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