IMPLICIT SCHEMES FOR THE SIMULATION OF UNDERWATER IMPLOSION AND EXPLOSION PROBLEMS? Charbel Farhat, Alex Main and Kevin Wang Department of Aeronautics and Astronautics Department of Mechanical Engineering Institute for Computational and Mathematical Engineering Stanford University
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IMPLICIT SCHEMES FOR THE SIMULATION
OF UNDERWATER IMPLOSION
AND EXPLOSION PROBLEMS?
Charbel Farhat, Alex Main and Kevin Wang
Department of Aeronautics and Astronautics
Department of Mechanical Engineering
Institute for Computational and Mathematical Engineering
Stanford University
Stanford, CA 94305
0.5
1.0
P = 1
r = 1
M = 0
g = 1.6
P = 0.1
r = 0.125
M = 0
g = 1.2
t1 = 1e1 s
5.0e01
Initial
CFL = 1
4.0e01
CFL= 20
3.0e01
CFL = 40
Density (kg/m3)
2.0e01
1.0e01
0.0e+00
5.0e01
6.0e01
7.0e01
X (m)
 Multifluid and multiphase flow problems
 Multifluid and multiphase fluidstructure interaction problems
1
Fj,j+1 = Fj+1/2 (nj,j+1) = (Fj + Fj+1 )  F’ j+1/2 (Wj+1 – Wj)
= Roe (Wj, Wj+1, gs, ps) (stiffened gas)
2
2
@(rf)
(ruf)
@
+ = 0
@t
@x
COMPUTATIONAL FRAMEWORK
j + 1/2
j
j + 1
(conservation form)
Wjn
W*n
W*n
Wj+1n
j  1
j  1/2
j
j + 1/2
j + 1
 Fj,j+1 = Roe (Wjn, W*n, EOSj)
Fj+1,j = Roe (Wj+1n, W*n, EOSj+1)
 W*n and W*n determined from the exact solution of local
twophase Riemann problems
C. Farhat, A. Rallu and S. Shankaran, "A HigherOrder Generalized
Ghost Fluid Method for the Poor for the ThreeDimensional
TwoPhase Flow Computation of Underwater Implosions",
Journal of Computational Physics, Vol. 227, pp. 76747700 (2008)
Wnpj+1
Wnpj
1
1
(RR(pI; pR,rR) 
RL(pI; pL,rL))
uI = (uL + uR) +
2
2
RL(pI; pL,rL) +
RR(pI; pR,rR) + uR – uL = 0
pI, rIL, rIR, uI
 Newton’s method
LOCAL RIEMANN SOLVER
rIL,pI,uI ,rIR
contact discontinuity
rarefaction
shock
t
gas
water
x
j
j + 1/2
j + 1
rLuL pL
rRuR pR
j  1/2
j + 1/2
j  1
j
j + 1
Fj,j+1 = Fj+1,j = Roe (Wjn, Wj+1n, EOSj= EOSj+1)
If fjnfj+1n< 0 then
Fj,j+1 = Roe (Wjn, WjRn(rIL, pI, uI), EOSj)
Fj+1,j = Roe (Wj+1n, W(j+1)Rn(rIR, pI, uI), EOSj+1)
Dt
~
 Wjn+1 = Wjn  (Fj,j+1  Fj,j1) (forward Euler)
Dx
~
 Unpack Wn+1 using fn and solve the levelset equation to get fn+1
 Pack Wpn+1 using fn+1 to get the updated solution Wn+1
j  1/2
j + 1/2
j  1
j
j + 1
Fj,j+1 = Fj+1,j = Roe (Wjn+1, Wj+1n+1,EOSj= EOSj+1)
If fjnfj+1n< 0 then
Fj,j+1 = Roe (Wjn+1, WjRn+1,EOSj)
Fj+1,j = Roe (Wj+1n+1, W(j+1)Rn+1, EOSj+1)
Dt
~
 Wjn+1 = Wjn  (Fj,j+1  Fj,j1) (backward Euler)
Dx
~
 Unpack Wn+1 using fn and solve the levelset equation to get fn+1
 Pack Wpn+1 using fn+1 to get the updated solution Wn+1
dFj,j+1 @Fj,j+1 @Fj,j+1@WjRn+1
+
=
dWjn+1@Wjn+1@WjRn+1 @Wjn+1
dFj,j+1 @Fj,j+1@WjRn+1
=
dWj+1n+1@WjRn+1 @Wj+1n+1
dFj+1,j @Fj+1,j @Fj+1,j@W(j+1)Rn+1
+
=
dWj+1n+1@Wj+1n+1 @W(j+1)Rn+1 @Wj+1n+1
dFj+1,j @Fj+1,j@W(j+1)Rn+1
=
dWjn+1@W(j+1)Rn+1 @Wjn+1
uL + FL(rL, pL;pI) = uIL
=
uIR = uR + FR(rR, pR; pI)
@FL @FL @FL
