Divergence-Free Smoothed Particle Hydrodynamics

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1 Copyright of figures and other materials in the paper belongs to original authors. Divergence-Free Smoothed Particle Hydrodynamics Bender et al. SCA 2015 Presented by MyungJin Choi

2 1. Introduction We introduce an efficient and stable implicit SPH solver Density & pressure solver Low volume compression (below 0.01%) Divergence-free velocity field Low volume compression and divergence-free velocity field Enforcing incompressibility and stability Position level Velocity level Our method offers the Following advantages: Low density & divergence error More stable Allowing large time step Decreasing the number of iterations MyungJin Choi # 2

3 2. Related Work EOS-based Approach Simulating Free Surface Flows with SPH [Monaghan / JCP 1994] Particle-Based Fluid Simulation for Interactive Applications [Muller et al. / SCA 2003] Adaptively Sampled Particle Fluids [Adams et al. / TOG 2007] p i = k( ρ i ρ ρ) Weakly Compressible SPH for Free Surface Flows [Becker and Teschner / SCA 2007], MyungJin Choi # 3

4 2. Related Work Pressure Solver Based Approach (1/2) Predictive-Corrective Incompressible SPH [Solenthaler and Pajarola / TOG 2009] Versatile Rigid-Fluid Coupling for Incompressible SPH [Akinci et al. / TOG 2012] Versatile Surface Tension and Adhesion for SPH Fluids [Akinci et al. / TOG 2013] Position Based Fluids [Macklin and Muller / TOG 2013] MyungJin Choi # 4

5 2. Related Work Pressure Solver Based Approach (2/2) Incompressible SPH using the Divergence-Free Condition [Kang and Sagong / CGF 2014] MyungJin Choi # 5

6 2. Related Work Pressure Poisson Equation Based Approach Two-way Coupled SPH and Particle Level Set Fluid Simulation [Lasasso et al. / TVCG 2008] An incompressible multi-phase SPH method [Hu et al. / JOCP 2007] Local Poisson SPH for Viscous Incompressible Fluids [He et al. / CGF 2012] MyungJin Choi # 6

7 3. Fluid Simulation We skip the viscous term on the right side of Navier-Stokes equation Use the XSPH variant Density can be approximated by using [Monaghan 1922] concept The pressure field is determined by the equation of state Special case: MyungJin Choi # 7

8 3.1 Simulation Step The density solver modify the velocities The divergence-free solver corrects the resulting velocities F adv contains gravity, surface tension and viscosity The time step size is determined by CFL condition t 0.4 d v max MyungJin Choi # 8

9 3.2 Divergence-Free Solver The Goal is Determination of the stiffness parameter: k i v and k j v MyungJin Choi # 9

10 3.2 Divergence-Free Solver Basic Knowledge (1/2) The divergence-free solver enforces the condition v = 0 The pressure force of particle i is determined by The pressure gradient can be driven as: ** p i = k ρ i ρ 0 p = k ρ t+i ρ 0 k ρ t ρ 0 = k ρ t+1 ρ t = k ρ MyungJin Choi # 10

11 3.2 Divergence-Free Solver Basic Knowledge (2/2) p The forces F j i can be driven as: Using the condition: Momentum Conservation MyungJin Choi # 11

12 3.2 Divergence-Free Solver How to drive the stiffness parameter k i v k i v can be driven as: Dρ i Dt = ρ i v i (the continuity equation) Dρ i Dt = ρ i 1 ρ i j m j v i v j W ij = j m j v i v j W ij (6) (Ihmsen et al. 2014b) Dρ i Dt = t j m j p F i p F j i W m i m ij (7) ( v i = t F i, v j m j = t F j i i p p m j ) (inserting equation (4), (5) in (7)) MyungJin Choi # 12

13 3.2 Divergence-Free Solver Algorithm Overview The solver performs at least one iteration If particle j is not dynamic, F j i p = 0 MyungJin Choi # 13

14 3.3 Constant Density Solver The Goal is Correction of the density error Using the stiffness parameter: k i and k j MyungJin Choi # 14

15 3.3 Constant Density Solver How to drive the stiffness parameter: k i k i can be driven as: With the equations (6) and (7) ρ i = ρ 0 Analogous to Equation (7), MyungJin Choi # 15

16 3.3 Constant Density Solver Algorithm Overview MyungJin Choi # 16

17 3.4 Kernel In our work we use the cubic spline kernel To speed up the simulation we use a precomputed lookup tables Both the kernel and its gradient we introduce a scalar function g(q) With W h q x with q = x h A lookup table is generated easily by a regular sampling MyungJin Choi # 17

18 4. Results Methods Neighborhood search: A Parallel SPH Implementation on Multi-core CPUs, [Ihmsen et al. / 2011] Rigid-fluid Coupling: Versatile Rigid-Fluid Coupling for Incompressible SPH, [Akinci et al. / 2012] Viscosity: Ghost SPH for Animating Water, [Schechter and Bridson / 2012] Surface tension: Versatile Surface Tension and Adhesion for SPH Fluids, [Akinci et al. / 2013] Time Step: CFL Condition MyungJin Choi # 18

19 4. Results Performance Comparison to other methods DFSPH solvers required only 4.5 and 2.8 iterations IISPH required 50.5 iterations For the sampling of the kernel function and its gradient we used 1000 sample points and measured an maximum local error of less than (Performance Table) (Speed up factor) MyungJin Choi # 19

20 4. Results Memory Requirements Per particle we only have to store the scalar value When performing a warm start, we have to store one additional scalar for each solver IISPH requires seven scalar values for the solver and one for the warm start MyungJin Choi # 20

21 4. Results Stability MyungJin Choi # 21

22 4. Results Full Video MyungJin Choi # 22

23 5. Conclusion and Future Work In this paper we presented a novel implicit SPH simulation method For incompressible fluids that prevents volume compression And enforce a divergence-free velocity field In SPH simulations the density near a free surface is underestimated which causes unnatural particle clustering artifacts Without pressure clamping more sophisticated solving algorithms like the conjugate gradient method We also plan to investigate If DFSPH can improve the stability of multi-phase simulations with high density contrasts. MyungJin Choi # 23

Divergence-Free Smoothed Particle Hydrodynamics

Divergence-Free Smoothed Particle Hydrodynamics Jan Bender Dan Koschier Graduate School CE TU Darmstadt Figure 1: Our new SPH method allows a stable simulation of incompressible fluids with high velocities