Smoothed Particle Hydrodynamics For Deformable Vessel Dynamics
Progress Report
Van Jones
2008/01/30
Research Goals
Near Term: Rigid-Body Dynamics with 3 Degrees of Freedom (2D) Calculation of Accurate Pressures/Forces on Boundaries
Long Term: Deformable-Body Dynamics with 6 Degrees of Freedom Non-uniform Time-stepping Dynamic Particle Refinement Parallelization + Integration with OpenFOAM
Code Status
• 2D Rigid Body Dynamics: In Progress• Surface Pressures/Forces: In Progress• Deformable Body Support: ---• Particle Refinement Control: ---• Non-Uniform Time-stepping: ---• 3D/6DOF: ---• Parallelization + Integration w/ OpenFOAM: ---
Theory - SPH Basics• Meshfree CFD Method• Particle Discretization of Continuous Fluid• Lagrangian Perspective• SPH Approximation of Fluid Field Calculations
Cubic Spline Kernel Function (Liu, Liu) SPH Form of N-S Governing Equations
Theory - SPH Basics (2)
• Integration over Support Domain Determines Fluid-Particle Properties
• Equation of State Used to Determine Pressure
Support Domain (Liu, Liu) Summation-Density Formulation and Tait Equation
Program Structure• Domain Particles Stored in Quadtree Data Structure
• Allows For Efficient Particle Support Domain Searching
Quadtree (Wikimedia Commons)
• Pressure Based Forces Depend on dW/dr• dW/dr approaches zero as r 0• Surface Penetration Prevention
– Virtual Particles– Inverse-Square Repulsive Boundary Force– Line-Boundary
Boundaries
Surface Pressures/Forces• Current: Code Estimates Forces/Pressures from Repulsion Force Only
– Fails to take into account momentum transfer from virtual particle interactions with fluid particles
• Desired: Corrected Smoothed Particle Method (CSPM)
– Provides Accurate Density/Pressure Calculations Even at Boundaries
Truncated Support Domain (Liu, Liu)
Example Simulations – Cylinder Drop
Example Simulations – Dam Break
Example Simulations – Airfoil
Future Work – Deformable Structures
• Soft-Couple Object Particles to Underlying Physics Model (Skeleton) – i.e. Simple 3DOF Beam Element
• Forces From Particles Transferred to “Skeleton”
• Skeleton Deformations Modify Particle Positions
Future Work – Non-uniform Timestep
• Fractional time-stepping in regions of high-resolution particles provides a more uniform CFL number
• Largest timesteps evaluated first, then positions/states are interpolated for adjacent particle timesteps
• Investigate possibility of continuous timestep variation (as opposed to fractional)
Future Work – Dynamic Particle Refinement
• Merge/Split particles to maintain volumes of increased/decreased particle resolution
• Investigate possibility of continuous particle refinement (vs. discreet merge/split) by particle to particle mass transport with spatial correction
References
• Liu, Liu Smoothed Particle Hydrodynamics – a meshfree particle method