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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

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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


MOTIVATION

Are explicit schemes really required

to perform such simulations?


SHOCK TUBE (GAS-GAS)

  • MURI Review, April 2007

0.5

1.0

P = 1

r = 1

M = 0

g = 1.6

P = 0.1

r = 0.125

M = 0

g = 1.2


POTENTIAL OF IMPLICIT SCHEMES

  • MURI Review, April 2007

t1 = 1e-1 s

5.0e-01

Initial

CFL = 1

4.0e-01

CFL= 20

3.0e-01

CFL = 40

Density (kg/m3)

2.0e-01

1.0e-01

0.0e+00

5.0e-01

6.0e-01

7.0e-01

X (m)


OBJECTIVES 2010

  • Develop fully-implicit schemes for the solution of:

- Multi-fluid and multi-phase flow problems

- Multi-fluid and multi-phase fluid-structure interaction problems

  • Implement them in AERO-F and AERO-F/AERO-S

  • Benchmark them using representative 3D underwater

  • implosion problems


1

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

  • MUSCL-based solver with Roe Flux

j + 1/2

j

j + 1

  • Interface capturing via the level-set equation

(conservation form)


FVM-ERS

  • FVM with exact local Riemann solver for multi-phase flows

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

two-phase Riemann problems

C. Farhat, A. Rallu and S. Shankaran, "A Higher-Order Generalized

Ghost Fluid Method for the Poor for the Three-Dimensional

Two-Phase Flow Computation of Underwater Implosions",

Journal of Computational Physics, Vol. 227, pp. 7674-7700 (2008)


Wnpj+1

Wnpj

  • Exact solution of the analytical problem

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

  • Wave structure and Riemann problem

rIL,pI,uI ,rIR

contact discontinuity

rarefaction

shock

t

gas

water

x

j

j + 1/2

j + 1

rLuL pL

rRuR pR


FVM-ERS (EXPLICIT)

  • GFMP with exact local Riemann solver

j - 1/2

j + 1/2

j - 1

j

j + 1

  • If fjnfj+1n> 0 then

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,j-1) (forward Euler)

Dx

~

- Unpack Wn+1 using fn and solve the level-set equation to get fn+1

- Pack Wpn+1 using fn+1 to get the updated solution Wn+1


FVM-ERS (IMPLICIT)

  • Implicit Extension of FVM-ERS method

j - 1/2

j + 1/2

j - 1

j

j + 1

  • If fjnfj+1n> 0 then

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,j-1) (backward Euler)

Dx

~

- Unpack Wn+1 using fn and solve the level-set equation to get fn+1

- Pack Wpn+1 using fn+1 to get the updated solution Wn+1


IMPLICIT FLUID-FLUID

  • Implicit scheme  nonlinear equation

  •  Jacobians of the two-phase

  • Riemann problem

dFj,j+1 @Fj,j+1 @Fj,[email protected]+1

+

=

[email protected][email protected]+1 @Wjn+1

dFj,j+1 @Fj,[email protected]+1

=

[email protected]+1 @Wj+1n+1

dFj+1,j @Fj+1,j @Fj+1,[email protected](j+1)Rn+1

+

=

[email protected]+1n+1 @W(j+1)Rn+1 @Wj+1n+1

dFj+1,j @Fj+1,[email protected](j+1)Rn+1

=

[email protected](j+1)Rn+1 @Wjn+1


STIFFENED GAS (SG)

  • Local two-phase Riemann solver for SG-SG requires the

  • solution of the equation

uL + FL(rL, pL;pI) = uIL

=

uIR = uR + FR(rR, pR; pI)

  • Differentiating the above contact equation gives

@FL @FL @FL

duL +

drL dpL dpI

+

+

@rL @[email protected]

@FR @FR @FR

= duR +

drR dpR dpI

+

+

@rR @[email protected]

dpI

dpI

dpI

dpI

dpI

dpI

, , , , ,

dpL

drL

dpR

drR

duL

duR


STIFFENED GAS

  • Since , the derivatives

  • can be obtained by straightforward differentiation

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

  • Finally, the derivatives of the interfacial densities

  • are obtained from the straightforward differentiation of

rIR = RR(rR,uI,pI)

rIL = RL(rL,uI,pI)


OTHER EOSs

  • Implicitization was also performed for

- Tait-Tait

- Tait-SG (which also implies Tait-PG)


p = A(1 - )e-R1+ B(1 - )e-R2 + wre

wr

wr

R1r0

R2r0

r0

r0

r

r

JWL EOS

  • Jones-Wilkins-Lee (JWL) equation of state for modeling

    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

  • Solution of exact Riemann problem involves a

  • system of two nonlinear equations

(1)

(2)

  • FL and GL depend on the nature of the interaction in the

  • phase modeled by the JWL EOS

  • shock algebraic equation

  • rarefaction differential equation


rIR,uIR ,pIR

t

rarefaction

c(r,p)

rR,uR ,pR

=

r

x

du

+

_

dr

= s

rw+1

p - Ae-R1+ Be-R2

r0

r0

r

r

SG-JWL RIEMANN SOLVER

  • Rarefaction wave in a JWL medium

(k)

  • The isentropic evolution in the

  • rarefaction fan between two

  • constant states is given by

(1)

