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 
Dui 
h
2

h h
2

u i 1  u i
2

Dui 
with
h
2

h h
2

2

Ri 
u i 1  u i
u i  u i 1
Dui
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 us  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