# Hexahedral Mesh vs. Tetrahedral: Comparing High-Quality Meshing

### Key Takeaways

• A tetrahedral mesh is a standard tool used to generate structured and unstructured meshes.

• One alternative to a tetrahedral mesh is to use a hexahedral mesh.

• Although cubic meshes are technically hexahedral, arbitrary hexahedral meshes can be used to build less computationally intensive simulation models. (Alt text: Hexahedral mesh vs. tetrahedral mesh)

When most engineering and mathematics students look at fluid dynamics simulation results, it's natural for them to pay attention to fluid flow results. None of these flow simulation results in complex systems would be possible without meshing methods. A mesh is used to model the shape and boundaries of the system while also defining the set of points where the main CFD equations will be solved with a numerical technique. Meshing has become such a prominent discussion factor in CFD that organizations like the international meshing roundtable have risen in importance.

Instead of using a typical square structured grid that is sometimes seen in more basic CFD simulations, modern CFD applications can be used to generate more complex meshes. These meshes can provide a closer approximation of the surface and boundary features in a system than a simple Cartesian mesh, yet they may do so with lower node/element density and lower simulation time. Two popular meshing styles are a hexahedral mesh and tetrahedral mesh, both of which can be used to build numerical models for arbitrary curved surfaces in CFD simulations. Hexahedral mesh generation involves applied mechanics, numerical methods, and some astounding computer science algorithm power to make the most of your next preprocessing requirements.

## Hexahedral Mesh vs. Tetrahedral Mesh

The goal in choosing a particular type of mesh , whether it is hexahedral or another type of mesh , is to find the best balance between simulation accuracy, computational time, convergence rate, and difficulty in generating a numerical model with a meshing technique. It is generally accepted that higher mesh densities, as well as mesh elements that match closer to system boundaries and flow surfaces, will produce more accurate results in a numerical simulation. In practical applications, however, we can't simply use a maximally dense mesh in every problem, nor can we perfectly match flow surfaces in every system. Most mesh generation methods these days will instead use a variety of hex mesh, surface mesh, and hybrid mesh to achieve mesh refinement. Proper utilization of prisms, blocks, quadrilaterals alongside hex mesh or hybrid mesh can help you achieve success.

For these two reasons, and with the goal of generally minimizing computational burden, meshes in 3D numerical models are developed by approximating with some standard mesh elements. The standard mesh elements used in finite element method (FEM) or finite volume method (FVM) simulations are shown below; these form the basis of meshes used to build CFD simulations that represent flow behavior in real systems. While technically a mesh element with any shape can be used to build a numerical simulation model, two of the most common 3D meshing elements are tetrahedra and hexahedra. The significant factors that affect the convergence of finite element solutions are the element's shape, distortion, polynomial order, and the specific form or CFD model being solved. The same ideas apply to mechanical simulations, particularly involving compression in complex systems.

Since these elements are important for reducing computational complexity and simulation time, when should you use a hexahedral mesh vs. tetrahedral mesh?

### When to Use a Tetrahedral Mesh

A tetrahedral mesh is a 3D generalization of a 2D triangular mesh. Tetrahedral elements are commonly constructed as equilateral, such as in systems with circular curvature, or they could be constructed as isosceles tetrahedra when a system has asymmetry. Tetrahedral elements could also be totally unstructured and adapted to arbitrary geometries with high accuracy. The accuracy would be much higher than a simple cubic grid applied in 3D or a square grid applied in 2D.

### When to Use a Hexahedral Mesh

A hexahedral mesh is a 3D generalization of a 2D quadrilateral mesh, but not necessarily with a cubic (i.e., Cartesian) arrangement of points. In other words, Cartesian meshes are a subset of hexahedral meshing, but not vice versa. Instead, a hexahedral mesh gives some freedom to approximate curves or bends in flat surfaces by varying the angles between quadrilateral faces along the mesh. An example showing a mesh along a flat surface with a curved edge is shown below; the resulting mesh captures the flat surface with perfect accuracy, thus node and element density can be kept low along that surface.

Along the curved region, the node density can be increased to capture flows along that particular surface with higher accuracy. This should illustrate the advantages of hexahedral meshes vs. tetrahedral elements for certain types of flows with small curvature. The computation time could be kept lower in certain systems because the total element count could be kept lower than would be the case with a tetrahedral mesh.

## Balance Accuracy and Simulation Time With a Hybrid Mesh

It is common to apply both tetrahedral and hexahedral elements in the same mesh to ensure high accuracy only where it is needed. It is common to place the tetrahedral elements where highly accurate flow results are required, such as along curved surfaces near the edge of a system boundary. Meanwhile, wall-bounded flows, symmetric surfaces, or larger flat surfaces can be better modeled with hexahedral elements, as complex flows may not occur there. Mixing these element types gives a better balance between high accuracy and low simulation time compared to only using one type of mesh element. Example of a mixed mesh style applied to watercraft

No matter what type of simulation you need to perform, you can compare results from hexahedral mesh vs. tetrahedral mesh models using the meshing tools in Pointwise and the complete set of fluid dynamics analysis and simulation tools in Fidelity from Cadence. You can rely on the utmost in mesh quality and structure for tetrahedral shape or tetrahderal arrangement and hexahedral arrangement. These two applications give systems designers everything they need to build and run CFD simulations with modern numerical approaches.