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 : 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 freeDensity invariantParticle 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 dont cause an abort of computation)