Efficient Octree Anchor Formation for AMR

• Octree anchor formation is a technique that defines spatial anchors within hierarchical data structures to ensure consistent mesh element mapping in distributed environments.
• Recursive algorithms and ghost layer construction optimize the partitioning and interface consistency, enabling efficient adaptive mesh refinement.
• The universal topology iterator and Lnodes approach provide unique node indexing and scalable parallel data assembly for large-scale simulations.

Octree anchor formation refers to the process of defining spatial or topological “anchors” within an octree data structure, typically to facilitate and formalize element-wise or region-wise correspondence in hierarchical, distributed, or adaptive mesh- and volume-based computational frameworks. In high-performance scientific computing, geometric modeling, and large-scale simulation, the efficient and scalable formation of such anchors is vital for tasks including mesh refinement, globally unique node numbering, adaptive domain partitioning, and parallel distributed assembly. Octree anchors are often associated with mesh elements, interfaces, basis function nodes, or representative geometric regions; their role is especially prominent in the context of distributed forests of octrees for adaptive mesh refinement (AMR).

1. Recursive Algorithms for Hierarchical and Topological Partitioning

Efficient octree anchor formation leverages recursive algorithms that simultaneously exploit hierarchical (tree-based) and topological (mesh adjacency) relationships. The canonical example is the recursive search and iterator routines formalized in the p4est library framework.

• The {Search} algorithm performs a recursive descent starting from the root, at each stage splitting the sorted leaves (stored in a flat array) into subarrays corresponding to octant children. The splitting is realized via binary search using helper functions such as {Split_array} and {Ancestor_id}; for an octant $o$ at level $\ell$, its domain is defined as

$$\operatorname{dom}(o) = \{ X \in \mathbb{R}^d : o.x_k < X_k < o.x_k + 2^{-o.\ell},\ 0 \leq k < d \}$$

• Recursion is “lazy,” as it prunes less-refined octants early (where matching is approximate), deferring expensive exact matching to the leaf level.
• The {Iterate_interior} algorithm implements recursive sweeps over the closed partition of the octree, traversing not only full cells (octants) but also their interfaces (faces, edges, corners). At each recursion, the “leaf support set” is assembled:

$$\operatorname{leaf}(c) = \{ o \in O : \overline{\operatorname{Dom}(o)} \cap \operatorname{Dom}(c) \neq \emptyset \}$$

The iterator passes these support sets to callbacks, which are responsible for performing topology-aware assembly operations and data anchoring at each mesh point or interface. This iterator directly underpins distributed nodal data mapping, providing the continuity and locality required for downstream solver assembly and parallel communication.

2. Distributed Ghost Layer Construction and Anchor Consistency

Anchor formation across process boundaries necessitates the construction of ghost layers. The ghost layer algorithm identifies those octants (“ghosts”) that reside on remote processes but are adjacent to locally owned domains.

• For a process $p$, the $k$-codimension ghost set relative to process $q$ is

$$G_{p \to q}^k = \{ o \in O : \exists c \in \operatorname{bound}(o),\ \dim(c) \geq d-k,\ \operatorname{Dom}(c) \cap \overline{\Omega_q} \neq \emptyset \}$$

where $\operatorname{bound}(o)$ enumerates the facets (faces, edges, corners, etc.) of $o$.

• The recursive “range-boundary intersection” test used for ghost construction avoids $O(P)$ all-to-all checks by computing, for a leaf interval $[f, l]$ and support octant $s$, the set

$$B_{(\cap)}(f, l, s) = \{ b \in B : \overline{\bigcup \operatorname{dom}([f, l])} \cap \operatorname{Dom}(s, b) \neq \emptyset \}$$

This intersection restricts attention to those boundary indices for which sharing is required, and can be parallelized via space-filling curve (SFC)–precomputed arrays.

• The resulting ghost layer provides a minimal set of remote entities necessary for ensuring that “anchors” (e.g., higher-order nodal representations or adjacency information) are formed consistently—meaning that interface nodes or data entities are properly shared and uniquely numbered across process partitions.

