Motivating Problem

1. Quantitative Methods in Defense & National Security“Understanding the Implications of Decoupling the Full Fluid Structure Interaction When Modeling Blast Waves Interacting with Structure”May 26, 2010

2. Motivating Problem

3. Example of a Blast-Structure Calculation using CTH Time=400 msec Time=0 msec Time=200 msec Top • Eulerian code CTH well suited for detonation • Blast-structure interaction for this problem is 3D in its behavior • Disparate length scales require Adaptive Mesh Refinement (AMR) • Detonation, mm scale • Air shock, mm scale • Boat, 8 ft wide • With half plane symmetry ~1000 cpu-hours

4. Calculating Damage in LS-DYNA with CTH Input CTH calculates blast in 3D Export correct pressure time histories from CTH LS-Dyna calculates hull deformation • CTH is an Eulerian code suited for shock physics • LS-Dyna is a Lagrangian code suited for structural response • Timescale for the explosive loading is small relative to the timescale for structural response • i.e. The boat hull does not deform during blast loading • This allows a combined code approach • CTH + LS-Dyna • Use boat test data for validation

5. Issues with Approach • Simulation is split up into 2 separate calculations which are run consecutively • This approach precludes mutual interaction (inaccurate) • This is inefficient from a parallelization standpoint • Requires loading histories on a rigid plate which is an engineering approximation • What if the geometry was much more complicated? • Not applicable for events involving long time behavior of structure (seconds) • E.g. Sympathetic Detonation due to Kinetic Trauma • Requires stand-alone routines to be written for specific application to allow CTH outputs to be read as LS-DYNA inputs  time-consuming • Simulation couples with commercial FEA solver • Scalability is limited by cost since licenses must be purchased for each processor • Source code is not available to allow for seamless coupling independent of specific application

6. Some Technical Background

7. The Finite Volume Method

8. An Aside: “The Small Cell Problem”

9. Approach

10. Our Approach (in the large) • Couple freely-available, scalable solvers using finite volume for the fluid (gas) and finite element for the solid structure focusing on the ability to handle complex geometry • Write an interface that passes information between the two solvers as part of a single simulation • Use embedded boundary techniques in conjunction with state-of-the-art numerical routines capable of dealing with the associated, so-called, “small cell problem”

11. Approach • Write 1D code to solve Euler equations with an ideal gas law • Modify 1D code to allow for small cells (irregular grid) • Implement new theoretical algorithm to deal with the small cell problem • Verify and validate standard code and newly enhanced code through suite of linear and nonlinear test problems • Apply code to fully coupled 1D fluid-structure interaction problem

12. 1D Code for Studying Complex Geometry • Written in C, ~1000 lines of code • Compatible with CLAWPACK 1D input decks • Allows for variable small-cell capability • Solves 1D inviscid Euler equations:

13. 1D Code for Studying Complex Geometry • Fully second order accurate (space and time) • MUSCL-Hancock • Use and in the Riemann solver • TVD-preserving Runge-Kutta scheme

14. 1D Code for Studying Complex Geometry • Slope reconstruction routine allows for irregular grids • Follows recent work of Berger • Uses least squares solution • Van Leer slope limiter is used on primitive variables with

15. 1D Code for Studying Complex Geometry • HLLC Riemann Solver • Work of Toro, Spruce, and Speares • Modifies HLL (Harten, Lax, and Van Leer) scheme by restoring missing contact and shear waves • Approximation for the intercell numerical flux is obtained directly in this approach • Shown to be effective in use with 1D inviscid Euler equations, for example • Roe Solver

16. (Post Processing) Cell Merging • Developed cell merging technique that occurs after the Riemann solve but still is conservative • Formulated and developed in 1D • Take a normal time step • After Riemann Solve, merge small cell with a non-ghost neighbor in a volume-weighted way • Perform irregular grid slope reconstruction on merged cell • Determine correct state values at centers of small cell and neighbor cell based on the new slope calculated • Successfully demonstrated when small cells are present

17. Verification of 1D Code • Constant input stays constant (regular and small cell) • Linear input stays linear (small cell)

18. Verification of 1D Code • Sinusoidal advection preserved (regular and small cell) • 200 cells on [-π,π] • Fixed dt=.0005; tfinal=0.5

19. Verification of 1D Code • Fully 2nd order accurate (regular grid)

20. Verification of 1D Code • Sod shock tube problem verified (regular grid) • 3000 cells

21. Verification (Courant Friedrichs)

22. Piston Problem • Simplest fluid-structure interaction problem (1D) • Subramaniam Paper (Intl J Impact Engineering, 2009) • Blast wave interacting with an elastic structure is analyzed in 1D within ALE framework • The effect of considering 2-way coupling in FSI is compared to 1-way coupling FSI • Builds on work of Blom (Comp. Meth. Appl. Mech. Engr., 1998) • Advocates monolithical FSI algorithm

23. Transient Dynamic Analysis of Elastic Structure • Consider a plate 4.5m in length, 2.25m wide, and 2.5cm thick • Assume fixed edges on the plate and pinned BCs • Can find equivalent structural mass and stiffness of the fundamental mode of vibration per unit cross sectional area: • Equation integrated with Newmark-Beta

24. Piston Problem • Structural displacement predicted by ignoring FSI is larger than the corresponding displacement considering FSI. Pressure history is qualitatively different as well:

25. Example Piston Problem Plots Pressure (1-way) Pressure (2-way)

26. Conclusions • Through an entirely different approach, have qualitatively verified the work of Subramaniam, et al., in 1D: • Distinct differences in pressure and displacement histories when comparing coupled and decoupled approaches • 1D case clearly demonstrates that the decoupling approach may be ill-advised for this class of problems (thin-walled structures)

27. Questions?