# A Solution Procedure to Improve 3D Solid Finite Element Analysis with an Enrichment Scheme
## 2 Finite Element Formulation Enriched by Cover Functions
The interpolation of the displacement of the enriched 3D solid finite elements is obtained by multiplying the shape functions with the cover functions defined in cover area \(C_{i}\) as follows [5; 13]:  
\[\mathbf{u}(r,s,t)=\sum_{i=1}^{n}h_{i}^{n}(r,s,t)\widetilde{\mathbf{u}}_{i} \text{ with }\widetilde{\mathbf{u}}_{i}=[\widetilde{u}_{i}\quad\widetilde{v}_{i} \quad\widetilde{w}_{i}]^{\text{T}}, \tag{3}\]  
where displacements \(\widetilde{u}_{i}\), \(\widetilde{v}_{i}\), and \(\widetilde{w}_{i}\) are cover functions in the \(x\)-, \(y\)- and \(z\)-directions, respectively, and the cover region \(C_{i}\) is the union of elements attached to node \(i\), as shown in Figure 3.  
The cover functions are expressed as  
\[\widetilde{u}_{i}=\mathbf{p}_{i}(\mathbf{x})\mathbf{u}_{i}^{u},\ \widetilde{v}_{i}=\mathbf{p}_{i}(\mathbf{x})\mathbf{u}_{i}^{v},\ \widetilde{w}_{i}=\mathbf{p}_{i}(\mathbf{x})\mathbf{u}_{i}^{w}\text{ in }C_{i} \tag{4}\]  
with  
\[\mathbf{p}_{i}(\mathbf{x})=[1\quad\bar{\xi}_{i}\quad\eta_{i}\quad\zeta_{i} \quad\bar{\zeta}_{i}^{2}\quad\bar{\zeta}_{i}\eta_{i}\quad\cdots\quad\bar{ \zeta}_{i}^{d}\quad\cdots],\]  
\[\xi_{i}=\frac{(x-x_{i})}{\chi_{i}},\ \eta_{i}=\frac{(y-y_{i})}{\chi_{i}},\ \zeta_{i}= \frac{(z-z_{i})}{\chi_{i}},\]  
\[h_{i}^{n}=(a_{i}+b_{i}r+c_{i}s+d_{i}t)/n, \tag{1}\]0  
\[h_{i}^{n}=(a_{i}+b_{i}r+c_{i}s+d_{i}t)/n, \tag{1}\]1  
in which \(\mathbf{p}(\mathbf{x})\) is a complete polynomial basis vector for node \(i\), the vectors \(\mathbf{u}_{i}^{u}\), \(\mathbf{u}_{i}^{v}\), and \(\mathbf{u}_{i}^{w}\) are the degrees of freedom (DOFs) for the complete polynomial bases corresponding to the displacements \(u\), \(v\), and \(w\), respectively, \(d\) is the order of complete polynomial bases, and \(\chi_{i}\) is the longest edge length of elements attached to node \(i\). \[h_{i}^{n}=(a_{i}+b_{i}r+c_{i}s+d_{i}t)/n, \tag{1}\]2 Cover regions corresponding to node \(i\): (**a**) 8-node hexahedral element mesh and (**b**) 4-node tetrahedral element mesh. By substituting (4) into (3), the interpolation of displacement for the enriched 3D solid elements is given by  
\[h_{i}^{n}=(a_{i}+b_{i}r+c_{i}s+d_{i}t)/n, \tag{1}\]3  
with  
\[\mathbf{H}_{i}^{n}(r,s,t)= \begin{bmatrix}\mathbf{h}_{i}^{n}(r,s,t)&\mathbf{0}& \mathbf{0}\\ \mathbf{0}&\mathbf{h}_{i}^{n}(r,s,t)&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{h}_{i}^{n}(r,s,t)\end{bmatrix} ,\,\mathbf{h}_{i}^ {n}=h_{i}^{n}\mathbf{p}_{i},\,\,\mathbf{u}_{i}= \begin{bmatrix}\mathbf{u}_{i}^{n }\\ \mathbf{u}_{i}^{n}\\ \mathbf{u}_{i}^{n}\end{bmatrix}  \tag{7}\]  
where \(\mathbf{u}_{i}\) and \(\mathbf{H}_{i}^{n}(r,s,t)\) are the nodal DOFs vector at node \(i\) and the corresponding interpolation matrix, respectively. Using the strain-displacement relation, the strain vector in the global Cartesian coordinate system is expressed as [1; 2; 3]  
\[\varepsilon(r,s,t)= \begin{bmatrix}\varepsilon_{xx}&\varepsilon_{yy}& \varepsilon_{zz}&2\varepsilon_{xy}&2\varepsilon_{yz}&2\varepsilon_{zx} \end{bmatrix} ^{\mathrm{T}}=\sum_{i=1}^{n}\mathbf{B}_{i}^{n}(r,s,t)\mathbf{u}_{i} \tag{8}\]  
where \(\mathbf{B}_{i}^{n}\) is the matrix for the relation between strain and displacement. In the local Cartesian coordinate system, the strain vector is given by  
\[\overline{\varepsilon}= \begin{bmatrix}\varepsilon_{\overline{xx}}&\varepsilon _{\overline{yy}}&\varepsilon_{\overline{zz}}&2\varepsilon_{\overline{xy}}&2 \varepsilon_{\overline{yz}}&2\varepsilon_{\overline{zz}}\end{bmatrix} ^{ \mathrm{T}}=(\mathbf{T}^{\varepsilon})^{-1}\varepsilon, \tag{9}\]  
with  
\[\mathbf{T}^{\varepsilon}=