3. Universal Mesh Topology Iterator and Node “Anchoring”

A universal topology iterator traverses the distributed forest, visiting every point in the global mesh partition $P$. Each callback provides not only the geometric and topological identification of the point but also the complete local support from adjacent leaves.

• If a point $c$ (e.g., a vertex, edge, or face) is “complete” (i.e., its local support set equals the global set), the iterator executes the callback directly; otherwise, it recursively subdivides $c$ using the child partition operator $\operatorname{part}(c)$.
• This mechanism is the core of anchor formation in the distributed sense: it enables the {Lnodes} algorithm (responsible for higher-order node numbering in $C^0$ finite element contexts) to assign a unique owner and index to every degree of freedom, regardless of mesh conformity or AMR nonconformities.
• The policy for anchor ownership is defined as $$\operatorname{owner}(c) = \min(\operatorname{leaf}(c))$$, and the mapping to node arrays can be constructed in double arrays $E_p$ and global arrays $N_p$. This ensures that nodal basis functions (the “anchors” for finite element solution space) are globally unique and support efficient scatter-gather operations.

4. Theoretical and Empirical Scalability

Recursive octree anchor algorithms presented in the context of distributed forests have been rigorously analyzed and empirically validated against large-scale hardware.

• For serial execution, recursive iterators exhibit $O(N)$ time for uniform or nearly uniform ($\Delta$-uniform) forests and $O(N \log N)$ for highly refined/unbalanced cases.
• In distributed memory, with an even partition of $N$ leaves over $P$ processes ($N = P N_p$), iterator complexity becomes $O(\log P + N_p)$, and ghost construction is $O\bigl((N/P)^{2/3} \log P\bigr)$.
• Experimental weak and strong scaling is demonstrated up to $458,\!752$ JUQUEEN supercomputer cores with negligible overhead in core mesh management tasks (searching, ghost assembly, topology iteration). This establishes anchor formation as non-bottleneck and robust to scale, crucial for PDE-based AMR applications that may involve $O(10^9)$–$O(10^{12})$ elements.

5. Applications in Adaptive Mesh Refinement and Finite Element Assembly

Octree anchor formation, via recursive, hierarchical, and topological routines, forms the backbone of adaptive mesh refinement and solver infrastructure.

• It enables the dynamic refinement and coarsening of mesh partitions, with all topological information (mandatory for node mapping, load balancing, adjacency queries) being “anchored” and numerically tractable as the mesh evolves.
• The {Lnodes} approach exemplifies the anchor formation process for constructing higher-order nodal basis functions, ensuring that global matrix and vector assembly in parallel is both correct and efficient.
• More generally, these routines are directly utilized in p4est and related AMR libraries for diverse applications: seismic wave modeling, mantle convection, fluid and structural simulation, and any PDE scenario where both adaptivity and parallelism are required.

6. Key Mathematical Language and Summary Table

The essential mathematical relationships and operator definitions framing octree anchor formation include:

Concept	Definition / Formula
Octant domain	$$\operatorname{dom}(o) = \{ X \in \mathbb{R}^d : o.x_i < X_i < o.x_i + 2^{-o.\ell} \}$$
Leaf support set	$$\operatorname{leaf}(c) = \{ o \in O : \overline{\operatorname{Dom}(o)} \cap \operatorname{Dom}(c) \neq \emptyset \}$$
Ghost set	$$G_{p \to q}^k = \{ o \in O : \exists c \in \operatorname{bound}(o),\ \dim(c) \geq d-k,\ \operatorname{Dom}(c) \cap \overline{\Omega_q} \neq \emptyset \}$$
Ghost intersection	$$B_{(\cap)}(f, l, s) = \{ b \in B : \overline{\bigcup \operatorname{dom}([f, l])} \cap \operatorname{Dom}(s, b) \neq \emptyset \}$$
Anchor ownership	$$\operatorname{owner}(c) = \min(\operatorname{leaf}(c))$$

The recursive exploitation of hierarchical (tree) and topological (mesh) structures, the construction and use of ghost layers, and the universal topology iterator form an integrated methodology enabling flexible, scalable, and robust octree anchor formation for mesh-based scientific computing. This approach underpins both the management and consistent global assembly of solution spaces in massively parallel PDE-based applications (Isaac et al., 2014).