Multi-scale Cycle Tracking in Dynamic Planar Graphs

TL;DR

This paper presents a nested tracking framework for analyzing cycles in 2D force networks within granular materials and can adapt concepts from nested tracking graphs originally developed for merge trees by leveraging the duality between this partitioning and the cycles.

Abstract

This paper presents a nested tracking framework for analyzing cycles in 2D force networks within granular materials. These materials are composed of interacting particles, whose interactions are described by a force network. Understanding the cycles within these networks at various scales and their evolution under external loads is crucial, as they significantly contribute to the mechanical and kinematic properties of the system. Our approach involves computing a cycle hierarchy by partitioning the 2D domain into segments bounded by cycles in the force network. We can adapt concepts from nested tracking graphs originally developed for merge trees by leveraging the duality between this partitioning and the cycles. We demonstrate the effectiveness of our method on two force networks derived from experiments with photoelastic disks.

Figures (9)

• Figure 1: (a) The photoelastic experimental setup. (b) An example of the stress fringe pattern image on the disks observed through polarizing filters. It is adapted from REN2020106263. (c) The arrangement of the photoelastic disks at one of the time steps during the experiment, and, (d) the corresponding derived force network, highlighting one cycle (blue disks) by a red square; the pink disks are the enclosed non-participating particles so-called rattlers.
• Figure 2: Dataset 0 - Sankey diagram visualization for one level. The ten columns represent different time steps (loading/unloading). The nodes in a single column indicate the respective cycles/segments. The width of the links represents the amount of overlap. The nodes and links are colored according to the spatial 2D color map, emphasizing the location of the cycles and helping to show how nearby cycles are developing over time.
• Figure 3: Nested Tracking Graph Visualization (Dataset 0): The graph displays ten time steps across two levels. Coarser-level tracking is shown in grey, while finer-level tracking uses a 2D colormap. Segments are connected if their overlap exceeds a threshold. As overlap values are computed separately for each level, the graph is not strictly nested. Link width reflects segment size between time steps. To emphasize nesting for the partitioning, nodes at coarser levels are scaled by a small factor.
• Figure 4: Dataset 0 - Sankey diagram visualization. The parameters for the visualization are similar to those in \ref{['fig:s3_sankey']} but here the node colors refer to the label of the parent node in the hierarchy, emphasizing groups of segments that have the same parent.
• Figure 5: Two-dimensional color map used to encode the spatial location of the centroid of the segments.