A Technique for Delayed Mesh Relaxation in Multi-Material ALE Applications ASME-PVP Conference - July 25-29 2004 K. Mahmadi, N. Aquelet, M. Souli.

A Technique for Delayed Mesh A Technique for Delayed Mesh Relaxation in Multi-Material ALE Relaxation in Multi-Material ALE

ApplicationsApplications

ASME-PVP Conference - July 25-29 2004

K. Mahmadi, N. Aquelet, M. Souli

The ChallengesThe Challenges

To apply a delayed mesh relaxation method to arbitrary Lagrangian Eulerian multi-material formulation to treat fast problems involving overpressure propagation such as detonations.

To define relaxation delay parameter for general applications of high pressures, because this parameter is a coefficient dependent.

The ProcessThe Process

• Introduction• Eulerian and ALE multi-material methods • Multi-material interface tracking

– VOF method

• Delayed mesh relaxation technique– Lagrangian phase– Mesh relaxation phase

• Numerical applications– Three-dimensional C-4 high explosive air blast– Three-dimensional C-4 high explosive air blast with reflection

• Conclusions

IntroductionIntroductionA problem of blast propagation Lagrangian

Formulation

The computational domain follows the fluid particle motion, which greatly simplifies the governing equations.

Advantages

The material may undergo large deformations that lead to severe mesh distortions and thereby accuracy losses and a reduction of the critical time step.

Drawbacks

Lagrangian schemes have proven very accurate as long as the mesh remains regular.

IntroductionIntroduction Multi-Material Eulerian Formulation

The mesh is fixed in space and the material passes through the element grid. The Eulerian formulation preserves the mesh regularity.

Advantages

The computational cost per cycle and the dissipation errors generated when treating the advective terms in the governing equations.

Drawbacks

IntroductionIntroduction Arbitrary Lagrangian Eulerian (ALE) Formulation

The principle of an ALE code is based on the independence of the finite element mesh movement with respect to the material motion. The freedom of moving the mesh offered by the ALE formulation enables a combination of advantages of Lagrangian and Eulerian methods.

Advantages

For transient problems involving high pressures, the ALE method will not allow to maintain a fine mesh in the vicinity of the shock wave for accurate solution.

Drawbacks

IntroductionIntroduction

The method aims at an as "Lagrange like" behavior as possible in the vicinity of shock fronts, while at the same time keeping the mesh distortions on an acceptable level.

The relaxation delay parameter must be defined for general applications of high pressures.

Delayed mesh Relaxation in ALE method

The method does not require to solve the equation systems and it is well suited for explicit time integration schemes.

IntroductionIntroduction

u = 0 Eulerian approach

u = v Lagrangian approach

ALE approach

v: Fluid particle velocity, u: Mesh velocity

Conservation of momentum

Conservation of mass

Conservation of energy

xuvvdivt i

ii

).()(.

xe

uvt

e

j

iiijij

.)(.

xv

uvtv

j

iiijij

i

).(.,

Equilibrium equations

Eulerian and ALE Multi-Material MethodEulerian and ALE Multi-Material Method Operator split

2 phases of calculations

Transport equation

Second step: Remap phase

Lagrangian

Vt

0

0.

Step n+1

Eulerian

ALE

Step n

jijitv

,

ijijte

First step: Lagrangian phase

Lagrangian

Multi-Material interface trackingMulti-Material interface tracking

In the Young technique, Volume fractions of either material for the cell and its eight surrounding cells are used to determine the slope of the interface.

VOF

VVelement

Fluid1

Delayed mesh relaxation techniqueDelayed mesh relaxation technique Lagrangian phase

mf

an

n

• Acceleration

tvxxnnn

Rn 1

211

• Lagrangian node coordinate

tavv nnnn21

21

21

)(21 1

21

tt nnn

• Material velocity

where

Mesh relaxation phase

• Reference system velocity

)( 1121

21

xxvv nn

Rnn

R

• Node coordinate after relaxation

txxxx nnn

Rnn

R 11111 )(

xn

R

1

is a node coordinate provided by a mesh relaxation algorithm, operating on the Lagrangian configuration at tn+1.

is a relaxation delay parameter.

Numerical applicationsNumerical applications

Jones Wilkins Lee equation of state

EV

VRR

BVRVR

Ap

2

21

1exp1exp1

A (Mbar) B (Mbar) R1 R2 E0 (Mbar)

5.98155 0.13750 4.5 1.5 0.32 0.087

C-4 high explosive JWL parameters

• Three dimensional C-4 high explosive air blast

Numerical applicationsNumerical applications• Three dimensional C-4 high explosive air blast

zoomModelin

g

Numerical applicationsNumerical applications• Three dimensional C-4 high explosive air blast with reflection

zoom

Modeling

Numerical applicationsNumerical applications

• Three dimensional C-4 high explosive air blast Pressure propagation

Numerical applicationsNumerical applications• Three dimensional C-4 high explosive air blast with reflection

Pressure propagation

Numerical applicationsNumerical applications

• Three dimensional C-4 high explosive air blast Pressure plot at 5 feet

Numerical applicationsNumerical applications

• Three dimensional C-4 high explosive air blast with reflectionPressure plot at 5 feet

Numerical applicationsNumerical applications• Three dimensional C-4 high explosive air blast

Overpressure according to relaxation parameter

With 28296 elements

Experimental overpressure = 3.40 bar

With 18864 elements

t0=1,58.10-2 µs

Numerical applicationsNumerical applications• Three dimensional C-4 high explosive air blast with reflection

Overpressure according to relaxation parameter

Experimental Overpresure=2.2 bar t0=2,1.10-2 µs

ConclusionsConclusions

Delaying the mesh relaxation makes the description of motion more "Lagrange like", contracting the mesh in the vicinity of the shock front.

In this study, the definition of the relaxation delay parameter has improved for general applications of shock wave: 0.001µs-1 0.1 µs-1.

Comparing numerical results using delayed mesh relaxation in ALE method to Lagrangian, Eulerian and classical ALE methods shows that this method is the best for problems involving high pressures.

This is beneficial for the numerical accuracy, in that dissipation and dispersion errors are reduced.