duL +
drL dpL dpI
+
+
@rL @pL@pI
@FR @FR @FR
= duR +
drR dpR dpI
+
+
@rR @pR@pI
dpI
dpI
dpI
dpI
dpI
dpI
, , , , ,
dpL
drL
dpR
drR
duL
duR
uI = uL + FL(rL, pL;pI) = uR + FR(rR, pR;pI)
duI
duI
duI
duI
duI
duI
, , , , ,
dpL
drL
dpR
drR
duL
duR
rIR, rIL
rIR = RR(rR,uI,pI)
rIL = RL(rL,uI,pI)
 TaitTait
 TaitSG (which also implies TaitPG)
p = A(1  )eR1+ B(1  )eR2 + wre
wr
wr
R1r0
R2r0
r0
r0
r
r
JWL EOS
explosive products of combustion (and in particular
Trinitrotoluene — a.k.a. TNT)
where A, B, R1, R2, w and r0 are material constants
 Highly nonlinear function p(r,e)
 Presence of exponentials
uL + FL(rL, pL;rIL) = uIL
=
uIR = uR + FR(rR, pR; rIR)
GL(rL, pL; rIL) = pIL
pIR= GR(rR, pR; rIR)
=
JWL EOS
(1)
(2)
rIR,uIR ,pIR
t
rarefaction
c(r,p)
rR,uR ,pR
=
r
x
du
+
_
dr
= s
rw+1
p  AeR1+ BeR2
r0
r0
r
r
SGJWL RIEMANN SOLVER
(k)
(1)
(2)
complex Riemann problem
 JWLJWL
 JWLSG (which also implies JWLPG)
 BackwardEuler
 Threepoint Backward Differencing Formula (3PBDF)
~
dWi
Win+1  Win
=
dt
Dt
~
dWi
a0Win+1 a1Win + a2Win1
=
dt
Dt
where are constants
a0, a1, a2
 When node i has changed phase between tn1 and tn,
replace Win1 by W(i1)Rn1, the exact solution of the twophase
Riemann problem on the upstream side of the interface
at node i1 and time tn1
n1
n
W(i1)Rn1
i+1
i2
i
i1
2fin+1  2fin
=
Dt
d fi
dt
3PBDF FOR LEVEL SET
1 dfin

2 dt
where the last term can be estimated from the fluxes
at tn
 Required to address phase change
timestep as necessary
r = 50 (kg/m3)
r = 1000.0 (kg/m3)
u= 0.0 (m/s)
u= 0.0 (m/s)
p= 105 (Pa)
p= 109 (Pa)
SHOCK TUBE PROBLEM
Air
Water
(0.5m, 0.5m)
Air (P = 105 Pa)
(0, 0)
Sensor
Water (P = 6.996 MPa)
z
(0.5m, 0.5m)
x
 All runs with I/O on were performed with equal amount of I/O
speedup factor ~ 8.74 (FE/BE)
~ 25 (3PBDF/RK2)
pI, rIRus
x = x(t)
contact discontinuity
not involved
rarefaction*
t
structure
fluid
x
i
j
Mij
Wnj
rRuR pR
EMBEDDED FSI FRAMEWORK
n
w(x,0) =W , if x ≥ 0
j
w
F
(w)
= 0
+
x
t
* could also be a shock
u(x(t), t) = u (Mij)∙ nG(Mij)
s
pI, rIRus
x = x(t)
contact discontinuity
not involved
rarefaction*
t
structure
fluid
x
i
j
Mij
Wnj
rRuR pR
ONESIDED RIEMANN PROBLEM
us = uR + R2(pI(2); pR , rR)
 Closed form Jacobians exist as well (SG, Tait)
G
Mij
i
j
fluid 1
fluid 2
Fij= Roe (us, pI(1), Wni , EOS(1), uij)
Fji= Roe (us, pI(2), Wnj , EOS(2), uji)
 SGstructure (which also implies PGstructure)
 Taitstructure remains to be done
n1
n
W(i1)Rn1
structure
i+1
i2
i
i1
air
( p = 14.5 psi )
water
( p = 1500 psi)
X 400
AEROF/DYNA3D
 AEROS (nonlinear): implicit midpoint rule timeintegrator
 Same amount of I/O performed in all runs with I/O on
speedup factor = 35.5 (FE/BE)
= 43.0 (RK2/3PBDF)
 Verification and validation using shock tube and Turner’s
implosion problems
 Achievement of speedup factors of 45
 Verification on a 2D implosion problem
 Achievement of speedup factor of 43 using the midpoint rule
in AEROS