Kd-Tree Based N-Body Simulations with Volume-Mass Heuristic on the GPU

Abstract

N-body simulations represent an important class of numerical simulations in order to study a wide range of physical phenomena for which researchers demand fast and accurate implementations. Due to the computational complexity, simple brute-force methods to solve the longdistance interaction between bodies can only be used for smallscale simulations. Smarter approaches utilize neighbor lists, tree methods or other hierarchical data structures to reduce the complexity of the force calculations. However, such data structures have complex building algorithms which hamper their parallelization for GPUs.In this paper, we introduce a novel method to effectively parallelize N-body simulations for GPU architectures. Our method is based on an efficient, three-phase, parallel Kd-tree building algorithm and a novel volume-mass heuristic to reduce the simulation time and increase accuracy.Experiments demonstrate that our approach is the fastest monopole implementation with an accuracy that is comparable with state of the art implementations (GADGET-2). In particular, we are able to reach a simulation speed of up to 3 Mparticles/s on a single GPU for the force calculation, while still having a relative force error below 0.4% for 99% of the particles. We also show competitive performance with existing GPU implementations, while our competitor shows worse accuracy behavior as well as a higher energy error during time integration.

G-ML-Octree: An Update-Efficient Index Structure for Simulating 3D Moving Objects Across GPUs

“…For example, for 2D simulations, the loose Quadtree structure is applied for indexing moving objects with extents in games [14], and Sim-tree is proposed for indexing vehicles for large-scale microscopic traffic simulations [18]. In the case of the simulations of 3D moving objects, the Octree structure [4], [15], [16] and the KD-tree [17] structure are used for indexing moving objects (moving particles) for the N-body simulation. In [34], a hybrid indexing structure of grid and loose Octree called as GLOctree is proposed to provide a general purpose spatial partitioning method for dynamic scenes.…”

“…Thus, we first measure the update performance of G-ML-Octree without load-balance scheme with 10 timesteps. For comparison, we used one GPUaided KD-tree [17] structure for the N-body simulation. For convenience, we denoted the GPU-aided KDtree as G-KDtree.…”

“…Indexing approaches based on tree structures are dominant for efficiently managing 3D moving objects in such simulations. These indexing approaches make use of tree structures, e.g., Octree [4], [15], [16] and KD-tree [17], to recursively divide up the simulation domain to sub-volumes and to index all moving objects in a time-step simulation. Thus, a time-step simulation with the tree-indexing structure contains two phases: the query phase and update phase [18].…”

“…Previous works on the uses of tree-indexing structures in simulations mainly focus on optimizing their query performances. These approaches exploit the parallelism of tree-indexing structures for processing queries with the techniques of high-performance computing (HPC) such as supercomputers [4], [5], General Purpose GPUs (GPGPUs) [17], [19] and Field-Programmable Gate Array (FPGA) [6]. Nevertheless, a new challenge arises with the volume of 3D moving objects becoming very large in current simulations.…”

A Standardised Benchmark for Assessing the Performance of Fixed Radius Near Neighbours

“…This searched area can be considered the agents neighbourhood and must be searched every timestep of a simulation to ensure agents have live information. Whilst various spatial data-structures such as kd-trees and R-trees are capable of providing efficient access to spatial neighbourhoods, in order to achieve high performance in a problem as general as FRNNs they must sacrifice accuracy [6]. The naive approach for carrying out a neighbourhood search is via a brute-force technique, individually considering whether each agent is located within the target neighbourhood.…”