Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods
This work addresses grid refinement challenges in computational methods like VEM and PolyDG, offering incremental improvements for domain-specific applications in finite element simulations.
The paper tackles the problem of polyhedral grid refinement by proposing two machine learning strategies: k-means clustering for point partitioning and convolutional neural networks for shape classification to define ad-hoc refinement criteria. The results show that these strategies preserve grid structure and quality while reducing computational cost and mesh complexity.
We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the "shape" of an element so that "ad-hoc" refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity.