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