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Quantitative Methods in Defense & National Security Jim Uschock “Understanding the Implications of Decoupling the Full Fluid Structure Interaction When Modeling Blast Waves Interacting with Structure” Naval Surface Warfare Center Dahlgren Division (540) 653-4463 jbu200@cims.nyu.edu May 26, 2010 Motivating Problem 2 Example of a Blast-Structure Calculation using CTH • • • 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) Top – Detonation, mm scale – Air shock, mm scale – Boat, 8 ft wide • With half plane symmetry ~1000 cpu-hours Time=0 msec Time=200 msec Time=400 msec 3 Calculating Damage in LS-DYNA with CTH Input • 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 CTH calculates blast in 3D Export correct pressure time histories from CTH – 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 LS-Dyna calculates hull deformation 4 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 5 Some Technical Background 6 The Finite Volume Method u t ( F ( u )) x 0 7 An Aside: “The Small Cell Problem” 8 Approach 9 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-theart numerical routines capable of dealing with the associated, socalled, “small cell problem” 10 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 11 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: t u x 0 u t u p x 0 E t Eu pu x 0 2 E p 1 1 u 2 2 12 1D Code for Studying Complex Geometry • Fully second order accurate (space and time) – MUSCL-Hancock i u u L i R i 1 2 (1 ) u i 1 / 2 u n i u n i 1 1 (1 ) u i 1 / 2 2 u L i u L i i 2 1 2 u i • Use u R i and u R i u R i 1 t 2 x 1 t 2 x f (u L i ) f (u i ) f (u L i ) f (u i ) R R L i 1 in the Riemann solver – TVD-preserving Runge-Kutta scheme Q Q tL ( Q ) * Q ** n n Q tL ( Q ) * * Q n 1 1 (Q Q ) n ** 2 13 1D Code for Studying Complex Geometry • Slope reconstruction routine allows for irregular grids – Follows recent work of Berger – Uses least squares solution u ( x ) u ( x i ) ( x x i ) ( R i ) u i D Li Dui h 2 h h 2 u i 1 u i 2 Dui with h 2 h h 2 2 Ri u i 1 u i u i u i 1 Dui h x i 1 x i h – Van Leer slope limiter is used on primitive variables 14 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 15 (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 16 Verification of 1D Code •Constant input stays constant (regular and small cell) •Linear input stays linear (small cell) 17 Verification of 1D Code •Sinusoidal advection preserved (regular and small cell) 2 sin( x ) p 2 u 1 • 200 cells on [-π,π] • Fixed dt=.0005; tfinal=0.5 18 Verification of 1D Code •Fully 2nd order accurate (regular grid) 19 Verification of 1D Code • Sod shock tube problem verified (regular grid) l p l 3; r p r 1 – 3000 cells 20 Verification (Courant Friedrichs) Pro Pio 8 Pio Po Pio 6 Po 21 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 22 Transient Dynamic Analysis of Elastic Structure m s us k s u s f s t p t P0 A • 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: m s 148 . 2 kg m 2 k s 1 . 397 10 6 N m 3 • Equation integrated with Newmark-Beta 23 Piston Problem • Structural displacement predicted by ignoring FSI is larger than the corresponding displacement considering FSI. Pressure history is qualitatively different as well: 24 Example Piston Problem Plots Pressure (1-way) Pressure (2-way) 25 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) 26 Questions? 27