(2)

complex Riemann problem

  • Algebraic entropy (s) formula for the JWL EOS

  • No obvious algebraic Riemann invariants for the JWL EOS

  • No analytical Jacobians of the invariants either


JWL EOS

  • Riemann invariants are tabulated for the explicit

  • time-stepping scheme

  • For implicit time-stepping where Jacobians are

  • required, they are currently computed on-line by

  • solving an ODE

  • Implicitization was performed for

- JWL-JWL

- JWL-SG (which also implies JWL-PG)


TIME-INTEGRATORS

  • AERO-F is equipped with two implicit time-integrators

- Backward-Euler

- Three-point Backward Differencing Formula (3PBDF)

  • Backward Euler estimates the time-derivative as follows

~

dWi

Win+1 - Win

=

dt

Dt

  • Node i has the same fluid ID at both tn and tn+1

  • and therefore the above estimation is meaningful


3PBDF

  • 3PBDF approximates the time-derivative as follows

~

dWi

a0Win+1 -a1Win + a2Win-1

=

dt

Dt

where are constants

a0, a1, a2

  • Problem: at tn-1, node i may have a different fluid ID

  • than at tn and tn+1: since the density can be

  • discontinuous across a fluid-fluid interface (contact

  • surface), Win-1 and Win are not necessarily related in

  • this case and therefore the above approximation is

  • invalid


3PBDF

  • Solution

- When node i has changed phase between tn-1 and tn,

replace Win-1 by W(i-1)Rn-1, the exact solution of the two-phase

Riemann problem on the upstream side of the interface

at node i-1 and time tn-1

n-1

n

W(i-1)Rn-1

i+1

i-2

i

i-1


2fin+1 - 2fin

=

Dt

d fi

dt

3PBDF FOR LEVEL SET

  • Similar issue arises when 3PBDF is applied to the level

  • set equation: after re-initialization, fn-1no longer exists!

  • Solution:

1 dfin

-

2 dt

where the last term can be estimated from the fluxes

at tn


LIMITATION

  • The fluid-fluid interface may cross no more than one cell

  • per time-step

- Required to address phase change

  • AERO-F enforces this condition by adjusting the

    time-step 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

  • Shock tube: air to the left, water to the right

  • Air modeled as a perfect gas (g = 1.4); water modeled

  • as a stiffened gas (g = 4.4, p = 6.0x 108)

Air

Water

  • Simulation up to t = 1 x 10-5 s




TURNER IMPLOSION PROBLEM

  • Implosion of a glass sphere (D = 0.0762 m)

(0.5m, 0.5m)

Air (P = 105 Pa)

(0, 0)

Sensor

Water (P = 6.996 MPa)

z

(0.5m, -0.5m)

x


TURNER IMPLOSION PROBLEM

  • Air modeled as a perfect gas (g = 1.4); water modeled

  • as a stiffened gas (g =7.15, p = 2.89 x 108 Pa)

  • 780,000 grid points

  • Simulation up to t = 0.5 ms


TURNER IMPLOSION PROBLEM

Explicit (RK2), CFL = 0.5

Implicit (3BDF), CFL = 100




TURNER IMPLOSION PROBLEM

  • CPU performance on 168 cores of a Linux cluster

- 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

  • Fluid-structure Riemann problem (1/2 problem)

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

ONE-SIDED RIEMANN PROBLEM

  • Fluid-shell problem

  • Closed form algebraic solution of the problem exists (SG, Tait)

us = uR + R2(pI(2); pR , rR)

- Closed form Jacobians exist as well (SG, Tait)


FLUX COMPUTATION

  • The flux across the face at Mij is given by

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)


EMBEDDED FSI (IMPLICIT)

  • Implicitization was performed for

- SG-structure (which also implies PG-structure)

- Tait-structure remains to be done

  • 3PBDF

n-1

n

W(i-1)Rn-1

structure

i+1

i-2

i

i-1


2D IMP45

  • Simplified IMP45 using a thin slice of the aluminum tube

air

( p = 14.5 psi )

water

( p = 1500 psi)

X 400

AERO-F/DYNA3D

  • Explicit = Explicit/Explicit simulation uses dt = 0.75 x 10-8

  • Implicit = Implicit/Explicit simulation uses dt = 3.0 x 10-6


2D IMP45

  • Pressure at t=0.4 ms

  • Solution

- Explicit (RK2)

- Implicit (3PBDF)

- Implicit (BE)


2D IMP45

  • Pressure at a sensing node


2D IMP45

Explicit (RK2), CFL=0.5

Implicit (3BDF), CFL=100


2D IMP45

  • CPU performance on 64 cores of a Linux cluster

- AERO-S (nonlinear): implicit midpoint rule time-integrator

- Same amount of I/O performed in all runs with I/O on

speedup factor = 35.5 (FE/BE)

= 43.0 (RK2/3PBDF)


SUMMARY

  • Implicitization of fluid-fluid schemes in AERO-F

  • Development of a new multi-phase flow solver based on the

  • three point backward difference formula

- Verification and validation using shock tube and Turner’s

implosion problems

- Achievement of speedup factors of 4-5

  • Implicitization of fluid-structure schemes using AERO-F

- Verification on a 2D implosion problem

- Achievement of speedup factor of 43 using the midpoint rule

in AERO-S


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