A Large Scale Comparison of Tetrahedral and Hexahedral Elements for Finite Element Analysis
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The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) and it is a staple in most engineering applications. The popularity of this approach led to the development of a large family of variants, and, while their theoretical properties (such as convergence rate, stability, etc.) are well studied, their practical performances have never been systematically studied over a large collection of 3D models. We introduce a large set of benchmark problems, starting from simple cases with an analytical solution, moving to classical experimental setups, and finally fabricating solutions for thousands of real-world 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 convection 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 the settings we tested. This unexpected result suggests that for static structural analysis, thermal analysis, and low reynolds number flows it is unnecessary to target automatic hex mesh generation, since superior results can be obtained with unstructured meshes, which can be created robustly and automatically with existing meshing algorithms. We release the experimental description, meshes, and reference implementation of our testing infrastructure, providing a standard benchmark. This enables statistically significant comparisons between different FE methods and discretization, which we believe will provide a guide in the development of new meshing and FEA techniques.
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