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

The 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 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

A problem of blast propagation Introduction A problem of blast propagation Lagrangian Formulation The computational domain follows the fluid particle motion, which greatly simplifies the governing equations. Advantages Lagrangian schemes have proven very accurate as long as the mesh remains regular. 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

Introduction Multi-Material Eulerian Formulation Advantages 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

Introduction 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

Introduction Delayed mesh Relaxation in ALE method 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 method does not require to solve the equation systems and it is well suited for explicit time integration schemes. The relaxation delay parameter must be defined for general applications of high pressures.

Equilibrium equations Introduction v: Fluid particle velocity, u: Mesh velocity Conservation of momentum Conservation of mass Conservation of energy Equilibrium equations ALE approach u = 0 Eulerian approach u = v Lagrangian approach

Eulerian and ALE Multi-Material Method Operator split Step n First step: Lagrangian phase Lagrangian Transport equation Second step: Remap phase Step n+1 Eulerian ALE 2 phases of calculations

Multi-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

Delayed mesh relaxation technique Mesh relaxation phase Reference system velocity Node coordinate after relaxation is a node coordinate provided by a mesh relaxation algorithm, operating on the Lagrangian configuration at tn+1.  is a relaxation delay parameter. Lagrangian phase Acceleration Lagrangian node coordinate Material velocity where

Numerical applications Three dimensional C-4 high explosive air blast Jones Wilkins Lee equation of state 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

Numerical applications Three dimensional C-4 high explosive air blast Modeling zoom

Numerical applications Three dimensional C-4 high explosive air blast with reflection Modeling zoom

Numerical applications Three dimensional C-4 high explosive air blast Pressure propagation

Numerical applications Three dimensional C-4 high explosive air blast with reflection Pressure propagation

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

Numerical applications Three dimensional C-4 high explosive air blast with reflection Pressure plot at 5 feet

Numerical applications Three dimensional C-4 high explosive air blast With 18864 elements Overpressure according to relaxation parameter  With 28296 elements Experimental overpressure = 3.40 bar  t0=1,58.10-2 µs

Numerical 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

Conclusions Delaying the mesh relaxation makes the description of motion more "Lagrange like", contracting the mesh in the vicinity of the shock front. This is beneficial for the numerical accuracy, in that dissipation and dispersion errors are reduced. 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.