The forest-of-octrees approach to parallel adaptive mesh re nement and coarsening
(AMR) has recently been demonstrated in the context of a number of large-scale PDE-based applications.
E cient reference software has been made freely available to the public both in the form
of the standalone p4est library and more indirectly by the general-purpose nite element library
deal.II, which has been equipped with a p4est backend.
Although linear octrees, which store only leaf octants, have an underlying tree structure by
de nition, it is not fully exploited in previously published mesh-related algorithms. This is because
the branches are not explicitly stored, and because the topological relationships in meshes, such as
the adjacency between cells, introduce dependencies that do not respect the octree hierarchy. In
this work we combine hierarchical and topological relationships between octants to design e cient
recursive algorithms that operate on distributed forests of octrees.
We present three important algorithms with recursive implementations. The rst is a parallel
search for leaves matching any of a set of multiple search criteria, such as leaves that contain points
or intersect polytopes. The second is a ghost layer construction algorithm that handles arbitrarily
re ned octrees that are not covered by previous algorithms, which require a 2:1 condition between
neighboring leaves. The third is a universal mesh topology iterator. This iterator visits every cell
in a partition, as well as every interface (face, edge and corner) between these cells. The iterator
calculates the local topological information for every interface that it visits, taking into account
the nonconforming interfaces that increase the complexity of describing the local topology. To
demonstrate the utility of the topology iterator, we use it to compute the numbering and encoding
of higher-order C0 nodal basis functions used for nite elements.
We analyze the complexity of the new recursive algorithms theoretically, and assess their performance,
both in terms of single-processor e ciency and in terms of parallel scalability, demonstrating
good weak and strong scaling up to 458k cores of the JUQUEEN supercomputer
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