1. Smoothed Particle
Hydrodynamics
(SPH)
Quirijn Frederix

2. Introduction: SPH

3. Table of contents
• Introduction
• The idea
• Kernel function
• Equations of motion
• Boundary conditions
• Problems
• Boundary particles
• Incompressible flow
• Particle clustering
• Example: Couette flow
• Conclusion

4. The idea
• Fluid is composed of many discrete particles (mesh-free)
• Lagrangian method
• Each Particle has certain velocity, density, thermal energy and
fixed mass
• Global conservation of mass automatically fulfilled

5. The idea: Kernel function
• Examples of Kernel functions:
• Guassian, infinetly differentiable but not compact supported:
• Compact supported splines with s = │r│/h :
• In some applications a varying smoothing
length, “h” can be preferable
(e.g. Astrophysics)

6. The idea: Interpolation
• It can be shown that:
• Example density:
Apply this to momentum equations

7. Equations of motion
•

8. Equations of motion
•

9. The complete method
•

10. Boundary conditions
• 3 types: Inflow, outflow and rigid wall
• No slip and free slip boundary conditions by:
• Ghost particles:
• Boundary forces (Lennard-Jones potential):

11. Table of contents
• Introduction
• The idea
• Kernel function
• Equations of motion
• Boundary conditions
• Problems
• Boundary particles
• Incompressible flow
• Particle clustering
• Example: Couette flow
• Conclusion

12. Problems: boundaries
• Calculating the density with the before mentioned formula:
• problematic for particles near the boundaries, where sphere
of influence partially falls outside the problem domain
• Lengthy calculations that have to be done before the
momentum equation can be solved
• Solution: use continuity equation to calculate ρ:
• All terms can now be calculated simultaneously

13. Problems: Incompressible flow
• When pressure is obtained through an explicit function of
density, local variations in the pressure gradient may force
particle motions due to local density gradients
• Approximate incompressible flow by slightly compressible
flow with high speed of sound, c
• But c not too high to have acceptable time step

14. Problem: Particle clustering
• Non-uniform particles can lead to ill-conditioned matrix in the
linear system
• Caused by “Kernel flaw”
• Influenced by Re
• Solution:
• Extra pressure term
• Remeshing
• Particle shifting

15. Problem: Particle clustering
• Comparison of methods
Divergence free
Density invariant
Particle shifting

16. Problem: Particle clustering
• Divergence free ISPH: Accurate but unstable
• Density invariant ISPH: Stable but less accurate
• Divergence free + Density invariant: Accurate and stable, not
efficient
• Particle shifting: Accurate and stable without loss of efficiency
• Simulation of lid-driven cavity flow:

17. Table of contents
• Introduction
• The idea
• Kernel function
• Equations of motion
• Boundary conditions
• Problems
• Boundary particles
• Incompressible flow
• Particle clustering
• Example: Couette flow
• Conclusion

18. Example: Couette flow
• Low order, compact support kernel since 2nd order derivative
not required, without loss of stability or accuracy
• Choose speed of sound such that density fluctuations are
limited to 3%

19. Example: Couette flow

20. Table of contents
• Introduction
• The idea
• Kernel function
• Equations of motion
• Boundary conditions
• Problems
• Boundary particles
• Incompressible flow
• Particle clustering
• Example: Couette flow
• Conclusion

21. Conclusion
• Advantages
•
•
•
•
Large dynamic range in resolution
Easily handle complex geometries and regions devoid of particles
Easy to implement, incredibly robust
Excellent conservation properties (linear + angular momentum)
• Disadvantages
• Numerical noise from approximation of kernel interpolation leads
to limited accuracy (mostly in 2D and 3D)
• Generally computationally slower compared to other mesh-based
techniques
• Too tobust (errors don’t cause an abort of computation)