A technique for regularizing the structure of a monotonic Lagrangian grid (Q1316063)

The authors describe and test a stochastic method for restructuring monotonic Lagrangian grids (MLGs) which are used in molecular dynamics and fluid dynamics for their efficiency on vector and parallel machines. On a set of points (nodes) in 3-dimensional space triple indices are to be assigned in a way which ensures that nearby nodes in Euclidean metric are also nearby in index space. This useful property for the detection of short range interactions is expected from monotonic Lagrangian grids. The latter are characterized by the monotonicity of each of the coordinates in their corresponding index. A simple method of constructing MLGs is given by iterative swapping of indices where the monotonicity is violated. Given the positions of the nodes, there exist in general different MLGs. The quality of a grid is measured by the average link lengths and average dot products. A stochastic method for restructuring a monotonic grid (SGR) is described in Section 2. The given positions are first randomly displaced, then the perturbed positions are ordered into an MLG. Starting from that grid, the original positions are reordered by the above relaxation algorithm. In Section 3 numerical tests on a \(20\times 20\times 20\) system of noninteracting nodes are carried out. The nodes start with random velocities from a uniform grid and are reflected from the boundary of the containing box. After each timestep the grid is restructured either by the classical reordering procedure or by SGR. The influences of the random perturbation, the frequency and the number of SGR steps are discussed in terms of mean values for 250 time steps of the above- mentioned quality measures. The optimal magnitude of the maximal displacement (uniform distribution) was found to be about 1/2 of the initial internode separation. The effects of increasing frequency and step number are monotonous, a saturation is reached for the latter after 10 iterations. Finally, advantages of the stochastic restructuring method in cases of highly distorted grid regions and an application in 2D simulation of shock propagation are mentioned.