In this paper, we present a novel algorithm that integrates deep learning with the polycube method (DL-Polycube) to generate high-quality hexahedral (hex) meshes, which are then used to construct volumetric splines for isogeometric analysis. Our DL-Polycube algorithm begins by establishing a connection between surface triangular meshes and polycube structures. We employ deep neural network to classify surface triangular meshes into their corresponding polycube structures. Following this, we combine the acquired polycube structural information with unsupervised learning to perform surface segmentation of triangular meshes. This step addresses the issue of segmentation not corresponding to a polycube while reducing manual intervention. Quality hex meshes are then generated from the polycube structures, with employing octree subdivision, parametric mapping and quality improvement techniques. The incorporation of deep learning for creating polycube structures, combined with unsupervised learning for segmentation of surface triangular meshes, substantially accelerates hex mesh generation. Finally, truncated hierarchical B-splines are constructed on the generated hex meshes. We extract trivariate B\'ezier elements from these splines and apply them directly in isogeometric analysis. We offer several examples to demonstrate the robustness of our DL-Polycube algorithm.

翻译：本文提出了一种将深度学习与多立方体方法相结合的新算法（DL-Polycube），用于生成高质量六面体网格，进而构建适用于等几何分析的体积样条。我们的DL-Polycube算法首先建立表面三角网格与多立方体结构之间的关联，利用深度神经网络将表面三角网格分类至对应的多立方体结构。随后，结合获取的多立方体结构信息与无监督学习技术，对三角网格进行表面分割。这一步骤解决了分割结果与多立方体不匹配的问题，同时减少了人工干预。接着通过八叉树细分、参数化映射及质量优化技术，从多立方体结构生成优质六面体网格。采用深度学习构建多立方体结构，并结合无监督学习进行表面三角网格分割，显著加速了六面体网格的生成过程。最后，在生成的六面体网格上构建截断分层B样条，从中提取三变量Bézier单元并直接应用于等几何分析。我们通过多个算例验证了DL-Polycube算法的鲁棒性。