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Particle-based fluid simulation for interactive applications Matthias Müller David Charypar Markus Gross 9557501 陳岳澤 Outline • Introduction • Navier-Stokes Equation • SPH (Smoothed Particle Hydrodynamics ) • Smooth Kernel • Marching Cubes • Result Introduction • Navier-Stokes Equation describe the motion of fluid substances such as liquids and gases • Use Smoothed Particle Hydrodynamics (SPH) to simulate fluids with free surfaces. • Interactive simulation (about 5 fps). Navier-Stokes Equation Conservation of momentum equation Three components: -1 – Pressure term – External force term – Viscosity term v: velocity, g: external force, : density, p: pressure, : viscosity coefficient Navier-Stokes Equation -2 • The acceleration ai of particle i is vi fi ai t i (fi is body force) • Using ai , we can get velocity and position of particle i SPH -1 • Originally developed for astrophysical problems (1977). • Interpolation method for particles. • Properties that are defined at discrete particles can be evaluated anywhere in space. • Uses smoothing kernels to distribute quantities. SPH -2 • Smoothing of attribute A mj: mass j : density Aj: quantity to be interpolated W: smoothing kernel h Particle density • Smoothing of attribute A • Particle density j s r m j W r rj , h m jW r rj , h j j j Pressure Term • Navier-Stokes Equation • Pressure Term Viscosity term • Navier-Stokes Equation • Viscosity Term External force term • Other external forces are directly applied to the particles. • Collisions: In case of collision the normal component of the velocity is flipped. Smoothing Kernel -1 • Has an impact on the stability and speed of the simulation. – ex: Avoid square-roots for distance computation. • Sample smoothing kernel: Smoothing Kernel all points inside a radius of ‘h’ are considered for “smoothing”. Thick line: the kernel Thin line: the gradient of kernel Dashed line: the laplacian of kernel -2 Smoothing Kernel -3 h • For n particles n2 potential interactions! • To reduce to linear complexity O(n2) define interaction cutoff distance h Smoothing Kernel -4 h h • Fill particles into grid with spacing h • Only search potential neighbors in adjacent cells Marching Cubes • To visualize the free surface -1 Marching Cubes -2 Result Interactive Simulation (5fps) 2200 particle Point Splatting Marching Cubs