Transcript 顏志翰
Visual Simulation of Smoke SIGGRAPH’01 Ronald Fedkiw, Jos Stam and Henrik Wann Jensen Stanford University & Alias|wavefront Outline Introduction Previous Work The Equations of Fluid Flow Self-Advection by Semi-Lagrangian Numerical Dissipation Results Introduction Ideally Looks good Fast simulation Looks good? Needs to look plausible Doesn’t need to be exactly correct doesn’t suffer from excessive numerical dissipation Previous Work Kajiya and Von Herzen (SIGGRAPH 84) N. Foster and D. Metaxas (SIGGRAPH 97) Improve Stam’99 (Stable Fluids) Handle moving boundaries Reduce numerical dissipation Add high quality volume rendering Method still fast but looks more “smoke-like” The Equations of Fluid Flow Review: Navier-Stokes Equations Incompressible Euler Equations self-advection Fluid velocity: u= <Vu , Vv , Vw > forces u 0 incompressible (Navier-Stokes without viscosity) Gradient Implement Additional Equations Density Temperature Buoyancy force (u ) t T (u )T t f z (T Tamb ) z Algorithm add forces t=0 self-advection project t = t + dt Helmholtz-Hodge Decomposition u 0 Self-Advection t (u ) t+dt Semi-Lagrangian solver (Courant, Issacson & Rees 1952) Semi- Lagrangian The semi-Lagrangian integrator itself is broken down into three main parts 1. Vector field extraction 2. Particle tracing 3. Interpolation Self-Advection t t+dt Semi-Lagrangian solver (Courant, Isaacson & Rees 1952) Self-Advection t For each u-component… t+dt Self-Advection t For each u-component… t+dt Self-Advection t Set interpolated value in new grid t+dt Self-Advection Repeat for all u-nodes Self-Advection Repeat for all v-nodes Numerical Dissipation ‘Stable Fluids’ method dampens the flow Typical with semi- Lagrangian methods Improve using “Vorticity Confinement” force Vorticity Confinement Basic idea: Add energy lost as an external force Use “Vorticity Confinement” force invented by John Steinhoff 1994 Vorticity Confinement : vorticity 1 u y u x 1 u ( ) 2 x y 2 Vorticity Confinement Vorticity location vector : Normalized vorticity Location vector : N Vorticity Confinement Spatial discretization: h Total Forces User supplied fields Buoyancy force New confinement force Results Results Any Problem?