八叉树分布式森林的递归算法
Recursive Algorithms for Distributed Forests of Octrees
摘要：
八叉树森林方法并行自适应网格细化和粗化（AMR）最近已被证明在一些大规模的PDE为基础的应用程序的背景下。虽然线性八叉树，它只存储叶八分区，有一个基本的树结构的定义，它不经常利用在以前公布的网格相关的算法。这是因为分支没有明确存储，并且因为网格中的拓扑关系（例如单元之间的邻接）引入了不遵守八叉树层次结构的依赖关系。在这项工作中，我们结合八叉树分支之间的层次和拓扑关系，设计高效的递归算法。我们提出了三个重要的算法与递归实现。第一种是并行搜索匹配一组多个搜索条件中的任何一个的叶子。第二种是幽灵层构造算法，该算法处理先前算法未覆盖的任意细化的八叉树，其需要相邻叶子之间的2：1条件。第三个是一个通用的网状拓扑迭代器。这个迭代器访问域分区中的每个单元，以及这些单元之间的每个接口（面、边和角）。迭代器计算它访问的每个接口的局部拓扑信息，同时考虑到增加描述局部拓扑复杂性的可扩展接口。为了证明拓扑迭代器的实用性，我们使用它来计算高阶$C^0$节点基函数的编号和编码。我们从理论上分析了新的递归算法的复杂性，并评估其性能，无论是在单处理器的效率和并行可扩展性方面，表现出良好的弱和强扩展到458k核心的JUQUEEN超级计算机。
英文摘要：
The forest-of-octrees approach to parallel adaptive mesh refinement and coarsening (AMR) has recently been demonstrated in the context of a number of large-scale PDE-based applications. Although linear octrees, which store only leaf octants, have an underlying tree structure by definition, it is not often 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 octree branches to design efficient recursive algorithms. We present three important algorithms with recursive implementations. The first is a parallel search for leaves matching any of a set of multiple search criteria. The second is a ghost layer construction algorithm that handles arbitrarily refined 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 domain 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 $C^0$ nodal basis functions. We analyze the complexity of the new recursive algorithms theoretically, and assess their performance, both in terms of single-processor efficiency and in terms of parallel scalability, demonstrating good weak and strong scaling up to 458k cores of the JUQUEEN supercomputer.
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