Crate bempp_octree

A Rust based Octree library

This library provides a single Octree data structure and utility routines to work with Morton keys used for indexing Octrees.

An Octree is a tree data structure in which each internal node has exactly eight children. Octrees are most often used to partition a three-dimensional space by recursively subdividing it into eight octants. Octrees are the 3D analog of quadtrees.

The library supports Octrees on single processes and distributed across multiple processes via MPI. Each node inside on Octree is indexed through a MortonKey. A Morton key is a 64 bit integer value that uniquely represents a node in an Octree.

The library is designed to only provide the Octree data structure itself by storing Morton keys, their relationship and the association of keys associated with leaf nodes to actual physical points.

It is up to the user to build algorithms around the data structure.

The Octrees provides by this library are adaptive and 2:1 balanced by default. This means that no neighbour of a node can be more than one level below or above the level of the node. This is a common property of Octrees used in scientific computing. The adaptivity ensures that refined where it is needed.

A heuristic strategic ensures that the tree is approximately load-balanced. The implementation of the load-balancing and the 2:1 balancing are modeled after the paper Bottom-Up Construction and 2:1 Balance Refinement of Linear Octrees in Parallel by Sundar et. al. The underlying implementation relies on parallel sorting of Morton keys, which is done using a simple bucket sort algorithm in this library.

Using the library.

A new Octree is generated from a list of points using the Octree::new function.

use bempp_octree::{Octree, generate_random_points};
use rand::prelude::*;
use rand_chacha::ChaCha8Rng;
use mpi::traits::Communicator;

let universe = mpi::initialize().unwrap();

let comm = universe.world();
let mut rng = ChaCha8Rng::seed_from_u64(comm.rank() as u64);
let npoints = 10000;
let max_level = 15;
let max_leaf_points = 50;

let points = generate_random_points(npoints, &mut rng, &comm);
let octree = Octree::new(&points, max_level, max_leaf_points, &comm);

In this code we first initialize MPI and generate random points in the unit cube. We then create an Octree with a maximum level of 15 and a maximum of 50 points per leaf node. Note that when the code is run with in debug mode a number of expensive assertion checks are performed during tree construction which cost noticeable time for larger trees and involve communication across MPI nodes. These checks are disabled in release mode.

An octree is constructed by definining and load balancing a coarse tree and then refining the coarse tree as long as the number of points per leaf node exceeds the maximum. Once sufficiently refined the tree is 2:1 balanced again. The octree data structure stores the coarse tree and the fine leaf nodes of the octree.

To now obtain the leaf nodes of the octree use

let leaf_keys = octree.leaf_keys();

We can get the leaf nodes and interior nodes by calling

let all_keys = octree.all_keys();

all_keys is a hash map marking a key as one of

• LocalLeaf: A leaf key that is stored on the local node.
• LocalInterior: An interior key that is stored on the local node.
• Ghost(rank): A ghost key that is adjacent to the local node and originates on the process with rank rank.
• Global: A global key that is stored on all processes. These are the ancestors of the coarse tree leafs.

An important property in octrees is the neighbour relationship. To get a hash map that stores the neighbours for each non-ghost key use

let neighbour_map = octree.neighbour_map();

The neighbour_map stores for each non-ghost key a vector of keys that are adjacent to the neighbour. Neighbours need not necesssarily be at the same level in the tree. Only for interior keys neighbours are guaranteed to be at the same level. For leaf keys neighbours may be one level higher or lower.

Each key in this library is a Morton Key. For details of Morton keys see the description in the morton module.

Re-exports

• pub use crate::geometry::PhysicalBox;
• pub use crate::geometry::Point;
• pub use crate::octree::Octree;
• pub use crate::tools::generate_random_points;
• pub use morton::MortonKey;

Modules

• constants
  Module wide constants.
• geometry
  Geometry information
• morton
  Routines for working with Morton keys.
• octree
  Definition of Octree.
• parsort
  Implementation of a parallel samplesort.
• tools
  Utility routines.
• types
  Basic types