The Barnes–Hut simulation (named after Josh Barnes and
Piet Hut) is an
approximation algorithm
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solu ...
for performing an
''n''-body simulation. It is notable for having
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
O(''n'' log ''n'') compared to a direct-sum algorithm which would be O(''n''
2).
The simulation volume is usually divided up into cubic cells via an
octree (in a three-dimensional space), so that only
particles from nearby cells need to be treated individually, and particles in distant cells can be treated as a single large particle centered at the cell's
center of mass (or as a low-order
multipole expansion). This can dramatically reduce the number of particle pair interactions that must be computed.
Some of the most demanding
high-performance computing projects do
computational astrophysics using the Barnes–Hut treecode algorithm,
such as
DEGIMA The DEGIMA (DEstination for Gpu Intensive MAchine) is a high performance computer cluster used for hierarchical N-body simulations at the Nagasaki Advanced Computing Center, Nagasaki University.
The system consists of a 144-node cluster of PCs conn ...
.
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]
Algorithm
The Barnes–Hut tree
In a three-dimensional
''n''-body simulation, the Barnes–Hut algorithm
recursively divides the ''n'' bodies into groups by storing them in an
octree (or a
quad-tree in a 2D simulation). Each
node
In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex).
Node may refer to:
In mathematics
* Vertex (graph theory), a vertex in a mathematical graph
* Vertex (geometry), a point where two or more curves, line ...
in this tree represents a region of the three-dimensional space.
The topmost node represents the whole space, and its eight children represent the eight
octants of the space. The space is recursively subdivided into octants until each subdivision contains 0 or 1 bodies (some regions do not have bodies in all of their octants).
There are two types of nodes in the octree: internal and external nodes. An external node has no children and is either empty or represents a single body. Each internal node represents the group of bodies beneath it, and stores the
center of mass and the total mass of all its children bodies.
Image:Barnes_hut_partikel.png, Particle distribution resembling two neighboring galaxies.
Image:Barnes_hut_tree.png, Complete Barnes–Hut tree. (Nodes that do not contain particles are not drawn)
Image:Barnes_hut_used_nodes.png, Nodes of the Barnes–Hut tree used for calculating the force acting on a particle at the point of origin.
File:Galaxy collision.ogv , ''n''-Body simulation based on the Barnes–Hut algorithm.
Calculating the force acting on a body
To calculate the
net force on a particular body, the nodes of the tree are traversed, starting from the root. If the center of mass of an internal node is sufficiently far from the body, the bodies contained in that part of the tree are treated as a single particle whose position and mass is respectively the center of mass and total mass of the internal node. If the internal node is sufficiently close to the body, the process is repeated for each of its children.
Whether a node is or isn't sufficiently far away from a body, depends on the quotient
, where ''s'' is the width of the region represented by the internal node, and ''d'' is the distance between the body and the node's center of mass. The node is sufficiently far away when this ratio is smaller than a threshold value ''θ''. The parameter ''θ'' determines the accuracy of the simulation; larger values of ''θ'' increase the speed of the simulation but decreases its accuracy. If ''θ'' = 0, no internal node is treated as a single body and the algorithm degenerates to a direct-sum algorithm.
See also
*
NEMO (Stellar Dynamics Toolbox)
*
Nearest neighbor search
Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function ...
*
Fast multipole method
References and sources
;References
;Sources
*
*
*
*
External links
Treecodes, J. BarnesHTML5/JavaScript Example Graphical Barnes–Hut SimulationPEPC – The Pretty Efficient Parallel Coulomb solver
an open-source parallel Barnes–Hut tree code with exchangeable interaction kernel for a multitude of applications
Parallel GPU N-body simulation program with fast stackless particles tree traversalBarnes-Hut galaxy simulator
at beltoforion.de
{{DEFAULTSORT:Barnes-Hut simulation
Simulation
Gravity
Physical cosmology
Numerical integration (quadrature)
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