Cell agglomeration strategy for cut cells in eXtended discontinuous Galerkin methods

TL;DR

This work tackles the ill-conditioning introduced by small cut cells in unfitted XDG discretizations of two-phase flows on Cartesian grids. It develops a graph-based cell agglomeration strategy that forms agglomeration groups, chains, and Levels, using a global coupling matrix $\bm{Q}$ to construct an agglomerated XDG space, and handles inter-processor communications via MPI. Implemented in the BoSSS framework, the method demonstrably reduces mass- and operator-condition numbers across 2D/3D immersed-boundary test cases (e.g., vanishing/colliding spheres, rotating popcorn/torus), stabilizing simulations under dynamic topologies. The approach preserves topology across time steps, accommodates dynamic interfaces, and remains parallelizable, offering a practical tool for robust high-fidelity multiphase simulations on unfitted meshes; future work will optimize memory usage and iterative solver performance.

Abstract

In this work, a cell agglomeration strategy for the cut cells arising in the extended discontinuous Galerkin (XDG) method is presented. Cut cells are a fundamental aspect of unfitted mesh approaches where complex geometries or interfaces separating sub-domains are embedded into Cartesian background grids to facilitate the mesh generation process. In such methods, arbitrary small cells occur due to the intersections of background cells with embedded geometries and lead to discretization difficulties due to their diminutive sizes. Furthermore, temporal evolutions of these geometries may lead to topological changes across different time steps. Both of these issues, i.e., small-cut cells and topological changes, can be addressed with a cell agglomeration technique. In this work, a comprehensive strategy for the typical issues associated with cell agglomeration in three-dimensional and multiprocessor simulations is provided. The proposed strategy is implemented into the open-source software package BoSSS and tested with 2- and 3-dimensional simulations of immersed boundary flows.

Figures (19)

• Figure 1: Cut cell agglomeration on the XDG space for an arbitrary interface $\mathfrak{I}$. The cut cells are indicated by $K_{i,\mathfrak{A}}$, $K_{i,\mathfrak{B}}$, $K_{i+1,\mathfrak{A}}$, $K_{i,1\mathfrak{B}}$. The small-cut cells are agglomerated to neighbor elements as $K_{i+1,\mathfrak{A}} \to K_{i,\mathfrak{A}}$ and $K_{i,\mathfrak{B}} \to K_{i+1,\mathfrak{B}}$.
• Figure 2: A sample illustration of agglomeration mapping on a regular mesh. Each tree represents an agglomeration group, with the first tree also representing an agglomeration pair. Final targets are displayed in red, while chain agglomeration edges and their source cells are displayed in blue. The black edges indicate the direct agglomeration pairs and the white vertices indicate their source cells.
• Figure 3: The formation of the agglomerated basis $\phi_{\mathrm{agg},m}$ (left) with respect to the original bases $\phi_{i,m}$ (right) with $\text{supp}(\phi_{i,m}) = K_i$.
• Figure 4: Unweighted (left) and weighted (right) chain agglomerations in a cut cell mesh. The arrows indicate the preferred way of agglomeration for the small-cut cells smaller than the half of background cell (shown with dots) in $\mathfrak{A}$. The subdomains $\mathfrak{A}$ and $\mathfrak{B}$ are shown in white and cyan, respectively.
• Figure 5: Illustration of an inter-processor agglomeration chain on a 2D domain decomposed into four regions. The top chain displays the sequential pairs, while the bottom chain features the level-reduced equivalent of the top chain. The red lines indicate the processor boundaries, whereas the arrows in the cells indicate the agglomeration pairs from source to target. Final targets are indicated by red vertices, while the source cells in direct and chain agglomerations are displayed with white and blue vertices, respectively.

Theorems & Definitions (6)

• Definition 1: Agglomeration group and mapping
• Definition 2: Agglomeration source and target
• Definition 3: Direct and chain agglomeration
• Definition 4: Agglomerated cell
• Definition 5: The agglomerated mesh and space
• Example 1