An Efficient Implementation of Edge-Based Discretization without Forming Dual Control Volumes

TL;DR

This work eliminates the need to construct median dual control volumes when implementing edge-based discretization on simplex grids by deriving formulations that compute lumped directed-area vectors ${f n}_{jk}$ and dual volumes without explicitly forming dual faces. For triangular and tetrahedral grids, the authors present an efficient algorithm that expresses dual-face contributions through element-face directed-area vectors, with boundary corrections applied separately, yielding substantial reductions in arithmetic operations, particularly in 3D. They also detail edge- and element-based strategies for computing dual volumes without median dual volumes, providing complexity analyses and practical performance results on large-scale meshes. The proposed approach simplifies implementation and accelerates grid-metric computations, paving the way for extending edge-based discretization to four dimensions (space-time) and beyond. Overall, the method achieves 1–2x speed-ups in 2D and 2–3x (often more in practice) in 3D, while offering a clear path to higher-dimensional applications.

Abstract

This paper shows that lumped directed-area vectors at edges and dual control volumes required to implement the edge-based discretization can be computed without explicitly defining the dual control volume around each node for triangular and tetrahedral grids. It is a simpler implementation because there is no need to form a dual control volume by connecting edge-midpoints, face centroids, and element centroids, and also reduces the time for computing lumped directed-area vectors for a given grid, especially for tetrahedral grids. The speed-up achieved by the proposed algorithm may not be large enough to greatly impact the overall simulation time, but the proposed algorithm is expected to serve as a major stepping stone towards extending the edge-based discretization to four dimensions and beyond (e.g., space-time simulations). Efficient algorithms for computing lumped directed-area vectors and dual volumes without forming dual volumes are presented, and their implementations are described and compared with traditional algorithms in terms of complexity as well as actual computing time for a given grid.

Figures (7)

• Figure 1: Median dual control volumes in two and three dimensions, constructed with edge midpoints and geometric centroids of triangular elements in two dimensions, and edge midpoints, geometric centroids of tetrahedral elements, and geometric centroids of triangular faces in three dimensions. Note that the center node $j$ in the tetrahedral grid is located inside the median control volume and is not seen in the figure.
• Figure 2: Lumped directed-area vector contributions from an element $E$ to the edge $\{ j,k \}$.
• Figure 3: Lumped directed-area vector contributions from two triangles to the edge $\{ j,k \}$.
• Figure 4: Boundary elements in two and three dimensions. Note that the directed-area vector ${\bf n}_B$ of a boundary element is pointing outward from the interior of a domain in both figures.
• Figure 5: Directed-area vectors needed to compute ${\bf n}_{jk}$ oriented from $j$ to $k$.