Optimizing Polygonal and Polyhedral Meshes for the Virtual Element Method: Reducing Mesh Size While Preserving Accuracy

Core Concepts

An optimization procedure can significantly reduce the number of elements in a polygonal or polyhedral mesh while preserving the accuracy and convergence rate of the Virtual Element Method.

Abstract

The authors present an optimization procedure for generic polygonal or polyhedral meshes, tailored for the Virtual Element Method (VEM). The key steps are:

Analyze the local quality of the mesh elements using a quality indicator specific to VEM.
Agglomerate groups of low-quality elements to optimize the global mesh quality.
The resulting discretization can remove up to 80% of the mesh elements, reducing the number of faces, edges, and vertices.
This leads to a drastic reduction in the total number of degrees of freedom, especially for high-order VEM formulations.
The VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable to the original meshes.
The optimization has a regularization effect, removing the most pathological elements and enabling convergence in cases where the original mesh caused the VEM to diverge.
The optimization of a real CAD model can be effectively used in the simulation of a time-dependent problem, leading to a significant reduction in computational time.

Stats

We can remove up to 80% of the mesh elements based on a user-set parameter.
For k=2, the optimized meshes save on average 33% in degrees of freedom compared to the original meshes.
For k=3, the optimized meshes save on average 53% in degrees of freedom compared to the original meshes.

Quotes

"We can remove up to 80% of the mesh elements, thus reducing the number of faces, edges, and vertices."
"The VEM convergence rate is preserved in the optimized meshes, and the approximation errors are comparable with those obtained with the original ones."
"The optimization has a regularization effect over low-quality meshes, removing the most pathological elements."

Key Insights Distilled From

Mesh Optimization for the Virtual Element Method: How Small Can an Agglomerated Mesh Become?

by Tommaso Sorg... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11484.pdf

Deeper Inquiries

How can the optimization algorithm be extended to handle anisotropic meshes or meshes with highly graded elements

To extend the optimization algorithm to handle anisotropic meshes or meshes with highly graded elements, we can modify the quality indicator used in the algorithm. Anisotropic meshes have elements with varying aspect ratios, which can affect the accuracy and efficiency of numerical simulations. By incorporating anisotropy metrics into the quality indicator, such as element aspect ratio or element skewness, the algorithm can prioritize the optimization of elements with extreme aspect ratios or skewness. This would ensure that the optimization process targets elements that contribute the most to the overall mesh quality degradation due to anisotropy. Additionally, for highly graded elements, where the size of elements varies significantly across the mesh, the quality indicator can be adjusted to consider element size variations. By assigning higher weights to elements with large size discrepancies, the algorithm can effectively optimize meshes with highly graded elements.

What are the limitations of the METIS partitioning algorithm used in the optimization, and how could they be addressed

The METIS partitioning algorithm used in the optimization process has certain limitations that can impact the effectiveness of the optimization. One limitation is the sensitivity of the partitioning results to the quality weights assigned to nodes and arcs in the graph representation of the mesh. If the quality weights are not accurately reflective of the mesh element quality, the partitioning may not result in an optimal reduction of elements. To address this limitation, a more sophisticated quality indicator can be developed to better capture the characteristics of the mesh elements, ensuring that the weights assigned to nodes and arcs are more representative of the mesh quality. Additionally, the METIS algorithm may struggle with highly irregular meshes or meshes with complex geometries, leading to suboptimal partitioning results. In such cases, alternative partitioning algorithms or customized partitioning strategies tailored to handle irregular meshes could be explored to improve the optimization process.

Can the optimization procedure be integrated into the mesh generation process to produce high-quality meshes directly, rather than as a post-processing step

Integrating the optimization procedure into the mesh generation process to produce high-quality meshes directly can offer several advantages. By incorporating the optimization algorithm as a step within the mesh generation pipeline, the generated meshes can be automatically optimized for the specific numerical method or simulation requirements. This integration can ensure that the meshes produced are not only geometrically sound but also optimized for computational efficiency and accuracy. Additionally, by optimizing the meshes during generation, the need for post-processing optimization steps is eliminated, streamlining the overall mesh generation and simulation workflow. This integration can lead to significant time savings and improved mesh quality, making the entire simulation process more efficient and effective.