[论文解读] A Large Scale Comparison of Tetrahedral and Hexahedral Elements for Finite Element Analysis

Teseo Schneider, Yixin Hu+4 more|arXiv (Cornell University)|Mar 22, 2019

Computational Geometry and Mesh Generation参考文献 90被引用 9

一句话总结

本研究在大规模3D几何基准测试中对比了四面体和六面体有限元，使用最先进的网格生成工具构建了非结构化四面体和结构化六面体网格。结果表明，在求解椭圆型PDE时，二次四面体单元始终优于六面体单元，提示在常见工程问题中，自动六面体网格生成往往并非必要。

ABSTRACT

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering applications. The popularity of FEM led to the development of a large family of variants, and, while their theoretical properties (such as convergence rate, stability, etc.) are understood well, their practical performance have not been systematically studied for large collections of automatically meshed 3D geometry. We introduce a set of benchmark problems, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and finally fabricating solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both unstructured (tetrahedral) and structured (hexahedral) meshes, and compare the performance of different element types for a wide spectrum of elliptic PDEs ranging from heat diffusion to fluid propagation. We observe that, while linear tetrahedral elements perform poorly, often leading to locking artefacts, quadratic tetrahedral elements outperform hexahedral elements in all settings we tested. This observation suggests that for most problems in static structural analysis, thermal analysis, and low reynolds number flows, it is unnecessary to target automatic hex mesh generation, since high-quality results can be obtained with unstructured meshes, which can be created robustly and automatically with existing meshing algorithms. We released the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure. This enables statistically significant comparisons between different FE methods, which we believe will provide a guide in the development of new meshing and FEA techniques.

研究动机与目标

• 系统评估四面体和六面体有限元在多种3D几何中的实际性能。
• 评估在真实世界、自动网格化问题中，六面体单元是否相对于四面体单元具有显著优势。
• 通过使用真实、自动生成的网格，建立具有统计显著性的FEM方法比较基准。
• 通过发布全面的测试基础设施，为未来网格生成与FEA技术的发展提供指导。

提出的方法

• 本研究构建了一个基准测试套件，涵盖从解析解到数千个真实世界、自动网格化的3D几何。
• 使用最先进的网格生成工具，为每种几何生成非结构化四面体和结构化六面体网格。
• 求解了涵盖热扩散、结构力学和低雷诺数流体流动的广泛椭圆型PDE，使用两种单元类型。
• 在所有测试案例中，对比了线性与二次四面体及六面体单元的性能。
• 测试基础设施及所有基准数据（包括网格和参考解）均已发布，以确保可复现性并支持进一步比较。

实验结果

研究问题

• RQ1 在实际、自动网格化的3D问题中，六面体单元是否相比四面体单元具有更高的精度？
• RQ2 在线性与二次四面体单元相对于六面体单元的性能，如何在多种PDE和几何中表现？
• RQ3 自动六面体网格生成所带来的计算成本与复杂性，是否被其在真实世界工程模拟中的性能提升所证明？
• RQ4 二次四面体单元是否能在静态结构、热力和流体流动分析中始终优于六面体单元？

主要发现

• 线性四面体单元表现出显著的锁定伪影，导致在许多测试案例中性能较差。
• 二次四面体单元在所有测试场景中（包括结构分析、热分析和低雷诺数流体模拟）均优于六面体单元。
• 二次四面体与六面体单元之间的性能差距在整个基准测试套件中保持一致，表明二次四面体具有稳健的优势。
• 结果表明，通过非结构化四面体网格可可靠地获得高质量的仿真结果，此类网格更易于自动生成且更具鲁棒性。