Generalities

1. From Lagrangian solver: - boundary location - V at boundary: From Eulerian solver: - extrapolate V (requires advection) - density & pressure (requires advection) From Lagrangian solver: - boundary location - V and pressure at boundary (+acceleration) From Eulerian solver: - extrapolate V (requires advection) - density (requires advection) Generalities • Fluid Solver: ENO LLF (2nd order) TVD R-K integration (3rd order) • Interface Coupling: Reflective Boundary Condition (RBC) • Interface Coupling: (modified) Ghost Fluid Method (mGFM) CIT ASCI Alliance Program

2. Scheme Symmetry • Do not enforce symmetry. • Verify symmetry by folding the solution around the axis. CIT ASCI Alliance Program

3. Mass/entropy conservation: 1D • Define |Ci | the measure of the cell Ci : • Initial conditions: • Coupling: RBC CIT ASCI Alliance Program

4. Mass/entropy conservation: 1D 1st order convergence • Case a) 2nd order convergence • Case b) CIT ASCI Alliance Program

5. Mass conservation: 2D • Shock running between two plates (RBC) Shock: M=3. Reflective wall Symmetrical plates CIT ASCI Alliance Program

6. Spring-mass-fluid 1D: equations • Equilibrium relation for a spring-mass-fluid system: - x the displacement w.r.t. initial position x0- m, k are mass and stiffness per unit area- g is the ratio of specific heats of a perfect gas- the flow is isentropic, with reservoir pressure P0 and sound speed c0 • Can be recast in non-dimensional form:Dxis defined as the grid resolution and CIT ASCI Alliance Program

7. Spring-mass-fluid 1D: results for RBC • Case a) 1st order convergence • Case b) Still 1st order, spurious oscillations disappearing with refinement CIT ASCI Alliance Program

8. Uniform State Vacuum Fluid-Piston Free Expansion • Examine the first time step. • dP/dx in rigid body depends on the spatial extent • The relevant parameter for the rigid body motion is m/A, independent of length. CIT ASCI Alliance Program

9. Boundary Condition and the GFM • Density: Discontinuous variable. • Density in Ghost Region extrapolated from Boundary • Trivial in 1D , Eikonal Equation in higher dimensions. • Speed: Taken from solid, 0 initially. • Pressure: • Pressure on left boundary is the initial pressure of fluid. • Not physical to use the stress inside the solid. • Constant extrapolation (dP/dx = 0) like density ? • Extrapolation based on acceleration of the boundary. Solid Fluid Extrapolation CIT ASCI Alliance Program

10. Analytical Profiles Analytical profiles obtained using the method of characteristics Gamma-law gas, Rigid Piston. CIT ASCI Alliance Program

11. Simulation Results for GFM • 4 Levels of Grid Refinement • 100, 200, 400, 800 cells. • Total spatial extent of the simulation: 1.0 m • dx = {0.01, 0.005, 0.0025, 0.00125} m • [dxrA/m] = {0.16, 0.08, 0.04, 0.02} • Constant CFL number used for all studies. • Total Simulation time: 0.5 ms (t = 2t) • Piston travels a total distance of 0.0744 m (~ ct) • Results: • P(t) plotted against analytical profile • V(t) plotted against analytical profile CIT ASCI Alliance Program

12. -2 x10 Pressure at the interface. Time (ms) 10 8 6 4 2 0 Error (%) 0.1 0.5 0.2 0.4 0.3 Time (ms) CIT ASCI Alliance Program

13. Speed of the piston. Error (%) 10 8 6 4 2 0 0.1 0.5 0.2 0.4 0.3 Time (ms) CIT ASCI Alliance Program

14. Convergence Results for GFM 10 1 1000 100 CIT ASCI Alliance Program

15. Conclusions & Path Forward • Coupling scheme is robust and 1st order accurate. • Several variations on similar coupling schemes were developed: need of “tougher” tests to select the fittest. • More verification is needed for wave transmission at solid-fluid interface: use of 1-D Lagrangian solid? • More tests is needed for non-commensurable Eulerian/Lagrangian grids (large difference in grid size): important for indipendent solid/fluid grid adaptivity. CIT ASCI Alliance Program