Finding Neighbors in a Forest: A b-tree for Smoothed Particle Hydrodynamics Simulations

Abstract

Finding the exact close neighbors of each fluid element in mesh-free computational hydrodynamical methods, such as the Smoothed Particle Hydrodynamics (SPH), often becomes a main bottleneck for scaling their performance beyond a few million fluid elements per computing node. Tree structures are particularly suitable for SPH simulation codes, which rely on finding the exact close neighbors of each fluid element (or SPH particle). In this work we present a novel tree structure, named \textit{$b$-tree}, which features an adaptive branching factor to reduce the depth of the neighbor search. Depending on the particle spatial distribution, finding neighbors using \tree has an asymptotic best case complexity of $O(n)$, as opposed to $O(n \log n)$ for other classical tree structures such as octrees and quadtrees. We also present the proposed tree structure as well as the algorithms to build it and to find the exact close neighbors of all particles. We assess the scalability of the proposed tree-based algorithms through an extensive set of performance experiments in a shared-memory system. Results show that b-tree is up to $12\times$ faster for building the tree and up to $1.6\times$ faster for finding the exact neighbors of all particles when compared to its octree form. Moreover, we apply b-tree to a SPH code and show its usefulness over the existing octree implementation, where b-tree is up to $5\times$ faster for finding the exact close neighbors compared to the legacy code.