Comprehensive Cell Agglomeration Strategy for Handling Small-Cut Cells and Topological Changes in Extended Discontinuous Galerkin Methods

Core Concepts

A comprehensive cell agglomeration strategy is presented to efficiently handle small-cut cells and topological changes in extended discontinuous Galerkin (XDG) methods, enabling stable and accurate numerical simulations on complex geometries.

Abstract

The content discusses a cell agglomeration strategy for the extended discontinuous Galerkin (XDG) method, which is used to handle complex geometries and interfaces in numerical simulations. The key points are: Small-cut cells: When an embedded geometry or interface intersects the background grid, it creates small-cut cells that can lead to discretization difficulties due to their diminutive sizes. Cell agglomeration is presented as a solution to address this small-cut problem. Topological changes: Temporal evolutions of the embedded geometries may lead to topological changes across different time steps, which can cause conceptual and computational difficulties. The proposed agglomeration strategy also aims to regulate these topological changes. Agglomeration strategy: The agglomeration strategy is designed to mitigate issues related to cut cells, such as agglomeration chains and implementation challenges in parallel simulations. It includes algorithms for source identification, target identification (direct and chain agglomeration), level determination, and agglomeration algebra. Implementation: The agglomeration strategy is implemented in the open-source software package BoSSS and tested with 2D and 3D simulations of immersed boundary flows. The proposed comprehensive cell agglomeration approach enables stable and accurate numerical simulations on complex geometries by effectively handling small-cut cells and topological changes in XDG methods.

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Cell agglomeration strategy for cut cells in eXtended discontinuous Galerkin methods

by Muhammed Top... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15285.pdf

Deeper Inquiries

How can the proposed cell agglomeration strategy be extended or adapted to handle more complex geometries or interface representations beyond the level-set function used in this work

The proposed cell agglomeration strategy can be extended or adapted to handle more complex geometries or interface representations by incorporating advanced techniques and algorithms. One approach could involve integrating machine learning algorithms to automatically identify optimal agglomeration pairs based on geometric features and cell characteristics. By training a model on a dataset of complex geometries and their corresponding agglomeration mappings, the algorithm can learn to predict the most suitable target cells for agglomeration in a given scenario. Furthermore, the strategy can be enhanced by incorporating adaptive mesh refinement techniques to dynamically adjust the agglomeration process based on the evolving geometry or interface. This adaptive approach would allow for more efficient and accurate agglomeration, particularly in cases where the geometry undergoes significant changes over time. Additionally, the strategy can be extended to handle multiple interfaces or overlapping geometries by developing algorithms that can differentiate between different interface types and appropriately agglomerate cells based on their interactions. This could involve implementing rules or criteria to prioritize certain interfaces over others, ensuring that the agglomeration process is tailored to the specific characteristics of each interface. Overall, by integrating advanced algorithms, adaptive techniques, and considerations for multiple interfaces, the cell agglomeration strategy can be adapted to effectively handle a wide range of complex geometries and interface representations in numerical simulations.

What are the potential trade-offs or limitations of the cell agglomeration approach compared to other techniques for addressing small-cut cells and topological changes, such as ghost penalty formulations or flux distribution methods

The cell agglomeration approach offers several advantages for addressing small-cut cells and topological changes, but it also has potential trade-offs and limitations compared to other techniques such as ghost penalty formulations or flux distribution methods. Trade-offs: Computational Cost: Cell agglomeration can be computationally expensive, especially in cases where complex chains of agglomeration are required. This can lead to increased computational overhead and slower simulation times compared to simpler techniques like ghost penalty formulations. Complexity: The implementation and management of agglomeration mappings, especially in the presence of dynamic interfaces or evolving geometries, can introduce complexity and potential errors in the simulation setup. Limitations: Accuracy: Depending on the criteria used for target identification and agglomeration, the cell agglomeration approach may not always provide the most accurate representation of the geometry, leading to potential discretization errors. Scalability: As the complexity of the geometry increases, the scalability of the cell agglomeration approach may become challenging, particularly in large-scale simulations with intricate geometries and multiple interfaces. Robustness: Cell agglomeration may struggle to handle certain types of geometries or interface representations, especially those with irregular shapes or overlapping boundaries, which can limit its applicability in certain scenarios. In comparison, techniques like ghost penalty formulations and flux distribution methods may offer simpler implementations and lower computational costs, but they may lack the flexibility and adaptability of the cell agglomeration approach in handling complex geometries and topological changes.

How could the cell agglomeration strategy be further optimized or parallelized to improve its computational efficiency for large-scale simulations on high-performance computing systems

To optimize and parallelize the cell agglomeration strategy for improved computational efficiency in large-scale simulations on high-performance computing systems, several approaches can be considered: Parallel Processing: Implementing parallel algorithms and data structures to distribute the agglomeration calculations across multiple processors can significantly improve computational efficiency. This can involve optimizing communication patterns, load balancing, and task scheduling to maximize parallel performance. Algorithmic Efficiency: Streamlining the agglomeration algorithm by reducing unnecessary computations, optimizing data access patterns, and minimizing memory overhead can enhance computational efficiency. This includes optimizing the selection criteria for target identification and agglomeration mapping to reduce the overall computational complexity. Adaptive Mesh Refinement: Incorporating adaptive mesh refinement techniques to dynamically adjust the agglomeration process based on the local mesh characteristics can improve efficiency. By refining the mesh only in regions where it is necessary, computational resources can be allocated more effectively. High-Performance Computing (HPC) Optimization: Leveraging HPC-specific optimization techniques such as vectorization, cache optimization, and GPU acceleration can further enhance the computational efficiency of the agglomeration strategy. Utilizing specialized hardware and optimizing code for parallel architectures can lead to significant performance improvements. By combining these approaches and continuously optimizing the implementation of the cell agglomeration strategy, it is possible to achieve enhanced computational efficiency for large-scale simulations on high-performance computing systems.