## 3 The Solid FEA Plugin
### Introducing the Components: Analysing a Cantilevering Beam
#### 3.4.2 Results from the Analysis  
Having described how each component works to establish an analysis, the section ends by looking at the results using different element types. As a reference, an analytical solution of the beam's tip deflection, \(\delta_{max}\), and maximum bending stresses, \(\sigma_{xx.max}\), are used. The formulas for calculating these values are given as:  
\[\delta_{max}=\frac{PL^{3}}{3 EI},\qquad\qquad\qquad\sigma_{xx.max}=\frac{6 PL}{ wh^{2}} \tag{1}\]  
where \(I=\frac{1}{2}wh^{3}\) is the second moment of inertia for a rectangular cross section. Inserting the relevant values into (1) results in a maximum deflection of 6.48 mm and maximum stress of 270 MPa. Below, Table 2 presents the the elements' normalised displacements  
Figure 8: The component that adds a middle node on all mesh edges for quadratic finite elements. Figure 7: Necessary components for establishing an FEA. The left column represents containers for native Rhino geometry, the middle column the components for creating loads and supports, while the right component assembles and analyses the mode. and bending stresses, with respect to the analytical values, for the minimum number of elements, \(n_{els}\), where the accuracy is at least 95% of the analytical solution. As expected, the linear tetrahedron element, Tet4, has the lowest accuracy and longest computational time. The extremely high computational time is caused by the number of elements being more than ten times greater than both the Tet10 and Hex8 elements. For a bending problem with linear stress distribution over the beam's height, there is a need for many linear elements as they only have the ability to represent constant stress within each element. It is worth noting that for a uniformly distributed load along the top of the beam, the Tet4 element is expected to perform even worse with the bending moment along the beam length varying with a quadratic instead of linear shape. Thus, an even higher number of elements is necessary to approximate the exact stress distributions within the beam. Using Tet10 elements, both the efficiency and accuracy drastically increase. Here, only 144 elements are necessary to achieve satisfying results, with a run time of 127 milliseconds. For the hexahedral elements, the same trend is visible. A finer mesh with more elements is necessary to achieve accurate results for the linear Hex8 element. The Hex20 only need 24 elements, but comes at the cost of more computational time. Figure 9 shows the convergence of displacement of the different element types toward the analytical solution. This plot further emphasises the need for more linear elements shown in Table 2. Both Tet10 and Hex20 can reproduce accurate solutions for a coarse mesh, whereas the linear elements have slower convergence towards the exact solution. As this plot shows, the user should pay extra attention when working with linear tetrahedral elements in problems where bending moments are present, such as this cantilever beam. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Element Type** & **Divisions** & \(n_{els}\) & \(\delta_{max}\) & \(\sigma_{xx}\) & **Run Time [ms]** \\ \hline Tet4 & 24-4-8 & 4608 & 0.95 & 0.89 & 5007 \\ Tet10 & 12-1-2 & 144 & 1.01 & 1.04 & 127 \\ Hex8 & 24-2-4 & 192 & 0.98 & 0.98 & 171 \\ Hex20 & 12-1-2 & 24 & 0.98 & 1.04 & 233 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Overview of the number of elements necessary for different element types to achieve an accuracy of at least 95% compared to the analytical solution of 6.48 mm. The run time for each analysis is included in the rightmost column. Note that \(n_{els}\) for Tet4- and 10 are obtained by multiplying the product of _Divisions_ with 6 as each hexahedron is divided into six tetrahedrons. Figure 9: Convergence plot of the available element types for the normalised displacements of the cantilever example. The analytical solution is plotted as a straight line, while the threshold value for Table 2 is dashed.