A vectorized “near neighbors” algorithm of order N using a monotonic logical grid

Abstract

In free Lagrangian representations of fluid dynamics, the fluid is assigned to discretized parcels which are defined throughout the flow by a large number of nodes moving with the local fluid velocity. These Lagrangian nodes define a finite difference or finite element grid for calculating fluid dynamic averages and driving gradients in the vicinity of the fluid parcels. Because the nodes move with the fluid, the convective terms in continuity equations governing the flow are transformed away. Thus unwanted numerical diffusion is reduced greatly or eliminated. The price for this improved numerical accuracy is having to compute derivatives in a complicated shifting geometry and having to keep track of which of the many Lagrangian nodes are nearby.

When N nodes move essentially randomly in space, N^*(N-1)/2 interactions might be important in determining how a given node moves. The exact positions and velocities of the neighboring nodes must be known. Knowing statistical averages and the general properties of the Lagrangian fluid parcels currently nearby does not provide enough data to compute local interactions accurately. At any instant a given Lagrangian node may interact strongly with only a few of the others. Unfortunately, keeping track of the other nodes with which it interacts or recomputing them each timestep is computationally very expensive. The goal is efficient, simple algorithms which select only the important near neighbor without having to check all N-squared interactions. Effort on the near neighbors problem has persisted in computational physics for several decades. To date the best algorithms scale nominally as N, or N log N rather than N², but they are scalar algorithms and address memory randomly using linked lists.

This report introduces a simple three-dimensional nearest-neighbors algorithm whose cost scales as N(1+ ε log N) with e a small coefficient, not as the square of N, and which vectorizes using data from contiguous memory locations. A compact data structure to store the object data, called a Monotonic Logical Grid (MLG), is defined dynamically so that nodes which are adjacent in real space automatically have close address indices in the MLG data arrays as well. As two nodes move past each other in space, their data are exchanged or “swapped” in the MLG data arrays