A Morton-Type Space-Filling Curve for Pyramid Subdivision and Hybrid Adaptive Mesh Refinement

TL;DR

This paper introduces the pyramid as a new functional element type; its primary purpose is to connect tetrahedral and hexahedral elements without hanging edges and proposes the necessary functional design and generalize the fundamental global parallel algorithms for refinement, coarsening, partitioning, and face ghost exchange to fully support this new element.

Abstract

The forest-of-refinement-trees approach allows for dynamic adaptive mesh refinement (AMR) at negligible cost. While originally developed for quadrilateral and hexahedral elements, previous work established the theory and algorithms for unstructured meshes of simplicial and prismatic elements. To harness the full potential of tree-based AMR for three-dimensional mixed-element meshes, this paper introduces the pyramid as a new functional element type; its primary purpose is to connect tetrahedral and hexahedral elements without hanging edges. We present a well-defined space-filling curve (SFC) for the pyramid and detail how the unique challenges on the element and forest level associated with the pyramidal refinement are resolved. We propose the necessary functional design and generalize the fundamental global parallel algorithms for refinement, coarsening, partitioning, and face ghost exchange to fully support this new element. Our demonstrations confirm the efficiency and scalability of this complete, hybrid-element dynamic AMR framework.

Figures (7)

• Figure 1: The forest-of-trees approach used in t8code. The trees $k_0$ and $k_1$ are uniformly partitioned over three processes $p_0$, $p_1$, $p_2$. Each process holds its local elements and the ghost elements. The black arrow indicates the order of the elements by the space-filling curve index. The array of elements shows the elements stored by $p_1$.
• Figure 1: (a): The vertices of a pyramid of type $6$ and its ten children. The vertex $\vec{c}_0$ is the anchor coordinate of the parent. The pyramid refines into six pyramids, one of them upside down, and four tetrahedra of two types. (b): The vertices of a pyramid of type $7$ and its ten children. (c): the two different types of the pyramids and the tetrahedra, respectively, that occur as children, partition the cube. Defining the refinement this way, the cube (c) is also a sub-cube of (a) and (b). Note that both types of pyramids share the same anchor coordinate. Only the pyramid in the middle switches the type, that is the center pyramid of (a) is a pyramid of type 7 and the center pyramid of (b) is a pyramid of type 6.
• Figure 1: The corners of a pyramid.
• Figure 1: The runtimes of the Adapt algorithm executed for four different problem sizes. We demonstrate near ideal strong and weak scaling for all types of elements.
• Figure 1: The mesh used for the benchmarks in Section \ref{['Sec:Sec:resultplane']}. It is a toy example of an airplane geometry with a single wing. The mesh consists of 69,431 tetrahedra, 3,800 hexahedra, 29,520 prisms and 3,120 pyramids. The mesh can be found in tetdata25. The elements near the "foil" are extruded prisms, the far field consist of tetrahedra. At the boundary, hexahedra and pyramids occur, resulting in a suitable hybrid CFD mesh.

Theorems & Definitions (15)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Theorem 3.5
• Proposition 3.6
• Proof 1: Proof of Proposition \ref{['Pro:sic-dim-mort']}
• Proof 2: Proof of Theorem \ref{['Thm:pyra-sfc']}
• Lemma 3.7
• Lemma 3.8