Presentation on theme: "ASME-PVP Conference - July"— Presentation transcript:
1ASME-PVP Conference - July 25-29 2004 A Technique for Delayed Mesh Relaxation in Multi-Material ALE ApplicationsK. Mahmadi, N. Aquelet, M. SouliASME-PVP Conference - July
2The ChallengesTo 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.
3The Process Introduction Eulerian and ALE multi-material methods Multi-material interface trackingVOF methodDelayed mesh relaxation techniqueLagrangian phaseMesh relaxation phaseNumerical applicationsThree-dimensional C-4 high explosive air blastThree-dimensional C-4 high explosive air blast with reflectionConclusions
4A problem of blast propagation IntroductionA problem of blast propagationLagrangian FormulationThe computational domain follows the fluid particle motion, which greatly simplifies the governing equations.AdvantagesLagrangian 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
5Introduction 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.AdvantagesThe computational cost per cycle and the dissipation errors generated when treating the advective terms in the governing equations.Drawbacks
6Introduction 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.AdvantagesFor 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
7Introduction 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.
8Equilibrium equations Introductionv: Fluid particle velocity, u: Mesh velocityConservation of momentumConservation of massConservation of energyEquilibrium equationsALE approachu = 0Eulerian approachu = vLagrangian approach
9Eulerian and ALE Multi-Material Method Operator splitStep nFirst step: Lagrangian phaseLagrangianTransport equationSecond step: Remap phaseStep n+1EulerianALE2 phases of calculations
10Multi-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
11Delayed mesh relaxation technique Mesh relaxation phaseReference system velocityNode coordinate after relaxationis a node coordinate provided by a mesh relaxation algorithm, operating on the Lagrangian configuration at tn+1. is a relaxation delay parameter.Lagrangian phaseAccelerationLagrangian node coordinateMaterial velocitywhere
12Numerical applications Three dimensional C-4 high explosive air blastJones Wilkins Lee equation of stateA (Mbar)B (Mbar)R1R2E0 (Mbar)18.104.22.1680.087C-4 high explosive JWL parameters
13Numerical applications Three dimensional C-4 high explosive air blastModelingzoom
14Numerical applications Three dimensional C-4 high explosive air blast with reflectionModelingzoom
15Numerical applications Three dimensional C-4 high explosive air blastPressure propagation
16Numerical applications Three dimensional C-4 high explosive air blast with reflectionPressure propagation
17Numerical applications Three dimensional C-4 high explosive air blastPressure plot at 5 feet
18Numerical applications Three dimensional C-4 high explosive air blast with reflectionPressure plot at 5 feet
19Numerical applications Three dimensional C-4 high explosive air blastWith elementsOverpressure according to relaxation parameter With elementsExperimental overpressure = 3.40 bar t0=1, µs
20Numerical applications Three dimensional C-4 high explosive air blast with reflectionOverpressure according to relaxation parameter Experimental Overpresure=2.2 bar t0=2, µs
21ConclusionsDelaying 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.