Transfer coefficient a lgorithm for small mass nodes in material point method

1. Transfer coefficient algorithm for small mass nodes in material point method Xia Ma, BalajiJayaraman, Paul T. Giguere and Duan Z. Zhang CartaBlanca Team Theoretical Division Fluid Dynamics and Solid Mechanics Group T-3, B216, Los Alamos National Laboratory

2. MPM on material interface • There have been ample applications for MPM in solid mechanics. • MPM has been regarded as a method to handle large material deformations. • Recently we have extended it for multi-material interactions. The formulation is similar, or an extension, to multiphase flow theory. • The numerical method to handle material interface becomes an important issue.

3. MPM Random Instability • We use one-D simulation to demonstrate the small mass instability associated with MPM method • Sometimes we observe instability in our MPM simulation and the instability is elusive. So we set-up a one-D case in a special way that the instability is not random with our control and we are for sure it is due to small mass nodes. Oscillation at the end of bar Elastic bar The last particle of the bar is placed very close to node 41 deliberately. MPM without DCA is un-stable for dt=3.2E-9s and become stable after lower the dt to 3E-9s which is an order of magnitude smaller then the CFL requirement.

4. Distribution Coefficients Formulation for Small Mass Nodes(1) • weak form of momentum equation (node based discretization, number of freedom is based on number of grid points, not on MP) Not small Node quantities are obtained from MP Small For node i=42, Mass is small Foce is not small The acceleration (dividing force by mass) is computed artificially big for node 42 and cause instability for MPM method. This instability happens usually in regions with large gradient of the volume faction which is typical near material interface.

5. Distribution Coefficients Formulation for Small Mass Nodes(2) • Introduce distribution coefficient to transfer a part of the force away from the small mass nodes to its neighbor nodes where the mass is not small in such a way that it conforms with the order of accuracy of original MPM. • In the region of large gradient of volume fraction , the error in the MPM is first order in space and this force is transferred within this order of error. • Should satisfy conservation of mass, momentum, and energy, which put some requirement on • Distribution coefficient , if lis not a small mass nodes to conserve second order in interior

6. Conservation laws • Mass conservation • Momentum conservation requires that be a constant. Combining with condition for the nodes with sufficient mass, we find . • Energy conservation to move particle by within an error of order for energy , we need to weight the node velocity by in calculating the stress terms.

7. Time scales affected by distribution coefficients • Let’s consider 1-D elastic bar in a cell with one material point at between nodes 0 and 1 . There is no material point beyond node 1. The momentum equation for node 1 is The solution has a time period If is small, the time period is also small, in turn a high intrinsic frequency. A small time step is thus needed for the original MPM to prevent a numerical instability. • With DCA, momentum equation is modified as the time period becomes , if we choose independent of . This suggest a general form of DCA as , where j is neighbor of l . node 0 1 j l j j j j

8. Numerical examples (1) This is a bench mark case for which there are available experimental data. Here we computed again with and without DCA to see the effect of the algorithm for small mass nodes for CartaBlanca. Air (Large density ratio) Tail positions Tungsten projectile Cylinder of Armor steel Nose position Time direction Projectile-target interaction

9. Numerical examples (2) Although DCA did not increase the accuracy, it is more robust and use less time An order of magnitude small than CFL requirement Overall the difference is small DCA: 25 min No DCA: 6 h 40min dt<1E-20 and exit

10. Numerical examples (3) To ensure the algorithm produces accurate results in cases with large ratio of boundary nodes to internal nodes, we no consider the vibration of a thin elastic shell (50m/s) Total energy Kinetic energy Analytic period is 29.27µs DCA with Δt=0.01µs, 29.28µs without DCA unstable for this Δt reduce Δt to 0.001µs, 29.28ms Small mass node are often near material boundary or interfaces

11. Numerical examples (4) • We studied the instability associated with MPM method in detail for one-D At the trough, last particle moves closer to node 41, reducing mass at 42 further, causing instability. MPM without DCA requires a time step an order of magnitude smaller than that based on CFL=0.75 for wave and material speed, which is dt=3E-8.

12. Conclusions • Small mass nodes often appear near material interfaces with large deformation of materials. • Such a small mass nodes often leads to tiny time steps. Often it results in instability and failure of numerical simulation. • The algorithm presented here satisfies mass and momentum conservation exactly. The energy conservation error is second order on time and in spatial discretization. • Numerical examples show the algorithm is very effective and can significantly increase the allowed size of time steps and make the simulation robust.