# A Solution Procedure to Improve 3D Solid Finite Element Analysis with an Enrichment Scheme
## 3 Improving 3D Solid FE Solutions
### Error Indicator and Cover Function Selection Scheme  
The indicator for error (\(M_{i}\)) and the normalized error (\(\hat{M}_{i}\)) are expressed as  
\[M_{i}=\frac{1}{2}\bigg{\{}\frac{I_{i}}{\overline{j}}+\bigg{(}\frac{\overline{ j}}{\overline{\tau}}\bigg{)}\frac{\overline{\tau}_{i}}{\overline{\tau}}\bigg{\}} \bigg{(}\frac{\chi_{i}}{L_{c}}\bigg{)} \tag{20}\]  
and  
\[\hat{M}_{i}=\frac{M_{i}}{M_{\max}},\ \overline{M}=\frac{1}{N}\sum_{i=1}^{N}M_{i} \tag{21}\]  
with  
\[\overline{J}=\frac{1}{N}\sum_{i=1}^{N}J_{i},\ \overline{\tau}=\frac{1}{N}\sum_{i =1}^{N}\overline{\tau}_{i}, \tag{22}\]  
where \(\overline{\tau}_{i}\) is the averaged stress at node \(i\), \(\chi_{i}\) is the longest edge length of elements including node \(i\), \(N\) is the number of nodes in the finite element model, \(M_{\max}\) is the maximum value of the error indicator, and \(J_{i}\) is the largest stress jump at node \(i\)[13; 25].  
The error indicator does not contain any artificial constants, and the normalized stress term (\(\overline{\tau}_{i}/\overline{\tau}\)) incorporated in the error indicator enables it to have a significant value in regions where high stress values occur [25]. A previous study [25] has confirmed that the error indicator in (20) converges as the actual errors converge.  
The values of \(\overline{J}\) and \(\overline{\tau}\) in (20) can be zero, but this is not practically possible. In such a case, using an adaptive enrichment scheme would not be required [2]. It is worth mentioning that any stress value can be used as an error indicator in (20). For verification purposes, von Mises stress is used in this paper.  
Then, the index (\(I_{i}\)) is defined to describe the arrangement of the error indicators (\(M_{i}\) with \(i=1,\ 2,\ \cdots,\ \ N\)) in ascending order, which is used in the cover selection scheme [25]. The ascending index (\(I_{i}\)) and normalized index (\(\hat{I}_{i}\)) are given by  
\[I_{i}\in\{1,\ 2,\ \cdots,\ \ N\}\ \text{with}\ M_{I_{i}}<M_{I_{i+1}}, \tag{23}\]  
and  
\[\hat{I}_{i}=I_{i}/N, \tag{24}\]  
in which \(N\) is the number of nodes in the finite element model.  
Considering the ad hoc problem [13; 24; 28] in Figure 4a, the error indicator and the index of the error indicator are expressed as given in Figure 5. The surface of the structure at \(y=-1\) is fixed, and the following body forces are applied to the entire structure:\[f_{x}^{B}=-\bigg{(}\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{xy}}{ \partial y}+\frac{\partial\tau_{xz}}{\partial z}\bigg{)},\;\;f_{y}^{B}=-\bigg{(} \frac{\partial\tau_{yx}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+ \frac{\partial\tau_{yz}}{\partial z}\bigg{)},\;\;f_{z}^{B}=-\bigg{(}\frac{ \partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{zy}}{\partial y}+\frac{ \partial\tau_{zz}}{\partial z}\bigg{)}, \tag{25}\]  
where the stress components are calculated from the displacement given by  
\[\begin{split} u&=(1-x^{2})^{2}(1-y^{2})^{2}(1-z^{2 })^{2}e^{my}\cos(mx)\sin(my)\cos(mz),\\ v&=(1-x^{2})^{2}(1-y^{2})^{2}(1-z^{2})^{2}e^{my} \sin(mx)\cos(my)\cos(mz),\\ w&=(1-x^{2})^{2}(1-y^{2})^{2}(1-z^{2})^{2}e^{my} \cos(mx)\cos(my)\sin(mz),\end{split} \tag{26}\]  
with \(m=5\). The finite element model for HEX8 is constructed by dividing the problem domain into cubes of side length \(2/NE\), resulting in a \(NE\times NE\times NE\) mesh. Similarly, the finite element model for TET4 is obtained by subdividing each HEX8 into six TET4s, resulting in a \(6\times NE\times NE\times NE\) mesh [24].