Properties of median-dual regions on triangulations in $\mathbb{R}^{4}$ with extensions to higher dimensions

TL;DR

The paper addresses efficiency in 4D space-time CFD by enabling node-centered edge-based schemes that rely on median-dual regions but avoid explicit dual-cell construction. It defines median-dual regions in $\mathbb{R}^d$ and derives a dimension-general hypervolume formula and a 4D directed-hyperarea formula, both expressed entirely in terms of simplex geometry. The hypervolume identity holds for all $d$, while the directed-hyperarea formula is proven for $d=4$, with detailed implementation and boundary considerations. The results enable on-the-fly residual calculations using precomputed edge-directed-hyperarea data, facilitating robust, conservation-compatible schemes on general triangulations and guiding future extensions to higher dimensions.

Abstract

Many time-dependent problems in the field of computational fluid dynamics can be solved in a four-dimensional space-time setting. However, such problems are computationally expensive to solve using modern high-order numerical methods. In order to address this issue, efficient, node-centered edge-based schemes are currently being developed. In these schemes, a median-dual tessellation of the space-time domain is constructed based on an initial triangulation. Unfortunately, it is not straightforward to construct median-dual regions or deduce their properties on triangulations for $d \geq 4$. In this work, we provide the first rigorous definition of median-dual regions on triangulations in any number of dimensions. In addition, we present the first methods for calculating the geometric properties of these dual regions. We introduce a new method for computing the hypervolume of a median-dual region in $\mathbb{R}^d$. Furthermore, we provide a new approach for computing the directed-hyperarea vectors for facets of a median-dual region in $\mathbb{R}^{4}$. These geometric properties are key for facilitating the construction of node-centered edge-based schemes in higher dimensions.

Figures (5)

• Figure 1: An example of the centroid-dual region not containing the associated node in 2D. The associated node is denoted by an open circle, and the region is denoted with a dotted-red line.
• Figure 2: An illustration comparing the median-dual region (dotted-blue line) to the centroid-dual region (dotted-red line, subfigure a), incenter-dual region (dotted-green line, subfigure b), and Voronoi region (dotted-magenta line, subfigure c) for a generic node of a Delaunay triangulation in 2D.
• Figure 3: An illustration showing the median-dual regions (dotted-blue lines), centroid-dual regions (dotted-red lines), incenter-dual regions (dotted-green lines), and Voronoi regions (dotted-magenta lines) for multiple nodes, on a Delaunay triangulation in 2D.
• Figure 4: A 3D illustration of a directed-hyperarea vector $\bm{n}_{jk}$ for the edge $\bm{p}_{k}-\bm{p}_{j}$ (left). The edge is shared by four tetrahedra $T_a, T_b, T_c$ and $T_d$. The lumped-normal vectors of the dual hypercuboid facets are summed together in order to obtain $\bm{n}_{jk}$ (right). More precisely, $\bm{n}_{jk} = \bm{n}(\mathcal{F}^{T_a}) + \bm{n}(\mathcal{F}^{T_b}) + \bm{n}(\mathcal{F}^{T_c}) + \bm{n}(\mathcal{F}^{T_d})$. The (partial) median-dual region around node $\bm{p}_{j}$ is highlighted in blue.
• Figure 5: A 3D illustration of the lumped-normal vector of the hypercuboid facet $\bm{n}(\mathcal{F}^{T})$ which contributes to directed-hyperarea vector $\bm{n}_{12}$ (left). The lumped-normal vector $\bm{n}(\mathcal{F}^{T})$ which contributes to $\bm{n}_{13}$ (center). The lumped-normal vector $\bm{n}(\mathcal{F}^{T})$ which contributes to $\bm{n}_{14}$ (right). These lumped-normal vectors are shown for a generic tetrahedron $T$ which belongs to $\mathbb{T}^{(3)}_{12}, \mathbb{T}^{(3)}_{13}$, and $\mathbb{T}^{(3)}_{14}$. The (partial) median-dual region around node $\bm{p}_{1}$ is highlighted in blue.

Theorems & Definitions (9)

• Definition 1: Median-Dual Region
• Lemma 2: Median-Dual Hypervolume
• Definition 3: Directed-Hyperarea Vectors
• Conjecture 4: Directed-Hyperarea Vector Identity in $\mathbb{R}^{d}$
• Theorem 5: Hypervolume Identity in $\mathbb{R}^{d}$
• Theorem 6: Directed-Hyperarea Vector Identity in $\mathbb{R}^{4}$
• Remark 7: Boundary Considerations
• Remark 8: Implementation Details
• Remark 9: Verification Details