a technique for delayed mesh relaxation in multi-material ale applications asme-pvp conference -...
TRANSCRIPT
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.