selle 2005]

컴퓨터 그래픽스 연구실 Computing Errors in the Advection Term The Equations of Flow Error compensation Scheme

컴퓨터 그래픽스 연구실 The Equations of Flow Navier-Stokes equation  Velocity vector field u = (u, v, w) (1) (2) : velocity : density : pressure : viscosity : external force : time step

컴퓨터 그래픽스 연구실 The Equations of Flow Standard fluid simulation process Force Self-advection of the velocity vector field diffusion projection Density advection along the velocity vector field Mass Conservation to counteract dissipation

컴퓨터 그래픽스 연구실 The Equations of Flow Advection step (3)

컴퓨터 그래픽스 연구실 Error Compensation Scheme Semi-Lagrangian scheme with error compensation that considers the time intervals before and after the advection step

컴퓨터 그래픽스 연구실 Semi-Lagrangian advection

컴퓨터 그래픽스 연구실 Semi-Lagrangian advection To find the velocity of a given voxel at time t+  t, we trace the velocity field backwards in time to time t, and take the velocity from there. A first order accurate backwards Euler time-step: Let d be any of the components of the fluid velocity.

컴퓨터 그래픽스 연구실 Numerical Dissipation Semi-Lagrangian advection (or in fact, just about any usable Eulerian method) has a flaw: “numerical dissipation” When we advect a field, the new values are smoothly interpolated at various points from the old values That interpolation smoothes the field

컴퓨터 그래픽스 연구실 Dissipation Example (1/3) Start with a function nicely sampled on a grid:

컴퓨터 그래픽스 연구실 Dissipation Example (2/3) The function moves to the left (perfect advection) and is resampled

컴퓨터 그래픽스 연구실 Dissipation Example (3/3) And now we interpolate new sample values, and ruin it all!

컴퓨터 그래픽스 연구실 The problem In the limit ∆x->0, this error goes to zero Problem: we can’t or won’t take the limit Ideally we want a grid with only just enough resolution to represent details we care about  We may be forced to use something even coarser if computer resources too limited Numerical dissipation very quickly smoothes them away

컴퓨터 그래픽스 연구실 The Symptoms For velocity fields:  It looks like fluids are too sticky (molasses) or implausible length scale (scale model)  Swirly turbulent detail quickly decays, left with just boring bulk motion For smoke concentration:  Smoke diffuses into thin air too fast, nice thin features vanish

컴퓨터 그래픽스 연구실 Error Compensation Scheme (4) (5) (6) (7)

컴퓨터 그래픽스 연구실 Error Compensation tt + 1 t’t’

컴퓨터 그래픽스 연구실 Backward error compensation algorithm

컴퓨터 그래픽스 연구실 Forward error correction method

컴퓨터 그래픽스 연구실 Forward error correction method tt + 1 t’t’

컴퓨터 그래픽스 연구실 Vortex Advection Based on Vorticity Confinement Using vortex advection based on vorticity confinement, we propose modified equations for developing a fluid simulation with a continuous vortex.  Vorticity Confinement  Vortex Advection Scheme

컴퓨터 그래픽스 연구실 Vorticity Confinement (1/4) (8) (9) (10)

컴퓨터 그래픽스 연구실 Vorticity Confinement (2/4) (Slide by Jos Stam, SIGGRAPH 2003) h = height of grid E = (constant) control the amount of small scale detail

컴퓨터 그래픽스 연구실 Vorticity Confinement (3/4) (Slide by Jos Stam, SIGGRAPH 2003) h = height of grid E = (constant) control the amount of small scale detail

컴퓨터 그래픽스 연구실 Vorticity Confinement (4/4) (Slide by Jos Stam, SIGGRAPH 2003) h = height of grid E = (constant) control the amount of small scale detail

컴퓨터 그래픽스 연구실 Vortex Advection Scheme (1/4) Two important properties of dynamic balance are:  It has an advection step with error compensation  It uses vortex advection Vortex advects along a velocity vector field

컴퓨터 그래픽스 연구실 Vortex Advection Scheme (2/4) We separate the vorticity field from the main velocity field New advection term (11) (12)

컴퓨터 그래픽스 연구실 Vortex Advection Scheme (3/4) ADVECT Velocity field Vorticity confinement field Vorticity Keeping vorticity

컴퓨터 그래픽스 연구실 Vortex Advection Scheme (4/4)

컴퓨터 그래픽스 연구실 Implementation -Simulation Steps- Determine the vorticity field Equation (7) Equation (12)

컴퓨터 그래픽스 연구실 Results and Discussion No swirling motionVortex is lost quickly Smoke keeps spinning w/o VC w/ VC w/ VA

컴퓨터 그래픽스 연구실 Conclusion We have proposed New method for persistent modeling for the unique features of smoke such as vortices