Three-dimensional meshing is always a tedious task for engineers. Three-dimensional meshing is always performed for the components where all three dimensions are comparable to each other. Body in white (BIW) and plastic components have meshed with shell elements whose third dimension is always negligible which is mainly the thickness. So, we generally mesh the 2D surface with shell elements and finally apply the necessary thickness to replicate the exact geometry.
The four main classifications of three-dimensional elements include Tetra, Penta, Hexahedral, and Pyramid elements. Out of these four, Tetra and Hexahedral elements are the ones predominantly used for 3D meshing. Linear tetra elements consist of 4 nodes whereas the Hexahedral element has 8 nodes, the Penta element has 6 nodes and the Pyramid element has 5 nodes per element. Whereas for quadratic elements, Tetra has 10 nodes, followed by a wedge with 15 nodes, Hex with 20 nodes, and Pyramid with 13 nodes per element.
Figure 2: Classification of 3D elements used in meshing.
There lies a notable difference between 2D and 3D elements. The 2D thin shell and 1D elements support 6 degrees of freedom. Whereas, solid elements like tetra and hexahedral do not support rotational degrees of freedom. These elements have only three translational degrees of freedom. Let us quickly jump into the main content, where we shall compare tetra and hexahedral elements. What is a sweepable body? One that has two opposing topologically similar faces can be treated as a sweepable body. The simple sweepable bodies that come to our minds are cylinders and cubes. Why did we start with the term sweepable bodies? Hexahedral elements are suitable elements for sweepable bodies. The meshing software explicitly asks the method of mesh to be followed for that specific component, where the user can assign the method of hexahedral meshing directly. In simple terms, hexahedral meshes are extruded forms of square-shaped elements, whereas tetras are tetragonal in shape as their name.
Figure 3 : Linear and quadratic tetra elements.
We are aware of the significance of shape functions and gauss points. If not, let us brush up. Linear elements always have nodes only at the vertices of elements. Thus, a linear tetra element has 4 nodes and a linear hexahedral element has 8 nodes per element. Apart from that, Gauss integration points are also present in every element. The response parameters such as displacement and strain are interpolated and calculated at Gauss integration points as well. Thus, the responses are calculated at nodes and Gauss points for an element. The responses are interpolated to the gauss points using functions called shape functions. Linear elements use linear shape functions, whereas quadratic elements use quadratic shape functions for interpolation of results, which yields more accurate results. A linear tetra element has 4 nodes and one gauss point at the center of the element. So, stress and strain are calculated only at a single point inside the element. And it is assumed that the stress and strain at any other location within the element are the same as that of the gauss integration point. These 4-node tetrahedral elements are also called constant strain elements ( CST ). A quadratic tetra element has four integration points, so the responses are calculated at 4 integration points within the element. Thus, quadratic tetra elements are far more accurate than linear tetra elements. In bending problems, linear tetra elements are prone to shear locking and are unable to capture the curvature of bending. This introduces shear stress in the element and thus increases the stiffness of the system. So lower order 4 node tetra elements are no longer used in linear static and dynamic analysis. The quadratic tetra elements are completely different from this property. Quadratic tetra elements are not prone to shear locking during bending due to the presence of mid-nodes at every edge of the element. Mesh transitions are also easy in the case of quadratic tetra elements. The linear hexahedral elements are better than linear tetra elements, but encounter problems such as shear locking during bending-dominated problems. Apart from that, both the linear and quadratic hexahedral elements do not support localized mesh refinement. Because to have a refined mesh at any critical region, the complete elements along the direction of sweep (both critical and non-critical regions) need to be refined. The same is not an issue in the case of tetra elements. Local refinement of mesh is possible in the case of both linear and quadratic tetra elements, which is an added advantage. Thus the use of tetra mesh can significantly reduce the computation time by meshing the non-critical areas with a coarse mesh and critical areas with a fine refined mesh. Smooth mesh transition is also easily achieved using tetra mesh.
Figure 4: Hexahedral meshing
So, it can be concluded that quadratic tetra element is a good option to capture all types of sweepable and non-sweep-able geometries with more accurate results than linear tetra elements. Though the hexahedral element provides more accurate results than tetra elements, the Hexahedral element fails to achieve local mesh refinement and thus increases the computational cost. Also, hexahedral elements fail to mesh non-sweep-able bodies as well.
