# A Solution Procedure to Improve 3D Solid Finite Element Analysis with an Enrichment Scheme
## 3 Improving 3D Solid FE Solutions
### Proximal Femur Problem  
We solved a proximal femur problem shown in Figure 19 as an application to biomechanics. The femur is subjected to a set of loads marked as red and black in Figure 19a. Loads represent the reaction force generated during walking. For the boundary conditions, the bottom surface of the femur marked as blue in Figure 19b is fixed. Taking into account the characteristics of the bone structure, the femur consists of two inhomogeneous and orthotropic materials as follows:  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Mesh**} & \multicolumn{2}{c}{**von Mises Stress**} \\ \cline{2-3}  & \multicolumn{2}{c}{**Standard Solution**} & **Adaptive Solution** \\ \hline Coarse mesh & 39.11\% & 19.32\% \\ Fine mesh & 23.73\% & 8.35\% \\ \hline Reference solution & \(\tau_{ref}=9.872\times 10^{1}\) N/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Relative errors in the maximum von Mises stress (\(e_{7}\)) for the connecting rod problem using coarse and fine meshes.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh**} & \multicolumn{2}{c}{**DOFs**} & \multicolumn{4}{c}{**Number of Nodes**} \\ \cline{2-7}  & \multicolumn{1}{c}{**Standard Solution**} & **Adaptive Solution** & **\(\mathbf{\alpha}\)** & **None** & **Linear Cover** & **Quadratic Cover** \\ \hline Coarse mesh & 3435 & 7179 & 0.19 & 898 & 104 & 104 \\ Fine mesh & 13,854 & 23,268 & 0.11 & 4178 & 262 & 262 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Supplementary information on the adaptive enrichment scheme for the connecting rod problem.  
Figure 19: Description of the proximal femur problem: (**a**) force boundary condition and (**b**) displacement boundary condition. The applied loads represent the reaction force generated during walking.  
For the cortical bone,  
\[\begin{split}& E_{1}=2065\rho^{3.09},\;\;E_{2}=E_{3}=2314\rho^{1.57}[ \text{MPa}]\\ & v_{12}=0.25,\;\;v_{23}=0.4,\;\;v_{31}=0.25\\ & G_{12}=\frac{G_{12\max}\rho^{2}}{\rho_{\max}^{2}},\;\;G_{23}= \frac{G_{23\max}\rho^{2}}{\rho_{\max}^{2}},\;\;G_{31}=\frac{G_{31\max}\rho^{2} }{\rho_{\max}^{2}}[\text{GPa}]\\ & G_{12\max}=6.58,\;\;G_{23\max}=5.71,\;\;G_{31\max}=7.11[\text{GPa }]\end{split}\, \tag{30}\]  
and for the cancellous bone,  
\[\begin{split}& E_{1}=1904\rho^{1.64},\;\;E_{2}=E_{3}=1157\rho^{1. 78}[\text{MPa}]\\ & v_{12}=0.25,\;\;v_{23}=0.4,\;\;v_{31}=0.25\\ & G_{12}=\frac{G_{12\max}\rho^{2}}{\rho_{\max}^{2}},\;\;G_{23}= \frac{G_{23\max}\rho^{2}}{\rho_{\max}^{2}},\;\;G_{31}=\frac{G_{31\max}\rho^{2} }{\rho_{\max}^{2}}[\text{GPa}]\\ & G_{12\max}=6.58,\;\;G_{23\max}=5.71,\;\;G_{31\max}=7.11[\text{GPa }]\end{split}\, \tag{31}\]  
where \(\rho\) is the apparent density obtained from the medical image-based voxel data. The distribution of the apparent density is given in Figure 20 with the mesh for the proximal femur problem. The mesh consists of a total of 10,543 elements (819 HEX8, 928 PYR5, and 8796 TET4). The reference solution was obtained using 10-node tetrahedral elements with the fine mesh.  
Figure 21 shows the distribution of the von Mises stress obtained from the standard and adaptive solutions including automatically adopted cover functions. The relative error in the maximum von Mises stress and supplementary information on the adaptive procedure are given in Table 8. The solution accuracy can be automatically improved even for inhomogeneous and orthotropic materials.