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.

我们提出了两种基于机器学习技术的新策略来处理多面体网格细化，可能会在自适应框架中使用。第一个使用 k-means 聚类算法来划分多面体的点以进行细化。该策略是众所周知的 Centroidal Voronoi Tessellation 的变体。第二个使用卷积神经网络对“形状”进行分类元素，以便可以定义“临时”细化标准。该策略可用于以较低的在线计算成本增强现有的细化策略，包括 k-means 策略。我们测试了所提出的算法，考虑了支持任意形状多面体元素的两个有限元方法族，即虚拟元素方法 (VEM) 和多边形不连续 Galerkin (PolyDG) 方法。我们证明这些策略确实保留了底层网格的结构和质量，降低了整体计算成本和网格复杂性。