Arbitrary-Lagrangian-Eulerian ALE Codes

Arbitrary-Lagrangian-Eulerian (ALE) codes dynamically position the mesh to optimize some feature of the solution. An ALE code has tremendous flexibility. It can treat part of the mesh in a Lagrangian fashion (mesh velocity equation to particle velocity), part of the mesh in an Eulerian fashion (mesh velocity equal to zero), and part in an intermediate fashion (arbitrary mesh velocity). All these techniques can be applied to different parts of the mesh at the same time as shown in Fig. 9.18. In particular, an element can be Lagrangian until the element distortion exceeds some criteria when the nodes are repositioned to minimize the distortion. 

We discuss, here, some examples of computational solutions to shock or impulsive loading problems. We consider, in turn, one-, two-, and three-dimensional simulations, and the role each typically plays in computational physics and mechanics investigations. 

Memory requirements for one-dimensional eontinuum dynamies ealeulations are minimal by the standards of eurrent hardware. Thus, sufTieiently fine zoning ean be used in sueh ealeulations to eapture details of material response and provide a rigorous test of fidelity for the numerieal models employed. The ability to use fine zoning also ensures that any diserepaneies between ealeulation and experiment ean be attributed, with eonsiderable eonfidenee, to Inadequaeies in the material response model. In faet, most desktop workstations have suffieient eomputing horsepower and memory to meet the eom-putating needs in one-dimensional material response studies. 

There are few problems of praetleal interest that ean be adequately approximated by one-dimensional simulations. As an example of sueh, eertain explosive blast problems are eoneerned with shoek attenuation and residual material stresses in nominally homogeneous media, and these ean be modeled as one-dimensional spherieally symmetrie problems. Simulations of planar impaet experiments, designed to produee uniaxial strain loading eonditions on a material sample, are also appropriately modeled with one-dimensional analysis teehniques. In faet, the prineipal use of one-dimensional eodes for the eomputational analyst is in the simulation of planar Impaet experiments for 

Test data are available for two experiments at different impact velocities in this configuration. In one of the tests the projectile impact velocity was 1.54 km/s, while in the second the impact velocity was 2.10 km/s. This test was simulated with the WONDY [60] one-dimensional Lagrangian wave code, and Fig. 9.21 compares calculated and measured particle velocity histories at the sample/window interface for the two tests [61]. Other test parameters are listed at the top of each plot in the figure. 

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